Found problems: 1311
2009 China Team Selection Test, 3
Consider function $ f: R\to R$ which satisfies the conditions for any mutually distinct real numbers $ a,b,c,d$ satisfying $ \frac {a \minus{} b}{b \minus{} c} \plus{} \frac {a \minus{} d}{d \minus{} c} \equal{} 0$, $ f(a),f(b),f(c),f(d)$ are mutully different and $ \frac {f(a) \minus{} f(b)}{f(b) \minus{} f(c)} \plus{} \frac {f(a) \minus{} f(d)}{f(d) \minus{} f(c)} \equal{} 0.$ Prove that function $ f$ is linear
2002 Romania National Olympiad, 3
Find all real numbers $a,b,c,d,e$ in the interval $[-2,2]$, that satisfy:
\begin{align*}a+b+c+d+e &= 0\\ a^3+b^3+c^3+d^3+e^3&= 0\\ a^5+b^5+c^5+d^5+e^5&=10 \end{align*}
Kvant 2024, M2799
Let $n$ be a positive integer. Ilya and Sasha both choose a pair of different polynomials of degree $n$ with real coefficients. Lenya knows $n$, his goal is to find out whether Ilya and Sasha have the same pair of polynomials. Lenya selects a set of $k$ real numbers $x_1<x_2<\dots<x_k$ and reports these numbers. Then Ilya fills out a $2 \times k$ table: For each $i=1,2,\dots,k$ he writes a pair of numbers $P(x_i),Q(x_i)$ (in any of the two possible orders) intwo the two cells of the $i$-th column, where $P$ and $Q$ are his polynomials. Sasha fills out a similar table. What is the minimal $k$ such that Lenya can surely achieve the goal by looking at the tables?
[i]Proposed by L. Shatunov[/i]
1999 Hungary-Israel Binational, 1
$ c$ is a positive integer. Consider the following recursive sequence: $ a_1\equal{}c, a_{n\plus{}1}\equal{}ca_{n}\plus{}\sqrt{(c^2\minus{}1)(a_n^2\minus{}1)}$, for all $ n \in N$.
Prove that all the terms of the sequence are positive integers.
2011 Mexico National Olympiad, 3
Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:
\[a_1^2 + a_1 - 1 = a_2\]
\[ a_2^2 + a_2 - 1 = a_3\]
\[\hspace*{3.3em} \vdots \]
\[a_{n}^2 + a_n - 1 = a_1\]
2004 Turkey Team Selection Test, 1
Find all possible values of $x-\lfloor x\rfloor$ if $\sin \alpha = 3/5$ and $x=5^{2003}\sin {(2004\alpha)}$.
2008 German National Olympiad, 3
Find all functions $ f$ defined on non-negative real numbers having the following properties:
(i) For all non-negative $ x$ it is $ f(x) \geq 0$.
(ii) It is $ f\left(1\right)\equal{}\frac 12$.
(iii) For all non-negative numbers $ x,y$ it is $ f\left( y \cdot f(x) \right) \cdot f(x) \equal{} f(x\plus{}y)$.
2006 Romania Team Selection Test, 1
Let $\{a_n\}_{n\geq 1}$ be a sequence with $a_1=1$, $a_2=4$ and for all $n>1$, \[ a_{n} = \sqrt{ a_{n-1}a_{n+1} + 1 } . \]
a) Prove that all the terms of the sequence are positive integers.
b) Prove that $2a_na_{n+1}+1$ is a perfect square for all positive integers $n$.
[i]Valentin Vornicu[/i]
1995 Moldova Team Selection Test, 4
Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying the following:
$i)$ $f(1)=1$;
$ii)$ $f(m+n)(f(m)-f(n))=f(m-n)(f(m)+f(n))$ for all $m,n \in \mathbb{Z}$.
2004 Germany Team Selection Test, 2
Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties:
(a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$.
(b) We have $f\left(2\right) = 0$.
(c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$.
[b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.
2012 Iran Team Selection Test, 2
Do there exist $2000$ real numbers (not necessarily distinct) such that all of them are not zero and if we put any group containing $1000$ of them as the roots of a monic polynomial of degree $1000$, the coefficients of the resulting polynomial (except the coefficient of $x^{1000}$) be a permutation of the $1000$ remaining numbers?
[i]Proposed by Morteza Saghafian[/i]
2007 Indonesia TST, 2
Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.
2005 Germany Team Selection Test, 1
Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$.
Hereby, $\mathbb{R}_+$ is the set of all positive real numbers.
[i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$.
A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.
1977 IMO Longlists, 50
Determine all positive integers $n$ for which there exists a polynomial $P_n(x)$ of degree $n$ with integer coefficients that is equal to $n$ at $n$ different integer points and that equals zero at zero.
