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ISI BStat - BMath Entrance 2023 problems and solutions.


This is a work in progress. Please come back for more solutions and discussions


Answer Key of Objective problems (Caution ... first please check the order of questions that we are using.)

Q1DQ2AQ3A
Q4BQ5BQ6B
Q7BQ8DQ9B
Q10CQ11DQ12D
Q13BQ14DQ15D
Q16AQ17CQ18D
Q19CQ20BQ21C
Q22BQ23AQ24D
Q25CQ26BQ27A
Q28AQ29AQ30C

Subjective


Problem 1

Determine all integers \( n>1 \) such that every power of \( n \) has an odd number of digits.

Problem 2

Let \(a_0=\frac{1}{2} \) and \( a_n \) be defined inductively by

$$ a_n=\sqrt{\frac{1+a_{n-1}}{2}}, n \geq 1. $$

(a) Show that for \(n=0,1,2, \ldots \),

\[ a_n = \cos \theta_n \qquad \textrm{for some} \qquad 0 < \theta_n < \frac{\pi}{2} \]

and determine \( \theta_n \).

(b) Using (a) or otherwise, calculate

\[ \lim_{n \to \infty} 4^n (1 - a_n) \]

Solution and Discussion

Problem 3

In a triangle \( A B C \), consider points \( D \) and \( E \) on \( A C \) and \( A B \), respectively, and assume that they do not coincide with any of the vertices \( A, B, C \). If the segments \( B D \) and \( C E \) intersect at \( F \), consider the areas \(w, x, y, z \) of the quadrilateral \( A E F D \) and the triangles \( B E F, B F C, C D F \), respectively.

(a) Prove that \( y^2>x z \).

(b) Determine \( w \) in terms of \( x, y, z \).

Solution and Discussion
Problem 4

Let \( n_1, n_2, \cdots, n_{51} \) be distinct natural numbers each of which has exactly 2023 positive integer factors. For instance, \( 2^{2022} \) has exactly 2023 positive integer factors \( 1,2,2^2, \cdots, 2^{2021}, 2^{2022} \). Assume that no prime larger than 11 divides any of the \( n_i \) 's. Show that there must be some perfect cube among the \( n_i \) 's. You may use the fact that \( 2023=7 \times 17 \times 17 \)

Solution and Discussion
Problem 5

There is a rectangular plot of size \( 1 \times n \). This has to be covered by three types of tiles - red, blue and black. The red tiles are of size \( 1 \times 1 \), the blue tiles are of size \( 1 \times 1\) and the black tiles are of size \( 1 \times 2 \). Let \( t_n \) denote the number of ways this can be done. For example, clearly \( t_1=2 \) because we can have either a red or a blue tile. Also, \( t_2=5 \) since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.

(a) Prove that \( t_{2 n+1}=t_n\left(t_{n-1}+t_{n+1}\right) \) for all \( n>1 \).

(b) Prove that \( t_n=\sum_{d \geq 0}\left(\begin{array}{c}n-d \\ d\end{array}\right) 2^{n-2 d} \) for all \( n>0 \).

Here,

$$ \left(\begin{array}{c} m \\ r \end{array}\right)= \begin{cases}\frac{m !}{r !(m-r) !}, & \text { if } 0 \leq r \leq m \\ 0, & \text { otherwise }\end{cases} $$

for integers \( m, r \).

Solution and Discussion
Problem 6

Let \( \left\{u_n\right\}_{n \geq 1} \) be a sequence of real numbers defined as \( u_1=1 \) and

$$ u_{n+1}=u_n+\frac{1}{u_n} \text { for all } n \geq 1 \text {. } $$

Prove that \( u_n \leq \frac{3 \sqrt{n}}{2} \) for all \( n \).

Solution and Discussion
Problem 7

(a) Let \( n \geq 1 \) be an integer. Prove that \( X^n+Y^n+Z^n \) can be written as a polynomial with integer coefficients in the variables \( \alpha=X+Y+Z \), \( \beta=X Y+Y Z+Z X \) and \( \gamma=X Y Z \).

(b) Let \( G_n=x^n \sin (n A)+y^n \sin (n B)+z^n \sin (n C) \), where \( x, y, z, A, B, C \) are real numbers such that \( A+B+C \) is an integral multiple of \( \pi \). <br>Using (a) or otherwise, show that if \( G_1=G_2=0 \), then \( G_n=0 \) for all positive integers \( n \).

Problem 8

Let \( f:[0,1] \rightarrow \mathbb{R}\) be a continuous function which is differentiable on \( (0,1) \). Prove that either \( f \) is a linear function \( f(x)=a x+b \) or there exists \( t \in(0,1) \) such that \( |f(1)-f(0)|<\left|f^{\prime}(t)\right| \).


