Found problems: 1269
2007 Olympic Revenge, 5
Find all functions $f\colon R \to R$ such that
\[f\left(x^{2}+yf(x)\right) = f(x)^{2}+xf(y)\]
for all reals $x,y$.
2009 Hong Kong TST, 2
Find the total number of solutions to the following system of equations:
$ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} \\
b(a + d)\equiv b \pmod{37} \\
c(a + d)\equiv c \pmod{37} \\
bc + d^2\equiv d \pmod{37} \\
ad - bc\equiv 1 \pmod{37} \end{array}$
2002 China Team Selection Test, 1
Find all natural numbers $n (n \geq 2)$ such that there exists reals $a_1, a_2, \dots, a_n$ which satisfy \[ \{ |a_i - a_j| \mid 1\leq i<j \leq n\} = \left\{1,2,\dots,\frac{n(n-1)}{2}\right\}. \]
Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9,\dots,n\}$. $A_i(i=1,2,\dots,20)$ contains eight numbers, three of which are chosen from $A$ and the other five numbers from $B$. $|A_i \cap A_j|\leq 2, 1\leq i<j\leq 20$. Find the minimum possible value of $n$.
2011 China Team Selection Test, 1
Let $n\geq 2$ be a given integer. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
\[f(x-f(y))=f(x+y^n)+f(f(y)+y^n), \qquad \forall x,y \in \mathbb R.\]
2000 China Team Selection Test, 2
[b]a.)[/b] Let $a,b$ be real numbers. Define sequence $x_k$ and $y_k$ such that
\[x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l, \quad y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots \]
Prove that
\[x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}\]
where $\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}$
[b]b.)[/b] Let $u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l} $. For positive integer $m,$ denote the remainder of $u_k$ divided by $2^m$ as $z_{m,k}$. Prove that $z_{m,k},$ $k = 0,1,2, \ldots$ is a periodic function, and find the smallest period.
1989 IMO Longlists, 54
Let $ n \equal{} 2k \minus{} 1$ where $ k \geq 6$ is an integer. Let $ T$ be the set of all $ n\minus{}$tuples $ (x_1, x_2, \ldots, x_n)$ where $ x_i \in \{0,1\}$ $ \forall i \equal{} \{1,2, \ldots, n\}$ For $ x \equal{} (x_1, x_2, \ldots, x_n) \in T$ and $ y \equal{} (y_1, y_2, \ldots, y_n) \in T$ let $ d(x,y)$ denote the number of integers $ j$ with $ 1 \leq j \leq n$ such that $ x_i \neq x_j$, in particular $ d(x,x) \equal{} 0.$ Suppose that there exists a subset $ S$ of $ T$ with $ 2^k$ elements that has the following property: Given any element $ x \in T,$ there is a unique element $ y \in S$ with $ d(x, y) \leq 3.$ Prove that $ n \equal{} 23.$
1992 IMTS, 2
Prove that if $a,b,c$ are positive integers such that $c^2 = a^2+b^2$, then both $c^2+ab$ and $c^2-ab$ are also expressible as the sums of squares of two positive integers.
2005 International Zhautykov Olympiad, 2
Let $ r$ be a real number such that the sequence $ (a_{n})_{n\geq 1}$ of positive real numbers satisfies the equation $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{m \plus{} 1} \leq r a_{m}$ for each positive integer $ m$. Prove that $ r \geq 4$.
2008 Argentina National Olympiad, 4
Find all real numbers $ x$ which satisfy the following equation:
$ [2x]\plus{}[3x]\plus{}[7x]\equal{}2008$.
Note: $ [x]$ means the greatest integer less or equal than $ x$.
1992 Bundeswettbewerb Mathematik, 4
For three sequences $(x_n),(y_n),(z_n)$ with positive starting elements $x_1,y_1,z_1$ we have the following formulae:
\[ x_{n+1} = y_n + \frac{1}{z_n} \quad y_{n+1} = z_n + \frac{1}{x_n} \quad z_{n+1} = x_n + \frac{1}{y_n} \quad (n = 1,2,3, \ldots)\]
a.) Prove that none of the three sequences is bounded from above.
b.) At least one of the numbers $x_{200},y_{200},z_{200}$ is greater than 20.
