Found problems: 2008
1980 IMO, 3
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
2006 ISI B.Stat Entrance Exam, 3
Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.
2011 China Team Selection Test, 2
Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.
1997 USAMO, 1
Let $p_1, p_2, p_3, \ldots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between 0 and 1. For positive integer $k$, define
\[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} \]
where $\{x\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \ldots$ eventually becomes 0.
2012 JBMO ShortLists, 2
On a board there are $n$ nails, each two connected by a rope. Each rope is colored in one of $n$ given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors.
a) Can $n$ be $6$ ?
b) Can $n$ be $7$ ?
2009 China Team Selection Test, 3
Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$
PEN D Problems, 18
Let $p$ be a prime number. Determine the maximal degree of a polynomial $T(x)$ whose coefficients belong to $\{ 0,1,\cdots,p-1 \}$, whose degree is less than $p$, and which satisfies \[T(n)=T(m) \; \pmod{p}\Longrightarrow n=m \; \pmod{p}\] for all integers $n, m$.
1996 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 8
Let $ x$ and $ y$ be positive integers. The least possible value of $ |11x^5 \minus{} 7y^3|$ is
A. 1
B. 2
C. 3
D. 4
E. None of these
2014 IberoAmerican, 1
$N$ coins are placed on a table, $N - 1$ are genuine and have the same weight, and one is fake, with a different weight. Using a two pan balance, the goal is to determine with certainty the fake coin, and whether it is lighter or heavier than a genuine coin. Whenever one can deduce that one or more coins are genuine, they will be inmediately discarded and may no longer be used in subsequent weighings. Determine all $N$ for which the goal is achievable. (There are no limits regarding how many times one may use the balance).
Note: the only difference between genuine and fake coins is their weight; otherwise, they are identical.
1971 IMO Longlists, 22
We are given an $n \times n$ board, where $n$ is an odd number. In each cell of the board either $+1$ or $-1$ is written. Let $a_k$ and $b_k$ denote them products of numbers in the $k^{th}$ row and in the $k^{th}$ column respectively. Prove that the sum $a_1 +a_2 +\cdots+a_n +b_1 +b_2 +\cdots+b_n$ cannot be equal to zero.
2010 CentroAmerican, 1
Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation
$n(S(n)-1)=2010.$
2017 Harvard-MIT Mathematics Tournament, 10
Let $\mathbb{N}$ denote the natural numbers. Compute the number of functions $f:\mathbb{N}\rightarrow \{0, 1, \dots, 16\}$ such that $$f(x+17)=f(x)\qquad \text{and} \qquad f(x^2)\equiv f(x)^2+15 \pmod {17}$$ for all integers $x\ge 1$.
2010 Princeton University Math Competition, 7
Let $f$ be a function such that $f(x)+f(x+1)=2^x$ and $f(0)=2010$. Find the last two digits of $f(2010)$.
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.)
2006 Polish MO Finals, 1
Given a triplet we perform on it the following operation. We choose two numbers among them and change them into their sum and product, left number stays unchanged. Can we, starting from triplet $(3,4,5)$ and performing above operation, obtain again a triplet of numbers which are lengths of right triangle?
PEN A Problems, 43
Suppose that $p$ is a prime number and is greater than $3$. Prove that $7^{p}-6^{p}-1$ is divisible by $43$.
2012 ITAMO, 2
Determine all positive integers that are equal to $300$ times the sum of their digits.
2012 Tuymaada Olympiad, 1
The vertices of a regular $2012$-gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$, then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled $A_1$ and $A_4$. How is the tenth vertex labeled?
[i]Proposed by A. Golovanov[/i]
2012 Online Math Open Problems, 25
Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1\le x_2\le\cdots\le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$, find the remainder when $m+n$ is divided by 1000.
[i]Victor Wang.[/i]
2002 IMO Shortlist, 1
What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]
2013 Princeton University Math Competition, 7
Evaluate \[\sqrt{2013+276\sqrt{2027+278\sqrt{2041+280\sqrt{2055+\ldots}}}}\]
1983 IMO Longlists, 22
Does there exist an infinite number of sets $C$ consisting of $1983$ consecutive natural numbers such that each of the numbers is divisible by some number of the form $a^{1983}$, with $a \in \mathbb N, a \neq 1?$
1987 IMO Shortlist, 23
Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$,
\[[r^m] \equiv -1 \pmod k .\]
[i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients.
[i]Proposed by Yugoslavia.[/i]
1996 Romania Team Selection Test, 11
Find all primes $ p,q $ such that $ \alpha^{3pq} -\alpha \equiv 0 \pmod {3pq} $ for all integers $ \alpha $.
2019 IMAR Test, 3
Consider a natural number $ n\equiv 9\pmod {25}. $ Prove that there exist three nonnegative integers $ a,b,c $ having the property that:
$$ n=\frac{a(a+1)}{2} +\frac{b(b+1)}{2} +\frac{c(c+1)}{2} $$