Found problems: 4776
2016 Ukraine Team Selection Test, 2
Find all functions from positive integers to itself such that $f(a+b)=f(a)+f(b)+f(c)+f(d)$ for all $c^2+d^2=2ab$
1974 USAMO, 3
Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemisphere of the given ball.
2012 Purple Comet Problems, 15
Let $N$ be a positive integer whose digits add up to $23$. What is the greatest possible product the digits of $N$ can have?
2001 Romania Team Selection Test, 4
Consider a convex polyhedron $P$ with vertices $V_1,\ldots ,V_p$. The distinct vertices $V_i$ and $V_j$ are called [i]neighbours[/i] if they belong to the same face of the polyhedron. To each vertex $V_k$ we assign a number $v_k(0)$, and construct inductively the sequence $v_k(n)\ (n\ge 0)$ as follows: $v_k(n+1)$ is the average of the $v_j(n)$ for all neighbours $V_j$ of $V_k$ . If all numbers $v_k(n)$ are integers, prove that there exists the positive integer $N$ such that all $v_k(n)$ are equal for $n\ge N$ .
2022 Junior Macedonian Mathematical Olympiad, P2
Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=3$. Prove the inequality
$$\frac{a^3}{a^2+1}+\frac{b^3}{b^2+1}+\frac{c^3}{c^2+1} \geq \frac{3}{2}.$$
[i]Proposed by Anastasija Trajanova[/i]
2015 Abels Math Contest (Norwegian MO) Final, 1b
Find all functions $f : R \to R$ such that $x^2f(yf(x))= y^2f(x)f(f(x))$ for all real numbers $x$ and $y$.
2013 NIMO Problems, 5
For every integer $n \ge 1$, the function $f_n : \left\{ 0, 1, \cdots, n \right\} \to \mathbb R$ is defined recursively by $f_n(0) = 0$, $f_n(1) = 1$ and \[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \] for each $1 \le k < n$. Let $S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N} (N)$. Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$. (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.)
[i]Proposed by Lewis Chen[/i]
2009 IMO Shortlist, 3
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
1999 Poland - Second Round, 5
Let $S = \{1,2,3,4,5\}$. Find the number of functions $f : S \to S$ such that $f ^{50}(x)= x$ for all $x \in S$.
1986 Traian Lălescu, 2.4
Prove that, if a continuous function has limits at $ \pm\infty , $ and these are equal, then it touches its maximum or minimum at one point.
2008 Bosnia And Herzegovina - Regional Olympiad, 4
$ n$ points (no three being collinear) are given in a plane. Some points are connected and they form $ k$ segments. If no three of these segments form triangle ( equiv. there are no three points, such that each two of them are connected) prove that $ k \leq \left \lfloor \frac {n^{2}}{4}\right\rfloor$
Gheorghe Țițeica 2025, P4
Let $n\geq 2$ and $\mathcal{M}$ be a subset of $S_n$ with at least two elements, and which is closed under composition. Consider a function $f:\mathcal{M}\rightarrow\mathbb{R}$ which satisfies $$|f(\sigma\tau)-f(\sigma)-f(\tau)|\leq 1,$$ for all $\sigma,\tau\in\mathcal{M}$. Prove that $$\max_{\sigma,\tau\in\mathcal{M}}|f(\sigma)-f(\tau)|\leq 2-\frac{2}{|\mathcal{M}|}.$$
2010 Mathcenter Contest, 1
A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition:
\[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\]
Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds:
\[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]
2005 India National Olympiad, 6
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.
2022 Ecuador NMO (OMEC), 2
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$
\[f(x + y)=f(f(x)) + y + 2022\]
1992 IMO Longlists, 24
[i](a)[/i] Show that there exists exactly one function $ f : \mathbb Q^+ \to \mathbb Q^+$ satisfying the following conditions:
[b](i)[/b] if $0 < q < \frac 12$, then $f(q)=1+f \left( \frac{q}{1-2q} \right);$
[b](ii)[/b] if $1 < q \leq 2$, then $f(q) = 1+f(q + 1);$
[b](iii)[/b] $f(q)f(1/q) = 1$ for all $q \in \mathbb Q^+.$
[i](b)[/i] Find the smallest rational number $q \in \mathbb Q^+$ such that $f(q) = \frac{19}{92}.$
2012 Pan African, 3
Find all real solutions $x$ to the equation $\lfloor x^2 - 2x \rfloor + 2\lfloor x \rfloor = \lfloor x \rfloor^2$.
2017 Thailand TSTST, 2
Let $f, g$ be bijections on $\{1, 2, 3, \dots, 2016\}$. Determine the value of $$\sum_{i=1}^{2016}\sum_{j=1}^{2016}[f(i)-g(j)]^{2559}.$$
2016 Romania National Olympiad, 4
Determine all functions $f: \mathbb R \to \mathbb R$ which satisfy the inequality
$$f(a^2) - f(b^2) \leq \left( f(a) + b\right)\left( a - f(b)\right),$$
for all $a,b \in \mathbb R$.
1965 IMO Shortlist, 2
Consider the sytem of equations
\[ a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0 \]\[a_{21}x_1+a_{22}x_2+a_{23}x_3 =0\]\[a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0 \] with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions:
a) $a_{11}, a_{22}, a_{33}$ are positive numbers;
b) the remaining coefficients are negative numbers;
c) in each equation, the sum ofthe coefficients is positive.
Prove that the given system has only the solution $x_1=x_2=x_3=0$.
1988 Czech And Slovak Olympiad IIIA, 1
Let $f$ be a representation of the set $M = \{1, 2,..., 1988\}$ into $M$. For any natural $n$, let $x_1 = f(1)$, $x_{n+1} = f(x_n)$. Find out if there exists $m$ such that $x_{2m} = x_m$.
2020 Israel Olympic Revenge, P1
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in \mathbb{R}$ one has
\[f(f(x)+y)=f(x+f(y))\]
and in addition the set $f^{-1}(a)=\{b\in \mathbb{R}\mid f(b)=a\}$ is a finite set for all $a\in \mathbb{R}$.
2014 Iran MO (2nd Round), 3
Members of "Professionous Riddlous" society have been divided into some groups, and groups are changed in a special way each weekend: In each group, one of the members is specified as the best member, and the best members of all groups separate from their previous group and form a new group. If a group has only one member, that member joins the new group and the previous group will be removed. Suppose that the society has $n$ members at first, and all the members are in one group. Prove that a week will come, after which number of members of each group will be at most $1+\sqrt{2n}$.
2011 BMO TST, 1
The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.
1975 Canada National Olympiad, 3
For each real number $ r$, $ [r]$ denotes the largest integer less than or equal to $ r$, e.g. $ [6] \equal{} 6, [\pi] \equal{} 3, [\minus{}1.5] \equal{} \minus{}2$. Indicate on the $ (x,y)$-plane the set of all points $ (x,y)$ for which $ [x]^2 \plus{} [y]^2 \equal{} 4$.