Singapore Mathematics Olympiad - 2019- Senior Years - Questions

Multiple Choice Questions

Problem 01:

The roots of the quadratic equation $x^2-7 m x+5 n=0$ are $m$ and $n$, where $m \neq 0$ and $n \neq 0$. Find a quadratic equation whose roots are $\frac{m}{n}$ and $\frac{n}{m}$.
(A) $6 x^2-37 x+1=0$
(B) $6 x^2-50 x-7=0$
(C) $6 x^2-50 x+7=0$
(D) $6 x^2-37 x+6=0$
(E) $x^2-37 x+1=0$

Problem 02:

Simplify
\[
(\sqrt{10}-\sqrt{2})^{\frac{1}{3}}(\sqrt{10}+\sqrt{2})^{\frac{7}{3}} .
\]
(A) $24+4 \sqrt{5}$
(B) $24+6 \sqrt{5}$
(C) $24+8 \sqrt{5}$
(D) $24+10 \sqrt{5}$
(E) $24+12 \sqrt{5}$

Problem 03:

Let $a=4^{3000}, b=6^{2500}$ and $c=7^{2000}$. Which of the following statement is true?
(A) $a<b<c$
(B) $a<c<b$
(C) $b<a<c$
(D) $c<a<b$
(E) $c<b<a$

Problem 04:

If $\log _{21} 3=x$, express $\log _7 9$ in terms of $x$.
(A) $\frac{2 x}{2-x}$
(B) $\frac{2 x}{1-x}$
(C) $\frac{2 x}{x-2}$
(D) $\frac{2 x}{x-1}$
(E) $\frac{x}{1-x}$

Problem 05:

Suppose that $\sin x=\frac{12}{13}$ and $\cos y=-\frac{4}{5}$, where $0^{\circ} \leq x \leq 90^{\circ}$ and $90^{\circ} \leq y \leq 180^{\circ}$. Find the value of $\cos (x+y)$.

(A) $-\frac{56}{65}$

(B) $\frac{56}{65}$

(D) $\frac{16}{65}$

(E) None of the above

Short Questions

Problem 06:

Find the largest positive integer (n) such that (n+8) is a factor of $n^3+13 n^2+40 n+40$.

Problem 07:

Suppose $\tan x=5$. Find the value of $\frac{6+\sin 2 x}{1+\cos 2 x}$.

Problem 08:

Suppose (x) and (y) are real numbers such that
\[
|x-y|+3 x-y=70, \text { and } \
|y-x|+3 y+x=50 .
\]

Find the maximum possible value of $x+2 y$.

Problem 09:

The coordinates of the vertices of a triangle $\triangle A B C$ are $A(6,0), B(0,8)$ and $C(x, y)$ such that $x^2-16 x+y^2-12 y+91=0$. Find the largest possible value of the area of the triangle $\triangle A B C$.

Problem 10:

In the figure below, $A D$ is perpendicular to the $B C, P Q$ is parallel to $B C$, and the triangle $\triangle P Q R$ is an equilateral triangle whose area in $meter ^2$ is equal to the length of $A D$ (in meter). Find the smallest possible value of the length of (B C).

Problem 11:

Find the value of $448\left(\frac{\sin 12^{\circ} \sin 39^{\circ} \sin 51^{\circ}}{\sin 24^{\circ}}\right)$.

Problem 12:

In the figure below, the chord $A F$ passes through the origin $O$ of the circle, and is perpendicular to the chord $B C$. It is given that $A B=17 \mathrm{~cm}, C D=5 \mathrm{~cm}$. Suppose $\frac{B E}{E D}=\frac{m}{n}$, where $m$ and $n$ are positive integers which are relatively prime. What is the value of $m+n$ ?

Problem 13:

Let $P(x)$ be the polynomial that results from the expansion of the following expression:
\[
\left(2 x^3+3 x^2+x\right)^5\left(\frac{x}{6}+\frac{1}{2}\right)^5 .
\]

Find the sum of the coefficients of $x^{2 k+1}$, where $k=0,1,2,3, \ldots, 9$.

