Found problems: 1782
1996 Turkey MO (2nd round), 3
Show that there is no function $f:{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$ such that $f(x+y)>f(x)(1+yf(x))$
for all $x,y\in {{\mathbb{R}}^{+}}$.
2007 Iran MO (2nd Round), 3
Farhad has made a machine. When the machine starts, it prints some special numbers. The property of this machine is that for every positive integer $n$, it prints exactly one of the numbers $n,2n,3n$. We know that the machine prints $2$. Prove that it doesn't print $13824$.
1988 IMO Longlists, 65
The Fibonacci sequence is defined by \[ a_{n+1} = a_n + a_{n-1}, n \geq 1, a_0 = 0, a_1 = a_2 = 1. \] Find the greatest common divisor of the 1960-th and 1988-th terms of the Fibonacci sequence.
1991 Arnold's Trivium, 84
Find the number of positive and negative squares in the canonical form of the quadratic form $\sum_{i<j}(x_i-x_j)^2$ in $n$ variables. The same for the form $\sum_{i<j}x_i x_j$.
2015 International Zhautykov Olympiad, 2
Let $ A_n $ be the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that every two neighbouring terms of each subsequence have different parity,and $ B_n $ the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that all the terms of each subsequence have the same parity ( for example,the partition $ {(1,4,5,8),(2,3),(6,9),(7)} $ is an element of $ A_9 $,and the partition $ {(1,3,5),(2,4),(6)} $ is an element of $ B_6 $ ).
Prove that for every positive integer $ n $ the sets $ A_n $ and $ B_{n+1} $ contain the same number of elements.
1998 IberoAmerican, 1
Given 98 points in a circle. Mary and Joseph play alternatively in the next way:
- Each one draw a segment joining two points that have not been joined before.
The game ends when the 98 points have been used as end points of a segments at least once. The winner is the person that draw the last segment. If Joseph starts the game, who can assure that is going to win the game.
2013 AMC 10, 20
The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
2009 China Western Mathematical Olympiad, 4
Prove that for every given positive integer $k$, there exist infinitely many $n$, such that $2^{n}+3^{n}-1, 2^{n}+3^{n}-2,\ldots, 2^{n}+3^{n}-k$ are all composite numbers.
2003 China Team Selection Test, 1
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.
2003 China Team Selection Test, 3
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
1987 China Team Selection Test, 3
Let $ G$ be a simple graph with $ 2 \cdot n$ vertices and $ n^{2}+1$ edges. Show that this graph $ G$ contains a $ K_{4}-\text{one edge}$, that is, two triangles with a common edge.
2012 All-Russian Olympiad, 3
On a circle there are $2n+1$ points, dividing it into equal arcs ($n\ge 2$). Two players take turns to erase one point. If after one player's turn, it turned out that all the triangles formed by the remaining points on the circle were obtuse, then the player wins and the game ends.
Who has a winning strategy: the starting player or his opponent?
1989 Flanders Math Olympiad, 4
Let $D$ be the set of positive reals different from $1$ and let $n$ be a positive integer.
If for $f: D\rightarrow \mathbb{R}$ we have $x^n f(x)=f(x^2)$,
and if $f(x)=x^n$ for $0<x<\frac{1}{1989}$ and for $x>1989$,
then prove that $f(x)=x^n$ for all $x \in D$.
2002 Balkan MO, 1
Consider $n$ points $A_1,A_2,A_3,\ldots, A_n$ ($n\geq 4$) in the plane, such that any three are not collinear. Some pairs of distinct points among $A_1,A_2,A_3,\ldots, A_n$ are connected by segments, such that every point is connected with at least three different points. Prove that there exists $k>1$ and the distinct points $X_1,X_2,\ldots, X_{2k}$ in the set $\{A_1,A_2,A_3,\ldots, A_n\}$, such that for every $i\in \overline{1,2k-1}$ the point $X_i$ is connected with $X_{i+1}$, and $X_{2k}$ is connected with $X_1$.
