Found problems: 4776
2023 JBMO TST - Turkey, 3
Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$f(x+f(x))=f(-x)$ and for all $x \leq y$ it satisfies $f(x) \leq f(y)$
2015 Iran MO (3rd round), 3
Does there exist an irreducible two variable polynomial $f(x,y)\in \mathbb{Q}[x,y]$ such that it has only four roots $(0,1),(1,0),(0,-1),(-1,0)$ on the unit circle.
2013 India IMO Training Camp, 1
Find all functions $f$ from the set of real numbers to itself satisfying
\[ f(x(1+y)) = f(x)(1 + f(y)) \]
for all real numbers $x, y$.
2008 Croatia Team Selection Test, 2
For which $ n\in \mathbb{N}$ do there exist rational numbers $ a,b$ which are not integers such that both $ a \plus{} b$ and $ a^n \plus{} b^n$ are integers?
1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 10
The minimal value of $ f(x) \equal{} \sqrt{a^2 \plus{} x^2} \plus{} \sqrt{(x\minus{}b)^2 \plus{} c^2}$ is
A. $ a\plus{}b\plus{}c$
B. $ \sqrt{a^2 \plus{} (b \plus{} c)^2}$
C. $ \sqrt{b^2 \plus{} (a\plus{}c)^2}$
D. $ \sqrt{(a\plus{}b)^2 \plus{} c^2}$
E. None of these
1999 China National Olympiad, 2
Let $a$ be a real number. Let $(f_n(x))_{n\ge 0}$ be a sequence of polynomials such that $f_0(x)=1$ and $f_{n+1}(x)=xf_n(x)+f_n(ax)$ for all non-negative integers $n$.
a) Prove that $f_n(x)=x^nf_n\left(x^{-1}\right)$ for all non-negative integers $n$.
b) Find an explicit expression for $f_n(x)$.
2013 Balkan MO, 3
Let $S$ be the set of positive real numbers. Find all functions $f\colon S^3 \to S$ such that, for all positive real numbers $x$, $y$, $z$ and $k$, the following three conditions are satisfied:
(a) $xf(x,y,z) = zf(z,y,x)$,
(b) $f(x, ky, k^2z) = kf(x,y,z)$,
(c) $f(1, k, k+1) = k+1$.
([i]United Kingdom[/i])
1991 Vietnam Team Selection Test, 2
For every natural number $n$ we define $f(n)$ by the following rule: $f(1) = 1$ and for $n>1$ then $f(n) = 1 + a_1 \cdot p_1 + \ldots + a_k \cdot p_k$, where $n = p_1^{a_1} \cdots p_k^{a_k}$ is the canonical prime factorisation of $n$ ($p_1, \ldots, p_k$ are distinct primes and $a_1, \ldots, a_k$ are positive integers). For every positive integer $s$, let $f_s(n) = f(f(\ldots f(n))\ldots)$, where on the right hand side there are exactly $s$ symbols $f$. Show that for every given natural number $a$, there is a natural number $s_0$ such that for all $s > s_0$, the sum $f_s(a) + f_{s-1}(a)$ does not depend on $s$.
STEMS 2023 Math Cat A, 4
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:
$\qquad\ (1) \; f(1) = 1.$
$\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$
$\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$
$\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$
Find the sum of all possible values of $f(2023)$.
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 $1+\sqrt{2n}$.
2006 China Northern MO, 4
Given a function $f(x)=x^{2}+ax+b$ with $a,b \in R$, if there exists a real number $m$ such that $\left| f(m) \right| \leq \frac{1}{4}$ and $\left| f(m+1) \right| \leq \frac{1}{4}$, then find the maximum and minimum of the value of $\Delta=a^{2}-4b$.
2010 Today's Calculation Of Integral, 549
Let $ f(x)$ be a function defined on $ [0,\ 1]$. For $ n=1,\ 2,\ 3,\ \cdots$, a polynomial $ P_n(x)$ is defined by $ P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}$. Prove that $ \lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx$.
