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CMI Entrance Exam

Mock test #2 ‘23: Full-syllabus test

Timings: 14:00-17:00 Hrs    Date: 7th April 2023

Part A: Short-answer type questions

Submission file: Write answers to all the ten questions on a single sheet of paper. Email a picture of your answer sheet. Name the file as PartA.jpg or PartA.png.

For this part, answers must be written without any explanation.

  1. How many five digit positive integers that are divisible by 3 can be formed using the digits \(0,1,2,3,4\) and 5, without any of the digits getting repeated?
    1. 216
    2. 96
    3. 120
    4. 625
  2. Suppose \(a_i\) and \(b_i\) are real numbers such that \(\sum_1^\infty a_i^2\) and \(\sum_1^\infty b_i^2\) converge. Which of these statements is/are true?
    1. The sequence \(\sum_1^\infty |a_i-b_i| \) converges.
    2. The sequence \(\sum_1^\infty |a_i-b_i|^{3/2} \) converges.
    3. The sequence \(\sum_1^\infty (a_i-b_i)^2 \) converges.
    4. The sequence \(\sum_1^\infty (a_i-b_i)^3 \) converges.

  3. Suppose \( S=\{1,2,3,4,5,6\} \). Find the number of pairs \( (A,B) \) that can be formed such that \(A \subseteq S\) and \(B\subset A\). Write only the answer.

  4. Let \( f(x)=1+ax+bx^{2}+3x^{3}\) be a polynomial where \(a\) and \(b\) are integers. Suppose \(f(x)\) has a rational root \(\frac{p}{q}\), where \(\text{gcd}(p,q)=1\). Which of the following statements are true?
    1. \(p\) must be even.
    2. \(q\) must be even.
    3. \(p\) must be odd.
    4. \(q\) must be odd.
  5. Positive integers \(x, y\) satisfy the following conditions: \[ \{\sqrt{x^2 + 2y}\}> \frac{2}{3}; \hspace{10mm} \{\sqrt{y^2 + 2x}\}> \frac{2}{3} \] where \( \{x\}\) denotes the fractional part of \(x\). For example, \( \{2.34\} = 0.34\). Which of the following is true?
    1. Either \(y = x + 1\) or \( x = y + 1\).
    2. \(y > x\).
    3. \(y = \lfloor 2x/3 \rfloor \).
    4. \(y = x \).
  6. Which of the following equations has the greatest number of real solutions?
    1. \(x^{3}=10-x\)
    2. \(x^{2}+5 x-7=x+8\)
    3. \(7 x+5=1-3 x\)
    4. \(e^{x}=x\)
  7. In how many regions is the plane divided when the following equations are graphed, not considering the axes? \[ y=x^{2} \] \[ y=2^{x} \]
    1. 3
    2. 4
    3. 5
    4. 6
  8. A positive integer \(n\) is such that \( \left(n 2^{n}-1\right) \) is divisible by 3. Which form does \(n\) take (for a positive integer \(k\))?
    1. \(6k+2\) or \(6k+4\)
    2. \(6k+3\) or \(6k+5\)
    3. \(6k+4\) or \(6k+5\)
    4. \(6k+1\) or \(6k+4\)
  9. On the real line place an object at 1. After every flip of a fair coin, move the object to the right by 1 unit if the outcome is Head and to the left by 1 unit if the outcome is Tail. Let \(n\) be a fixed positive integer. Game ends when the object reaches either 0 or \(n\). Let \( P(n) \) denote the probability of the object reaching \(n\). Write down the value of \( P(3) \).
  10. If \(1, w_{1}, w_{2}, w_{3}, w_{4}, w_{5}\) are distinct roots of \(x^{6}-1\), then
    1. \(1+w_{i}+w_{i}^{2}+w_{i}^{3}+w_{i}^{4}+w_{i}^{5}=0\) for \(i=1,2,3,4,5\).
    2. \(1+w_{i}^{2}+w_{i}^{4}+w_{i}^{6}+w_{i}^{8}+w_{i}^{10}=0\) for \(i=1,2,3,4,5\).
    3. \(1+w_{i}^{3}+w_{i}^{6}+w_{i}^{9}+w_{i}^{12}+w_{i}^{15}=0\) for \(i=1,2,3,4,5\).
    4. \(1+w_{i}^{5}+w_{i}^{10}+w_{i}^{15}+w_{i}^{20}+w_{i}^{25}=0\) for \(i=1,2,3,4,5\).

