Found problems: 1269
2009 Croatia Team Selection Test, 1
Determine the lowest positive integer n such that following statement is true:
If polynomial with integer coefficients gets value 2 for n different integers,
then it can't take value 4 for any integer.
2007 Hong Kong TST, 5
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 5
The sequence $\{a_{n}\}$ is defined by $a_{1}=0$ and $(n+1)^{3}a_{n+1}=2n^{2}(2n+1)a_{n}+2(3n+1)$ for all integers $\geq 1$. Show that infintely many members of the sequence are positive integers.
2010 Contests, 1
Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations
\[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]
1999 China Team Selection Test, 2
For a fixed natural number $m \geq 2$, prove that
[b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\]
[b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.
1989 IMO Longlists, 25
The integers $ c_{m,n}$ with $ m \geq 0, \geq 0$ are defined by
\[ c_{m,0} \equal{} 1 \quad \forall m \geq 0, c_{0,n} \equal{} 1 \quad \forall n \geq 0,\]
and
\[ c_{m,n} \equal{} c_{m\minus{}1,n} \minus{} n \cdot c_{m\minus{}1,n\minus{}1} \quad \forall m > 0, n > 0.\]
Prove that \[ c_{m,n} \equal{} c_{n,m} \quad \forall m > 0, n > 0.\]
1985 IMO Longlists, 12
Find the maximum value of
\[\sin^2 \theta_1+\sin^2 \theta_2+\cdots+\sin^2 \theta_n\]
subject to the restrictions $0 \leq \theta_i , \theta_1+\theta_2+\cdots+\theta_n=\pi.$
1989 IMO Longlists, 90
Find the set of all $ a \in \mathbb{R}$ for which there is no infinite sequene $ (x_n)_{n \geq 0} \subset \mathbb{R}$ satisfying $ x_0 \equal{} a,$ and for $ n \equal{} 0,1, \ldots$ we have \[ x_{n\plus{}1} \equal{} \frac{x_n \plus{} \alpha}{\beta x_n \plus{} 1}\] where $ \alpha \beta > 0.$
2005 German National Olympiad, 6
The sequence $x_0,x_1,x_2,.....$ of real numbers is called with period $p$,with $p$ being a natural number, when for each $p\ge2$, $x_n=x_{n+p}$.
Prove that,for each $p\ge2$ there exists a sequence such that $p$ is its least period
and $x_{n+1}=x_n-\frac{1}{x_n}$ $(n=0,1,....)$
2007 Federal Competition For Advanced Students, Part 2, 1
Let $ M$ be the set of all polynomials $ P(x)$ with pairwise distinct integer roots, integer coefficients and all absolut values of the coefficients less than $ 2007$. Which is the highest degree among all the polynomials of the set $ M$?
2008 China Girls Math Olympiad, 6
Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]
Determine real number $ a$ such that if $ x_1 > a$, then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$, then the sequence is not monotonic.
1985 IberoAmerican, 1
Find all the triples of integers $ (a, b,c)$ such that:
\[ \begin{array}{ccc}a\plus{}b\plus{}c &\equal{}& 24\\ a^{2}\plus{}b^{2}\plus{}c^{2}&\equal{}& 210\\ abc &\equal{}& 440\end{array}\]
2008 ISI B.Stat Entrance Exam, 10
Two subsets $A$ and $B$ of the $(x,y)$-plane are said to be [i]equivalent[/i] if there exists a function $f: A\to B$ which is both one-to-one and onto.
(i) Show that any two line segments in the plane are equivalent.
(ii) Show that any two circles in the plane are equivalent.