2001 Italy TST, 4
We are given $2001$ balloons and a positive integer $k$. Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most $k$ balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of $k$ such that, whatever the initial sizes are, it is possible to make all the balloons have equal size after a finite number of steps.
2001 Italy TST, 4
We are given $2001$ balloons and a positive integer $k$. Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most $k$ balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of $k$ such that, whatever the initial sizes are, it is possible to make all the balloons have equal size after a finite number of steps.
2004 IberoAmerican, 3
Let $ n$ and $ k$ be positive integers such as either $ n$ is odd or both $ n$ and $ k$ are even. Prove that exists integers $ a$ and $ b$ such as $ GCD(a,n) \equal{} GCD(b,n) \equal{} 1$ and $ k \equal{} a \plus{} b$
2014 Romania Team Selection Test, 4
Let $f$ be the function of the set of positive integers into itself, defi ned by $f(1) = 1$,
$f(2n) = f(n)$ and $f(2n + 1) = f(n) + f(n + 1)$. Show that, for any positive integer $n$, the
number of positive odd integers m such that $f(m) = n$ is equal to the number of positive
integers[color=#0000FF][b] less or equal to [/b][/color]$n$ and coprime to $n$.
[color=#FF0000][mod: the initial statement said less than $n$, which is wrong.][/color]
2004 All-Russian Olympiad, 2
In the cabinet 2004 telephones are located; each two of these telephones are connected by a cable, which is colored in one of four colors. From each color there is one cable at least. Can one always select several telephones in such a way that among their pairwise cable connections exactly 3 different colors occur?
2004 Bulgaria National Olympiad, 4
In a word formed with the letters $a,b$ we can change some blocks: $aba$ in $b$ and back, $bba$ in $a$ and backwards. If the initial word is $aaa\ldots ab$ where $a$ appears 2003 times can we reach the word $baaa\ldots a$, where $a$ appears 2003 times.
2014 Online Math Open Problems, 25
Kevin has a set $S$ of $2014$ points scattered on an infinitely large planar gameboard. Because he is bored, he asks Ashley to evaluate \[ x = 4f_4 + 6f_6 + 8f_8 + 10f_{10} + \cdots \] while he evaluates \[ y = 3f_3 + 5f_5+7f_7+9f_9 + \cdots, \] where $f_k$ denotes the number of convex $k$-gons whose vertices lie in $S$ but none of whose interior points lie in $S$.
However, since Kevin wishes to one-up everything that Ashley does, he secretly positions the points so that $y-x$ is as large as possible, but in order to avoid suspicion, he makes sure no three points lie on a single line. Find $\left\lvert y-x \right\rvert$.
[i]Proposed by Robin Park[/i]
2012 Today's Calculation Of Integral, 786
For each positive integer $n$, define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$
(1) Find $H_1(x),\ H_2(x),\ H_3(x)$.
(2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction.
(3) Let $a$ be real number. For $n\geq 3$, express $S_n(a)=\int_0^a xH_n(x)e^{-x^2}dx$ in terms of $H_{n-1}(a),\ H_{n-2}(a),\ H_{n-2}(0)$.
(4) Find $\lim_{a\to\infty} S_6(a)$.
If necessary, you may use $\lim_{x\to\infty}x^ke^{-x^2}=0$ for a positive integer $k$.
2022 Bulgaria EGMO TST, 6
Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold:
(a) the union of any two white subsets is white;
(b) the union of any two black subsets is black;
(c) there are exactly $N$ white subsets.
2012 Brazil National Olympiad, 3
Find the least non-negative integer $n$ such that exists a non-negative integer $k$ such that the last 2012 decimal digits of $n^k$ are all $1$'s.
PEN Q Problems, 5
(Eisentein's Criterion) Let $f(x)=a_{n}x^{n} +\cdots +a_{1}x+a_{0}$ be a nonconstant polynomial with integer coefficients. If there is a prime $p$ such that $p$ divides each of $a_{0}$, $a_{1}$, $\cdots$,$a_{n-1}$ but $p$ does not divide $a_{n}$ and $p^2$ does not divide $a_{0}$, then $f(x)$ is irreducible in $\mathbb{Q}[x]$.