Olympiad problems

2015 Dutch IMO TST, 5

For a positive integer $n$, we dene $D_n$ as the largest integer that is a divisor of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$. 1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer. 2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.

2009 IMAC Arhimede, 5

Tags: diophantine equation , diophantine , exponential equation , exponential , number theory

Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .

2007 Abels Math Contest (Norwegian MO) Final, 4

Tags: exponential , divisible , number theory

Let $a, b$ and $c$ be integers such that $a + b + c = 0$. (a) Show that $a^4 + b^4 + c^4$ is divisible by $a^2 + b^2 + c^2$. (b) Show that $a^{100} + b^{100} + c^{100}$ is divisible by $a^2 + b^2 + c^2$. .

2009 Belarus Team Selection Test, 2

Tags: algebra , constant , exponential

Find all $n \in N$ for which the value of the expression $x^n+y^n+z^n$ is constant for all $x,y,z \in R$ such that $x+y+z=0$ and $xyz=1$. D. Bazylev

2007 Dutch Mathematical Olympiad, 4

Tags: number theory , perfect square , exponential

Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square. (And prove that your answer is correct.)

2015 Hanoi Open Mathematics Competitions, 13

Tags: number theory , relatively prime , exponential

Let $m$ be given odd number, and let $a, b$ denote the roots of equation $x^2 + mx - 1 = 0$ and $c = a^{2014} + b^{2014}$ , $d =a^{2015} + b^{2015}$ . Prove that $c$ and $d$ are relatively prime numbers.

2023 Romania National Olympiad, 1

Tags: algebra , exponential , exponential equation

Solve the following equation for real values of $x$: \[ 2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x. \]

1976 IMO Shortlist, 11

Tags: number theory , digit , decimal representation , exponential

Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.

2009 Federal Competition For Advanced Students, P1, 1

Tags: inequalities , factorial , exponential , exponential inequality , diophantine , induction , algebra

Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$ .

2015 Abels Math Contest (Norwegian MO) Final, 4

Tags: exponential , perfect square , number theory

a. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is even. b. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is odd

2015 Balkan MO Shortlist, A5

Tags: algebra , inequalities , exponential inequality , exponential , exponential equation

Let $m, n$ be positive integers and $a, b$ positive real numbers different from $1$ such thath $m > n$ and $$\frac{a^{m+1}-1}{a^m-1} = \frac{b^{n+1}-1}{b^n-1} = c$$. Prove that $a^m c^n > b^n c^{m}$ (Turkey)

1996 Tuymaada Olympiad, 7

Tags: equation , algebra , exponential , rational , exponential equation

In the set of all positive real numbers define the operation $a * b = a^b$ . Find all positive rational numbers for which $a * b = b * a$.

2024 VJIMC, 1

Tags: inequalities , calculus , exponential , function

Suppose that $f:[-1,1] \to \mathbb{R}$ is continuous and satisfies \[\left(\int_{-1}^1 e^xf(x) dx\right)^2 \ge \left(\int_{-1}^1 f(x) dx\right)\left(\int_{-1}^1 e^{2x}f(x) dx\right).\] Prove that there exists a point $c \in (-1,1)$ such that $f(c)=0$.

2019 Korea USCM, 5

Tags: logarithm , sequence , series , exponential

A sequence $\{a_n\}_{n\geq 1}$ is defined by a recurrence relation $$a_1 = 1,\quad a_{n+1} = \log \frac{e^{a_n}-1}{a_n}$$ And a sequence $\{b_n\}_{n\geq 1}$ is defined as $b_n = \prod\limits_{i=1}^n a_i$. Evaluate an infinite series $\sum\limits_{n=1}^\infty b_n$.

2015 Finnish National High School Mathematics Comp, 3

Tags: number theory , factorial , factor , exponential , divide

Determine the largest integer $k$ for which $12^k$ is a factor of $120! $

2021 JBMO Shortlist, N7

Tags: game , number theory , shortlist , exponential

Alice chooses a prime number $p > 2$ and then Bob chooses a positive integer $n_0$. Alice, in the first move, chooses an integer $n_1 > n_0$ and calculates the expression $s_1 = n_0^{n_1} + n_1^{n_0}$; then Bob, in the second move, chooses an integer $n_2 > n_1$ and calculates the expression $s_2 = n_1^{n_2} + n_2^{n_1}$; etc. one by one. (Each player knows the numbers chosen by the other in the previous moves.) The winner is the one who first chooses the number $n_k$ such that $p$ divides $s_k(s_1 + 2s_2 + · · · + ks_k)$. Who has a winning strategy? Proposed by [i]Borche Joshevski, Macedonia[/i]

2010 Belarus Team Selection Test, 7.3

Tags: algebra , inequalities , exponential inequality , exponential

Prove that all positive real $x, y, z$ satisfy the inequality $x^y + y^z + z^x > 1$. (D. Bazylev)

2014 Federal Competition For Advanced Students, P2, 3

Tags: geometric inequality , inequalities , triangle inequality , exponential

(i) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also the inequalities $a^2 + b^2> c^2, b^2 + c^2> a^2$ and $a^2 + c^2> b^2$ ? (ii) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also for all positive natural $n$ the inequalities $a^n + b^n> c^n, b^n + c^n> a^n$ and $a^n + c^n> b^n$ ?