Proving Cyclic Quadrilaterals and Right Angles: A Problem from the Singapore Math Olympiad

In this video, we explore a challenging geometry problem from the Singapore Math Olympiad (Senior Section, Round 2). The problem involves a square, a randomly chosen point on one of its sides, and various perpendiculars and intersections leading to the proof of a right angle. Let’s break down the key concepts used to arrive at the solution.

Problem Overview:

We are given a square $ABCD$ and a point $E$ on side $CD$.
We draw $\triangle AEB$ and drop perpendiculars from points $A$ and $B$ onto the opposite sides $AE$ and $BE$, with the feet of these perpendiculars being points $F$ and $G$, respectively.
We then join $DF$ and $CG$, which intersect at point $H$.
The goal is to prove that the $\angle AHB$ is $90{^\circ}$.

Watch the Video

Key Concepts Used:

Cyclic Quadrilaterals and Concyclic Points: The proof involves showing that multiple points lie on a common circle. The cyclic nature of quadrilaterals helps us determine certain angle relationships that are key to solving the problem.
Angle Chasing: The solution involves a careful examination of angles formed by different line segments and using the properties of cyclic quadrilaterals to establish relationships between them.
Concyclic Conditions: Since points $A$, $F$, $O$, $G$, and $B$ (where $O$ is the center of the square) are concyclic, it follows that the angles subtended by these points must add up to $180{^\circ}$. This property helps prove that $H$ lies on the same circle.

Step-by-Step Proof Summary:

Establish Concyclic Points: We first construct the circumcircle of $\triangle AFB$ and show that it passes through point $G$, making $A$, $F$, $O$, $G$, and $B$ concyclic.
Use Angle Properties: By analyzing the angles subtended by the chords, we establish that the angles at the circumference involving these points are equal, ensuring concyclicity.
Prove Point $H$ Lies on the Circle: By showing that points $F$, $D$, $E$, and $O$ are concyclic, and performing a similar analysis on the other side of the square, we conclude that point $H$ must also lie on the circumcircle.
Conclude with the Right Angle: Since point $H$ lies on the circle whose diameter is segment $AB$, the $\angle AHB$ must be $90{^\circ}$ by the inscribed angle theorem.

This solution beautifully illustrates how advanced geometry concepts like cyclic quadrilaterals, concyclicity, and angle chasing can be used to solve complex problems involving right angles and perpendiculars.

Motivation and Exploration:
The video also discusses the motivation behind defining certain points and relationships, such as the center of the square. Experimentation, including transformations like inversion, can often reveal hidden properties and relationships in geometry. This problem is an excellent example of how problem-solving in geometry is as much about exploration and insight as it is about formal methods.

Singapore Math Olympiad Past Years Questions- Geometry (Junior)

Problem 02 - SMO Year 2023:

How many non-congruent triangles with integer side lengths have perimeter 7 ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Problem 04 - SMO Year 2023:

The following diagram shows two semicircles whose diameters lie on the same line. (A B) is a chord of the larger semicircle that is tangent to the smaller semicircle at the point $\mathrm{C}$ and is parallel to the diameter $\mathrm{DE}$ of the larger semicircle. If $|\mathrm{AB}|=16 \mathrm{~cm}$, what is the area of the shaded region in $\mathrm{cm}^2$ ?

(A) $8 \pi$
(B) $16 \pi$
(C) $32 \pi$
(D) $48 \pi$

Problem 15 - SMO Year 2023:

In the following diagram, $\mathrm{ABCD}$ is a square of side $16 \mathrm{~cm}$. $\mathrm{E}$ lies on $\mathrm{CD}$ such that $|\mathrm{DE}|=) (4 \mathrm{~cm} . \mathrm{M}) and (\mathrm{N}$ lie on $\mathrm{AD}$ and $\mathrm{BC}$ respectively such that $\mathrm{MN}$ is perpendicular to $\mathrm{BE} . \mathrm{X}$ is the intersection of $\mathrm{MN}$ and $\mathrm{BE}). If (|\mathrm{MX}|=11 \mathrm{~cm}$ and $|\mathrm{BN}|=x \mathrm{~cm}$, what is the value of (x) ?

Problem 18 - SMO Year 2023:

The sum of all the interior angles except one of a convex polygon is $2023^{\circ}$. What is the number of sides of this polygon? (A polygon is convex if every interior angle is between $0^{\circ}$ and $180^{\circ}).$

Problem 21 - SMO Year 2023:

In the following diagram, (A B) is parallel to $\mathrm{DC},|\mathrm{AB}|=6 \mathrm{~cm},|\mathrm{AD}|=17 \mathrm{~cm},|\mathrm{DC}|=10 \mathrm{~cm}$ and angle $\mathrm{DAB}=90^{\circ}$. $\mathrm{E}$ lies on $\mathrm{AD}$ such that $\mathrm{BE}$ is perpendicular to $\mathrm{EC}$. If the area of triangle $\mathrm{BEC}=k \mathrm{~cm}^2$, what is the largest possible value of (k) ?

Problem 11- SMO Year 2022

The following diagram shows a star that is cut out from a square with sides of length 30 . What is the area enclosed by the star?

Problem 14 - SMO Year 2022

In the following diagram, a white square and four grey squares of equal size are drawn in a circle such that both dashed lines form diameters of the circle. If the diameter has length 60 , find the smallest possible value for the total area of the five squares.

Problem 19 - SMO Year 2022

In the following diagram, (A C D) is a triangle such that $|A B|=|B C|, \angle A B D=45^{\circ}) and (\angle B D C=15^{\circ}$. If $\angle A D B=x^{\circ}$, what is the value of $x$ ?

Problem 20 - SMO Year 2022

What is the area of a triangle with side lengths
$\sqrt{6^2+7^2}, \sqrt{12^2+7^2} \text { and } \sqrt{6^2+14^2} \text { ? }$

Problem 25 - SMO Year 2022

The following diagram shows a rectangle that is partitioned into 17 squares. If the two smallest squares, shaded in grey, have sides of length 2 , what is the area of the rectangle?

Problem 04 - SMO Year 2021:

In the diagram, six circles are tangent to each other. If the radius of the largest circle is 1 and the radii of the four medium sized circles are equal, what is the radius of the smallest circle?

(A) $\sqrt{2}-1$

(B) $3-2 \sqrt{2}$

(D) $6-4 \sqrt{2}$

(E) None of the above

Problem 16 - SMO Year 2021:

In the following diagram, $A B C D$ is a quadrilateral inscribed in a circle with center $O$. If $|A B|=|B C|=6,|A D|=14$ and $C D$ is a diameter, what is the length of $|C D|$ ?

Problem 17 - SMO Year 2021:

The diagram below shows a piece of cardboard in the shape of an equilateral triangle with side length $36 \mathrm{~cm}$. Six perpendicular cuts of length $2 \sqrt{3} \mathrm{~cm}$ are made to remove the corners in order to fold the cardboard into a tray whose base is an equilateral triangle and height is $2 \sqrt{3} \mathrm{~cm}$. What is the volume of the tray in $\mathrm{cm}^3$ ?

Problem 11 - SMO Year 2020:

Let (A B C) be a triangle where (D) is the midpoint of (B C) and (E) lies on (A C) such that (A E: E C=3: 1). Let (F) be the intersection of (A D) and (B E). If the area of (A B C) is 280 , what is the area of triangle (B F D) ?

Problem 17 - SMO Year 2020:

In the following diagram, (A B C D) is a rectangle where (E) and (F) are points on (B C) and (C D) respectively. The area of triangle (A E F), denoted ([A E F]), is 2037 . If
$[
[A E C F]=2[A B E]=3[A D F],
]$
what is the area of the rectangle (A B C D) ?

