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

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

(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 \(a+b\).