2011 Kazakhstan National Olympiad, 6
Given a positive integer $n$. One of the roots of a quadratic equation $x^{2}-ax +2 n = 0$ is equal to
$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}$. Prove that $2\sqrt{2n}\le a\le 3\sqrt{n}$
2010 China Team Selection Test, 3
Given integer $n\geq 2$ and real numbers $x_1,x_2,\cdots, x_n$ in the interval $[0,1]$. Prove that there exist real numbers $a_0,a_1,\cdots,a_n$ satisfying the following conditions:
(1) $a_0+a_n=0$;
(2) $|a_i|\leq 1$, for $i=0,1,\cdots,n$;
(3) $|a_i-a_{i-1}|=x_i$, for $i=1,2,\cdots,n$.
1984 IMO Longlists, 27
The function $f(n)$ is defined on the nonnegative integers $n$ by: $f(0) = 0, f(1) = 1$, and
\[f(n) = f\left(n -\frac{1}{2}m(m - 1)\right)-f\left(\frac{1}{2}m(m+ 1)-n\right)\]
for $\frac{1}{2}m(m - 1) < n \le \frac{1}{2}m(m+ 1), m \ge 2$. Find the smallest integer $n$ for which $f(n) = 5$.
1984 IMO Longlists, 45
Let $X$ be an arbitrary nonempty set contained in the plane and let sets $A_1, A_2,\cdots,A_m$ and $B_1, B_2,\cdots, B_n$ be its images under parallel translations. Let us suppose that $A_1\cup A_2 \cup \cdots\cup A_m \subset B_1 \cup B_2 \cup\cdots\cup B_n$ and that the sets $A_1, A_2,\cdots,A_m$ are disjoint. Prove that $m \le n$.
1990 Vietnam National Olympiad, 2
Suppose $ f(x)\equal{}a_0x^n\plus{}a_1x^{n\minus{}1}\plus{}\ldots\plus{}a_{n\minus{}1}x\plus{}a_n$ ($ a_0\neq 0$) is a polynomial with real coefficients satisfying $ f(x)f(2x^2) \equal{} f(2x^3 \plus{} x)$ for all $ x \in\mathbb{R}$. Prove that $ f(x)$ has no real roots.
1987 China National Olympiad, 1
Let $n$ be a natural number. Prove that a necessary and sufficient condition for the equation $z^{n+1}-z^n-1=0$ to have a complex root whose modulus is equal to $1$ is that $n+2$ is divisible by $6$.
2012 Turkmenistan National Math Olympiad, 4
Solve: \[ \begin{cases}x_{2}x_{3}x_{4}\cdots x_{n}=a_{1}x_{1}\\ x_{1}x_{3}x_{4}\cdots x_{n}=a_{2}x_{2}\\x_{1}x_{2}x_{4}\cdots x_{n}=a_{3}x_{3}\\ \ldots\\x_{1}x_{2}x_{3}\cdots x_{n-1}=a_{n-1}x_{n-1} \end{cases} \]
2003 All-Russian Olympiad, 4
A sequence $(a_n)$ is defined as follows: $a_1 = p$ is a prime number with exactly $300$ nonzero digits, and for each $n \geq 1, a_{n+1}$ is the decimal period of $1/a_n$ multiplies by $2$. Determine $a_{2003}.$
2001 India National Olympiad, 6
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(x +y) = f(x) f(y) f(xy)$ for all $x, y \in \mathbb{R}.$
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?
2010 Paenza, 2
A polynomial $f$ with integer coefficients is written on the blackboard. The teacher is a mathematician who has $3$ kids: Andrew, Beth and Charles. Andrew, who is $7$, is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains:
[list]$f(7) = 77$
$f(b) = 85$, where $b$ is Beth's age,
$f(c) = 0$, where $c$ is Charles' age.[/list]
How old is each child?
2012 International Zhautykov Olympiad, 3
Let $P, Q,R$ be three polynomials with real coefficients such that \[P(Q(x)) + P(R(x))=\text{constant}\] for all $x$. Prove that $P(x)=\text{constant}$ or $Q(x)+R(x)=\text{constant}$ for all $x$.
2014 Dutch IMO TST, 5
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$