Olympiad problems

2023 Junior Balkan Team Selection Tests - Moldova, 11

Tags: number theory

Find all prime $x,y$ and $z,$ such that $x^5 +y^3 -(x+y)^2=3z^3$

2017 Simon Marais Mathematical Competition, B2

Tags: modular arithmetic , diophantine equation , number theory

Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square. [i]Proposed by P. Boyvalenkov[/i]

2014 Saudi Arabia Pre-TST, 2.1

Tags: divide , divisible , number theory

Prove that $2014$ divides $53n^{55}- 57n^{53} + 4n$ for all integer $n$.

1985 Bulgaria National Olympiad, Problem 6

Tags: algebra , number theory , sequence

Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit.

1983 Poland - Second Round, 4

Tags: number theory , divisor , sum

Let $ a(k) $ be the largest odd number by which $ k $ is divisible. Prove that $$ \sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).$$

2022 MOAA, 10

Tags: number theory , combinatorics

Three integers $A, B, C$ are written on a whiteboard. Every move, Mr. Doba can either subtract $1$ from all numbers on the board, or choose two numbers on the board and subtract $1$ from both of them whilst leaving the third untouched. For how many ordered triples $(A, B, C)$ with $1 \le A < B < C\le 20$ is it possible for Mr. Doba to turn all three of the numbers on the board to $0$?

1997 Italy TST, 3

Tags: number theory

Determine all triples $(x,y, p)$ with $x$, $y$ positive integers and $p$ a prime number verifying the equation $p^x -y^p = 1$.

2023 LMT Fall, 4

Tags: number theory

Fred chooses a positive two-digit number with distinct nonzero digits. Laura takes Fred’s number and swaps its digits. She notices that the sum of her number and Fred’s number is a perfect square and the positive difference between them is a perfect cube. Find the greater of the two numbers.

2017 All-Russian Olympiad, 7

Tags: number theory , algebra

There is number $N$ on the board. Every minute Ivan makes next operation: takes any number $a$ written on the board, erases it, then writes all divisors of $a$ except $a$( Can be same numbers on the board). After some time on the board there are $N^2$ numbers. For which $N$ is it possible?

2022 Cyprus TST, 2

Tags: perfect square , number theory , pell equation

Let $n, m$ be positive integers such that \[n(4n+1)=m(5m+1)\] (a) Show that the difference $n-m$ is a perfect square of a positive integer. (b) Find a pair of positive integers $(n, m)$ which satisfies the above relation. Additional part (not asked in the TST): Find all such pairs $(n,m)$.

1993 Tournament Of Towns, (387) 5

Tags: number theory , sum of digits

Let $S(n)$ denote the sum of digits of $n$ (in decimal representation). Do there exist three different natural numbers $n$, $p$ and $q$ such that $$n +S(n) = p + S(p) = q + S(q)?$$ (M Gerver)

2015 Romania Team Selection Tests, 2

Tags: binomial coefficient , number theory

Given an integer $k \geq 2$, determine the largest number of divisors the binomial coefficient $\binom{n}{k}$ may have in the range $n-k+1, \ldots, n$ , as $n$ runs through the integers greater than or equal to $k$.

2009 Kosovo National Mathematical Olympiad, 2

Tags: number theory

Let $p$ be a prime number and $n$ a natural one. How many natural numbers are between $1$ and $p^n$ that are relatively prime with $p^n$?

2025 Harvard-MIT Mathematics Tournament, 5

Tags: algebra , number theory

Let $\mathcal{S}$ be the set of all nonconstant polynomials $P$ with integer coefficients satisfying $P(\sqrt{3}+\sqrt{2})=P(\sqrt{3}-\sqrt{2}).$ If $Q$ is an element of $\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10)-Q(0).$

2001 Tournament Of Towns, 2

Tags: greatest common divisor , number theory

In three piles there are $51, 49$, and $5$ stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get $105$ piles with one stone in each?

2014 Bosnia and Herzegovina Junior BMO TST, 1

Tags: number theory , digit , divisible

Let $x$, $y$ and $z$ be nonnegative integers. Find all numbers in form $\overline{13xy45z}$ divisible with $792$, where $x$, $y$ and $z$ are digits.

2011 India IMO Training Camp, 2

Tags: number theory

Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.

1964 IMO, 1

Tags: number theory

(a) Find all positive integers $ n$ for which $ 2^n\minus{}1$ is divisible by $ 7$. (b) Prove that there is no positive integer $ n$ for which $ 2^n\plus{}1$ is divisible by $ 7$.

2012 Korea National Olympiad, 1

Tags: induction , modular arithmetic , number theory

$ p >3 $ is a prime number such that $ p | 2^{p-1} -1 $ and $ p \not | 2^x - 1 $ for $ x = 1, 2, \cdots , p-2 $. Let $ p = 2k+3 $. Now we define sequence $ \{ a_n \} $ as \[ a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \ a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 ) \] Prove that there exist $2k$ consecutive terms of sequence $ a_{x+1} , a_{x+2} , \cdots , a_{x+2k} $ such that for all $ 1 \le i < j \le 2k $, $ a_{x+i} \not \equiv a_{x+j} \ (mod \ p) $.

2017 Czech-Polish-Slovak Junior Match, 3

Tags: digit , divisible , number theory

How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?

2014 Greece National Olympiad, 2

Tags: fraction , number theory

Find all the integers $n$ for which $\frac{8n-25}{n+5}$ is cube of a rational number.

2021 BMT, 14

Tags: algebra , number theory

Given an integer $c$, the sequence $a_0, a_1, a_2, ...$ is generated using the recurrence relation $a_0 = c$ and $a_i = a^i_{i-1} + 2021a_{i-1}$ for all $i \ge 1$. Given that $a_0 = c$, let $f(c)$ be the smallest positive integer $n$ such that $a_n - 1$ is a multiple of $47$. Compute $$\sum^{46}_{k=1} f(k).$$

2000 Turkey MO (2nd round), 2

Tags: polynomial , number theory , relatively prime , algebra

Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that \[P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).\]

1999 Kazakhstan National Olympiad, 6

Tags: number theory with sequences , sequence , divisible , remainder , number theory

In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$.

2019 Portugal MO, 2

Tags: number theory , digit

A five-digit integer is said to be [i]balanced [/i]i f the sum of any three of its digits is divisible by any of the other two. How many [i]balanced [/i] numbers are there?