Problem 18 - SMO Year 2020:

In the following diagram, (A B C D) is a square of side length (64 . E) is the midpoint of (A B), (F) is the midpoint of (E C) and (G) is the midpoint of (F D). What is the area enclosed by the quadrilateral (A E F G) ?

Problem 5 - SMO Year 2019:

In a quadrilateral $A B C D$, the diagonals $A C$ and $B D$ intersect at the point $O$. Suppose that $\angle B A D+\angle A C B=180^{\circ},|B C|=3,|A D|=4,|A C|=5$ and $|A B|=6$. What is the value of $\frac{|O D|}{|O B|}$ ?

(A) $\frac{2}{3}$
(B) $\frac{8}{9}$
(C) $\frac{9}{10}$

Problem 6 - SMO Year 2019

In the following diagram, all lines are straight. What is the value $\left(\right.$ in $\left.^{\circ}\right)$ of
\[
\angle a+\angle b+\angle c+\angle d+\angle e+\angle f+\angle g+\angle h+\angle i+\angle j ?
\]

Problem 11 - SMO Year 2019

In the following diagram, $A B C D$ is a rectangle with $|A B|=4$ and $|B C|=6$. Points $E$ and $F$ lie on the sides $B C$ and $A D$ respectively such that $|B E|=|F D|=2$. Points $G$ and $H$ lie on the sides $A B$ and $C D$ respectively such that $|A G|=|C H|=1$. Suppose $P$ lies on $E F$, such that the quadrilateral $B G P E$ has area 5 . What is the area of the quadrilateral $F D H P ?$

Problem 13 - SMO Year 2019

In the following diagram, $A B C D, A C F E$ and $E C H G$ are all rectangles. If $|A B|=6$ and $|B C|=8$, what is area of the rectangle $E C H G$ ?

Problem 23 - SMO year 2019

$A B C D$ is a square sheet of paper with sides of length 6 . The paper is folded along a crease line $E F$ so that points $A$ and $B$ now lie on $A^{\prime}$ and $B^{\prime}$ respectively as indicated in the diagram. If $H$ is the intersection of $A^{\prime} B^{\prime}$ and $B C$, what is the perimeter of the triangle $A^{\prime} C H$?

Problem 24 - SMO Year 2019

In the following diagram, $\angle B E D=30^{\circ}$ and $\angle D B E=15^{\circ}$. If $|C D|=|D E|$, what is the value of $x$ ?

Problem 25 - SMO Year 2019

In the following diagram, $P_1, P_2, \cdots, P_8$ are points on $\triangle A B C$ such that
$
\left|A P_1\right|=\left|A P_8\right|=\left|P_i P_{i+1}\right|, \text { for all } i=1,2, \cdots, 7 .
$

What is the value $in ({ }^{\circ}) $ of $\angle B A C$ ?

Singapore Mathematics Olympiad - 2021- Senior Years - Questions

Multiple Choice Questions

Problem 01:

Let $p$ be a real number such that the equation $x^2-10 x=p$ has no real solution. of the following is true?
(A) $0<p<25$
(B) $p = 25$
(C) $p>25$
(D) $p<-25$
(E) $-25<p<0$

Problem 02:

Which of the following is the largest?
(A) $\tan 50^{\circ}+\sin 50^{\circ}$
(B) $\tan 50^{\circ}+\cos 50^{\circ}$
(C) $\sin 50^{\circ}+\cos 50^{\circ}$
(D) $\tan 50^{\circ}+\sin ^2 50^{\circ}$
(E) $\sin ^2 50^{\circ}+\cos ^2 50^{\circ}$

Problem 03:

Find the value of $2021^{\left(\log {2021} 2020\right)\left(\log {2020} 2019\right)\left(\log _{2019} 2018\right)}$.
(A) 2018
(B) 2019
(C) 2020
(D) 2021
(E) None of the above

Problem 04:

Suppose $\sin \theta=\frac{n-3}{n+5}$ and $\cos \theta=\frac{4-2 n}{n+5}$ for some integer $n$. Find the maximum value of $160 \tan ^2 \theta$.
(A) 80
(B) 90
(C) 100
(D) 120
(E) None of the above

Problem 05:

Select all the inequalities which hold for all real values of (x) and (y).

(i) $x \leq x^2+y^2$,
(ii) $x y \leq x^2+y^2$,
(iii) $x-y \leq x^2+y^2$,
(iv) $y+x y \leq x^2+y^2$,
(v) $x+y-1 \leq x^2+y^2 $.
(A) (i)
(B) (i) and (iii)
(C) (iii) and (iv)
(D) (ii)
(E) (ii) and (v)

Short Questions

Problem 06:

Let $x$ be the integer such that $x=5 \sqrt{2+4 \log _x 5}$. Determine the value of $x$.

Problem 07:

If $\cos A-\cos B=\frac{1}{2}$ and $\sin A-\sin B=-\frac{1}{4}$, find the value of $100 \sin (A+B)$.

Problem 08:

Find the constant in the expansion of $\left(\sqrt[3]{x}+\frac{1}{\sqrt{x}}\right)^6\left(\sqrt{x}+\frac{1}{x}\right)^{10}$.

Problem 09:

A quadratic polynomial $P(x)=a x^2+b x+c$, where $a \neq 0$, has the following properties:
$P(n)=\frac{1}{n^2} \text { for all } n=-1,2,3$. Determine the smallest positive value of $k$, where $k \neq 2,3$, such that $P(k)=\frac{1}{k^2}$.

Problem 10:

The figure below shows a triangle $A B C$ such that $A D$ and $B E$ are altitudes to the sides $B C$ and $C A$ respectively. The lines $A D$ and $B E$ intersect at $H$. Determine the area in $\mathrm{cm}^2$ of the triangle $A B C$ if $A H=50 \mathrm{~cm}, D H=18 \mathrm{~cm}$ and $B H=E H$.

Problem 11:

In the figure below, $\angle G C B=\angle A C E=\angle D F E=90^{\circ}$, and $\angle G B C=\angle E A C=$ $\angle E D F=\theta^{\circ}$. Also, $G B=6 \mathrm{~cm}, A E=10 \mathrm{~cm}$ and $D E=8 \mathrm{~cm}$. Let $\mathcal{S}$ denote the sum of the areas of the triangles $A B C$ and $C D E$. Find the maximum possible value of $\mathcal{S}$ in $\mathrm{cm}^2$ .

Problem 12:

Find the sum of all the solutions to the equation $\sqrt[3]{x-110}-\sqrt[3]{x-381}=1$ .

Problem 13:

If $f(x)=\left(2 x+4+\frac{x-2}{x+3}\right)^2$, where $-2 \leq x \leq 2$, find the maximum value of $f(x)$.

Problem 14:

Given that $D=\sqrt{\sqrt{x^2+(y-1)^2}+\sqrt{(x-1)^2+y^2}}$ for real values of (x) and (y), find the minimum value of $D^8$.

Problem 15:

Find the minimum value of $\frac{8}{\sin 2 \theta}+12 \tan \theta$, where $0<\theta<\frac{\pi}{2}$.

Problem 16:

Determine the largest angle $\theta$ (in degree), where $0^{\circ} \leq \theta \leq 360^{\circ}$, such that $\sin \left(\theta+18^{\circ}\right)+\sin \left(\theta+162^{\circ}\right)+\sin \left(\theta+234^{\circ}\right)+\sin \left(\theta+306^{\circ}\right)=1+\cos \left(\theta+60^{\circ}\right)+\cos \left(\theta+300^{\circ}\right)$.

Problem 17:

Let $O$ be the circumcentre of the triangle $A B C$ and that $\angle A B C=30^{\circ}$. Let (D) be a point on the side (B C) such that the length of $A D$ is the same as the radius of the circle. Determine the value of $\angle A D O$ (in degree) if $\angle O A B=10^{\circ}$.