1986 China Team Selection Test, 2
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:
[b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$
[b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.
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 $f$.
2005 China Western Mathematical Olympiad, 6
In isosceles right-angled triangle $ABC$, $CA = CB = 1$. $P$ is an arbitrary point on the sides of $ABC$. Find the maximum of $PA \cdot PB \cdot PC$.
2007 Today's Calculation Of Integral, 182
Find the area of the domain of the system of inequality
\[y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0. \]
2014 Brazil Team Selection Test, 4
Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation
\[ f(f(f(n))) = f(n+1 ) +1 \]
for all $ n\in \mathbb{Z}_{\ge 0}$.
2006 AMC 12/AHSME, 18
The function $ f$ has the property that for each real number $ x$ in its domain, $ 1/x$ is also in its domain and
\[ f(x) \plus{} f\left(\frac {1}{x}\right) \equal{} x.
\]What is the largest set of real numbers that can be in the domain of $ f$?
$ \textbf{(A) } \{ x | x\ne 0\} \qquad \textbf{(B) } \{ x | x < 0\} \qquad \textbf{(C) }\{ x | x > 0\}\\
\textbf{(D) } \{ x | x\ne \minus{} 1 \text{ and } x\ne 0 \text{ and } x\ne 1\} \qquad \textbf{(E) } \{ \minus{} 1,1\}$
2024 Vietnam Team Selection Test, 1
Let $P(x) \in \mathbb{R}[x]$ be a monic, non-constant polynomial. Determine all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(P(x))+y+2023f(y))=P(x)+2024f(y),$$
for all reals $x,y$.
2009 IMS, 5
Suppose that $ f: \mathbb R^2\rightarrow \mathbb R$ is a non-negative and continuous function that $ \iint_{\mathbb R^2}f(x,y)dxdy\equal{}1$. Prove that there is a closed disc $ D$ with the least radius possible such that $ \iint_D f(x,y)dxdy\equal{}\frac12$.
2019 LIMIT Category B, Problem 6
Let $f(x)=a_0+a_1|x|+a_2|x|^2+a_3|x|^3$, where $a_0,a_1,a_2,a_3$ are constant. Then
$\textbf{(A)}~f(x)\text{ is differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$
$\textbf{(B)}~f(x)\text{ is not differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$
$\textbf{(C)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0$
$\textbf{(D)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0,a_3=0$
2010 District Olympiad, 4
Consider the sequence $ a_n\equal{}\left|z^n\plus{}\frac{1}{z^n}\right|\ ,\ n\ge 1$, where $ z\in \mathbb{C}^*$ is given.
i) Prove that if $ a_1>2$, then:
\[ a_{n\plus{}1}<\frac{a_n\plus{}a_{n\plus{}2}}{2}\ ,\ (\forall)n\in \mathbb{N}^*\]
ii) Prove that if there is a $ k\in \mathbb{N}^*$ such that $ a_k\le 2$, then $ a_1\le 2$.
1983 IMO Longlists, 68
Three of the roots of the equation $x^4 -px^3 +qx^2 -rx+s = 0$ are $\tan A, \tan B$, and $\tan C$, where $A, B$, and $C$ are angles of a triangle. Determine the fourth root as a function only of $p, q, r$, and $s.$
2002 Greece National Olympiad, 3
In a triangle $ABC$ we have $\angle C>10^0$ and $\angle B=\angle C+10^0.$We consider point $E$ on side $AB$ such that $\angle ACE=10^0,$ and point $D$ on side $AC$ such that $\angle DBA=15^0.$ Let $Z\neq A$ be a point of interection of the circumcircles of the triangles $ABD$ and $AEC.$Prove that $\angle ZBA>\angle ZCA.$
2012 Today's Calculation Of Integral, 846
For $a>0$, let $f(a)=\lim_{t\rightarrow +0} \int_{t}^{1} |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$, find the minimum value of $f(a)$.