This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 2008

2017 Ukraine Team Selection Test, 12

Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that \[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \] and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$. [i]Proposed by Victor Wang[/i]

2011 IMO Shortlist, 5

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$. [i]Proposed by Mahyar Sefidgaran, Iran[/i]

2014 Contests, 2

Do there exist positive integers $a$ and $b$ such that $a^n+n^b$ and $b^n+n^a$ are relatively prime for all natural $n$?

1975 IMO, 4

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

1982 IMO Longlists, 7

Find all solutions $(x, y) \in \mathbb Z^2$ of the equation \[x^3 - y^3 = 2xy + 8.\]

2011 ELMO Shortlist, 1

Prove that $n^3-n-3$ is not a perfect square for any integer $n$. [i]Calvin Deng.[/i]

PEN H Problems, 37

Prove that for each positive integer $n$ there exist odd positive integers $x_n$ and $y_n$ such that ${x_{n}}^2 +7{y_{n}}^2 =2^n$.

2010 Contests, 1

Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that: \[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\] Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$

2010 N.N. Mihăileanu Individual, 4

If $ p $ is an odd prime, then the following characterization holds. $$ 2^{p-1}\equiv 1\pmod{p^2}\iff \sum_{2=q}^{(p-1)/2} q^{p-2}\equiv -1\pmod p $$ [i]Marius Cavachi[/i]

2003 India IMO Training Camp, 10

Let $n$ be a positive integer greater than $1$, and let $p$ be a prime such that $n$ divides $p-1$ and $p$ divides $n^3-1$. Prove that $4p-3$ is a square.

2014 Czech and Slovak Olympiad III A, 1

Let be $n$ a positive integer. Denote all its (positive) divisors as $1=d_1<d_2<\cdots<d_{k-1}<d_k=n$. Find all values of $n$ satisfying $d_5-d_3=50$ and $11d_5+8d_7=3n$. (Day 1, 1st problem author: Matúš Harminc)

2008 Macedonia National Olympiad, 4

We call an integer $ n > 1$ [i]good[/i] if, for any natural numbers $ 1 \le b_1, b_2, \ldots , b_{n\minus{}1} \le n \minus{} 1$ and any $ i \in \{0, 1, \ldots , n \minus{} 1\}$, there is a subset $ I$ of $ \{1, \ldots , n \minus{} 1\}$ such that $ \sum_{k\in I} b_k \equiv i \pmod n$. (The sum over the empty set is zero.) Find all good numbers.

2011 Lusophon Mathematical Olympiad, 2

A non-negative integer $n$ is said to be [i]squaredigital[/i] if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.

2008 Bosnia and Herzegovina Junior BMO TST, 2

Let $ x,y,z$ be positive integers. If $ 7$ divides $ (x\plus{}6y)(2x\plus{}5y)(3x\plus{}4y)$ than prove that $ 343$ also divides it.

2012 Iran MO (3rd Round), 5

Let $p$ be a prime number. We know that each natural number can be written in the form \[\sum_{i=0}^{t}a_ip^i (t,a_i \in \mathbb N\cup \{0\},0\le a_i\le p-1)\] Uniquely. Now let $T$ be the set of all the sums of the form \[\sum_{i=0}^{\infty}a_ip^i (0\le a_i \le p-1).\] (This means to allow numbers with an infinite base $p$ representation). So numbers that for some $N\in \mathbb N$ all the coefficients $a_i, i\ge N$ are zero are natural numbers. (In fact we can consider members of $T$ as sequences $(a_0,a_1,a_2,...)$ for which $\forall_{i\in \mathbb N}: 0\le a_i \le p-1$.) Now we generalize addition and multiplication of natural numbers to this set so that it becomes a ring (it's not necessary to prove this fact). For example: $1+(\sum_{i=0}^{\infty} (p-1)p^i)=1+(p-1)+(p-1)p+(p-1)p^2+...$ $=p+(p-1)p+(p-1)p^2+...=p^2+(p-1)p^2+(p-1)p^3+...$ $=p^3+(p-1)p^3+...=...$ So in this sum, coefficients of all the numbers $p^k, k\in \mathbb N$ are zero, so this sum is zero and thus we can conclude that $\sum_{i=0}^{\infty}(p-1)p^i$ is playing the role of $-1$ (the additive inverse of $1$) in this ring. As an example of multiplication consider \[(1+p)(1+p+p^2+p^3+...)=1+2p+2p^2+\cdots\] Suppose $p$ is $1$ modulo $4$. Prove that there exists $x\in T$ such that $x^2+1=0$. [i]Proposed by Masoud Shafaei[/i]

1990 IMO Longlists, 79

Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

2013 Online Math Open Problems, 42

Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$. [i]Victor Wang[/i]

1982 IMO Longlists, 15

Show that the set $S$ of natural numbers $n$ for which $\frac{3}{n}$ cannot be written as the sum of two reciprocals of natural numbers ($S =\left\{n |\frac{3}{n} \neq \frac{1}{p} + \frac{1}{q} \text{ for any } p, q \in \mathbb N \right\}$) is not the union of finitely many arithmetic progressions.

2011 ELMO Shortlist, 1

Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$; b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$. Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if \[\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).\] [i]Evan O'Dorney.[/i]

2003 IMO Shortlist, 4

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

2008 Romanian Master of Mathematics, 3

Let $ a>1$ be a positive integer. Prove that every non-zero positive integer $ N$ has a multiple in the sequence $ (a_n)_{n\ge1}$, $ a_n\equal{}\left\lfloor\frac{a^n}n\right\rfloor$.

2009 Iran Team Selection Test, 11

Let $n$ be a positive integer. Prove that \[ 3^{\dfrac{5^{2^n}-1}{2^{n+2}}} \equiv (-5)^{\dfrac{3^{2^n}-1}{2^{n+2}}} \pmod{2^{n+4}}. \]

2014 South East Mathematical Olympiad, 3

Let $p$ be a primes ,$x,y,z $ be positive integers such that $x<y<z<p$ and $\{\frac{x^3}{p}\}=\{\frac{y^3}{p}\}=\{\frac{z^3}{p}\}$. Prove that $(x+y+z)|(x^5+y^5+z^5).$

2010 Bulgaria National Olympiad, 1

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 $k \geq 3 \ ?$

1997 IMO Shortlist, 15

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.