Found problems: 15925
2011 Baltic Way, 5
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
\[f(f(x))=x^2-x+1\]
for all real numbers $x$. Determine $f(0)$.
2022 Israel TST, 2
Let $f: \mathbb{Z}^2\to \mathbb{R}$ be a function.
It is known that for any integer $C$ the four functions of $x$
\[f(x,C), f(C,x), f(x,x+C), f(x, C-x)\]
are polynomials of degree at most $100$. Prove that $f$ is equal to a polynomial in two variables and find its maximal possible degree.
[i]Remark: The degree of a bivariate polynomial $P(x,y)$ is defined as the maximal value of $i+j$ over all monomials $x^iy^j$ appearing in $P$ with a non-zero coefficient.[/i]
I Soros Olympiad 1994-95 (Rus + Ukr), 11.2
Find the smallest positive $x$ for which holds the inequality
$$\sin x \le \sin (x+1)\le \sin (x+2)\le sin (x+3)\le \sin (x+4) .$$
1996 IMO Shortlist, 5
Let $ p,q,n$ be three positive integers with $ p \plus{} q < n$. Let $ (x_{0},x_{1},\cdots ,x_{n})$ be an $ (n \plus{} 1)$-tuple of integers satisfying the following conditions :
(a) $ x_{0} \equal{} x_{n} \equal{} 0$, and
(b) For each $ i$ with $ 1\leq i\leq n$, either $ x_{i} \minus{} x_{i \minus{} 1} \equal{} p$ or $ x_{i} \minus{} x_{i \minus{} 1} \equal{} \minus{} q$.
Show that there exist indices $ i < j$ with $ (i,j)\neq (0,n)$, such that $ x_{i} \equal{} x_{j}$.
2020 Moldova Team Selection Test, 11
Find all functions $f:[-1,1] \rightarrow \mathbb{R},$ which satisfy
$$f(\sin{x})+f(\cos{x})=2020$$
for any real number $x.$
1999 Tournament Of Towns, 1
A father and his son are skating around a circular skating rink. From time to time, the father overtakes the son. After the son starts skating in the opposite direction, they begin to meet five times more often. What is the ratio of the skating speeds of the father and the son?
(Tairova)
1984 Iran MO (2nd round), 2
Consider the function
\[f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).\]
Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.
2015 Argentina National Olympiad Level 2, 4
Let $N$ be the number of ordered lists of $9$ positive integers $(a,b,c,d,e,f,g,h,i)$ such that
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}+\frac{1}{g}+\frac{1}{h}+\frac{1}{i}=1.$$
Determine whether $N$ is even or odd.
2013 Korea - Final Round, 2
Find all functions $ f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions.
(a) $ f(x) \ge 0 $ for all $ x \in \mathbb{R} $.
(b) For $ a, b, c, d \in \mathbb{R} $ with $ ab + bc + cd = 0 $, equality $ f(a-b) + f(c-d) = f(a) + f(b+c) + f(d) $ holds.
2008 Harvard-MIT Mathematics Tournament, 1
Positive real numbers $ x$, $ y$ satisfy the equations $ x^2 \plus{} y^2 \equal{} 1$ and $ x^4 \plus{} y^4 \equal{} \frac {17}{18}$. Find $ xy$.
1984 IMO Longlists, 6
Let $P,Q,R$ be the polynomials with real or complex coefficients such that at least one of them is not constant. If $P^n+Q^n+R^n = 0$, prove that $n < 3.$
2013 Gulf Math Olympiad, 1
Let $a_1,a_2,\ldots,a_{2n}$ be positive real numbers such that $a_ja_{n+j}=1$ for the values $j=1,2,\ldots,n$.
