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.