Olympiad problems

ICMC 6, 2

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f'(x) > f(x)>0$ for all real numbers $x$. Show that $f(8) > 2022f(0)$. [i]Proposed by Ethan Tan[/i]

ICMC 6, 4

Tags: ICMC , college contests , number theory , sum of divisors , Perfect Square

Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square? [i]Proposed by Dylan Toh[/i]

2023 SG Originals, Q5

Tags: geometry , combinatorics , ICMC , college contests , expected value , area of a triangle

A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$? [i]Proposed by Dylan Toh[/i]

ICMC 5, 5

Tags: 3d , Vectors , dot product , vector , linear algebra , college contests , ICMC

A [i]tanned vector[/i] is a nonzero vector in $\mathbb R^3$ with integer entries. Prove that any tanned vector of length at most $2021$ is perpendicular to a tanned vector of length at most $100$. [i]Proposed by Ethan Tan[/i]

ICMC 5, 2

Tags: table , combinatorics , number theory , college contests , ICMC

Find all integers $n$ for which there exists a table with $n$ rows, $2022$ columns, and integer entries, such that subtracting any two rows entry-wise leaves every remainder modulo $2022$. [i]Proposed by Tony Wang[/i]

ICMC 4, 4

Tags: analysis , ICMC , college contests , geometry

Let $\mathbb R^2$ denote the Euclidean plane. A continuous function $f : \mathbb R^2 \to \mathbb R^2$ maps circles to circles. (A point is not a circle.) Prove that it maps lines to lines. [i]Proposed by Tony Wang[/i]

ICMC 4, 3

Tags: expectation , ICMC , college contests , Inequality

Let $f,g,h : \mathbb R \to \mathbb R$ be continuous functions and $X$ be a random variable such that $E(g(X)h(X))=0$ and $E(g(X)^2) \neq 0 \neq E(h(X)^2)$. Prove that $$E(f(X)^2) \geq \frac{E(f(X)g(X))^2}{E(g(X)^2)} + \frac{E(f(X)h(X))^2}{E(h(X)^2)}.$$ You may assume that all expected values exist. [i]Proposed by Cristi Calin[/i]

ICMC 4, 3

Tags: calculus , integration , college contests , ICMC , sin

Let $\displaystyle s_n=\int_0^1 \text{sin}^n(nx) \,dx$. (a) Prove that $s_n \leq \dfrac 2n$ for all odd $n$. (b) Find all the limit points of the sequence $s_1, s_2, s_3, \dots$. [i]Proposed by Cristi Calin[/i]

ICMC 6, 1

Tags: college contests , combinatorial geometry , ICMC

Two straight lines divide a square of side length $1$ into four regions. Show that at least one of the regions has a perimeter greater than or equal to $2$. [i]Proposed by Dylan Toh[/i]

2023 SG Originals, Q3

Tags: ICMC , combinatorics , Fibonacci , college contests

Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started. [i]Proposed by Dylan Toh[/i]

ICMC 5, 3

Tags: ICMC , geometry , combinatorial geometry , 3D geometry , college contests , symmetry

A set of points has [i]point symmetry[/i] if a reflection in some point maps the set to itself. Let $\cal P$ be a solid convex polyhedron whose orthogonal projections onto any plane have point symmetry. Prove that $\cal P$ has point symmetry. [i]Proposed by Ethan Tan[/i]

ICMC 7, 5

Tags: number theory , polynomial , ICMC

[list=a] [*]Is there a non-linear integer-coefficient polynomial $P(x)$ and an integer $N{}$ such that all integers greater than $N{}$ may be written as the greatest common divisor of $P(a){}$ and $P(b){}$ for positive integers $a>b$? [*]Is there a non-linear integer-coefficient polynomial $Q(x)$ and an integer $M{}$ such that all integers greater than $M{}$ may be written as $Q(a) - Q(b)$ for positive integers $a>b$? [/list][i]Proposed by Dylan Toh[/i]

2023 SG Originals, Q4

Tags: ICMC , college contests , number theory , sum of divisors , Perfect Square

Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square? [i]Proposed by Dylan Toh[/i]

ICMC 4, 2

Tags: number theory , ICMC , college contests , mod p

Let $p > 3$ be a prime number. A sequence of $p-1$ integers $a_1,a_2, \dots, a_{p-1}$ is called [i]wonky[/i] if they are distinct modulo $p$ and $a_ia_{i+2} \not\equiv a_{i+1}^2 \pmod p$ for all $i \in \{1, 2, \dots, p-1\}$, where $a_p = a_1$ and $a_{p+1} = a_2$. Does there always exist a wonky sequence such that $$a_1a_2, \qquad a_1a_2+a_2a_3, \qquad \dots, \qquad a_1a_2+\cdots +a_{p-1}a_1,$$ are all distinct modulo $p$? [i]Proposed by Harun Khan[/i]

ICMC 6, 3

Tags: ICMC , college contests , probability , combinatorics , random process

The numbers $1, 2, \dots , n$ are written on a blackboard and then erased via the following process:[list] [*] Before any numbers are erased, a pair of numbers is chosen uniformly at random and circled. [*] Each minute for the next $n -1$ minutes, a pair of numbers still on the blackboard is chosen uniformly at random and the smaller one is erased. [*] In minute $n$, the last number is erased. [/list] What is the probability that the smaller circled number is erased before the larger? [i]Proposed by Ethan Tan[/i]

ICMC 5, 4

Tags: ICMC , number theory , combinatorics

Fix a set of integers $S$. An integer is [i]clean[/i] if it is the sum of distinct elements of $S$ in exactly one way, and [i]dirty[/i] otherwise. Prove that the set of dirty numbers is either empty or infinite. [i]Note:[/i] We consider the empty sum to equal $0$. [i]Proposed by Tony Wang and Ethan Tan[/i]

ICMC 4, 4

Tags: combinatorics , positive rationals , college contests , ICMC

Does there exist a set $\mathcal{R}$ of positive rational numbers such that every positive rational number is the sum of the elements of a unique finite subset of $\mathcal{R}$? [i]Proposed by Tony Wang[/i]