Objective


The following notations are used in the question paper:

\( \mathbb{R}\) is the set of real numbers,

\( \mathbb{C}\) is the set of complex numbers,

\( \mathbb{Z} \) is the set of integers,

\( \left(\begin{array}{l}n \\ r\end{array}\right)=\frac{n !}{r !(n-r) !} \) for all \( n=1,2,3, \ldots \) and \( r=0,1, \ldots, n \).

Problem 1

For a real number \( x \),

$$x^3-7 x+6>0$$

if and only if

(A) \( x \)>2

(B) -3<\( x \)<1

(C) \( x \)<-3 or 1<\( x \)<2

(D) -3< \( x \)<1 or \( x \)>2

Problem 2

The number of consecutive zeroes adjacent to the digit in the unit's place of \( 401^{50} \) is

(A) 3

(B) 4

(C) 49

(D) 50

Problem 3

Consider a right-angled triangle \( \triangle A B C \) whose hypotenuse \( A C \) is of length 1. The bisector of \( \angle A C B \) intersects \( A B \) at \( D \). If \( B C \) is of length \( x \), then what is the length of \( C D \)?

(A) \( \sqrt{\frac{2 x^2}{1+x}}\)

(B) \( \frac{1}{\sqrt{2+2 x}}\)

(C) \( \sqrt{\frac{x}{1+x}}\)

(D) \( \frac{x}{\sqrt{1-x^2}}\)

Problem 4

Define a polynomial \( f(x) \)

$$f(x)=\left|\begin{array}{lll}1 & x & x \\x & 1 & x \\x & x & 1\end{array}\right|$$

for all \( x \in \mathbb{R}\), where the right hand side above is a determinant. Then the roots of \( f(x) \) are of the form

(A) \( \alpha, \beta \pm i \gamma \) where \( \alpha, \beta, \gamma \in \mathbb{R}, \gamma \neq 0 \) and \( i \) is a square root of \( -1\)

(B) \( \alpha, \alpha, \beta \) where \( \alpha, \beta \in \mathbb{R}\) are distinct.

(C) \( \alpha, \beta, \gamma \) where \( \alpha, \beta, \gamma \in \mathbb{R} \) are all distinct.

(D) \( \alpha, \alpha, \alpha \) for some \( \alpha \in \mathbb{R} \).

Problem 5

Let \( S \) be the set of those real numbers \( x \) for which the identity

$$\sum_{n=2}^{\infty} \cos ^n x=(1+\cos x) \cot ^2 x$$

is valid, and the quantities on both sides are finite. Then

(A) \( S \) is the empty set.

(B) \( S=\{x \in \mathbb{R}: x \neq n \pi \) for all \( n \in \mathbb{Z}\} \).

(C) \( S=\{x \in \mathbb{R}: x \neq 2 n \pi \) for all \( n \in \mathbb{Z}\} \).

(D) \( S^{\prime}=\{x \in \mathbb{R}: x \neq(2 n+1) \pi \) for all \( n \in \mathbb{Z}\} \)

Problem 6

Let

$$\begin{gathered}S=\left\{\left(\theta \sin \frac{\pi \theta}{1+\theta}, \frac{1}{\theta} \cos \frac{\pi \theta}{1+\theta}\right): theta \in \mathbb{R}, \theta>0\right\} \\T=\left\{(x, y): x \in \mathbb{R}, y \in \mathbb{R}, x y=\frac{1}{2}\right\}\end{gathered}$$

How many elements does \( S \cap T \) have?

(A) 9.

(B) 1.

(C) 2.

(D) 3

Problem 7

How many numbers formed by rearranging the digits of 234578 are divisible by 55?

(A) 0

(B) 12

(C) 36

(D) \( 72 \quad 2 \)

Solution and Discussion
Problem 8

The limit

$$\lim _{n \rightarrow \infty} n^{-\frac{3}{2}}\left((n+1)^{(n+1)}(n+2)^{(n+2)} \ldots(2 n)^{(2 n)}\right)^{\frac{1}{n^2}}$$

equals

(A) 0

(B) 1

(C) \( e^{-\frac{1}{4}} \).

(D) \( 4 e^{-\frac{3}{4}} \).

Problem 9

Consider a triangle with vertices \( (0,0),(1,2) \) and \( (-4,2) \). Let \( A \) be the area of the triangle and \( B \) be the area of the circumcircle of the triangle. Then \( \frac{B}{A} \) equals

(A) \( \frac{\pi}{2} \).

(B) \( \frac{5 \pi}{4} \).

(C) \( \frac{3}{\sqrt{2}} \pi \).

(D) \( 2 \pi \).

Problem 10

The value of

$$\sum_{k=0}^{202}(-1)^k\left(\begin{array}{c}202 \\k\end{array}\right) \cos \left(\frac{k \pi}{3}\right)$$

equals

(A) \( \sin \left(\frac{202}{3} \pi\right) \).

