Found problems: 15925
2021 CMIMC, 1.5
Suppose $f$ is a degree 42 polynomial such that for all integers $0\le i\le 42$,
$$f(i)+f(43+i)+f(2\cdot43+i)+\cdots+f(46\cdot43+i)=(-2)^i$$
Find $f(2021)-f(0)$.
[i]Proposed by Adam Bertelli[/i]
2013 Saudi Arabia BMO TST, 3
Let $T$ be a real number satisfying the property:
For any nonnegative real numbers $a, b, c,d, e$ with their sum equal to $1$, it is possible to arrange them around a circle such that the products of any two neighboring numbers are no greater than $T$.
Determine the minimum value of $T$.
2017 Irish Math Olympiad, 5
The sequence $a = (a_0, a_1,a_2,...)$ is defined by $a_0 = 0, a_1 =2$ and $$a_{n+2} = 2a_{n+1} + 41a_n$$Prove that $a_{2016}$ is divisible by $2017.$
2002 IMO Shortlist, 1
Find all functions $f$ from the reals to the reals such that
\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]
for all real $x,y$.
2013 South East Mathematical Olympiad, 1
Let $a,b$ be real numbers such that the equation $x^3-ax^2+bx-a=0$ has three positive real roots . Find the minimum of $\frac{2a^3-3ab+3a}{b+1}$.
2008 Germany Team Selection Test, 1
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
\[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1
\]
for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$
[i]Author: Nikolai Nikolov, Bulgaria[/i]
2021 Kyiv City MO Round 1, 8.1
Find all positive integers $n$ that can be subtracted from both the numerator and denominator of the fraction $\frac{1234}{6789}$, to get, after the reduction, the fraction of form $\frac{a}{b}$, where $a, b$ are single digit numbers.
[i]Proposed by Bogdan Rublov[/i]
1975 Spain Mathematical Olympiad, 6
Let $\{x_n\}$ and $\{y_n\}$ be two sequences of natural numbers defined as follow:
$x_1 = 1, \,\,\, x_2 = 1, \,\,\, x_{n+2} = x_{n+1} + 2x_n$ for $n = 1, 2, 3, ...$
$y_1 = 1, \,\,\, y_2 = 7, \,\,\, y_{n+2} = 2y_{n+1} + 3y_n$ for $n = 1, 2, 3, ...$
Prove that, except for the case $x_1 = y_1 = 1$, there is no natural value that occurs in the two sequences.
2018 Dutch IMO TST, 1
(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
(b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
2011 Philippine MO, 4
Find all (if there is one) functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x\in\mathbb{R}$,
\[f(f(x))+xf(x)=1.\]
2014 Contests, 1
Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .
2007 Indonesia TST, 3
Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that:
(i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and
(ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.
1990 Austrian-Polish Competition, 9
$a_1, a_2, ... , a_n$ is a sequence of integers such that every non-empty subsequence has non-zero sum. Show that we can partition the positive integers into a finite number of sets such that if $x_i$ all belong to the same set, then $a_1x_1 + a_2x_2 + ... + a_nx_n$ is non-zero.
2021 Iberoamerican, 4
Let $a,b,c,x,y,z$ be real numbers such that
\[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \]
Show that $a^2+b^2+c^2=x^2+y^2+z^2$.
MOAA Team Rounds, 2019.6
Let $f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil$. Suppose $x, y$ are chosen independently uniformly at random from the interval $(0, 1]$. Let $p$ be the probability that $f(x, y) < 6$. If $p$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$, compute $m + n$.
(Note: $\lfloor x\rfloor $ is defined as the greatest integer less than or equal to $x$ and $\lceil x \rceil$ is defined as the least integer greater than or equal to$ x$.)
2012 Singapore Senior Math Olympiad, 5
For $a,b,c,d \geq 0$ with $a + b = c + d = 2$, prove
\[(a^2 + c^2)(a^2 + d^2)(b^2 + c^2)(b^2 + d^2) \leq 25\]
2020-21 IOQM India, 11
Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and }x^3+bx+a \text{ have at least a common real zero .}\}$
How many elements are there in $S$?
2025 Harvard-MIT Mathematics Tournament, 4
Let $\lfloor z \rfloor$ denote the greatest integer less than or equal to $z.$ Compute $$\sum_{j=-1000}^{1000} \left\lfloor \frac{2025}{j+0.5}\right\rfloor.$$
2015 Belarus Team Selection Test, 3
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
1987 IMO Longlists, 64
Let $r > 1$ be a real number, and let $n$ be the largest integer smaller than $r$. Consider an arbitrary real number $x$ with $0 \leq x \leq \frac{n}{r-1}.$ By a [i]base-$r$ expansion[/i] of $x$ we mean a representation of $x$ in the form
\[x=\frac{a_1}{r} + \frac{a_2}{r^2}+\frac{a_3}{r^3}+\cdots\]
where the $a_i$ are integers with $0 \leq a_i < r.$
You may assume without proof that every number $x$ with $0 \leq x \leq \frac{n}{r-1}$ has at least one [i]base-$r$ expansion[/i].
Prove that if $r$ is not an integer, then there exists a number $p$, $0 \leq p \leq \frac{n}{r-1}$, which has infinitely many distinct [i]base-$r$ expansions[/i].
2013 USA Team Selection Test, 2
Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and
\[x^3(y^3+z^3)=2012(xyz+2).\]
1967 IMO Longlists, 37
Prove that for arbitrary positive numbers the following inequality holds
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]
2022 Korea National Olympiad, 7
Suppose that the sequence $\{a_n\}$ of positive reals satisfies the following conditions:
[list]
[*]$a_i \leq a_j$ for every positive integers $i <j$.
[*]For any positive integer $k \geq 3$, the following inequality holds:
$$(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k$$
[/list]
Prove that $\{a_n\}$ is constant.
2004 Croatia Team Selection Test, 1
Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$
2016 Iran Team Selection Test, 4
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.