Part B: Subjective questions

Submission files: Each question in this part must be answered on a page of its own. Name the files as B1.jpg, B2.jpg, etc. In case you have multiple files for the same question, say B4, name the corresponding files as B4-1.jpg, B4-2.jpg, etc.

Clearly explain your entire reasoning. No credit will be given without reasoning. Partial solutions may get partial credit.

B1. A frog is intially at the origin of a co-ordinate plane. It wants to reach a destination point by making a sequence of jumps. The frog can jump from point \( (x_1,y_1) \) to point \((x_2,y_2)\) if the points are exactly 5 units apart in distance or, in other words, if they satisfy: \[ \sqrt{ (x_1-x_2)^2 + (y_1-y_2)^2 } = 5 \] Note that the frog's coordinates before and after the jump are integers. For example, the frog can jump to \( (3,4) \) or \( (-3,4) \) or \( (5,0) \), etc. from the origin. Similarly, the frog can reach either the point \( (10,0) \) or \( (7,7) \) in two jumps.
(i) True or False: The frog can reach any given point with integer coordinates in a finite number of jumps. [2 points]
(ii) What is the minimum number of jumps required to reach the point \( (180,180) \) [8 points]?
For both the parts (a) and (b), explain your reasoning.
Source: Slight variation of Problem A1, Putnam 2021.

B2. Prove that every positive rational number can be written in the form \( a/b \) such that both \(a\) and \(b\) are products of factorials of primes. For example, \[ \frac{10}{9} = \frac{2!\cdot 5!}{3!\cdot 3! \cdot 3!}. \] Note that the primes appearing in \(a\) or \(b\) need not be distinct. [10 points]
Source: 2009 Putnam.

B3. Consider the following two functions defined on the interval \( [0,1] \): \[ f(x)=\sin \left(\frac{\pi \sin \frac{\pi x}{2}}{2}\right) \] and \[ g(x)=\frac{2}{\pi} \arcsin \left(\frac{2}{\pi} \arcsin x\right) \] (i) Show that the graphs of \( f(x) \) and \(g(x)\) are symmetric about the line \(y=x\). [3 points]
(ii) Let \(0 < a < 1\) be a fixed number. Show that there are at least two values of \(x\) in the interval \( (0,1) \) such that: \[ \int_0^x(f(t)+g(t)-2 t) = \frac{(a-g(a))(f(a)-a)}{2} \] [7 points]
Source: Simon Marais '22.

B4. Determine whether the series \[ \sum_{n=1}^{\infty} \frac{1}{n^{1+[\sin n\rceil}} \] is convergent or divergent. Explain your reasoning. Here \(\lceil x\rceil\) denotes the least integer greater than or equal to \(x\). [10 points]
Source: Simon Marais '22.

B5. Point \(T\) is chosen in the plane of a rhombus \(ABCD\) so that \(\angle ATC + \angle BTD = 180^\circ\), and circumcircles of triangles \(ATC\) and \(BTD\) are tangent to each other.
(i) Suppose \( O_1 \) and \( O_2 \) are the circumcenters of \(\triangle ATC\) and \(\triangle BTD\), respectively. Prove that quadrilaterals \( ABO_1O_2 \) and \( DCO_2O_1\) are cyclic. [5 points]
(ii) Show that \(T\) is equidistant from diagonals of \(ABCD\). [5 points]
Proposed by Fedir Yudin. https://artofproblemsolving.com/community/c6h3046178p27437928

B6. Let \(A_{1}=\{1\}, A_{n+1}=\left\{3 k, 3 k+1: k \in A_{n}\right\}\) for all \(n \geq 1\) and \(A=\bigcup_{n=1}^{\infty} A_{n}\).
(i) What is the set \(A_3\)? [2 points]
(ii) Can 2017 be written as the sum of two elements of \(A\)? Explain your reasoning. [8 points]

Source: B1, 2017 Madhava Competition.