CMI Mock Test — Paper I

Question 1   [4 marks]

Evaluate each of the following statements. Label each as True (T) or False (F).

1. If the one-sided limits $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ both exist and are equal, then the two-sided limit $\lim_{x \to a} f(x)$ must exist.

2. If a function $f : [a, b] \to \mathbb{R}$ is continuous on a closed and bounded interval, it must attain an absolute maximum and an absolute minimum value within that interval.

3. The function $f(x) = |x|$ possesses a well-defined, finite derivative at every real value of $x$ except exactly at $x = 0$.

4. Every function that is continuous on the open interval $(0, \infty)$ is necessarily uniformly continuous on that interval.

Question 2   [4 marks]

Evaluate each of the following statements. Label each as True (T) or False (F).

1. If a function of two variables $f(x, y)$ is continuous in each variable separately, then $f(x, y)$ is continuous as a function of both variables jointly.

2. If a function $f$ is differentiable everywhere on $\mathbb{R}$, its derivative function $f'$ must be continuous everywhere on $\mathbb{R}$.

3. If a differentiable function $f$ has a strictly positive derivative at a point $x = c$ (i.e., $f'(c) > 0$), then $f$ must be strictly increasing on some open neighbourhood of $c$.

4. If $\lim_{x \to \infty} f(x)$ exists and is finite, and if $f$ is differentiable for all $x > 0$, then it necessarily follows that $\lim_{x \to \infty} f'(x) = 0$.

Question 3   [4 marks]

Evaluate each of the following statements. Label each as True (T) or False (F).

1. Any polynomial $p(x)$ with real coefficients and odd degree must possess at least one real root.

2. In any arbitrary ring $R$, for any two elements $a, b \in R$, the binomial expansion $(a+b)^2 = a^2 + 2ab + b^2$ holds true.

3. If an $n \times n$ matrix $A$ over $\mathbb{R}$ has $0$ as an eigenvalue, then $\det(A) = 0$ and $A$ is not invertible.

4. Every non-constant polynomial with real coefficients can be factored completely into a product of linear and irreducible quadratic factors over $\mathbb{R}$.

Question 4   [4 marks]

Evaluate each of the following statements. Label each as True (T) or False (F).

1. In any finite undirected graph where every vertex has exactly degree 3, the total number of vertices must be even.

2. If 101 pigeons are distributed into 10 pigeonholes, the pigeonhole principle guarantees that at least one hole contains exactly 11 pigeons.

3. For any positive integer $n$, the total number of compositions of $n$ (ordered partitions into positive integers) is exactly $2^{n-1}$.

4. The total number of distinct subsets of an $n$-element set is exactly $2^n$.

Question 5   [4 marks]

Evaluate each of the following statements. Label each as True (T) or False (F).

1. For any non-equilateral triangle, the orthocenter, centroid, and circumcenter are always collinear.

2. In any cyclic quadrilateral, the sum of the products of the lengths of opposite sides equals the product of the lengths of its diagonals.

3. The Simson line corresponding to a point $P$ on the circumcircle of $\triangle ABC$ always passes directly through the orthocenter of $\triangle ABC$.

4. In any parallelogram, the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of its two diagonals.

Question 6   [4 marks]

Evaluate each of the following statements. Label each as True (T) or False (F).

1. If a real-valued function is continuous and strictly bounded on $[0, \infty)$, it must be uniformly continuous on $[0, \infty)$.

2. If $f$ is a continuous real-valued function on $[0,1]$ such that $\int_0^1 f(x)\, x^n\, dx = 0$ for every non-negative integer $n \ge 0$, then $f(x) = 0$ for all $x \in [0,1]$.

3. There exists a real-valued function that is continuous at every point on $\mathbb{R}$ but differentiable at no point on $\mathbb{R}$.

4. The infinite series $\displaystyle\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$ is absolutely convergent.

Question 7   [4 marks]

Evaluate each of the following statements. Label each as True (T) or False (F).

1. Every composite positive integer can be expressed in the form $xy + xz + yz + 1$ for some positive integers $x, y, z$.

2. If $p$ is a prime number, then $(p-1)! + 1$ is an integer multiple of $p$.

3. The number of strictly positive divisors of a positive integer $n$ is odd if and only if $n$ is a perfect square.

4. If $p > 5$ is prime, the sequence $1, a, a^2, \ldots, a^{p-5}$ can always be rearranged to form an arithmetic progression modulo $p$ for some suitably chosen integer $a$.

Question 8   [4 marks]

Evaluate each of the following statements. Label each as True (T) or False (F).

1. If $b = \sup A$ for a non-empty subset $A \subset \mathbb{R}$, then for every $\varepsilon > 0$, there exists $x \in A$ such that $b - \varepsilon < x \le b$.