(C) \( \cos \left(\frac{202}{3} \pi\right) \).

(B) \( -\sin \left(\frac{202}{3} \pi\right) \).

(D) \( \cos ^{202}\left(\frac{\pi}{3}\right) \).

Problem 11

For real numbers \( a, b, c, d, a^{\prime}, b^{\prime}, c^{\prime}, d^{\prime} \), consider the system of equations

$$\begin{aligned}a x^2+a y^2+b x+c y+d & =0, \\a^{\prime} x^2+a^{\prime} y^2+b^{\prime} x+c^{\prime} y+d^{\prime} & =0\end{aligned}$$

If \( S \) denotes the set of all real solutions \( (x, y) \) of the above system of equations, then the number of elements in \( S \) can never be

(A) 0

(B) 1

(C) 2

(D) 3

Problem 12

Let \( f, g \) be continuous functions from \( [0, \infty) \) to itself,

$$h(x)=\int_{2^x}^{3^x} f(t) d t, x>0,$$

and

$$F(x)=\int_0^{h(x)} g(t) d t, x>0$$

If \( F^{\prime}\) is the derivative of \( F \), then for \( x>0 \),

(A) \( F^{\prime}(x)=g(h(x)) \).

(B) \( F^{\prime}(x)=g(h(x))\left[f\left(3^x\right)-f\left(2^x\right)\right] \).

(C) \( F^{\prime}(x)=g(h(x))\left[x 3^{x-1} f\left(3^x\right)-x 2^{x-1} f\left(2^x\right)\right] \).

(D) \( F^{\prime}(x)=g(h(x))\left[3^x f\left(3^x\right) \ln 3-2^x f\left(2^x\right) \ln 2\right] \).

Problem 13

Suppose \( x \) and \( y \) are positive integers. If \( 4 x+3 y \) and \( 2 x+4 y \) are divided by 7 , then the respective remainders are 2 and 5. If \( 11 x+5 y \) is divided by 7, then the remainder equals

(A) 0

(B) 1

(C) 2

(D) 3

Problem 14

Suppose \( F: \mathbb{R} \rightarrow \mathbb{R} \)is a continuous function which has exactly one local maximum. Then which of the following is true?

(A) \( F \) cannot have a local minimum.

(B) \( F \) must have exactly one local minimum.

(C) \( F \) must have at least two local minima.

(D) \( F \) must have either a global maximum or a local minimum.

Problem 15

Suppose \( z \in \mathbb{C} \) is such that the imaginary part of \( z \) is non-zero and \( z^{25}=1 \). Then

$$\sum_{k=0}^{2023} z^k$$

equals

(A) 0

(B) 1

(C) \( -1-z^{24} \).

(D) \( -z^{24} \).

Problem 16

The limit

$$\lim _{x \rightarrow 0} \frac{1}{x}\left(\cos (x)+\cos \left(\frac{1}{x}\right)-\cos (x) \cos \left(\frac{1}{x}\right)-1\right)$$

(A) equals 0

(B) equals \( \frac{1}{2} \),

(C) equals 1

(D) does not exist.

Problem 17

Let \( n \) be a positive integer having 27 divisors including 1 and \( n \), which are denoted by \( d_1, \ldots, d_{27} \). Then the product of \( d_1, d_2, \ldots, d_{27}\) equals

(A) \( n^{13} \).

(B) \( n^{14} \).

(C) \( n^{\frac{27}{2}} \),

(D) \( 27 n \).

Problem 18

If \( (f:[0, \infty) \rightarrow \mathbb{R} \) is a continuous function such that

$$f(x)+\ln 2 \int_0^x f(t) d t=1, x \geq 0,$$

then for all \( x \geq 0 \),

(A) \( f(x)=e^x \ln 2 \).

(B) \( f(x)=e^{-x} \ln 2 \).

(C) \( f(x)=2^x \).

(D) \( f(x)=\left(\frac{1}{2}\right)^x \).

Problem 19

If \( [x] \) denotes the largest integer less than or equal to \( x \), then

$$\left[(9+\sqrt{80})^{20}\right]$$

equals

(A) \( (9+\sqrt{80})^{20}-(9-\sqrt{80})^{20} \).

(B) \( (9+\sqrt{80})^{20}+(9-\sqrt{80})^{20}-20 \).

(C) \( (9+\sqrt{80})^{20}+(9-\sqrt{80})^{20}-1 \).

(D) \( (9-\sqrt{80})^{20} \).