Problem 18:

A function $f$ satisfies $f(x) f(x+1)=x^2+3 x$ for all real numbers $x$. If $f(1)+f(2)=\frac{25}{6}) and (0<f(1)<2$, determine the value of $11 \times f(10)$.

Problem 19:

Find the value of

$\frac{1}{\sin ^2 0.5^{\circ}}-\tan ^2 0.5^{\circ}+\frac{1}{\sin ^2 1.5^{\circ}}-\tan ^2 1.5^{\circ}+\frac{1}{\sin ^2 2.5^{\circ}}-\tan ^2 2.5^{\circ}+\cdots+\frac{1}{\sin ^2 179.5^{\circ}}-\tan ^2 179.5^{\circ}$ .

Problem 20:

Let $a_1, a_2, a_3$ be three distinct integers where $1000>a_1>a_2>a_3>0$. Suppose there exist real numbers $x, y, z$ such that
$\left(a_1-a_2\right) y+\left(a_1-a_3\right) z=a_1+a_2+a_3 $
$ \left(a_1-a_2\right) x+\left(a_2-a_3\right) z=a_1+a_2+a_3 $
$\left(a_1-a_3\right) x+\left(a_2-a_3\right) y=a_1+a_2+a_3$ .

Find the largest possible value of $x+y+z$.

Problem 21:

The figure below shows a circle centred at $O$ with radius $555 \mathrm{~cm}$. If $O A=O B$ and $\frac{R A}{A S}+\frac{R B}{B T}=\frac{13}{6}$, find $O A$ (in cm).

Problem 22:

Find the number of real solutions $x, y$ of the system of equations

$x^3+y^3+y^2 =0, $
$x^2+x^2 y+x y^2 =0$ .

Problem 23:

The following $3 \times 5$ rectangle consists of $151 \times 1$ squares. Determine the number of ways in which 9 out of the 15 squares are to be coloured in black such that every row and every column has an odd number of black squares.

Problem 24:

Let $n$ be a positive integer such that $\frac{2021 n}{2021+n}$ is also a positive integer. Determine the smallest possible value of (n).

Problem 25:

Determine the number of 5-digit numbers with the following properties:
(i) All the digits are non-zero;
(ii) The digits can be repeated;
(iii) The difference between consecutive digits is exactly 1 .

Singapore Mathematics Olympiad - 2020- Junior Years - Questions

Multiple Choice Questions

Problem 01:

Let $x=2^{300}, y=3^{200}$ and $z=6^{100}$. Which of the following is true?
(A) $x>y>z$
(B) $x>z>y$
(C) $y>z>x$
(D) $y>x>z$
(E) $z>x>y$

Problem 02:

An expensive painting was stolen and the police rounded up five suspects Alfred, Boris, Chucky, Dan and Eddie. These were the statements that were recorded.
Alfred: "Either Boris or Dan stole the painting."
Boris: "I think Dan or Eddie is the guilty party."
Chucky: "It must be Dan."
Dan: "Boris or Eddie did it!"
Eddie: "I am absolutely sure the thief is Alfred."
The police knew that only one of the five suspects stole the painting and that all five were lying. Who stole the painting?
(A) Alfred
(B) Boris
(C) Chucky
(D) Dan
(E) Eddie

Problem 03:

The following diagram shows a system of balances hanging from the ceiling with three types of weights. The balances tip down to the heavier side. If we use $\square<\triangle$ to represent $\square$ is lighter than $\triangle$, which of the following is true?

(A) $\square<0<\triangle$
(B) $\square<\triangle<0$
(C) $\triangle<\square<0$
(D) $\triangle<0<\square$

Problem 04:

The integer 6 has exactly four positive factors, namely $1,2,3$ and $6$ . Likewise, the integer $8$ has exactly four positive factors, $1,2,4$ and $8$ . How many integers from $9$ to $50$ (inclusive) have exactly four positive factors?
(A) $10$
(B) $11$
(C) $12$
(D) $13$
(E) $14$

Problem 05:

A company sells a bottle of juice at $\$ 1$ per bottle and offers to give you a free bottle of juice if you recycle four empty bottles. Assuming there is no limit to the offer and you can recycle bottles that you receive from the offer, what is the largest number of bottles you can eventually consume if you have $\$ 43$ ?
(A) 53
(B) 54
(C) 55
(D) 56
(E) 57

Short Questions

Problem 06:

Let $n$ be a positive integer. Suppose the lowest common multiple of $4,5$ and $n$ is $2020$ . What is the sum of the smallest possible value of $n$ and the largest possible value of $n$ ?

Problem 07:

When the five-digit integer $\overline{2 x 6 y x}$ is divided by the four-digit integer $\overline{5 y 27}$, the quotient is 4 and remainder is $\overline{x 106}$, which is a four-digit integer. What is the value of the digit $x$ ?

Problem 08:

How many multiples of 11 are there in the sequence $1100,1101,1102, \ldots, 2019,2020$ ?

Problem 09:

A quadruple $(a, b, c, d)$ of positive integers is skewed if the median and mode of $a, b, c, d$ are equal, but strictly greater than the mean of $a, b, c, d$. How many skewed $(a, b, c, d)$ of positive integers are there that satisfy $a \leq b \leq c \leq d$ and $a+b+c+d=40$ ?

Problem 10:

Let $T_n$ be a linear sequence given by
\[
T_1=10, T_2=13, T_3=16, \ldots,
\]
and $S_n$ be a quadratic sequence given by
\[
S_1=10, S_2=23, S_3=39, \ldots
\]

What is the value of $S_{101}-T_{101}$ ?

Problem 11:

Let $A B C$ be a triangle where $D$ is the midpoint of $B C$ and $E$ lies on $A C$ such that $A E: E C=3: 1$. Let $F$ be the intersection of $A D$ and $B E$. If the area of $A B C$ is $280$ , what is the area of triangle $B F D$ ?

Problem 12:

In the figure below, each distinct letter represents a unique distinct digit such that the arithmetic holds. If the letter K represents 6 , what number does SHAKE represent?

Problem 13:

The diagram below shows a shape formed from four semi-circles whose radii are distinct integers. If the shape has perimeter of $18 \pi$ and encloses an area of $k \pi$, where $k$ is a prime, what is the value of $k$ ?

Problem 14:

If the sum of the two diagonals of a kite is $100$ , what is the largest possible area of the kite?

Problem 15:

What is the smallest five-digit integer of the form $\overline{5 x 20 y}$ that is divisible by $33$ ?

Problem 16:

What is the value of $2020^3-9(2020)(2017)-2017^3$ ?

Problem 17:

In the following diagram, $A B C D$ is a rectangle where $E$ and $F$ are points on $B C$ and $C D$ respectively. The area of triangle $A E F$, denoted $[A E F]$, is $2037$ . If
\[
[A E C F]=2[A B E]=3[A D F],
\]
what is the area of the rectangle $A B C D$ ?

Problem 18:

In the following diagram, $A B C D$ is a square of side length $64$ . $E$ is the midpoint of $A B$, $F$ is the midpoint of $E C$ and $G$ is the midpoint of $F D$. What is the area enclosed by the quadrilateral $A E F G$ ?

Problem 19:

Let $X=1234 \cdots 78798081$ be the integer that consists of all the integers from $1$ to $81$ written from left to right. What is the remainder of $X$ when divided by $2020$ ?

Problem 20:

Let $A=\frac{1}{7} \times 3.14 \dot{1} \dot{5}$, where $3.14 \dot{1} \dot{5}$ is the rational number with recurring digits $15$ . In other words,
\[
3.14 \dot{1} \dot{5}=3.14+0.0015+0.000015+0.00000015+\cdots \] Suppose that $A=\frac{m}{n}$, where $m$ and $n$ are positive integers with no common factors larger than $1$ . What is the value of $m+n$ ?

