Found problems: 15925
2003 Baltic Way, 1
Find all functions $f:\mathbb{Q}^{+}\rightarrow \mathbb{Q}^{+}$ which for all $x \in \mathbb{Q}^{+}$ fulfil
\[f\left(\frac{1}{x}\right)=f(x) \ \ \text{and} \ \ \left(1+\frac{1}{x}\right)f(x)=f(x+1). \]
2006 Bosnia and Herzegovina Team Selection Test, 6
Let $a_1$, $a_2$,...,$a_n$ be constant real numbers and $x$ be variable real number $x$. Let $f(x)=cos(a_1+x)+\frac{cos(a_2+x)}{2}+\frac{cos(a_3+x)}{2^2}+...+\frac{cos(a_n+x)}{2^{n-1}}$. If $f(x_1)=f(x_2)=0$, prove that $x_1-x_2=m\pi$, where $m$ is integer.
2004 India IMO Training Camp, 3
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that
\[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.
2006 Turkey MO (2nd round), 3
Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.
2003 China Team Selection Test, 3
The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.
2018 ELMO Shortlist, 1
Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$?
[i]Proposed by Daniel Liu[/i]
Mathley 2014-15, 4
Let $S_k$ be the set of all triplets of real numbers $(a, b, c)$ satisfying $a <k (b + c)$, $b <k (c + a)$, and $c <k (a + b)$. For what value of $k$ then $S_k$ is a subset of $\{(a, b, c) | ab + bc + ca> 0\}$ ?
Michel Bataille, France
2022 Pan-African, 3
Let $n$ be a positive integer, and $a_1, a_2, \dots, a_{2n}$ be a sequence of positive real numbers whose product is equal to $2$. For $k = 1, 2, \dots, 2n$, set $a_{2n + k} = a_k$, and define
$$
A_k = \frac{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + n - 2}}{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + 2n - 2}}.
$$
Suppose that $A_1, A_2, \dots, A_{2n}$ are pairwise distinct; show that exactly half of them are less than $\sqrt{2} - 1$.
1987 Traian Lălescu, 1.3
Let be three polynomials of degree two $ p_1,p_2,p_3\in\mathbb{R} [X] $ and the function
$$ f:\mathbb{R}\longrightarrow\mathbb{R} ,\quad f(x)=\max\left( p_1(x),p_2(x),p_3(x)\right) . $$
Then, $ f $ is differentiable if and only if any of these three polynomials dominates the other two.
1988 Greece Junior Math Olympiad, 4
i) If $b^2+c^2=a^2, \,\,\,\, b\ne \pm c$ , calculate the expression $\frac{b^3+c^3}{b+c}+\frac{b^3-c^3}{b-c}$.
ii) If $a+\frac{1}{a}=k, a\ne 0$, find the expression $a^4+\frac{1}{a^4}$ in terms of $k$.
1978 Czech and Slovak Olympiad III A, 2
Determine (at least one) pair of real numbers $k,q$ such that the inequality
\[2\left|\sqrt{1-x^2}-kx-q\right|\le\sqrt2-1\]
holds for all $x\in[0,1].$
1990 Greece National Olympiad, 3
Find all functions $f: \mathbb{R}\to\mathbb{R}$ that satisfy $y^2f(x)(f(x)-2x)\le (1-xy)(1+xy) $ for any $x,y \in\mathbb{R}$.
1998 Croatia National Olympiad, Problem 1
Which number is greater:
$$A=\frac{2.00\ldots04}{1.00\ldots04^2+2.00\ldots04},\text{ or }B=\frac{2.00\ldots02}{1.00\ldots02^2+2.00\ldots02},$$where each of the numbers above contains $1998$ zeros?
2013 QEDMO 13th or 12th, 4
Let $a> 0$ and $f: R\to R$ a function such that $f (x) + f (x + 2a) + f (x + 3a) + f (x + 5a) = 1$ for all $x\in R$ . Show that $f$ is periodic, that is, that there is some $b> 0$ for which $f (x) = f (x + b)$ for every $x \in R$ holds. Find the smallest such $b$, which works for all these functions .
2020 Thailand TSTST, 1
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(\max \left\{ x, y \right\} + \min \left\{ f(x), f(y) \right\}) = x+y $$ for all $x,y \in \mathbb{R}$.
2001 District Olympiad, 2
Let $K$ commutative field with $8$ elements. Prove that $(\exists)a\in K$ such that $a^3=a+1$.
[i]Mircea Becheanu[/i]
2008 Argentina National Olympiad, 2
In every cell of a $ 60 \times 60$ board is written a real number, whose absolute value is less or equal than $ 1$. The sum of all numbers on the board equals $ 600$.
Prove that there is a $ 12 \times 12$ square in the board such that the absolute value of the sum of all numbers on it is less or equal than $ 24$.
1992 USAMO, 2
Prove
\[ \frac{1}{\cos 0^\circ \cos 1^\circ} + \frac{1}{\cos 1^\circ \cos 2^\circ} + \cdots + \frac{1}{\cos 88^\circ \cos 89^\circ} = \frac{\cos 1^\circ}{\sin^2 1^\circ}. \]
2006 Iran MO (3rd Round), 2
Find all real polynomials that \[p(x+p(x))=p(x)+p(p(x))\]
1991 Greece Junior Math Olympiad, 4
Let $x+y=a$ and $xy=b$. Calculate exression $ x^4+y^4$ in terms of $a$ and $b$.
2013 Balkan MO Shortlist, A6
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])
BIMO 2021, 1
Find all continuous functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all real numbers $ x, y $ $$ f(x^2+f(y))=f(f(y)-x^2)+f(xy) $$
[Extra: Can you solve this without continuity?]
2005 Junior Balkan Team Selection Tests - Moldova, 8
The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter.
Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.
1980 Canada National Olympiad, 4
A gambling student tosses a fair coin. She gains $1$ point for each head that turns up, and gains $2$ points for each tail that turns up. Prove that the probability of the student scoring [i]exactly[/i] $n$ points is $\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)$.
2016 Silk Road, 1
Let $a,b$ and $c$ be real numbers such that $| (a-b) (b-c) (c-a) | = 1$. Find the smallest value of the expression $| a | + | b | + | c |$. (K.Satylhanov )