Problems

I’ve written a few problems for various math competitions in the past; here are some of my favorites, with some notes.

National exams

#1 (2021 Korea Winter Program Practice Test 1, Problem 1). Is there an infinite set of positive integers $S$ satisfying the following condition?

For every two pairwise distinct elements $a, b$ of $S$, there exists an odd positive integer $k$ such that $a$ divides $b^k-1$.

#2 (2022 Korea Winter Program Practice Test 1, Problem 2). You are given a positive integer $n (\ge 2)$ and $n$ polynomials $P_1(x), P_2(x), \ldots, P_n(x)$, each of which has real coefficients and a positive leading coefficient. Provided that the polynomials aren’t all identical, prove the inequality

$$ \deg \left(P_1^n+P_2^n+\cdots+P_n^n-nP_1P_2\cdots P_n \right) \ge (n-2) \max_{1\le i\le n} \deg P_i $$

and determine when the equality holds.

Online exams

#3 (MOMO 2020 Problem 4; read more about the exam here¹). Suppose that there exist a nonempty set $X \subset \mathbb{R}$ and a function $f: X \rightarrow X$ such that for $x, y \in X$, $f(x) +y \in X$ if and only if $x \neq y$.

Prove that there is a real constant $C$ for which $f(x) = - x + C$ for all $x \in X$.

#4 (MOMO 2020 Problem 3). Let $\mathcal{S}(n)$ denote the sum of digits of a positive integer $n$ when written in base ten. Given a positive integer $k$, determine all tuples of pairwise distinct positive integers $(n_1, \cdots, n_k)$ satisfying the following conditions:

• $n_1, n_2, \cdots, n_k$ are not multiples of $10$;

• all but finitely many positive integers can be expressed in the form $\mathcal{S}(n_1t)+\cdots +\mathcal{S}(n_kt)$ for some positive integer $t$.

#5 (MOMO 2020 Problem 6). Call a set of unit squares a Thumbs-up sign if it can be obtained by cutting out a unit square from a corner of a $2 \times m$ rectangle. For example, a single unit square and a L-tromino are both Thumbs-up signs.

Given an integer $n \ge 1$, what is the minimum number of Thumbs-up signs required to cover a $2n \times 2n$ square so that every unit square in the $2n \times 2n$ square is covered by exactly one Thumbs-up sign?

#6 (2016). Let $p$ be an odd prime. For each integer $k$ coprime with $p$, define $f(k)$ as the smallest nonnegative integer such that $k$ divides $pf(k)+1$.

Compute $\left\lfloor \sum_{k=1}^{p-1} \frac{f(k)}{k} \right\rfloor$ in terms of $p$.

and other miscellaneous exams…

#7 (SMO² 2019). Let $ABC$ be a scalene triangle with orthocenter $H$. Let $\omega_B, \Omega_B$ be the excircles of triangle $AHB$ corresponding to vertices $A$ and $H$, respectively. Define $\omega_C$ and $\Omega_C$ similarly as the excircles of triangle $AHC$ corresponding to vertices $A$ and $H$.

Prove that an external tangent to $\omega_B$ and $\omega_C$ and an external tangent to $\Omega_B$ and $\Omega_C$ meet on line $AH$.

#8 (2019; school math club entrance exam). Let $p$ be a odd prime. Prove that among the divisors of $p-1$, the number of quadratic residues modulo $p$ is no less than that of non-quadratic residues modulo $p$.

#8* (2019; harder variant with the same idea) Let $\Omega(x)$ be an integer-coefficient polynomial with a positive leading coefficient. If $\Omega(0) \neq 0$ and $\Omega(3) \equiv 3 \pmod 4$, then prove that there exists some odd prime $p$ such that

• $\Omega(p) \neq 0$, and
• among the positive divisors of $|\Omega(p)|$, the number of quadratic residues and non-quadratic residues modulo $p$ are equal.

#9 (SMO 2018). Find all integers $n$ for which there exists a set $S$ of positive integers satisfying the following conditions:

(1) $S$ contains at least four elements;

(2) for every $a \in S$, there exist elements $b, c \in S$ (not necessarily different) such that $a = b^n - c^n$.