Problem 21:

Ali and Barry went running on a standard $400$ metre track. They started simultaneously at the same location on the track but ran in opposite directions. Coincidentally, after $24$ minutes, they ended at the same location where they started. Ali completed $12$ rounds of the track in those $24$ minutes while Barry completed $10$ rounds. How many times did Ali and Barry pass each other during the run? (Exclude from your answer the times that they met at the start of the of run and when they completed the run after $24$ minutes.)

Problem 22:

If we have
\[
(1-3 x)+(1-3 x)^2+\cdots+(1-3 x)^{100}=a_0+a_1 x+a_2 x^2+\cdots+a_{100} x^{100},
\]
for some integers $a_0, a_1, \ldots, a_{100}$, what is the value of
\[
\left|\frac{a_1}{3}+\frac{a_2}{3^2}+\cdots+\frac{a_{100}}{3^{100}}\right| ?
\]

Problem 23:

What is the value of $\sqrt{27-\sqrt{704}}+\sqrt{27+\sqrt{704}}$ ?

Problem 24:

If $m>n$ are positive integers satisfying
\[
\left(m^2-n^2\right)^2=1+80 n,
\]
what is the smallest possible value of $mn$ ?

Problem 25:

What is the value of
\[\left(\frac{1}{2}+\frac{1}{3}\right+\frac{1}{4}+\frac{1}{5}+\ldots+\frac{1}{37}\right)+\left(\frac{2}{3}+\frac{2}{4}+\frac{2}{5}+\ldots+\frac{2}{37}\right)+\left(\frac{3}{4}+\frac{3}{5}+\frac{3}{6}+\ldots+\frac{3}{37}\right)+\ldots+\left(\frac{35}{36}+\frac{35}{37}\right)+\frac{36}{37}=?\]

Singapore Mathematics Olympiad - 2020- Senior Years - Questions

Multiple Choice Questions

Problem 01:

Let $b$ be a positive integer. If the minimum possible value of the quadratic function $5 x^2+b x+506$ is $6$ , find the value of $b$.
(A) 90
(B) 100
(C) 110
(D) 120
(E) 130

Problem 02:

Which of the following is equal to $\sqrt{5+\sqrt{3}}+\sqrt{5-\sqrt{3}}$ ?
(A) $\sqrt{10-\sqrt{22}}$
(B) $\sqrt{10+\sqrt{22}}$
(C) $\sqrt{10-2 \sqrt{22}}$
(D) $\sqrt{10+2 \sqrt{22}}$
(E) None of the above

Problem 03:

Simplify
$\log 8 5 \cdot\left(\log _5 3+\log {25} 9+\log _{125} 27\right)$ .
(A) $\log _2 3$
(B)$\log _3 2$
(C) $\log _2 9$
(D) $\log _3 16$
(E) $\log _2 27$

Problem 04:

Let $a=50^{\frac{1}{505}}, b=10^{\frac{1}{303}}$ and $c=6^{\frac{1}{202}}$. Which of the following is true?
(A) $a<b<c$
(B) $a<c<b$
(C) $b<a<c$
(D) $b<c<a$
(E) $c<b<a$

Problem 05:

Let $p=\log _{10}(\sin x), q=(\sin x)^{10}, r=10^{\sin x}$, where $0<x<\frac{\pi}{2}$. Which of the following is true? following is true?
(A) $p<q<r$
(B) $p<r<q$
(C) $q<r<p$
(D) $q<p<r$
(E) $r<p<q$

Short Questions

Problem 06:

Find the minimum possible value of $|x-10|-|x-20|+|x-30|$, where $x$ is any real number.

Problem 07:

Parallelogram $A B C D$ has sides $A B=39 \mathrm{~cm}$ and $B C=25 \mathrm{~cm}$. Find the length of diagonal $A C$ in $\mathrm{cm}$ if diagonal $B D=34 \mathrm{~cm}$.

Problem 08:

Suppose $\sin 45^{\circ}-x$=$-\frac{1}{3}$, where $45^{\circ}<x<90^{\circ}$. Find $6 \sin x-\sqrt{2})^2$.

Problem 09:

If $8 \cos x-8 \sin x=3$, find the value of $55 \tan x+\frac{55}{\tan x}$.

Problem 10:

Find the number of ordered pairs $x, y$, where $x$ and $y$ are integers, such that
$x^2+y^2-20 x-14 y+140<0$ .

Problem 11:

The figure below shows a right-angled triangle $A B C$ such that $\angle B A C=90^{\circ}, \angle A B C=$ $30^{\circ}$ and $A B=48 \mathrm{~cm}$. Let $P$ be a point on side $A B$ such that $C P$ is the angle bisector of $\angle A C B$ and $Q$ be a point on side $B C$ such that line $A Q$ is perpendicular to line $C P$. Determine the length of $P Q$.

Problem 12:

In the figure below, the point $O$ is the center of the circle, $A D$ and $B C$ intersect at $E$, and $\angle A E B=70^{\circ}, \angle A O B=62^{\circ}$. Find the angle $\angle O C D\left(\right.)$ in degree $\left.{ }^{\circ}\right)$.

Problem 13:

Find the value of $\frac{4 \cos 43^{\circ}}{\sin 73^{\circ}}-\frac{12 \sin 43^{\circ}}{\sqrt{3} \sin 253^{\circ}}$.

Problem 14:

If $\frac{x^2}{5}+\frac{y^2}{7}=1$, find the largest possible value of $(x+y)^2$.

Problem 15:

Find the coefficient of $x^6$ in the expansion of $\left(1+x+2 x^2\right)^7$.

Problem 16:

Suppose $3 x-y)^2+\sqrt{x+38+14 \sqrt{x-11}}+|z+x-y|=7$. Find the value of $|x+y+z|$.

Problem 17:

Suppose there are real numbers (x, y, z) satisfying the following equations: $x+y+z=60, x y-z^2=900$ Find the maximum possible value of $|z|$.

Problem 18:

Find the sum $\sum_{k=1}^{16} \log _2\left(\sqrt{\sin ^2 \frac{k \pi}{8}+1}-\sin \frac{k \pi}{8}\right)$

Problem 19:

Let $a, b$ be positive real numbers, where $a>b$. Suppose there exists a real number (x) such that $\left(\log _2 a x\right)\left(\log _2 b x\right)+25=0$. Find the minimum possible value of $\frac{a}{b}$.

Problem 20:

The figure below shows a rectangle (A B C D) such that the diagonal $A C=20 \mathrm{~cm}$. Let (P) be a point on side $C D$ such that $B P$ is perpendicular to diagonal $A C$. Find the area of rectangle $A B C D$ $in (\mathrm{cm}^2) $ if $B P=15 \mathrm{~cm}$.

Problem 21:

Find the smallest positive integer that is greater than the following expression:
($\sqrt{7}+\sqrt{5})^4$.

Problem 22:

Find the number of non-congruent right-angled triangles such that the length of all their sides are integers and that the hypotenuse has a length of $65 \mathrm{~cm}$.

Problem 23:

There are 6 couples, each comprising a husband and a wife. Find the number of ways to divide the 6 couples into 3 teams such that each team has exactly 4 members, and that the husband and the wife from the same couple are in different teams.

Problem 24:

The digit sum of a number, say 987 , is the sum of its digits, $9+8+7=24$. Let (A) be the digit sum of $2020^{2021}$, and let (B) be the digit sum of (A). Find the digit sum of (B).

Problem 25:

$40=2 \times 2 \times 2 \times 5$ is a positive divisor of 1440 that is a product of 4 prime numbers. $48=2 \times 2 \times 2 \times 2 \times 3$ is a positive divisor of 1440 that is a product of 5 prime numbers. Find the sum of all the positive divisors of 1440 that are products of an odd number of prime numbers.