2010 Contests, 4
Let $a_n$ and $b_n$ to be two sequences defined as below:
$i)$ $a_1 = 1$
$ii)$ $a_n + b_n = 6n - 1$
$iii)$ $a_{n+1}$ is the least positive integer different of $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$.
Determine $a_{2009}$.
2000 Baltic Way, 12
Let $x_1,x_2,\ldots x_n$ be positive integers such that no one of them is an initial fragment of any other (for example, $12$ is an initial fragment of $\underline{12},\underline{12}5$ and $\underline{12}405$). Prove that
\[\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}<3. \]
2009 Hong Kong TST, 1
Let $ f: Z \to Z$ be such that $ f(1) \equal{} 1, f(2) \equal{} 20, f(\minus{}4) \equal{} \minus{}4$ and $ f(x\plus{}y) \equal{} f(x) \plus{}f(y)\plus{}axy(x\plus{}y)\plus{}bxy\plus{}c(x\plus{}y)\plus{}4 \forall x,y \in Z$, where $ a,b,c$ are constants.
(a) Find a formula for $ f(x)$, where $ x$ is any integer.
(b) If $ f(x) \geq mx^2\plus{}(5m\plus{}1)x\plus{}4m$ for all non-negative integers $ x$, find the greatest possible value of $ m$.
2006 Iran MO (3rd Round), 5
A calculating ruler is a ruler for doing algebric calculations. This ruler has three arms, two of them are sationary and one can move freely right and left. Each of arms is gradient. Gradation of each arm depends on the algebric operation ruler does. For eaxample the ruler below is designed for multiplying two numbers. Gradations are logarithmic.
[img]http://aycu05.webshots.com/image/5604/2000468517162383885_rs.jpg[/img]
For working with ruler, (e.g for calculating $x.y$) we must move the middle arm that the arrow at the beginning of its gradation locate above the $x$ in the lower arm. We find $y$ in the middle arm, and we will read the number on the upper arm. The number written on the ruler is the answer.
1) Design a ruler for calculating $x^{y}$. Grade first arm ($x$) and ($y$) from 1 to 10.
2) Find all rulers that do the multiplication in the interval $[1,10]$.
3) Prove that there is not a ruler for calculating $x^{2}+xy+y^{2}$, that its first and second arm are grade from 0 to 10.
2006 Junior Balkan Team Selection Tests - Moldova, 3
Determine all 2nd degree polynomials with integer coefficients of the form $P(X)=aX^{2}+bX+c$, that satisfy: $P(a)=b$, $P(b)=a$, with $a\neq b$.
1987 India National Olympiad, 5
Find a finite sequence of 16 numbers such that:
(a) it reads same from left to right as from right to left.
(b) the sum of any 7 consecutive terms is $ \minus{}1$,
(c) the sum of any 11 consecutive terms is $ \plus{}1$.
2012 Iran MO (3rd Round), 5
We call the three variable polynomial $P$ cyclic if $P(x,y,z)=P(y,z,x)$. Prove that cyclic three variable polynomials $P_1,P_2,P_3$ and $P_4$ exist such that for each cyclic three variable polynomial $P$, there exists a four variable polynomial $Q$ such that $P(x,y,z)=Q(P_1(x,y,z),P_2(x,y,z),P_3(x,y,z),P_4(x,y,z))$.
[i]Solution by Mostafa Eynollahzade and Erfan Salavati[/i]
2011 Singapore Senior Math Olympiad, 3
Find all positive integers $n$ such that
\[\cos\frac{\pi}{n}\cos\frac{2\pi}{n}\cos\frac{3\pi}{n}=\frac{1}{n+1}\]
2000 Romania Team Selection Test, 1
Let $n\ge 2$ be a positive integer. Find the number of functions $f:\{1,2,\ldots ,n\}\rightarrow\{1,2,3,4,5 \}$ which have the following property: $|f(k+1)-f(k)|\ge 3$, for any $k=1,2,\ldots n-1$.
[i]Vasile Pop[/i]
2011 India National Olympiad, 6
Find all functions $f:\mathbb{R}\to \mathbb R$ satisfying
\[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\]
For all $x,y\in\mathbb R$.
2006 IberoAmerican Olympiad For University Students, 6
Let $x_0(t)=1$, $x_{k+1}(t)=(1+t^{k+1})x_k(t)$ for all $k\geq 0$; $y_{n,0}(t)=1$, $y_{n,k}(t)=\frac{t^{n-k+1}-1}{t^k-1}y_{n,k-1}(t)$ for all $n\geq 0$, $1\leq k \leq n$.
Prove that $\sum_{j=0}^{n-1}(-1)^j x_{n-j-1}(t)y_{n,j}(t)=\frac{1-(-1)^n}{2}$ for all $n\geq 1$.