Found problems: 15925
2023 ELMO Shortlist, A1
Find all polynomials \(P(x)\) with real coefficients such that for all nonzero real numbers \(x\), \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\]
[i]Proposed by Holden Mui[/i]
STEMS 2021-22 Math Cat A-B, A3 B1
Find all functions $f :\mathbb{N} \rightarrow \mathbb{N}$ such that
$f(m + f(n)f(m)) = nf(m) + m$
holds for all $m,n \in \mathbb{N}$.
2001 Korea Junior Math Olympiad, 2
$n$ is a product of some two consecutive primes. $s(n)$ denotes the sum of the divisors of $n$ and $p(n)$ denotes the number of relatively prime positive integers not exceeding $n$. Express $s(n)p(n)$ as a polynomial of $n$.
2003 China Team Selection Test, 2
Given an integer $a_1$($a_1 \neq -1$), find a real number sequence $\{ a_n \}$($a_i \neq 0, i=1,2,\cdots,5$) such that $x_1,x_2,\cdots,x_5$ and $y_1,y_2,\cdots,y_5$ satisfy $b_{i1}x_1+b_{i2}x_2+\cdots +b_{i5}x_5=2y_i$, $i=1,2,3,4,5$, then $x_1y_1+x_2y_2+\cdots+x_5y_5=0$, where $b_{ij}=\prod_{1 \leq k \leq i} (1+ja_k)$.
2014 BMT Spring, 16
Evaluate $$\sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \min (n, k) \left( \frac12 \right)^n\left( \frac13 \right)^k$$
1999 Harvard-MIT Mathematics Tournament, 7
Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$, Anne and Carl in $5$. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?
1999 Bosnia and Herzegovina Team Selection Test, 1
Let $a$, $b$ and $c$ be lengths of sides of triangle $ABC$. Prove that at least one of the equations $$x^2-2bx+2ac=0$$ $$x^2-2cx+2ab=0$$ $$x^2-2ax+2bc=0$$ does not have real solutions
1989 China National Olympiad, 3
Let $S$ be the unit circle in the complex plane (i.e. the set of all complex numbers with their moduli equal to $1$).
We define function $f:S\rightarrow S$ as follow: $\forall z\in S$,
$ f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \dots,$
$f^{(k)}(z)=f(f^{(k-1)}(z)) (k>1,k\in \mathbb{N}), \dots$
We call $c$ an $n$-[i]period-point[/i] of $f$ if $c$ ($c\in S$) and $n$ ($n\in\mathbb{N}$) satisfy:
$f^{(1)}(c) \not=c, f^{(2)}(c) \not=c, f^{(3)}(c) \not=c, \dots, f^{(n-1)}(c) \not=c, f^{(n)}(c)=c$.
Suppose that $f(z)=z^m$ ($z\in S; m>1, m\in \mathbb{N}$), find the number of $1989$-[i]period-point[/i] of $f$.
1973 IMO, 2
$G$ is a set of non-constant functions $f$. Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$. If $f$ and $g$ are in $G$, then so is $fg$, where $fg$ is defined by $fg(x)=f(g(x))$. If $f$ is in $G$, then so is the inverse $f^{-1}$. If $f(x)=ax+b$, then $f^{-1}(x)= \frac{x-b}{a}$. Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$. Prove that all the functions in $G$ have a common fixed point.
2017 Balkan MO Shortlist, A3
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f:\mathbb{N}\longrightarrow\mathbb{N}$ such that
\[n+f(m)\mid f(n)+nf(m)\]
for all $m,n\in \mathbb{N}$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
VI Soros Olympiad 1999 - 2000 (Russia), 10.4
Solve the equation $$16x^3 = (11x^2 + x -1)\sqrt{x^2 - x + 1}.$$
2022 OlimphÃada, 4
Let $a_1,a_2,\dots$ be a sequence of integers satisfying $a_1=2$ and:
$$a_n=\begin{cases}a_{n-1}+1, & \text{ if }n\ne a_k \text{ for some }k=1,2,\dots,n-1; \\ a_{n-1}+2, & \text{ if } n=a_k \text{ for some }k=1,2,\dots,n-1. \end{cases}$$
Find the value of $a_{2022!}$.
2002 Austria Beginners' Competition, 1
We calculate the sum of $7$ natural consecutive pairs (e.g. $2+4+6+8+10+12+14$) and we will call the result $A$, then the sum of the next $7$ consecutive pairs (in the example, $16+ 18+...$) and its result we will call $B$, and finally we calculate the sum of the following $7$ consecutive pairs and its result we will call $C$. Can the product $ABC$ be $2002^3$?
2002 AIME Problems, 15
Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6),$ and the product of the radii is $68.$ The x-axis and the line $y=mx$, where $m>0,$ are tangent to both circles. It is given that $m$ can be written in the form $a\sqrt{b}/c,$ where $a,$ $b,$ and $c$ are positive integers, $b$ is not divisible by the square of any prime, and $a$ and $c$ are relatively prime. Find $a+b+c.$
2023 Mongolian Mathematical Olympiad, 1
Find all functions $f : \mathbb{R} \to \mathbb{R}$ and $h : \mathbb{R}^2 \to \mathbb{R}$ such that \[f(x+y-z)^2=f(xy)+h(x+y+z, xy+yz+zx)\] for all real numbers $x,y,z$.
VI Soros Olympiad 1999 - 2000 (Russia), 10.1
Find all real functions of a real numbers, such that for any $x$, $y$ and $z$ holds the equality $$ f(x)f(y)f(z)-f(xyz)=xy+yz+xz+x+y+z.$$
2024 Chile TST Ibero., 4
Prove that if \( a \), \( b \), and \( c \) are positive real numbers, then the following inequality holds:
\[
\frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6.
\]
2018 Hanoi Open Mathematics Competitions, 5
Let $f$ be a polynomial such that, for all real number $x$, $f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019$.
Compute $f(2018)$.
1968 Kurschak Competition, 1
In an infinite sequence of positive integers every element (starting with the second) is the harmonic mean of its neighbors. Show that all the numbers must be equal.
2018 District Olympiad, 1
Show that $$\sqrt{n + \left[ \sqrt{n} +\frac12\right]}$$ is an irrational number, for every positive integer $n$.
1987 ITAMO, 5
Let $a_1,a_2,...$ and $b_1,b_2,..$. be two arbitrary infinite sequences of natural numbers.
Prove that there exist different indices $r$ and $s$ such that $a_r \ge a_s$ and $b_r \ge b_s$.
2011 Saint Petersburg Mathematical Olympiad, 1
$f(x),g(x)$ - two square trinomials and $a,b,c,d$ - some reals. $f(a)=2,f(b)=3,f(c)=7,f(d)=10$ and $g(a)=16,g(b)=15,g(c)=11$ Find $g(d)$
2009 India IMO Training Camp, 5
Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients.
We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that
$ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$.
Prove that there exists $ a,b,c\in\mathbb{C}$ such that
$ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.
2005 Federal Math Competition of S&M, Problem 3
Determine all polynomials $p$ with real coefficients for which $p(0)=0$ and
$$f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,$$where $f(n)=\lfloor p(n)\rfloor$.
2022 Saudi Arabia BMO + EGMO TST, 2.4
Find all functions $f : R \to R$ such that
$$2f(x)f(x + y) -f(x^2) =\frac{x}{2}(f(2x) + 4f(f(y)))$$
for all $x, y \in R$.