Singapore Mathematics Olympiad - 2021- Junior Years - Questions

Multiple Choice Questions

Problem 01:

Let $a$ and $b$ be real numbers satisfying $a<0<b$. Which of the following is not true?

(A)$a^2 b<0$
(B) $a b^2<0$ (C) $\frac{a}{b}>0$
(D) $b-a>0$
(E) $|a-b|>0$

Problem 02:

The following diagram shows a system of balances hanging from the ceiling with three types of weights. The balances tip down to the heavier side. If we use $\square<\triangle$ to represent is lighter than $\triangle$, which of the following is true?

(A) $\square<\circ<\triangle$
(B) $\square<\triangle<\circ$
(C) $\triangle<\square<\circ$
(D) $\triangle<\circ<\square$
(E) $\circ<\square<\triangle$

Problem 03:

Let $x=2^{20} \cdot 3^5, y=2^5 \cdot 5^{10}$ and $z=7^{10}$. Which of the following is true?
(A) $x>y>z$
(B) $x>z>y$
(C) $y>z>x$
(D) $y>x>z$
(E) $z>x>y$

Problem 04:

In the diagram, six circles are tangent to each other. If the radius of the largest circle is $1$ and the radii of the four medium sized circles are equal, what is the radius of the smallest circle?

(A) $\sqrt{2}-1$

(B) $3-2 \sqrt{2}$

(D) $6-4 \sqrt{2}$

(E) None of the above

Problem 05:

Which of the following is closest to the value of
\[\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+\cdots+\frac{1}{\sqrt{2021}+\sqrt{2020}}\]
(A) $10$
(B) $20$
(C) $30$
(D) $40$
(E) $50$

Short Questions

Problem 06:

Let $x$ be a positive integer. Suppose that the lowest common multiple of $x$ and $14$ is $42$ and the lowest common multiple of $x$ and $33$ is $66$ . What is the value of $x$ ?

Problem 07:

What are the last four digits of the sum
$1+22+333+4444+\cdots+\underbrace{999999999}_{\text {nine } 9 \mathrm{~s}} \text { ? A205 }$. Give your answer as a 4-digit number.

Problem 08:

How many distinct triples of positive integers $(a, b, c)$ satisfy $1 \leqslant a \leqslant b \leqslant c$ and
$\frac{1}{a_3}+\frac{1}{b_3}+\frac{1}{c_3}=1 $ ?

Problem 09:

Given five consecutive positive integers, if the product of the largest and the smallest integer is $2021$ , what is the sum of the five integers?

Problem 10:

The numbers from $1$ to $2021$ are concatenated from left to right and the result is read as an integer $12345678910111213 \cdots 201920202021$ .What is the remainder when this number is divided by $6 $?

Problem 11:

In the figure below, each distinct letter represents a unique distinct digit such that the arithmetic holds. If $S$ represents $6 $and $E$ represents $8$,-what number does SIX represent?

Problem 12:

What is the value of
$\sqrt{(219)(220)(221)(222)+1} = ?$

Problem 13:

Let $A, B, \ldots, I$ be unknowns satisfying

$A+B+C=1$,
$ B+C+D=2$,
$C+D+E=3$,
$ D+E+F=4$,
$E+F+G=5$,
$ F+G+H=6$,
$G+H+I=7$ .
What is the value of $A+E+I$ ?

Problem 14:

If $x$ is a 3-digit number, we define $M(x)$ and $m(x)$ respectively as the largest and smallest positive number that can be formed by rearranging the three digits of (x). For example, if $x=123$, then $M(123)=321$ and $m(123)=123$. If $y=898$, then $M(898)=988$ and $m(898)=889$.
Given that $z$ is a 3-digit number that satisfies $z=M(z)-m(z)$, what is the value of $z$ ?

Problem 15:

How many integers $k$ are there such that the quadratic equation $k x^2+20 x+20-k=0$ has only integer solutions?

Problem 16:

In the following diagram, $A B C D$ is a quadrilateral inscribed in a circle with center $O$. If $|A B|=|B C|=6,|A D|=14$ and $C D$ is a diameter, what is the length of $|C D|$ ?

Problem 17:

Problem 18:

What is the value of $\lfloor\sqrt{45+\sqrt{2021}}-\sqrt{45-\sqrt{2021}}\rfloor$ ?

Problem 19:

Let $x$ be the positive real number that satisfies $\sqrt{x^2-4 x+5}+\sqrt{x^2+4 x+5}=3 x $.

What is the value of $\left\lfloor 10^4 x^2\right\rfloor$ ?

Problem 20:

What is the number of positive integers $c$ such that the equation $x^2-2021 x+100 c=0$ has real roots?

Problem 21:

In chess, two queens are said to be attacking each other if they are positioned in the same row, column or diagonal on a chessboard. How many ways are there to place two identical queens in a $4 \times 4$ chessboard such that they do not attack each other?
$\frac{1}{2} \times \frac{1}{4} \times 401 \times 403 x \times 801=$

Problem 22:

Let $A=\frac{3}{2} \times \frac{5}{4} \times \frac{7}{6} \times \cdots \times \frac{801}{800}$. What is the value of $\left\lfloor\frac{A}{10}\right\rfloor$ ?

Problem 23:

A $3 \times 3$ grid is filled with the integers $1$ to $9$ . An arrangement is nicely ordered if the integers in each horizontal row is increasing from left to right and the integers in each vertical column is increasing from top to bottom. Two examples of nicely ordered arrangements are given in the diagram below. What is the total number of distinct nicely ordered arrangements?

Problem 24:

A class has exactly $50$ students and it is known that $40$ students scored $A$ in English, $45$ scored $A$ in Mathematics and $42$ scored $A$ in Science. What is the minimum number students who scored $A$ in all three subjects?

Problem 25:

Suppose a positive integer $x$ satisfies the following equation
$\sqrt[5]{x+76638}-\sqrt[5]{x-76637}=5 $.

What is the value of $x$ ?

Singapore Mathematics Olympiad - 2023- Junior Years - Questions

Multiple Choice Questions

Problem 01:

In a certain company, one-third of the employees have a university degree. Half of the employees who have a university degree are men and $40 \%$ of the employees who do not have a university degree are women. If 102 of the employees are men, how many of the employees are women?
(A) 58
(B) 78
(C) 98
(D) 108
(E) 118

Problem 02:

How many non-congruent triangles with integer side lengths have perimeter 7 ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Problem 03:

The median and mean of five distinct numbers, (4,7,10,11, N), are equal. Find the sum of all possible values of (N).
(A) 18
(B) 21
(C) 26
(D) 29
(E) 35

Problem 04:

(A) $8 \pi$
(B) $16 \pi$
(C) $32 \pi$
(D) $48 \pi$
(E) $64 \pi$

Problem 05:

The graph (not drawn to scale) of (y=a x^2+b x+c) with the maximum point at (x=-1) is shown below. Which of the following must be true?

(I) (a-b+c>0)
(II) (a+b-c>0)
(III) (a b c>0)
(IV) (a b c<0)
(A) (I) and (IV)
(B) (II) and (IV)
(C) (I) and (III)
(D) (II) and (III)
(E) (I)

Short Questions

Problem 06:

The product of the ages of three adults is 26390 . Find the sum of their ages. (A person is an adult if he or she is at least 21 years old.)

Problem 07:

Let (n) be a positive integer such that (n+11) is a factor of (n^2+121). Find the largest possible value of (n).

Problem 08:

Find the largest integer less than or equal to $(3+\sqrt{5})^3$.

Problem 09:

The product of the two-digit number $\overline{x 4}$ and the three-digit number $\overline{3 y z}$ is 7656 . Find the value of (x+y+z).

Problem 10:

If (x) and (y) are real numbers such that (x+y=12) and (x y=10), find the value of (x^4+y^4).