Problem 20

Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a twice differentiable one-to-one function. If \( f(2)=2, f(3)=-8 \) and

$$\int_2^3 f(x) d x=-3$$

then

$$\int_{-8}^2 f^{-1}(x) d x$$

equals

(A) -25

(B) 25

(C) -31

(D) 31

Problem 21

Three left brackets and three right brackets have to be arranged in such a way that if the brackets are serially counted from the left, then the number of right brackets counted is always less than or equal to the number of left brackets counted. In how many ways can this be done?

(A) 3

(B) 4

(C) 5

(D) 6

Problem 22

The limit

$$\lim _{n \rightarrow \infty}\left(2^{-2^{n+1}}+2^{-2^{n-1}}\right)^{2^{-n}}$$

equals

(A) 1

(B) \( \frac{1}{\sqrt{2}} \)

(C) 0

(D) \( \frac{1}{4} \)

Problem 23

In the following figure, \( O A B \) is a quarter-circle. The unshaded region is a circle to which \( O A \) and \( C D \) are tangents.

If \( C D \) is of length 10 and is parallel to \( O A \), then the area of the shaded region in the above figure equals

(A) \( 25 \pi \).

(B) \( \pi \).

(C) \( 75 \pi \)

(D) \( 100 \pi \).

Problem 24

Consider the function \( f: \mathbb{C} \rightarrow \mathbb{C} \) defined by

$$f(a+i b)=e^a(\cos b+i \sin b), a, b \in \mathbb{R}$$

where \( i \) is a square root of -1. Then

(A) \( f \) is one-to-one and onto.

(B) \( f \) is one-to-one but not onto.

(C) \( f \) is onto but not one-to-one.

(D) \( f \) is neither one-to-one nor onto.

Problem 25

Suppose \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) is a non-decreasing function. Consider the following two cases:

Case 1. \( f(0)=2, f(10)=8 \),

Case 2. \( f(0)=-2, f(10)=12 \).

In which of the above cases it is necessarily true that there exists an \( n \) with \( f(n)=n \) ?

(A) In both cases.

(B) In neither case.

(C) In Case 1. but not necessarily in Case 2.

(D) In Case 2. but not necessarily in Case 1.

Problem 26

Suppose that \( f(x)=a x^3+b x^2+c x+d\) where \( a, b, c, d \) are real numbers with \( a \neq 0 \). The equation \( f(x)=0 \) has exactly two distinct real solutions. If \( f^{\prime}(x) \) is the derivative of \( f(x) \), then which of the following is a possible graph of \( f^{\prime}(x) \)?

Problem 27

Suppose \( a, b, c \in \mathbb{R} \) and

$$f(x)=a x^2+b x+c, x \in \mathbb{R}$$

If \( 0 \leq f(x) \leq(x-1)^2 \) for all \( x \), and \( f(3)=2 \), then

(A) \( a=\frac{1}{2}, b=-1, c=\frac{1}{2} \).

(B) \( a=\frac{1}{3}, b=-\frac{1}{3}, c=0 \).

(C) \( a=\frac{2}{3}, b=-\frac{5}{3}, c=1 \).

(D) \( a=\frac{3}{4}, b=-2, c=\frac{5}{4} \).

Problem 28

The polynomial \( x^{10}+x^5+1 \) is divisible by

(A) \( x^2+x+1 \).

(B) \( x^2-x+1 \).

(C) \( x^2+1 \).

(D) \( x^5-1 \).

Problem 29

As in the following figure, the straight line \( O A \) lies in the second quadrant of the \( (x, y) \)-plane and makes an angle \( \theta \) with the negative half of the \( x \)-axis, where \( 0<\theta<\frac{\pi}{2} \).

The line segment \( C D \) of length 1 slides on the \( (x, y) \)-plane in such a way that \( C \) is always on \( O A \) and \( D \) on the positive side of the \( x \)-axis. The locus of the mid-point of \( C D \) is

(A) \( x^2+4 x y \cot \theta+y^2\left(1+4 \cot ^2 \theta\right)=\frac{1}{4} \).

(B) \( x^2+y^2=\frac{1}{4}+\cot ^2 \theta \).

(C) \( x^2+4 x y \cot \theta+y^2=\frac{1}{4} \).

(D) \( x^2+y^2\left(1+4 \cot ^2 \theta\right)=\frac{1}{4} \).

Problem 30

How many functions \( f:\{1,2, \ldots, 10\} \rightarrow\{1, \ldots, 2000\} \), which satisfy

$$f(i+1)-f(i) \geq 20, \text { for all } 1 \leq i \leq 9,$$

are there?

(A) \( 10 !\left(\begin{array}{c}1829 \\ 10\end{array}\right) \)

(B) \( 11 !\left(\begin{array}{c}1830 \\ 11\end{array}\right) \)

(C) \( \left(\begin{array}{c}1829 \\ 10\end{array}\right) \)

(D) \( \left(\begin{array}{c}1830 \\ 11\end{array}\right) \)


This is a work in progress. Please come back for more solutions and discussions


Answer Key of Objective problems (Caution ... first please check the order of questions that we are using.)