Problem 14:

Find the value of the following expression:
\[
\frac{2\left(1^2+2^2+3^2+\ldots+49^2+50^2\right)+(1 \times 2)+(2 \times 3)+(3 \times 4)+\ldots+(48 \times 49)+(49 \times 50)}{100} .
\]

Problem 15:

Let $M$ be the maximum possible value of $\frac{15 x^2-x^3-39 x-54}{x+1}$, where $x$ is a positive integer. Find the value of $9 M$.

Problem 16:

Find the maximum possible value of $x+y+z$ where $x, y, z$ are integers satisfying the following system of equations:
\[
x^2 z+y^2 z+8 x y=200 \
2 x^2+2 y^2+x y z=50 .
\]

Problem 17:

Find the remainder when $10^{43}$ is divided by $126$ .

Problem 18:

Suppose $\left(\log _2 x\right)^2$+$4\left(\log _2\left(\log _3 y\right)\right)^2$ = $4\left(\log _2 x\right)\left(\log _2\left(\log _3 y\right)\right)$. If $x = 49$ and $y$ is a positive integer, find $y$.

Problem 19:

The figure below shows a rectangle $A B C D$ with $A B=16 \mathrm{~cm}) and (B C=15 \mathrm{~cm}$. Let $P$ be a point on the side $B C$ such that $B P=7 \mathrm{~cm}$, and let $Q$ be a point on the side $C D$ such that $C Q=6 \mathrm{~cm}$.
Find the length of $A R$ $in (\mathrm{cm})$, where $R$ is the foot of the perpendicular from $A$ to $P Q$.

Problem 20:

A sequence $x_0, x_1, x_2, x_3, \ldots$ of integers satisfies the following conditions: $x_0=1$, and for any positive integer $n \geq 1,\left|x_n-1\right|=\left|x_{n-1}+2\right|$. Find the maximum possible value of $2019-\left(x_1+x_2+\cdots+x_{2018}\right)$.

Problem 21:

Consider a square $A B C D$ on the $x y$-plane where the coordinates of its vertices are given by $A(13,0), B(23,13), C(10,23)$ and $D(0,10)$. A lattice point is a point with integer coordinates. Find the number of lattice points in the interior of the square.

Problem 22:

Eleven distinct chemicals $C_1, C_2, \ldots, C_{11}$ are to be stored in three different warehouses. Each warehouse stores at least one chemical. A pair $C_i, C_j$ of chemicals, where $i \neq j$, is either compatible or incompatible. Any two incompatible chemicals cannot be stored in the same warehouse. However, a pair of compatible chemicals may or may not be stored in the same warehouse. Find the maximum possible number of pairs of incompatible chemicals that can be found among the stored chemicals.

Problem 23:

Let $k$ be a positive integer and let the function $f$ be defined as follows:
\[
f(x)=\frac{\pi^x}{\pi^x+\pi^{2 k-x}} .
\]

Suppose the function $g(k)$ is defined as follows:
\[
g(k)=f(0)+f\left(\frac{k}{2019}\right)+f\left(\frac{2 k}{2019}\right)+f\left(\frac{3 k}{2019}\right)+\ldots+f\left(\frac{4037 k}{2019}\right)+f(2 k) .
\]

Find the greatest positive integer $n$ such that $g(k) \geq n$ for all $k \geq 1$.

Problem 24:

Some students sat for a test. The first group of students scored an average of 91 marks and were given Grade A. The second group of students scored an average of 80 marks and were given Grade B. The last group of students scored an average of 70 marks and were given Grade $\mathrm{C}$. The numbers of students in all three groups are prime numbers and the total score of all the students is 1785 . Determine the total number of students.

Problem 25:

Suppose (a) and (b) are positive integers satisfying
$a^2-2 b^2=1$ .

If $500<a+b<1000$, find $a+b$.

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:

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 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 - 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.