Problem 11:

Let (a) and (b) be two positive integers such that $a^2+4 b^2 \leq 1105$. Find the largest possible value of the product (a b).

Problem 12:

Find the value of the integer (n) such that the following equation holds:
$[
\frac{\sqrt{5}+n \sqrt{3}-2 \sqrt{2}}{(\sqrt{5}+\sqrt{3})(\sqrt{3}-\sqrt{2})}=\sqrt{5}+\sqrt{2} .
]$

Problem 13:

The lengths of the sides of a triangle are $ \log_{10} 12$, $\log_{10} 75$ and $\log _{10}n$ where (n) is a positive integer. Find the number of possible values for (n).

Problem 14:

Let (x) be a real number. What is the minimum value of the following expression?
$[
\frac{20 x^2+10 x+3}{2 x^2+x+1}
]$

Problem 15:

Problem 16:

Find the smallest positive integer (k) such that every subset of ${1,2, \ldots, 2023}$ with exactly (k) integers contains three integers (a, b, c) with (a<b<c) and (a b=c).

Problem 17:

The difference between the least common multiple (LCM) and highest common factor (HCF) of a positive integer (n) and 18 is 627 . Find the value of (n).

Problem 18:

Problem 19:

If $\sqrt{19-8 \sqrt{3}}$ is a root of the equation (x^2-a x+b=0) where (a) and (b) are rational numbers, find the value of (a+b).

Problem 20:

Find the smallest positive integer (n) such that the equation (x^2+y^3=n^4) has a solution in positive integers (x) and (y).

Problem 21:

Problem 22:

The angle between the hour-hand and the minute-hand of a clock is denoted by $\theta^{\circ}$ where $0 \leq \theta \leq 180$. If the clock now shows (3: y) (that is, 3 o'clock and (y) minutes) and $\theta=175$, find the value of (y), where (y) is an integer and $0 \leq y<60$.

Problem 23:

The diagram below shows a river with two bridges at $\mathrm{A}$ and $\mathrm{B}$ that are $2 \mathrm{~km}$ apart. The water flows from west to east at a constant speed.

A man rowed a boat upstream. He dropped his water bottle at $\mathrm{A}$ but he only noticed it at $\mathrm{C}$ after rowing for another 20 minutes upstream from $\mathrm{A}$. He immediately turned back and rowed his boat downstream, and was able to pick up his water bottle at B. Assume that the man rowed his boat at a constant speed.
Find the speed of the water flowing in the river in $\mathrm{km} / \mathrm{h}$.

Problem 24:

Four positive integers (x, y, z) and (w) satisfy the following equations:
$[
\begin{aligned}
& x y+x+y=104 \
& y z+y+z=146 \
& z w+z+w=524
\end{aligned}
]$

If the product $x y z w=2^7 \times 3^2 \times 5 \times 7$, find the value of (x+y+z+w).

Problem 25:

In the following diagram, $\mathrm{ABC}$ is a triangle. Points $\mathrm{E}, \mathrm{D}$ and $\mathrm{F}$ lie on the side $\mathrm{BC}$ and divide the side into four equal parts. $\mathrm{M}$ is the midpoint of $\mathrm{AB}$ and $\mathrm{CM}$ intersects the line segments $\mathrm{AE}, \mathrm{AD}$ and $\mathrm{AF}$ into the ratio (x: y: z: u) where $x \geq y \geq z \geq$ and (x, y, z) and (u) are integers. Find the least possible value of (x+y+z+u).

Singapore Mathematics Olympiad - 2023- Senior Years - Questions

Multiple Choice Questions

Problem 01:

Find the value of $m$ such that $2 x^2+3 x+m$ has a minimum value of $9$ .
(A) $\frac{9}{8}$
(B) $-\frac{9}{8}$
(C) $\frac{81}{8}$
(D) $-\frac{81}{8}$
(E) $\frac{63}{8}$

Problem 02:

Which of the following is true?
(A) $\sin \left(105^{\circ}\right)-\cos \left(105^{\circ}\right)=\frac{\sqrt{3}}{2}$
(B) $\sin \left(105^{\circ}\right)-\cos \left(105^{\circ}\right)=\frac{\sqrt{3}}{\sqrt{2}}$
(C) $\sin \left(105^{\circ}\right)+\cos \left(105^{\circ}\right)=\frac{1}{2}$
(D) $\sin \left(105^{\circ}\right)+\cos \left(105^{\circ}\right)=\frac{1}{\sqrt{3}}$
(E) None of the above.

Problem 03:

If $\log {\sqrt{2}} x=10-3 \log {\sqrt{2}} 10$, find $x$.
(A) 0.32
(B) 0.032
(C) 0.0032
(D) 0.64
(E) 0.064

Problem 04:

If $(x-5)^2+(y-5)^2=5^2$, find the smallest value of $(x+5)^2+(y+5)^2$.
(A) $225-100 \sqrt{2}$
(B) $225+100 \sqrt{2}$
(C) $225 \sqrt{2}$
(D) $100 \sqrt{2}$
(E) None of the above

Problem 05:

Suppose $\cos \left(180^{\circ}+x\right)=\frac{4}{5}$, where $90^{\circ}<x<180^{\circ}$. Find $\tan (2 x)$.

(A) $\frac{24}{7}$
(B) $\frac{7}{24}$
(C) $-\frac{24}{7}$
(D) $-\frac{7}{24}$
(E) $-\frac{24}{25}$

Short Questions

Problem 06:

Suppose the roots of $x^2+11 x+3=0$ are $p$ and $q$, and the roots of $x^2+B x-C=0$ are $p+1$ and $q+1$. Find $C$.

Problem 07:

If the smallest possible value of $(A-x)(23-x)(A+x)(23+x)$ is $-(48)^2$, find the value of $A$ if $A>0$.

Problem 08:

Find the smallest positive odd integer greater than 1 that is a factor of
$(2023)^{2023}+(2026)^{2026}+(2029)^{2029}$ .

Problem 09:

Find the remainder of $7^{2023}+9^{2023}$ when divided by $64$ .

Problem 10:

Let $x, y, z>1$, and let $A$ be a positive number such that $\log x A=30, \log _y A=50$ and $\log {x y}(A z)=150$. Find

$\left(\log _A z\right)^2$.

Problem 11:

Find the largest integer that is less than
$\text { - } \frac{3^{10}-2^{10}}{10 !}\left(\frac{1}{1 ! 9 ! 2}+\frac{1}{2 ! 8 ! 2^2}+\frac{1}{3 ! 7 ! 2^3}+\cdots+\frac{1}{9 ! 1 ! 2^9}\right)^{-1}$ .

Here, $n !=n \cdot(n-1) \cdots 3 \cdot 2 \cdot 1$. For example, $5 !=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=120$.

Problem 12:

Consider the following simultaneous equations:

$ x y^2+x y z=91$
$ x y z-y^2 z=72$,

where $x, y$, and $z$ are positive integers. Find the maximum value of $x z$.

Problem 13:

Let $x$ be a real number such that
$\frac{\sin ^4 x+\cos ^4 x}{\sin ^2 x+\cos ^4 x}=\frac{8}{11}$ .
Assuming $\sin ^2 x>\frac{1}{2}$, find the value of $\sqrt{28}\left(\sin ^4 x-\cos ^4 x\right)$.

Problem 14:

A sequence $a_1, a_2, \ldots$, is defined by
$a_1=5, a_2=7, a_{n+1}=\frac{a_n+1}{a_{n-1}} \text { for } n \geq 2$ .
Find the value of $100 \times a_{2023}$

Problem 15:

Let $C$ be a constant such that the equation $5 \cos x+6 \sin x-C=0$ have two distint roots $a$ and $b$, where $0<b<a<\pi$. Find the value of $61 \times \sin (a+b)$.

