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 .