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:

Watch the Video

Key Concepts Used:

Step-by-Step Proof Summary:

  1. 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.
  2. 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.
  3. 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.
  4. 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- Algebra (Senior)

Problem 01: (Year 2023, 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: (Year 2023, 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 03: (Year 2023, 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 04: (Year 2023, 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 05: (Year 2023, 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 06: (Year 2023, 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 07: (Year 2023, 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 08: (Year 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 09: (Year 2023, 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 10: (Year 2023, 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 11: (Year 2023, 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 01: (Year 2022, 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 07: (Year 2022, 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 12: (Year 2022, 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: (Year 2022, 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 16: (Year 2022, 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: (Year 2022, 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 20: (Year 2022, Problem 20)

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

Problem 24: (Year 2022, 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 01: (Year 2021, 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 07: (Year 2021, 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 12: (Year 2021, Problem 12)

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

Problem 13: (Year 2021, 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: (Year 2021, 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 18: (Year 2021, 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 20: (Year 2021, 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 22: (Year 2021, 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 24: (Year 2021, 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: (Year 2020, 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 26: (Year 2020, 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) \(r<p<q\)

Problem 27: (Year 2020, 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 28: (Year 2020, Problem 14)

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

Problem 29: (Year 2020, Problem 15)

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

Problem 30: (Year 2020, 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 31: (Year 2020, 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 32: (Year 2020, 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 33: (Year 2019, 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 34: (Year 2019, Problem 08)

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

Problem 35: (Year 2019, 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 36: (Year 2019, 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 37: (Year 2019, 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 38: (Year 2019, 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 39: (Year 2019, 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 40: (Year 2019, 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 41: (Year 2019, 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 Math Olympiad Past Years Questions- Number Theory (Senior)

Problem 01: (Year 2023, Problem 04)

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 02: (Year 2023, Problem 08)

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

Problem 03: (Year 2023, Problem 09)

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

Problem 04: (Year 2023, 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 05: (Year 2023, 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 06: (Year 2023, Problem 19)

Find the remainder when \(3^{2023}\) is divided by \(215\) .

Problem 07: (Year 2023, Problem 20)

Find the sum of the prime divisors of \(64000027\) .

Problem 08: (Year 2022, 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 09: (Year 2022, 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 10: (Year 2022, 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 11: (Year 2022, 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 12: (Year 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 13: (Year 2022, 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 14: (Year 2022, Problem 23)

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

Problem 15: (Year 2021, 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 16: (Year 2021, 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)

Problem 17: (Year 2021, Problem 06)

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

Problem 18: (Year 2021, 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 19: (Year 2021, 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 .

Problem 20: (Year 2020, 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 21: (Year 2020, 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 22: (Year 2020, 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 23: (Year 2020, Problem 06)

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

Problem 24: (Year 2020, 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 25: (Year 2020, Problem 21)

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

Problem 26: (Year 2020, 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 27: (Year 2020, 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.

Problem 28: (Year 2019, 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 29: (Year 2019, 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 30: (Year 2019, 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 31: (Year 2019, 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 32: (Year 2019, 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 33: (Year 2019, Problem 17)

Find the remainder when \(10^{43}\) is divided by \(126\) .

Singapore Math Olympiad Past Years Questions- Geometry (Senior)

Problem 01: (Year 2023, 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 02: (Year 2023, 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) \(-\frac{24}{25}\)

Problem 03: (Year 2023, 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 04: (Year 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 05: (Year 2023,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 06: (Year 2023, 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 07: (Year 2023,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 08:(Year 2023, Problem 23)

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 09: (Year 2023, Problem 25)

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

Problem 10: (Year 2022, 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 11: (Year 2022, 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) \(-\frac{56}{81} \sqrt{2}\)

Problem 12: (Year 2022, 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 13: (Year 2022, 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 14: (Year 2022,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 15: (Year 2022, 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 16: (Year 2022, 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 17: (Year 2022, 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.

Problem 18: (Year 2021, 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 19: (Year 2021, 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 20: (Year 2021, 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 21:(Year 2021, 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 22:(Year 2021, Problem 12)

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 23: (Year 2021, Problem 15)

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

Problem 24: (Year 2021, 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 25:(Year 2021, 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 26: (Year 2021, 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 27:(Year 2021, 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 28:(Year 2020, Problem 7)

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 29: (Year 2020, Problem 08)

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

Problem 30: (Year 2020, Problem 09)

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

Problem 31: (Year 2020, 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 32:(Year 2020, 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 33: (Year 2020, 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 34: (Year 2020, 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 35:(Year 2020, 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 36: (Year 2019, 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) \(\frac{56}{65}\)

(C) \(-\frac{16}{65}\)

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

(E) None of the above

Problem 37: (Year 2019, Problem 07)

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

Problem 38:(Year 2019, 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 39:(Year 2019, 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 40: (Year 2019, Problem 11)

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

Problem 41:(Year 2019, 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 42:(Year 2019, 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 43:(Year 2019, 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.

Singapore Math Olympiad Past years Questions- Combinatorics (Senior)

Problem 01: (Year 2023, 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 02: (Year 2021, 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 03: (Year 2020, 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 04: (Year 2019, 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 05: (Year 2020, 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.