Found problems: 1269
1993 Polish MO Finals, 1
Find all rational solutions to: \begin{eqnarray*} t^2 - w^2 + z^2 &=& 2xy \\ t^2 - y^2 + w^2 &=& 2xz \\ t^2 - w^2 + x^2 &=& 2yz . \end{eqnarray*}
2009 Mediterranean Mathematics Olympiad, 1
Determine all integers $n\ge1$ for which there exists $n$ real numbers $x_1,\ldots,x_n$ in the closed interval $[-4,2]$ such that the following three conditions are fulfilled:
- the sum of these real numbers is at least $n$.
- the sum of their squares is at most $4n$.
- the sum of their fourth powers is at least $34n$.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
2006 China Team Selection Test, 1
Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties:
(1) $f(0)=0$ and $f(1)=1$; and.
(2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.
1994 USAMO, 5
Let $\, |U|, \, \sigma(U) \,$ and $\, \pi(U) \,$ denote the number of elements, the sum, and the product, respectively, of a finite set $\, U \,$ of positive integers. (If $\, U \,$ is the empty set, $\, |U| = 0, \, \sigma(U) = 0, \, \pi(U) = 1$.) Let $\, S \,$ be a finite set of positive integers. As usual, let $\, \binom{n}{k} \,$ denote $\, n! \over k! \, (n-k)!$. Prove that \[ \sum_{U \subseteq S} (-1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S) \] for all integers $\, m \geq \sigma(S)$.
2008 District Round (Round II), 1
Let $n$ be an integer greater than $1$.Find all pairs of integers $(s,t)$ such that equations:
$x^n+sx=2007$
and
$x^n+tx=2008$
have at least one common real root.
1993 USAMO, 3
Consider functions $\, f: [0,1] \rightarrow \mathbb{R} \,$ which satisfy
(i) $f(x) \geq 0 \,$ for all $\, x \,$ in $\, [0,1],$
(ii) $f(1) = 1,$
(iii) $f(x) + f(y) \leq f(x+y)\,$ whenever $\, x, \, y, \,$ and $\, x + y \,$ are all in $\, [0,1]$.
Find, with proof, the smallest constant $\, c \,$ such that
\[ f(x) \leq cx \]
for every function $\, f \,$ satisfying (i)-(iii) and every $\, x \,$ in $\, [0,1]$.
2014 Postal Coaching, 4
Given arbitrary complex numbers $w_1,w_2,\ldots,w_n$, show that there exists a positive integer $k\le 2n+1$ for which $\text{Re} (w_1^k+w_2^k+\cdots+w_n^k)\ge 0$.
2007 German National Olympiad, 6
For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$
2014 Dutch IMO TST, 1
Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \]
Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.
2007 Regional Competition For Advanced Students, 2
Find all tuples $ (x_1,x_2,x_3,x_4,x_5)$ of positive integers with $ x_1>x_2>x_3>x_4>x_5>0$ and
$ {\left \lfloor \frac{x_1+x_2}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_2+x_3}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_3+x_4}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_4+x_5}{3} \right \rfloor }^2 = 38.$
1976 IMO Longlists, 50
Find a function $f(x)$ defined for all real values of $x$ such that for all $x$,
\[f(x+ 2) - f(x) = x^2 + 2x + 4,\]
and if $x \in [0, 2)$, then $f(x) = x^2.$
2008 China Second Round Olympiad, 3
For all $k=1,2,\ldots,2008$,$a_k>0$.Prove that iff $\sum_{k=1}^{2008}a_k>1$,there exists a function $f:N\rightarrow R$ satisfying
(1)$0=f(0)<f(1)<f(2)<\ldots$;
(2)$f(n)$ has a finite limit when $n$ approaches infinity;
(3)$f(n)-f({n-1})=\sum_{k=1}^{2008}a_kf({n+k})-\sum_{k=0}^{2007}a_{k+1}f({n+k})$,for all $n=1,2,3,\ldots$.