[list]
a. Prove that either the average of the numbers $a_1,a_2,\ldots,a_n$ is at least 1 or the average of
the numbers $a_{n+1},a_{n+2},\ldots,a_{2n}$ is at least 1.
b. Assuming that $n\ge2$, prove that there exist two distinct numbers $j,k$ in the set $\{1,2,\ldots,2n\}$ such that
\[|a_j-a_k|<\frac{1}{n-1}.\]
[/list]
2022 MMATHS, 3
There are $522$ people at a beach, each of whom owns a cat, a dog, both, or neither. If $20$ percent of cat-owners also own a dog, $70$ percent of dog-owners do not own a cat, and $50$ percent of people who don’t own a cat also don’t own a dog, how many people own neither type of pet?
India EGMO 2024 TST, 4
Let $N \geq 3$ be an integer, and let $a_0, \dots, a_{N-1}$ be pairwise distinct reals so that $a_i \geq a_{2i}$ for all $i$ (indices are taken $\bmod~ N$). Find all possible $N$ for which this is possible.
[i]Proposed by Sutanay Bhattacharya[/i]
2022 Germany Team Selection Test, 1
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
2010 Indonesia TST, 1
Find all triplets of real numbers $(x, y, z)$ that satisfies the system of equations
$x^5 = 2y^3 + y - 2$
$y^5 = 2z^3 + z - 2$
$z^5 = 2x^3 + x - 2$
2003 Bosnia and Herzegovina Junior BMO TST, 1
Non-zero real numbers $a, b, c$ satisfy the condition $\frac{1}{a}+\frac{2}{b}+\frac{3}{c}= 0$.
Determine the value of $w =\frac{3b + 2c}{6a}+\frac{2c + 6a}{3b}+\frac{6a + 3b}{2c}$
.
2018 Purple Comet Problems, 7
Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac14$ of the way to his destination, and in $45$ minutes he will be exactly $\frac13$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.
2005 MOP Homework, 3
Determine all polynomials $P(x)$ with real coeffcients such that $(x^3+3x^2+3x+2)P(x-1)=(x^3-3x^2+3x-2)P(x)$.
2019 Malaysia National Olympiad, 1
Evaluate the following sum
$$\frac{1}{\log_2{\frac{1}{7}}}+\frac{1}{\log_3{\frac{1}{7}}}+\frac{1}{\log_4{\frac{1}{7}}}+\frac{1}{\log_5{\frac{1}{7}}}+\frac{1}{\log_6{\frac{1}{7}}}-\frac{1}{\log_7{\frac{1}{7}}}-\frac{1}{\log_8{\frac{1}{7}}}-\frac{1}{\log_9{\frac{1}{7}}}-\frac{1}{\log_{10}{\frac{1}{7}}}$$
2020 Kürschák Competition, P2
Find all functions $f\colon \mathbb{Q}\to \mathbb{R}_{\geq 0}$ such that for any two rational numbers $x$ and $y$ the following conditions hold
[list]
[*] $f(x+y)\leq f(x)+f(y)$,
[*]$f(xy)=f(x)f(y)$,
[*]$f(2)=1/2$.
[/list]
2009 Polish MO Finals, 6
Let $ n$ be a natural number equal or greater than 3 . A sequence of non-negative numbers $ (c_0,c_1,\ldots,c_n)$ satisfies the condition: $ c_{p}c_{s}\plus{}c_{r}c_{t}\equal{} c_{p\plus{}r}c_{r\plus{}s}$ for all non-negative $ p,q,r,s$ such that $ p\plus{}q\plus{}r\plus{}s\equal{}n$. Determine all possible values of $ c_2$ when $ c_1\equal{}1$.
2013 Online Math Open Problems, 48
$\omega$ is a complex number such that $\omega^{2013} = 1$ and $\omega^m \neq 1$ for $m=1,2,\ldots,2012$. Find the number of ordered pairs of integers $(a,b)$ with $1 \le a, b \le 2013$ such that \[ \frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3} \] is the root of some polynomial with integer coefficients and leading coefficient $1$. (Such complex numbers are called [i]algebraic integers[/i].)
[i]Victor Wang[/i]
2011 ELMO Problems, 4
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
2011 Romanian Master of Mathematics, 1
Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing.
[i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]