Found problems: 1311
2002 Romania National Olympiad, 2
Given real numbers $a,c,d$ show that there exists at most one function $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies:
\[f(ax+c)+d\le x\le f(x+d)+c\quad\text{for any}\ x\in\mathbb{R}\]
2012 Rioplatense Mathematical Olympiad, Level 3, 4
Find all real numbers $x$, such that:
a) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2012x \rfloor = 2013$
b) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2013x \rfloor = 2014$
2012 Kyoto University Entry Examination, 3
When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$
Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$
30 points
2011 Iran MO (3rd Round), 5
$f(x)$ is a monic polynomial of degree $2$ with integer coefficients such that $f(x)$ doesn't have any real roots and also $f(0)$ is a square-free integer (and is not $1$ or $-1$). Prove that for every integer $n$ the polynomial $f(x^n)$ is irreducible over $\mathbb Z[x]$.
[i]proposed by Mohammadmahdi Yazdi[/i]
2010 Indonesia TST, 3
Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1\equal{}1$, $ a_2\equal{}\dfrac{4}{3}$, and \[ a_{n\plus{}1}\equal{}\sqrt{1\plus{}a_na_{n\minus{}1}}, \quad \forall n \ge 2.\] Prove that for all $ n \ge 2$, \[ a_n^2>a_{n\minus{}1}^2\plus{}\dfrac{1}{2}\] and \[ 1\plus{}\dfrac{1}{a_1}\plus{}\dfrac{1}{a_2}\plus{}\dots\plus{}\dfrac{1}{a_n}>2a_n.\]
[i]Fajar Yuliawan, Bandung[/i]
1993 Baltic Way, 8
Compute the sum of all positive integers whose digits form either a strictly increasing or strictly decreasing sequence.
2006 ISI B.Math Entrance Exam, 3
Find all roots of the equation :-
$1-\frac{x}{1}+\frac{x(x-1)}{2!} - \cdots +(-1)^n\frac{x(x-1)(x-2)...(x-n+1)}{n!}=0$.
2009 Ukraine National Mathematical Olympiad, 2
Find all functions $f : \mathbb Z \to \mathbb Z$ such that
\[f (n |m|) + f (n(|m| +2)) = 2f (n(|m| +1)) \qquad \forall m,n \in \mathbb Z.\]
[b]Note.[/b] $|x|$ denotes the absolute value of the integer $x.$
1999 Romania Team Selection Test, 11
Let $a,n$ be integer numbers, $p$ a prime number such that $p>|a|+1$. Prove that the polynomial $f(x)=x^n+ax+p$ cannot be represented as a product of two integer polynomials.
[i]Laurentiu Panaitopol[/i]
2010 China Western Mathematical Olympiad, 5
Let $k$ be an integer and $k > 1$. Define a sequence $\{a_n\}$ as follows:
$a_0 = 0$,
$a_1 = 1$, and
$a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$.
Determine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \not= m)$ and positive integers $p,q$ such that $a_l + ka_p = a_m + ka_q$.
2010 China Girls Math Olympiad, 7
For given integer $n \geq 3$, set $S =\{p_1, p_2, \cdots, p_m\}$ consists of permutations $p_i$ of $(1, 2, \cdots, n)$. Suppose that among every three distinct numbers in $\{1, 2, \cdots, n\}$, one of these number does not lie in between the other two numbers in every permutations $p_i$ ($1 \leq i \leq m$). (For example, in the permutation $(1, 3, 2, 4)$, $3$ lies in between $1$ and $4$, and $4$ does not lie in between $1$ and $2$.) Determine the maximum value of $m$.
2006 Iran MO (3rd Round), 3
Find all real $x,y,z$ that \[\left\{\begin{array}{c}x+y+zx=\frac12\\ \\ y+z+xy=\frac12\\ \\ z+x+yz=\frac12\end{array}\right.\]
2002 Romania National Olympiad, 2
Find all real polynomials $f$ and $g$, such that:
\[(x^2+x+1)\cdot f(x^2-x+1)=(x^2-x+1)\cdot g(x^2+x+1), \]
for all $x\in\mathbb{R}$.