Q1DQ2AQ3A
Q4BQ5BQ6B
Q7BQ8DQ9B
Q10CQ11DQ12D
Q13BQ14DQ15D
Q16AQ17CQ18D
Q19CQ20BQ21C
Q22BQ23AQ24D
Q25CQ26BQ27A
Q28AQ29AQ30C

Subjective


Problem 1

Determine all integers \( n>1 \) such that every power of \( n \) has an odd number of digits.

Problem 2

Let \(a_0=\frac{1}{2} \) and \( a_n \) be defined inductively by

$$ a_n=\sqrt{\frac{1+a_{n-1}}{2}}, n \geq 1. $$

(a) Show that for \(n=0,1,2, \ldots \),

\[ a_n = \cos \theta_n \qquad \textrm{for some} \qquad 0 < \theta_n < \frac{\pi}{2} \]

and determine \( \theta_n \).

(b) Using (a) or otherwise, calculate

\[ \lim_{n \to \infty} 4^n (1 - a_n) \]

Solution and Discussion

Problem 3

In a triangle \( A B C \), consider points \( D \) and \( E \) on \( A C \) and \( A B \), respectively, and assume that they do not coincide with any of the vertices \( A, B, C \). If the segments \( B D \) and \( C E \) intersect at \( F \), consider the areas \(w, x, y, z \) of the quadrilateral \( A E F D \) and the triangles \( B E F, B F C, C D F \), respectively.

(a) Prove that \( y^2>x z \).

(b) Determine \( w \) in terms of \( x, y, z \).

Solution and Discussion
Problem 4

Let \( n_1, n_2, \cdots, n_{51} \) be distinct natural numbers each of which has exactly 2023 positive integer factors. For instance, \( 2^{2022} \) has exactly 2023 positive integer factors \( 1,2,2^2, \cdots, 2^{2021}, 2^{2022} \). Assume that no prime larger than 11 divides any of the \( n_i \) 's. Show that there must be some perfect cube among the \( n_i \) 's. You may use the fact that \( 2023=7 \times 17 \times 17 \)

Solution and Discussion
Problem 5

There is a rectangular plot of size \( 1 \times n \). This has to be covered by three types of tiles - red, blue and black. The red tiles are of size \( 1 \times 1 \), the blue tiles are of size \( 1 \times 1\) and the black tiles are of size \( 1 \times 2 \). Let \( t_n \) denote the number of ways this can be done. For example, clearly \( t_1=2 \) because we can have either a red or a blue tile. Also, \( t_2=5 \) since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.

(a) Prove that \( t_{2 n+1}=t_n\left(t_{n-1}+t_{n+1}\right) \) for all \( n>1 \).

(b) Prove that \( t_n=\sum_{d \geq 0}\left(\begin{array}{c}n-d \\ d\end{array}\right) 2^{n-2 d} \) for all \( n>0 \).

Here,

$$ \left(\begin{array}{c} m \\ r \end{array}\right)= \begin{cases}\frac{m !}{r !(m-r) !}, & \text { if } 0 \leq r \leq m \\ 0, & \text { otherwise }\end{cases} $$

for integers \( m, r \).

Solution and Discussion
Problem 6

Let \( \left\{u_n\right\}_{n \geq 1} \) be a sequence of real numbers defined as \( u_1=1 \) and

$$ u_{n+1}=u_n+\frac{1}{u_n} \text { for all } n \geq 1 \text {. } $$

Prove that \( u_n \leq \frac{3 \sqrt{n}}{2} \) for all \( n \).

Solution and Discussion
Problem 7

(a) Let \( n \geq 1 \) be an integer. Prove that \( X^n+Y^n+Z^n \) can be written as a polynomial with integer coefficients in the variables \( \alpha=X+Y+Z \), \( \beta=X Y+Y Z+Z X \) and \( \gamma=X Y Z \).

(b) Let \( G_n=x^n \sin (n A)+y^n \sin (n B)+z^n \sin (n C) \), where \( x, y, z, A, B, C \) are real numbers such that \( A+B+C \) is an integral multiple of \( \pi \). <br>Using (a) or otherwise, show that if \( G_1=G_2=0 \), then \( G_n=0 \) for all positive integers \( n \).

Problem 8

Let \( f:[0,1] \rightarrow \mathbb{R}\) be a continuous function which is differentiable on \( (0,1) \). Prove that either \( f \) is a linear function \( f(x)=a x+b \) or there exists \( t \in(0,1) \) such that \( |f(1)-f(0)|<\left|f^{\prime}(t)\right| \).