1981 Vietnam National Olympiad, 1
Solve the system of equations
\[x^2 + y^2 + z^2 + t^2 = 50;\]
\[x^2 - y^2 + z^2 - t^2 = -24;\]
\[xy = zt;\]
\[x - y + z - t = 0.\]
1988 Federal Competition For Advanced Students, P2, 6
Determine all monic polynomials $ p(x)$ of fifth degree having real coefficients and the following property: Whenever $ a$ is a (real or complex) root of $ p(x)$, then so are $ \frac{1}{a}$ and $ 1\minus{}a$.
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]
2012 South africa National Olympiad, 6
Find all functions $f:\mathbb{N}\to\mathbb{R}$ such that
$f(km)+f(kn)-f(k)f(mn)\ge 1$
for all $k,m,n\in\mathbb{N}$.
1993 Irish Math Olympiad, 2
Let $ a_i,b_i$ $ (i\equal{}1,2,...,n)$ be real numbers such that the $ a_i$ are distinct, and suppose that there is a real number $ \alpha$ such that the product $ (a_i\plus{}b_1)(a_i\plus{}b_2)...(a_i\plus{}b_n)$ is equal to $ \alpha$ for each $ i$. Prove that there is a real number $ \beta$ such that $ (a_1\plus{}b_j)(a_2\plus{}b_j)...(a_n\plus{}b_j)$ is equal to $ \beta$ for each $ j$.
1978 IMO Longlists, 29
Given a nonconstant function $f : \mathbb{R}^+ \longrightarrow\mathbb{R}$ such that $f(xy) = f(x)f(y)$ for any $x, y > 0$, find functions $c, s : \mathbb{R}^+ \longrightarrow \mathbb{R}$ that satisfy $c\left(\frac{x}{y}\right) = c(x)c(y)-s(x)s(y)$ for all $x, y > 0$ and $c(x)+s(x) = f(x)$ for all $x > 0$.
1995 South africa National Olympiad, 3
Suppose that $a_1,a_2,\dots,a_n$ are the numbers $1,2,3,\dots,n$ but written in any order. Prove that
\[(a_1-1)^2+(a_2-2)^2+\cdots+(a_n-n)^2\]
is always even.
2008 Bundeswettbewerb Mathematik, 1
Determine all real $ x$ satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion.
2012 ISI Entrance Examination, 4
Prove that the polynomial equation $x^{8}-x^{7}+x^{2}-x+15=0$ has no real solution.
1995 China Team Selection Test, 3
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.)
Does the same result hold with a degree 3 or degree 5 polynomial?
1995 Turkey MO (2nd round), 3
Let $A$ be a real number and $(a_{n})$ be a sequence of real numbers such that $a_{1}=1$ and \[1<\frac{a_{n+1}}{a_{n}}\leq A \mbox{ for all }n\in\mathbb{N}.\]
$(a)$ Show that there is a unique non-decreasing surjective function $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $1<A^{k(n)}/a_{n}\leq A$ for all $n\in \mathbb{N}$.
$(b)$ If $k$ takes every value at most $m$ times, show that there is a real number $C>1$ such that $Aa_{n}\geq C^{n}$ for all $n\in \mathbb{N}$.
2013 ELMO Problems, 6
Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$?
[i]Proposed by David Yang[/i]
2011 Postal Coaching, 6
Prove that there exist integers $a, b, c$ all greater than $2011$ such that
\[(a+\sqrt{b})^c=\ldots 2010 \cdot 2011\ldots\]
[Decimal point separates an integer ending in $2010$ and a decimal part beginning with $2011$.]
2002 China Team Selection Test, 2
For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always:
\[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]
2009 German National Olympiad, 6
Let a sequences: $ x_0\in [0;1],x_{n\plus{}1}\equal{}\frac56\minus{}\frac43 \Big|x_n\minus{}\frac12\Big|$. Find the "best" $ |a;b|$ so that for all $ x_0$ we have $ x_{2009}\in [a;b]$