2018 IFYM, Sozopol, 1
$A = \{a_1, a_2, . . . , a_k\}$ is a set of positive integers for which the sum of some (we can have only one number too) different numbers from the set is equal to a different number i.e. there $2^k - 1$ different sums of different numbers from $A$. Prove that the following inequality holds:
$\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}<2$
2011 Morocco National Olympiad, 2
Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$
($p$ and $q$ are two real parameters).
2009 Polish MO Finals, 3
Let $P,Q,R$ be polynomials of degree at least $1$ with integer coefficients such that for any real number $x$ holds: $P(Q(x))\equal{}Q(R(x))\equal{}R(P(x))$. Show that the polynomials $P,Q,R$ are equal.
1988 USAMO, 5
A polynomial product of the form \[(1-z)^{b_1}(1-z^2)^{b_2}(1-z^3)^{b_3}(1-z^4)^{b_4}(1-z^5)^{b_5}\cdots(1-z^{32})^{b_{32}},\] where the $b_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving $z$ to a power larger than $32$, what is left is just $1-2z$. Determine, with proof, $b_{32}$.
1982 IMO Longlists, 18
You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that
\[(a + ab^{-1} a)^{-1}+ (a + b)^{-1} = a^{-1},\]
where $x^{-1}$ is the element for which $x^{-1}x = xx^{-1} = e$, where $e$ is the element of the system such that for all $a$ the equality $ea = ae = a$ holds.
2000 Vietnam National Olympiad, 1
Given a real number $ c > 0$, a sequence $ (x_n)$ of real numbers is defined by $ x_{n \plus{} 1} \equal{} \sqrt {c \minus{} \sqrt {c \plus{} x_n}}$ for $ n \ge 0$. Find all values of $ c$ such that for each initial value $ x_0$ in $ (0, c)$, the sequence $ (x_n)$ is defined for all $ n$ and has a finite limit $ \lim x_n$ when $ n\to \plus{} \infty$.
1990 IMO Longlists, 54
Let $M = \{1, 2, \ldots, n\}$ and $\phi : M \to M$ be a bijection.
(i) Prove that there exist bijections $\phi_1, \phi_2 : M \to M$ such that $\phi_1 \cdot \phi_2 = \phi , \phi_1^2 =\phi_2^2=E$, where $E$ is the identity mapping.
(ii) Prove that the conclusion in (i) is also true if $M$ is the set of all positive integers.
1988 China Team Selection Test, 2
Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying
(i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$.
(ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.
1987 IMO Longlists, 63
Compute $\sum_{k=0}^{2n} (-1)^k a_k^2$ where $a_k$ are the coefficients in the expansion
\[(1- \sqrt 2 x +x^2)^n =\sum_{k=0}^{2n} a_k x^k.\]
1977 Polish MO Finals, 3
Consider the polynomial $W(x) = (x - a)^kQ(x)$, where $a \neq 0$, $Q$ is a nonzero polynomial, and $k$ a natural number. Prove that $W$ has at least $k + 1$ nonzero coefficients.
2007 Czech and Slovak Olympiad III A, 6
Find all pariwise distinct real numbers $x,y,z$ such that $\left\{\frac{x-y}{y-z},\frac{y-z}{z-x},\frac{z-x}{x-y} \right\} = \{x,y,z\}$. (It means, those three fractions make a permutation of $x, y$, and $z$.)
2004 India IMO Training Camp, 3
Suppose the polynomial $P(x) \equiv x^3 + ax^2 + bx +c$ has only real zeroes and let $Q(x) \equiv 5x^2 - 16x + 2004$. Assume that $P(Q(x)) = 0$ has no real roots. Prove that $P(2004) > 2004$
1988 IMO Longlists, 15
Let $1 \leq k \leq n.$ Consider all finite sequences of positive integers with sum $n.$ Find $T(n,k),$ the total number of terms of size $k$ in all of the sequences.