Objective


The following notations are used in the question paper:

\( \mathbb{R}\) is the set of real numbers,

\( \mathbb{C}\) is the set of complex numbers,

\( \mathbb{Z} \) is the set of integers,

\( \left(\begin{array}{l}n \\ r\end{array}\right)=\frac{n !}{r !(n-r) !} \) for all \( n=1,2,3, \ldots \) and \( r=0,1, \ldots, n \).

Problem 1

For a real number \( x \),

$$x^3-7 x+6>0$$

if and only if

(A) \( x \)>2

(B) -3<\( x \)<1

(C) \( x \)<-3 or 1<\( x \)<2

(D) -3< \( x \)<1 or \( x \)>2

Problem 2

The number of consecutive zeroes adjacent to the digit in the unit's place of \( 401^{50} \) is

(A) 3

(B) 4

(C) 49

(D) 50

Problem 3

Consider a right-angled triangle \( \triangle A B C \) whose hypotenuse \( A C \) is of length 1. The bisector of \( \angle A C B \) intersects \( A B \) at \( D \). If \( B C \) is of length \( x \), then what is the length of \( C D \)?

(A) \( \sqrt{\frac{2 x^2}{1+x}}\)

(B) \( \frac{1}{\sqrt{2+2 x}}\)

(C) \( \sqrt{\frac{x}{1+x}}\)

(D) \( \frac{x}{\sqrt{1-x^2}}\)

Problem 4

Define a polynomial \( f(x) \)

$$f(x)=\left|\begin{array}{lll}1 & x & x \\x & 1 & x \\x & x & 1\end{array}\right|$$

for all \( x \in \mathbb{R}\), where the right hand side above is a determinant. Then the roots of \( f(x) \) are of the form

(A) \( \alpha, \beta \pm i \gamma \) where \( \alpha, \beta, \gamma \in \mathbb{R}, \gamma \neq 0 \) and \( i \) is a square root of \( -1\)

(B) \( \alpha, \alpha, \beta \) where \( \alpha, \beta \in \mathbb{R}\) are distinct.

(C) \( \alpha, \beta, \gamma \) where \( \alpha, \beta, \gamma \in \mathbb{R} \) are all distinct.

(D) \( \alpha, \alpha, \alpha \) for some \( \alpha \in \mathbb{R} \).

Problem 5

Let \( S \) be the set of those real numbers \( x \) for which the identity

$$\sum_{n=2}^{\infty} \cos ^n x=(1+\cos x) \cot ^2 x$$

is valid, and the quantities on both sides are finite. Then

(A) \( S \) is the empty set.

(B) \( S=\{x \in \mathbb{R}: x \neq n \pi \) for all \( n \in \mathbb{Z}\} \).

(C) \( S=\{x \in \mathbb{R}: x \neq 2 n \pi \) for all \( n \in \mathbb{Z}\} \).

(D) \( S^{\prime}=\{x \in \mathbb{R}: x \neq(2 n+1) \pi \) for all \( n \in \mathbb{Z}\} \)

Problem 6

Let

$$\begin{gathered}S=\left\{\left(\theta \sin \frac{\pi \theta}{1+\theta}, \frac{1}{\theta} \cos \frac{\pi \theta}{1+\theta}\right): theta \in \mathbb{R}, \theta>0\right\} \\T=\left\{(x, y): x \in \mathbb{R}, y \in \mathbb{R}, x y=\frac{1}{2}\right\}\end{gathered}$$

How many elements does \( S \cap T \) have?

(A) 9.

(B) 1.

(C) 2.

(D) 3

Problem 7

How many numbers formed by rearranging the digits of 234578 are divisible by 55?

(A) 0

(B) 12

(C) 36

(D) \( 72 \quad 2 \)

Solution and Discussion
Problem 8

The limit

$$\lim _{n \rightarrow \infty} n^{-\frac{3}{2}}\left((n+1)^{(n+1)}(n+2)^{(n+2)} \ldots(2 n)^{(2 n)}\right)^{\frac{1}{n^2}}$$

equals

(A) 0

(B) 1

(C) \( e^{-\frac{1}{4}} \).

(D) \( 4 e^{-\frac{3}{4}} \).

Problem 9

Consider a triangle with vertices \( (0,0),(1,2) \) and \( (-4,2) \). Let \( A \) be the area of the triangle and \( B \) be the area of the circumcircle of the triangle. Then \( \frac{B}{A} \) equals

(A) \( \frac{\pi}{2} \).

(B) \( \frac{5 \pi}{4} \).

(C) \( \frac{3}{\sqrt{2}} \pi \).

(D) \( 2 \pi \).

Problem 10

The value of

$$\sum_{k=0}^{202}(-1)^k\left(\begin{array}{c}202 \\k\end{array}\right) \cos \left(\frac{k \pi}{3}\right)$$

equals

(A) \( \sin \left(\frac{202}{3} \pi\right) \).