Problem 16:

In the diagram below, $C E$ is tangent to the circle at point $D, A D$ is a diameter of the circle, and $A B C, A F E$ are straight lines. It is given that $\frac{A B}{A C}=\frac{16}{41}$ and $\frac{A F}{A E}=\frac{49}{74}$. Let $\tan (\angle C A E)=\frac{m}{n}$, where $m, n$ are positive integers and $\frac{m}{n}$ is a fraction in its lowest form. Find the sum $m+n$.

Problem 17:

In the diagram below, $A B$ is a diameter of the circle with centre $O, M N$ is a chord of the circle that intersects $A B$ at $P, \angle B O N$ and $\angle M O A$ are acute angles, $\angle M P A=45^{\circ}$, $M P=\sqrt{56}$, and $N P=12$. Find the radius of the circle.

Problem 18:

Let $f(x)=\cos ^2\left(\frac{\pi x}{2}\right)$. Find the value of
$f \left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+\cdots+f\left(\frac{2021}{2023}\right)+f\left(\frac{2022}{2023}\right)$ .

Problem 19:

Find the remainder when $3^{2023}$ is divided by $215$ .

Problem 20:

Find the sum of the prime divisors of $64000027$ .

Problem 21:

Let $\triangle A B C$ be an equilateral triangle. $D, E, F$ are points on the sides such that
$B D: D C=C E: E A=A F: F B=2: 1$ .
Suppose the area of the triangle bounded by $A D, B E$ and $C F$ is $2023$ . Find the area of $\triangle A B C$.

Problem 22:

Find the number of possible ways of arranging $m$ ones and $n$ zeros in a row such that there are in total $2 k+1$ strings of ones and zeros. For example, $1110001001110001$ consists of 4 strings of ones and 3 strings of zeros.

Problem 23:

Suppose that there exist numbers $a, b, c$ and a function $f$ such that for any real numbers $x$ and $y$,
$f(x+y)+f(x-y)=2 f(x)+2 f(y)+a x+b y+c$ .
It is given that
$f(2)=3, \quad f(3)=-5, \quad$and $\quad f(5)=7$ . Find the value of $f(123)$.

Problem 24:

Let $f$ be a function such that for any nonzero number $x$,
$6 x f(x)+5 x^2 f(1 / x)+10=0$ .
Find the value of $f(10)$.

Problem 25:

Find the number of triangles such that all the sides are integers and the area equals the perimeter (in number).

Singapore Mathematics Olympiad - 2022- Senior Years - Questions

Multiple Choice Questions

Problem 01:

Suppose the roots of $\frac{x^2}{2}+m x+n=0$ are $\frac{m}{2}$ and $\frac{n}{3}$. Find the smallest value of $mn$.
(A) -1080
(B) -90
(C) 0
(D) 90
(E) 1080

Problem 02:

Which of the following is true?
(A) $\sqrt[6]{\frac{1}{333}}<\sqrt[3]{\frac{1}{18}}<\sqrt{\frac{1}{7}}$
(B) $\sqrt[3]{\frac{1}{18}}<\sqrt[6]{\frac{1}{333}}<\sqrt{\frac{1}{7}}$
(C) $\sqrt[3]{\frac{1}{18}}<\sqrt{\frac{1}{7}}<\sqrt[6]{\frac{1}{333}}$
(D) $\sqrt{\frac{1}{7}}<\sqrt[6]{\frac{1}{333}}<\sqrt[3]{\frac{1}{18}}$
(E) None of the above.

Problem 03:

Suppose $\sqrt{\left(\log {377 \times 377} 2022\right)\left(\log {377} 2022\right)}=\log _k 2022$. Find $k$.
(A) $\sqrt{337}$
(B) $337^{\sqrt{2}}$
(C) $337 \sqrt{2}$
(D) $\sqrt{337}^{\sqrt{2}}$
(E) $\sqrt{337 \times 2}$

Problem 04:

Suppose $y=\cos ^2 x-7 \cos x+25$, where $x$ is any real number. Find the range of $y$.
(A) $17 \leq y \leq 33$
(B) $18 \leq y \leq 33$
(C) $19 \leq y \leq 33$
(D) $20 \leq y \leq 33$
(E) None of the above

Problem 05:

Suppose $\sin \left(180^{\circ}+x\right)=-\frac{7}{9}$, where $450^{\circ}<x<540^{\circ}$. Find $\sin (2 x)$.
(A) $\frac{49}{81} \sqrt{2}$
(B) $\frac{56}{81} \sqrt{2}$
(C) $-\frac{56}{81}$
(D) $-\frac{49}{81} \sqrt{2}$
(E) $-\frac{56}{81} \sqrt{2}$

Short Questions

Problem 06:

Find the value of
$\left(\frac{\cos 10^{\circ}+\cos 50^{\circ}+\cos 70^{\circ}+\cos 110^{\circ}}{\cos 20^{\circ}}\right)^8$ .

Problem 07:

Suppose $x^{20}+\frac{x^{10}}{2}-\frac{3^{2 x}}{9}+\frac{1}{16}=0$ for some positive real number $x$. Find the value of

$4 \cdot 3^x-12 x^{10}$.

Problem 08:

How many positive integers less than or equal to 2022 cannot be expressed as
$\lfloor 2 x+1\rfloor+\lfloor 5 x+1\rfloor$ for some real number $x$ ? Here, $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For example, $\lfloor-2.1\rfloor=-3,\lfloor 3.9\rfloor=3$.

Problem 09:

Suppose
$y=\frac{\tan ^2 x-\tan x+\sqrt{33}}{\tan ^2 x+\tan x+1}$,
where $-90^{\circ}<x<90^{\circ}$. Find the maximum possible value of $\sqrt{33}(y-5)$.

Problem 10:

In the figure below, $P Q R S$ is a square inscribed in a circle. Let $W$ be a point on the arc $P Q$ such that $W S=\sqrt{20}$. Find $(W P+W R)^2$.

Problem 11:

The figure below shows a quadrilateral $A B C D$ such that $A C=B D$ and $P$ and $Q$ are the midpoints of the sides $A D$ and $B C$ respectively. The lines $P Q$ and $A C$ meet at $R$ and the lines $B D$ and $A C$ meet at (S). If $\angle P R C=130^{\circ}$, find the angle $\angle D S C$ in ${ }^{\circ})$.

Problem 12:

How many distinct terms are there if $\left(x^2+y^2\right)^{11}\left(x^{11}+y^{11}\right)^9$ is algebraically expanded and simplified?

Problem 13:

If $\sqrt{x^2+7 x-4}+\sqrt{x^2-x+4}=x-1$, find the value of $3 x^2+14 x$.

Problem 14:

Let $k=-1+\sqrt{2022^{1 / 5}-1}$, and let $f(x)=\left(k^2+2 k+2\right)^{10 x}$. Find the value of $\log _{2022} f(2022)$.

Problem 15:

Find the smallest odd integer $N$, where $N>2022$, such that when $1808,2022$ and $N$ are each divided by a positive integer $p$, where $p>1$, they all leave the same remainder.

Problem 16:

If $\frac{12}{x}+\frac{48}{y}=1$, where $x$ and $y$ are positive real numbers, find the smallest possible value of $x+y$.

Problem 17:

Find the largest value of $40 x+60 y$ if $x-y \leq 2,5 x+y \geq 5$ and $5 x+3 y \leq 15$.

Problem 18:

Suppose
$\cos x-\cos y =\frac{1}{2}$,
$\sin x-\sin y =-\frac{1}{3}$

If $\sin (x+y)=\frac{m}{n})$, where $\frac{m}{n}$ is expressed as a fraction in its lowest terms, find the value of $m+n$.

Problem 19:

For some positive integer $n$, the number $n^3-3 n^2+3 n$ has a units digit of $6$ . Find the product of the last two digits of the number $7(n-1)^{12}+1$.

