Olympiad problems

2005 Germany Team Selection Test, 1

[b](a)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ ends with the string $2004$, followed by a number of digits from the set $\left\{0;\;4\right\}$ ? [b](b)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ starts with the string $2004$ ?

2021 China Team Selection Test, 3

Tags: number theory , sum of digits , decimal representation , China TST , China , number theory proposed

Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$ where $S(n)$ denotes sum of digits of decimal representation of $n$.

2014 Taiwan TST Round 1, 2

Tags: algebra , polynomial , modular arithmetic , induction , function , strong induction , number theory proposed

For a fixed integer $k$, determine all polynomials $f(x)$ with integer coefficients such that $f(n)$ divides $(n!)^k$ for every positive integer $n$.

2012 Canadian Mathematical Olympiad Qualification Repechage, 5

Tags: number theory proposed , number theory

Given a positive integer $n$, let $d(n)$ be the largest positive divisor of $n$ less than $n$. For example, $d(8) = 4$ and $d(13) = 1$. A sequence of positive integers $a_1, a_2,\dots$ satisfies \[a_{i+1} = a_i +d(a_i),\] for all positive integers $i$. Prove that regardless of the choice of $a_1$, there are inﬁnitely many terms in the sequence divisible by $3^{2011}$.

2012 Puerto Rico Team Selection Test, 6

Tags: MATHCOUNTS , number theory proposed , number theory

The increasing sequence $1; 3; 4; 9; 10; 12; 13; 27; 28; 30; 31, \ldots$ is formed with positive integers which are powers of $3$ or sums of different powers of $3$. Which number is in the $100^{th}$ position?

2000 Turkey MO (2nd round), 1

Tags: algebra , polynomial , geometric series , number theory proposed , number theory

Let $p$ be a prime number. $T(x)$ is a polynomial with integer coefficients and degree from the set $\{0,1,...,p-1\}$ and such that $T(n) \equiv T(m) (mod p)$ for some integers m and n implies that $ m \equiv n (mod p)$. Determine the maximum possible value of degree of $T(x)$

2011 Regional Competition For Advanced Students, 1

Tags: number theory proposed , number theory , AUT , Austria

Let $p_1, p_2, \ldots, p_{42}$ be $42$ pairwise distinct prime numbers. Show that the sum \[\sum_{j=1}^{42}\frac{1}{p_j^2+1}\] is not a unit fraction $\frac{1}{n^2}$ of some integer square number.

2012 India IMO Training Camp, 2

Tags: pigeonhole principle , number theory proposed , number theory

Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p\in P}p\right)-1$ are also in $S$. Determine all possible such sets $S$.

2014 European Mathematical Cup, 4

Tags: function , induction , number theory proposed , number theory

Find all functions $f$ from positive integers to themselves such that: 1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$ 2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.

2008 China Team Selection Test, 3

Tags: analytic geometry , geometry , modular arithmetic , trapezoid , induction , trigonometry , number theory proposed

Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)

2010 Bulgaria National Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

Does there exist a number $n=\overline{a_1a_2a_3a_4a_5a_6}$ such that $\overline{a_1a_2a_3}+4 = \overline{a_4a_5a_6}$ (all bases are $10$) and $n=a^k$ for some positive integers $a,k$ with $k \geq 3 \ ?$

2011 All-Russian Olympiad, 4

Tags: number theory , relatively prime , number theory proposed

Do there exist any three relatively prime natural numbers so that the square of each of them is divisible by the sum of the two remaining numbers?

1993 Baltic Way, 2

Tags: modular arithmetic , number theory proposed , number theory

Do there exist positive integers $a>b>1$ such that for each positive integer $k$ there exists a positive integer $n$ for which $an+b$ is a $k$-th power of a positive integer?

2014 Contests, 1

Tags: quadratics , number theory proposed , number theory

Find all pairs of non-negative integers $(x,y)$ such that \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]

2007 ITAMO, 5

Tags: number theory proposed , number theory

The sequence of integers $(a_{n})_{n \ge 1}$ is defined by $a_{1}= 2$, $a_{n+1}= 2a_{n}^{2}-1$. Prove that for each positive integer n, $n$ and $a_{n}$ are coprime.

2021 Baltic Way, 20

Tags: number theory , number theory proposed , Divisibility , least common multiple

Let $n\ge 2$ be an integer. Given numbers $a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\}$ such that $\operatorname{lcm}(a_i,a_j)>2n$ for all $1\le i<j\le n$, prove that $$a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).$$

2007 Turkey Team Selection Test, 2

Tags: number theory proposed , number theory

A number $n$ is satisfying the conditions below i) $n$ is a positive odd integer; ii) there are some odd integers such that their squares' sum is equal to $n^{4}$. Find all such numbers.

2021 Baltic Way, 19

Tags: number theory , number theory proposed , Polynomials

Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.

2002 Irish Math Olympiad, 2

Tags: number theory proposed , number theory

Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that: $ (i)$ $ a\plus{}c\equal{}d;$ $ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$ $ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$. Determine $ n$.

1976 Miklós Schweitzer, 3

Tags: inequalities , function , logarithms , number theory proposed , number theory

Let $ H$ denote the set of those natural numbers for which $ \tau(n)$ divides $ n$, where $ \tau(n)$ is the number of divisors of $ n$. Show that a) $ n! \in H$ for all sufficiently large $ n$, b)$ H$ has density $ 0$. [i]P. Erdos[/i]

2011 Iran MO (3rd Round), 1

Tags: algorithm , number theory proposed , number theory

Suppose that $S\subseteq \mathbb Z$ has the following property: if $a,b\in S$, then $a+b\in S$. Further, we know that $S$ has at least one negative element and one positive element. Is the following statement true? There exists an integer $d$ such that for every $x\in \mathbb Z$, $x\in S$ if and only if $d|x$. [i]proposed by Mahyar Sefidgaran[/i]

2007 Czech-Polish-Slovak Match, 2

Tags: number theory proposed , number theory

The Fibonacci sequence is deﬁned by $a_1=a_2=1$ and $a_{k+2}=a_{k+1}+a_k$ for $k\in\mathbb N.$ Prove that for any natural number $m,$ there exists an index $k$ such that $a_k^4-a_k-2$ is divisible by $m.$

1964 IMO, 1

Tags: number theory proposed , number theory , IMO , IMO 1964

(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$.

2008 China Western Mathematical Olympiad, 1

Tags: geometry , geometric transformation , reflection , analytic geometry , number theory proposed , number theory

Four frogs are positioned at four points on a straight line such that the distance between any two neighbouring points is 1 unit length. Suppose the every frog can jump to its corresponding point of reflection, by taking any one of the other 3 frogs as the reference point. Prove that, there is no such case that the distance between any two neighbouring points, where the frogs stay, are all equal to 2008 unit length.

1949 Miklós Schweitzer, 6

Tags: number theory , greatest common divisor , number theory proposed

Let $ n$ and $ k$ be positive integers, $ n\geq k$. Prove that the greatest common divisor of the numbers $ \binom{n}{k},\binom{n\plus{}1}{k},\ldots,\binom{n\plus{}k}{k}$ is $ 1$.