(C) \( \cos \left(\frac{202}{3} \pi\right) \).

(B) \( -\sin \left(\frac{202}{3} \pi\right) \).

(D) \( \cos ^{202}\left(\frac{\pi}{3}\right) \).

Problem 11

For real numbers \( a, b, c, d, a^{\prime}, b^{\prime}, c^{\prime}, d^{\prime} \), consider the system of equations

$$\begin{aligned}a x^2+a y^2+b x+c y+d & =0, \\a^{\prime} x^2+a^{\prime} y^2+b^{\prime} x+c^{\prime} y+d^{\prime} & =0\end{aligned}$$

If \( S \) denotes the set of all real solutions \( (x, y) \) of the above system of equations, then the number of elements in \( S \) can never be

(A) 0

(B) 1

(C) 2

(D) 3

Problem 12

Let \( f, g \) be continuous functions from \( [0, \infty) \) to itself,

$$h(x)=\int_{2^x}^{3^x} f(t) d t, x>0,$$

and

$$F(x)=\int_0^{h(x)} g(t) d t, x>0$$

If \( F^{\prime}\) is the derivative of \( F \), then for \( x>0 \),

(A) \( F^{\prime}(x)=g(h(x)) \).

(B) \( F^{\prime}(x)=g(h(x))\left[f\left(3^x\right)-f\left(2^x\right)\right] \).

(C) \( F^{\prime}(x)=g(h(x))\left[x 3^{x-1} f\left(3^x\right)-x 2^{x-1} f\left(2^x\right)\right] \).

(D) \( F^{\prime}(x)=g(h(x))\left[3^x f\left(3^x\right) \ln 3-2^x f\left(2^x\right) \ln 2\right] \).

Problem 13

Suppose \( x \) and \( y \) are positive integers. If \( 4 x+3 y \) and \( 2 x+4 y \) are divided by 7 , then the respective remainders are 2 and 5. If \( 11 x+5 y \) is divided by 7, then the remainder equals

(A) 0

(B) 1

(C) 2

(D) 3

Problem 14

Suppose \( F: \mathbb{R} \rightarrow \mathbb{R} \)is a continuous function which has exactly one local maximum. Then which of the following is true?

(A) \( F \) cannot have a local minimum.

(B) \( F \) must have exactly one local minimum.

(C) \( F \) must have at least two local minima.

(D) \( F \) must have either a global maximum or a local minimum.

Problem 15

Suppose \( z \in \mathbb{C} \) is such that the imaginary part of \( z \) is non-zero and \( z^{25}=1 \). Then

$$\sum_{k=0}^{2023} z^k$$

equals

(A) 0

(B) 1

(C) \( -1-z^{24} \).

(D) \( -z^{24} \).

Problem 16

The limit

$$\lim _{x \rightarrow 0} \frac{1}{x}\left(\cos (x)+\cos \left(\frac{1}{x}\right)-\cos (x) \cos \left(\frac{1}{x}\right)-1\right)$$

(A) equals 0

(B) equals \( \frac{1}{2} \),

(C) equals 1

(D) does not exist.

Problem 17

Let \( n \) be a positive integer having 27 divisors including 1 and \( n \), which are denoted by \( d_1, \ldots, d_{27} \). Then the product of \( d_1, d_2, \ldots, d_{27}\) equals

(A) \( n^{13} \).

(B) \( n^{14} \).

(C) \( n^{\frac{27}{2}} \),

(D) \( 27 n \).

Problem 18

If \( (f:[0, \infty) \rightarrow \mathbb{R} \) is a continuous function such that

$$f(x)+\ln 2 \int_0^x f(t) d t=1, x \geq 0,$$

then for all \( x \geq 0 \),

(A) \( f(x)=e^x \ln 2 \).

(B) \( f(x)=e^{-x} \ln 2 \).

(C) \( f(x)=2^x \).

(D) \( f(x)=\left(\frac{1}{2}\right)^x \).

Problem 19

If \( [x] \) denotes the largest integer less than or equal to \( x \), then

$$\left[(9+\sqrt{80})^{20}\right]$$

equals

(A) \( (9+\sqrt{80})^{20}-(9-\sqrt{80})^{20} \).

(B) \( (9+\sqrt{80})^{20}+(9-\sqrt{80})^{20}-20 \).

(C) \( (9+\sqrt{80})^{20}+(9-\sqrt{80})^{20}-1 \).

(D) \( (9-\sqrt{80})^{20} \).