2. Between any two distinct real numbers $a$ and $b$, there exists at least one rational number.

3. Every non-empty subset of integers that is bounded above in $\mathbb{R}$ possesses a maximum element.

4. If a sequence of real numbers $(x_n)$ is bounded above, it is guaranteed to contain a convergent subsequence.

Question 9   [4 marks — 2 each]

Solve the following two independent sub-parts. State only the numerical answer; no proof required.

(a) Determine the total number of distinct ways to rearrange the letters of the word MATHEMATICS.

(b) Let $S$ be the set of all 5-digit numbers containing each of the digits $1, 3, 5, 7, 9$ exactly once. Compute: $\frac{\text{sum of all elements of } S}{11111}$

Question 10   [4 marks — 2 each]

Solve the following two independent sub-parts. State only the numerical answer; no proof required.

(a) For a positive integer $n$, define: $f_n(x) = \cos(x)\cos(2x)\cos(3x)\cdots\cos(nx)$ Find the smallest positive integer $n$ such that $|f_n''(0)| > 2023$.

(b) Determine the unique positive integer $n$ for which the Diophantine equation $2a^n + 3b^n = 4c^n$ has at least one solution in positive integers $a, b, c$ with $\gcd(a, b, c) = 1$.

Question 1   [12 marks]

Let the sequence of real numbers $(x_n)$ be defined by $x_0 = 1$ and the recurrence: $x_{n+1} = \ln(e^{x_n} - x_n) \quad \text{for all } n \ge 0$

(a) (4 marks) Prove that $(x_n)$ is strictly decreasing and bounded below, and determine $\displaystyle\lim_{n \to \infty} x_n$.

(b) (8 marks) Prove that the infinite series $\displaystyle\sum_{n=0}^{\infty} x_n$ converges absolutely.

Question 2   [14 marks]

Let $n > 1$ be a positive integer. A rearrangement $a_1, a_2, \ldots, a_n$ of the sequence $1, 2, \ldots, n$ is called nice if, for every $k \in \{2, 3, \ldots, n\}$, the partial sum $\displaystyle\sum_{i=1}^{k} a_i$ is not divisible by $k$.

(a) (6 marks) Prove that if $n$ is odd, no nice rearrangement exists.

(b) (8 marks) Prove that if $n$ is even, a nice rearrangement always exists by explicitly constructing one and verifying the modular condition for all $k$.

Question 3   [14 marks]

Let $A_1, A_2, A_3$ be three distinct points on a circle $\Gamma$ of radius $R$. For $i \in \{1,2,3\}$, let $\tau_i$ denote the counter-clockwise rotation centred at $A_i$ by the interior angle of $\triangle A_1 A_2 A_3$ at vertex $A_i$. (Indices are taken mod 3, so $A_4 = A_1$.)

For an arbitrary point $P$ in the plane, define: $P_i = \tau_{i+2}\!\bigl(\tau_i\!\bigl(\tau_{i+1}(P)\bigr)\bigr)$

(a) (6 marks) Geometrically describe the location of $P_i$ relative to $P$. What is the total rotation angle of the composite transformation, and what does it imply?

(b) (8 marks) Using the result from (a), prove that the circumradius of $\triangle P_1 P_2 P_3$ is at most $R$.

Question 4   [12 marks]

In a mathematics competition, 16 students take a test of multiple-choice questions, each question having exactly 4 distinct choices. After grading, the following unusual property is discovered: any two distinct students share at most one common answer across the entire test.

(a) (4 marks) Model this scenario using an incidence matrix. Define the matrix dimensions precisely, state what each entry represents, and express the “at most one common answer” condition as a linear-algebraic constraint on the matrix.

(b) (8 marks) Using double counting on the number of pairs of students sharing an answer, prove rigorously that the maximum possible number of questions on this test is $\mathbf{5}$.

Question 5   [14 marks]

Suppose $P(x) = a_1 x + a_2 x^2 + \cdots + a_m x^m$ is a polynomial with integer coefficients such that $a_1$ is odd. Define the formal power series: $e^{P(x)} = \sum_{k=0}^{\infty} b_k\, x^k$

(a) (4 marks) Show that $b_0 = 1$ and derive a recursive formula expressing $b_k$ in terms of $b_{k-1}, b_{k-2}, \ldots$ and the coefficients $a_i$.

(b) (10 marks) Prove that $b_k$ is strictly non-zero for all $k \ge 0$.

Question 6   [14 marks]

Let $p(x)$ be a polynomial with real coefficients. Suppose there exists a polynomial $q(x)$ with real coefficients such that: $p(p(x)) - x = (p(x) - x)^2\, q(x) \quad \text{for all } x \in \mathbb{R}$

(a) (4 marks) Let $f(x) = p(x) - x$. Rewrite the functional equation purely in terms of $f$. (Hint: Substitute $p(x) = x + f(x)$ into $p(p(x))$.)

(b) (10 marks) By analysing the degree of $f(x)$ and using formal differentiation of the identity from (a), prove that the only polynomials $p(x)$ satisfying the given equation are of the form: $p(x) = \pm x + c$ for some constant $c \in \mathbb{R}$.