Problem 20:

Find the largest positive integer $n$ for which $\frac{20 n+2020}{3 n-6}$ is a positive integer.

Problem 21:

In the $x y$-coordinate system, there are two circles passing through the point $11,3 \sqrt{3}$, and each of these circles is tangent to both the $x$-axis and the line $y=\sqrt{3} x$. Let $S$ be the sum of the radii of the two circles. Find $\sqrt{3} S$.

Problem 22:

Let $P$ and $Q$ be the points $20(\sqrt{5}-1), 0$ and $0,10(\sqrt{5}-1)$ on the $x y$-plane. Let $R$ be the point $a, b$. If $\angle P R Q$ is a right angle, find the maximum possible value of $b$.

Problem 23:

How many positive integers $n$ do not satisfy the inequality $n^{\frac{1}{3} \log _{20} n}>\sqrt{n}$ ?

Problem 24:

Let $f(x)$ be a function such that $3 f\left(x^2\right)+f(13-4 x)=3 x^2-4 x+293$
for all real number $x$. Find the value of $f(1)$.

Problem 25:

Find the largest positive integer $M$ such that $\cos ^2 x-\sin ^2 x+\sin x=\frac{M}{888}$ has a real solution.

Singapore Mathematics Olympiad - 2022- Junior Years - Questions

Multiple Choice Questions

Problem 01

Which of the five numbers, $30^2, 10^3, 5^4, 4^5$ or $3^6$, is the largest?

(A) $30^2$
(B) $10^3$
(C) $5^4$
(D) $4^5$
(E) $3^6$

Problem 02

Adele has a fruit basket containing one apple, two bananas and three oranges, while Betty has a fruit basket containing two apples and three oranges. Assuming that Adele and Betty each randomly picked a fruit from her own basket, what is the probability that together they picked an apple and an orange?

(A) $\frac{1}{10}$
(B) $\frac{3}{10}$
(C) $\frac{1}{5}$
(D) $\frac{6}{11}$
(E) None of the above

Problem 03

The following diagram shows a system of balances hanging from the ceiling with three types of weights. The balances tip down to the heavier side. If we use $\square<\Delta$ to represent $\square$ is lighter than $\Delta$, which of the following is true?

(A) $\circ< \square<\Delta$
(B) $\circ<\Delta< \square$

(E) Impossible to tell

Problem 04

A shop sells two types of buns, with either cream or jam filling, which are indistinguishable until someone bites into the buns. Four mathematicians visited the shop and ordered (not necessarily in that sequence): three cream buns, two cream buns and one jam bun, one cream and two jam buns, and three jam buns. Each knew precisely what the others had ordered. Unfortunately, the shop owner mixed up the orders and gave each mathematician the wrong order!
The mathematicians started eating, all still unaware of the mixup, until the shop owner ran over to inform them of the mistake. Mathematician A said: "I ate two buns and both had cream filling. So, if my order was wrong, I now know what type my third bun is." Mathematician B then said: "I only ate one bun and it had cream filling. Based on what A said and since I remember A's order, I now know what type my other two buns are." Finally, Mathematician C said: "I have not started eating but I must have received three jam buns." Which of the following statements about Mathematician D is correct?

(A) D ordered two cream and one jam but received three jam buns.
(B) D ordered one cream and two jam but received two cream and one jam buns.
(C) D ordered three cream but received one cream and two jam buns.
(D) D ordered three jam but received three cream buns.
(E) None of the above

Problem 05

If (n) can take the value of any positive integer, how many distinct values can the expression
$[
\left\lfloor\frac{n}{4}\right\rfloor-\left\lfloor\frac{n}{7}\right\rfloor-\left\lfloor\frac{n}{14}\right\rfloor-\left\lfloor\frac{n}{28}\right\rfloor
]$
take?
(A) 1
(B) 2
(C) 3
(D) 4
(E) infinitely many

Short Questions

Problem 06

Let (n) be a positive integer. If the lowest common multiple of (n) and 12 is 60 , and the highest common factor of (n) and 36 is 4 , what is the value of (n) ?

Problem 07

The digits (1,2,3,4,5) and 6 are arranged to form two positive integers with each digit appearing exactly once. How many ways can this be done if the sum of the two integers is 570 ?

Problem 08

Consider the following product of two mixed fractions
$m\frac{6}{7} \times n \frac{1}{3}=23$,
where (m) and (n) are positive integers. What is the value of (m+n) ?

Problem 09

What are the last four digits of the sum
$222+2022+20022+\cdots+2 \underbrace{0000000000}_{\text {ten } 0 s} 22$ ?

Give your answer as a 4-digit number.

Problem 10

If (a) and (b) are distinct solutions to the equation
$x^2+10 x+20=0$,
what is the value of $a^4+b^4$ ?

Problem 11

The following diagram shows a star that is cut out from a square with sides of length 30 . What is the area enclosed by the star?

Problem 12

If (x) is a real number, how many solutions are there to the equation

$(3 x+2)^{x+5}=1 \text { ? }$

Problem 13

In the figure below, each distinct letter represents a unique distinct digit such that the arithmetic holds. If $\mathrm{W}$ represents 5 , what number does TROOP represent?

Problem 14

Problem 15

In the following diagram, (A B C) is a triangle and (D) lies on (A B) such that the ratio of $|A D|:|D B|=3: 4$. Points (E) and (G) lie on (A C) and (B C) respectively such that (D B G E) is a parallelogram and (F) is the intersection of (D C) and (G E). If the ratio of the area of $\triangle E D F$ to the area of (\triangle A B C) is (m: n), where the highest common factor of (m) and (n) is 1 , what is the value of $m+n$ ?

Problem 16

Eggs in a certain supermarket are sold only in trays containing exactly 10,12 or 30 eggs per tray. It is thus impossible to buy exactly 14 eggs or any odd number of eggs. However, it is possible to buy exactly 78 eggs using four trays of 12 and one tray of 30 . What is the largest even number of eggs that is impossible to be bought from this supermarket?

Problem 17

How many integers (n) are there in ${1,2, \ldots, 2022}$ such that at least one of the digits of $n$ is ' 2 '?

Problem 18

What is the smallest (k) such that every subset of ${1,2, \ldots, 99}$ with exactly $k$ integers contains at least two distinct integers (x) and (y), such that $|x-y| \leqslant 2$ ?

Problem 19

In the following diagram, (A C D) is a triangle such that $|A B|=|B C|, \angle A B D=45^{\circ}) and (\angle B D C=15^{\circ}$. If $\angle A D B=x^{\circ}$, what is the value of $x$ ?

Problem 20

What is the area of a triangle with side lengths
$\sqrt{6^2+7^2}, \sqrt{12^2+7^2} \text { and } \sqrt{6^2+14^2} \text { ? }$

Problem 21

If $x=\sqrt[3]{4}+\sqrt[3]{2}+1)$, what is the value of
$2022+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3} ?$

Problem 22

If we have
$\frac{\sqrt{15}+\sqrt{35}+\sqrt{21}+5}{\sqrt{3}+2 \sqrt{5}+\sqrt{7}}=\frac{a \sqrt{7}+b \sqrt{5}+c \sqrt{3}}{2}$
for some integers (a, b, c). What is the value of (a+b+c) ?

Problem 23

How many integers (n) are there in ${1,2, \ldots, 2022}$ such that $\lfloor\sqrt[3]{n}\rfloor$ is a factor of $n$ ?

Problem 24

If (x) and (y) can take any real values, what is the smallest possible value of the expression
$2 x^2+4 x y+5 y^2+4 x+10 y+13 ?$

Problem 25

The following diagram shows a rectangle that is partitioned into 17 squares. If the two smallest squares, shaded in grey, have sides of length 2 , what is the area of the rectangle?