Problem 20

Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a twice differentiable one-to-one function. If \( f(2)=2, f(3)=-8 \) and

$$\int_2^3 f(x) d x=-3$$

then

$$\int_{-8}^2 f^{-1}(x) d x$$

equals

(A) -25

(B) 25

(C) -31

(D) 31

Problem 21

Three left brackets and three right brackets have to be arranged in such a way that if the brackets are serially counted from the left, then the number of right brackets counted is always less than or equal to the number of left brackets counted. In how many ways can this be done?

(A) 3

(B) 4

(C) 5

(D) 6

Problem 22

The limit

$$\lim _{n \rightarrow \infty}\left(2^{-2^{n+1}}+2^{-2^{n-1}}\right)^{2^{-n}}$$

equals

(A) 1

(B) \( \frac{1}{\sqrt{2}} \)

(C) 0

(D) \( \frac{1}{4} \)

Problem 23

In the following figure, \( O A B \) is a quarter-circle. The unshaded region is a circle to which \( O A \) and \( C D \) are tangents.

If \( C D \) is of length 10 and is parallel to \( O A \), then the area of the shaded region in the above figure equals

(A) \( 25 \pi \).

(B) \( \pi \).

(C) \( 75 \pi \)

(D) \( 100 \pi \).

Problem 24

Consider the function \( f: \mathbb{C} \rightarrow \mathbb{C} \) defined by

$$f(a+i b)=e^a(\cos b+i \sin b), a, b \in \mathbb{R}$$

where \( i \) is a square root of -1. Then

(A) \( f \) is one-to-one and onto.

(B) \( f \) is one-to-one but not onto.

(C) \( f \) is onto but not one-to-one.

(D) \( f \) is neither one-to-one nor onto.

Problem 25

Suppose \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) is a non-decreasing function. Consider the following two cases:

Case 1. \( f(0)=2, f(10)=8 \),

Case 2. \( f(0)=-2, f(10)=12 \).

In which of the above cases it is necessarily true that there exists an \( n \) with \( f(n)=n \) ?

(A) In both cases.

(B) In neither case.

(C) In Case 1. but not necessarily in Case 2.

(D) In Case 2. but not necessarily in Case 1.

Problem 26

Suppose that \( f(x)=a x^3+b x^2+c x+d\) where \( a, b, c, d \) are real numbers with \( a \neq 0 \). The equation \( f(x)=0 \) has exactly two distinct real solutions. If \( f^{\prime}(x) \) is the derivative of \( f(x) \), then which of the following is a possible graph of \( f^{\prime}(x) \)?

Problem 27

Suppose \( a, b, c \in \mathbb{R} \) and

$$f(x)=a x^2+b x+c, x \in \mathbb{R}$$

If \( 0 \leq f(x) \leq(x-1)^2 \) for all \( x \), and \( f(3)=2 \), then

(A) \( a=\frac{1}{2}, b=-1, c=\frac{1}{2} \).

(B) \( a=\frac{1}{3}, b=-\frac{1}{3}, c=0 \).

(C) \( a=\frac{2}{3}, b=-\frac{5}{3}, c=1 \).

(D) \( a=\frac{3}{4}, b=-2, c=\frac{5}{4} \).

Problem 28

The polynomial \( x^{10}+x^5+1 \) is divisible by

(A) \( x^2+x+1 \).

(B) \( x^2-x+1 \).

(C) \( x^2+1 \).

(D) \( x^5-1 \).

Problem 29

As in the following figure, the straight line \( O A \) lies in the second quadrant of the \( (x, y) \)-plane and makes an angle \( \theta \) with the negative half of the \( x \)-axis, where \( 0<\theta<\frac{\pi}{2} \).

The line segment \( C D \) of length 1 slides on the \( (x, y) \)-plane in such a way that \( C \) is always on \( O A \) and \( D \) on the positive side of the \( x \)-axis. The locus of the mid-point of \( C D \) is

(A) \( x^2+4 x y \cot \theta+y^2\left(1+4 \cot ^2 \theta\right)=\frac{1}{4} \).

(B) \( x^2+y^2=\frac{1}{4}+\cot ^2 \theta \).

(C) \( x^2+4 x y \cot \theta+y^2=\frac{1}{4} \).

(D) \( x^2+y^2\left(1+4 \cot ^2 \theta\right)=\frac{1}{4} \).

Problem 30

How many functions \( f:\{1,2, \ldots, 10\} \rightarrow\{1, \ldots, 2000\} \), which satisfy

$$f(i+1)-f(i) \geq 20, \text { for all } 1 \leq i \leq 9,$$

are there?

(A) \( 10 !\left(\begin{array}{c}1829 \\ 10\end{array}\right) \)

(B) \( 11 !\left(\begin{array}{c}1830 \\ 11\end{array}\right) \)

(C) \( \left(\begin{array}{c}1829 \\ 10\end{array}\right) \)

(D) \( \left(\begin{array}{c}1830 \\ 11\end{array}\right) \)

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