2010 Romanian Master of Mathematics, 4
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]
1950 Miklós Schweitzer, 4
Find the polynomials $ f(x)$ having the following properties:
(i) $ f(0) \equal{} 1$, $ f'(0) \equal{} f''(0) \equal{} \cdots \equal{} f^{(n)}(0) \equal{} 0$
(ii) $ f(1) \equal{} f'(1) \equal{} f''(1) \equal{} \cdots \equal{} f^{(m)}(1) \equal{} 0$
2006 Austrian-Polish Competition, 7
Find all nonnegative integers $m,n$ so that \[\sum_{k=1}^{2^{m}}\lfloor \frac{kn}{2^{m}}\rfloor\in \{28,29,30\}\]
1992 Dutch Mathematical Olympiad, 5
We consider regular $ n$-gons with a fixed circumference $ 4$. Let $ r_n$ and $ a_n$ respectively be the distances from the center of such an $ n$-gon to a vertex and to an edge.
$ (a)$ Determine $ a_4,r_4,a_8,r_8$.
$ (b)$ Give an appropriate interpretation for $ a_2$ and $ r_2$
$ (c)$ Prove that $ a_{2n}\equal{}\frac{1}{2} (a_n\plus{}r_n)$ and $ r_{2n}\equal{}\sqrt{a_2n r_n}.$
$ (d)$ Define $ u_0\equal{}0, u_1\equal{}1$ and $ u_n\equal{}\frac{1}{2}(u_{n\minus{}2}\plus{}u_{n\minus{}1})$ for $ n$ even or $ u_n\equal{}\sqrt{u_{n\minus{}2} u_{n\minus{}1}}$ for $ n$ odd. Determine $ \displaystyle\lim_{n\to\infty}u_n$.
2001 Baltic Way, 11
The real-valued function $f$ is defined for all positive integers. For any integers $a>1, b>1$ with $d=\gcd (a, b)$, we have
\[f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right) \]
Determine all possible values of $f(2001)$.
2008 Saint Petersburg Mathematical Olympiad, 1
Replacing any of the coefficients of quadratic trinomial $f(x)=ax^2+bx+c$ with an $1$ will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point.
EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet.
Note: fresh translation
2010 China Girls Math Olympiad, 1
Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of
\[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\]
Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$
2011 All-Russian Olympiad Regional Round, 10.1
Two runners started a race simultaneously. Initially they ran on the street toward the stadium and then 3 laps on the stadium. Both runners covered the whole distance at their own constant speed. During the whole race the first runner passed the second runner exactly twice. Prove that the speed of the first runner is at least double the speed of the second runner.
(Author: I. Rubanov)
2008 Vietnam Team Selection Test, 2
Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$, we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$.
2012 Waseda University Entrance Examination, 2
Consider a sequence $\{a_n\}_{n\geq 0}$ such that $a_{n+1}=a_n-\lfloor{\sqrt{a_n}}\rfloor\ (n\geq 0),\ a_0\geq 0$.
(1) If $a_0=24$, then find the smallest $n$ such that $a_n=0$.
(2) If $a_0=m^2\ (m=2,\ 3,\ \cdots)$, then for $j$ with $1\leq j\leq m$, express $a_{2j-1},\ a_{2j}$ in terms of $j,\ m$.
(3) Let $m\geq 2$ be integer and for integer $p$ with $1\leq p\leq m-1$, let $a\0=m^2-p$. Find $k$ such that $a_k=(m-p)^2$, then
find the smallest $n$ such that $a_n=0$.
1982 IMO Longlists, 47
Evaluate $\sec'' \frac{\pi}4 +\sec'' \frac{3\pi}4+\sec'' \frac{5\pi}4+\sec'' \frac{7\pi}4$. (Here $\sec''$ means the second derivative of $\sec$).
2006 Taiwan National Olympiad, 2
Find all reals $x$ satisfying $0 \le x \le 5$ and
$\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor$.