Found problems: 1311
2011 All-Russian Olympiad, 2
Nine quadratics, $x^2+a_1x+b_1, x^2+a_2x+b_2,...,x^2+a_9x+b_9$ are written on the board. The sequences $a_1, a_2,...,a_9$ and $b_1, b_2,...,b_9$ are arithmetic. The sum of all nine quadratics has at least one real root. What is the the greatest possible number of original quadratics that can have no real roots?
2008 All-Russian Olympiad, 2
Petya and Vasya are given equal sets of $ N$ weights, in which the masses of any two weights are in ratio at most $ 1.25$. Petya succeeded to divide his set into $ 10$ groups of equal masses, while Vasya succeeded to divide his set into $ 11$ groups of equal masses. Find the smallest possible $ N$.
1997 Iran MO (3rd Round), 3
Let $S = \{x_0, x_1,\dots , x_n\}$ be a finite set of numbers in the interval $[0, 1]$ with $x_0 = 0$ and $x_1 = 1$. We consider pairwise distances between numbers in $S$. If every distance that appears, except the distance $1$, occurs at least twice, prove that all the $x_i$ are rational.
2010 Contests, 2
For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]
1991 Dutch Mathematical Olympiad, 4
Three real numbers $ a,b,c$ satisfy the equations $ a\plus{}b\plus{}c\equal{}3, a^2\plus{}b^2\plus{}c^2\equal{}9, a^3\plus{}b^3\plus{}c^3\equal{}24.$ Find $ a^4\plus{}b^4\plus{}c^4$.
2008 Romanian Master of Mathematics, 2
Prove that every bijective function $ f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $ f\equal{}u\plus{}v$ where $ u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.
2011 Romanian Masters In Mathematics, 1
Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing.
[i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]
2014 IMS, 1
Let $A$ be a subset of the irrational numbers such that the sum of any two distinct elements of it be a rational number. Prove that $A$ has two elements at most.
2000 Baltic Way, 11
A sequence of positive integers $a_1,a_2,\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$, then $a_m$ is a divisor of $a_n$ and $a_m<a_n$. Find the least possible value of $a_{2000}$.
1998 Romania Team Selection Test, 1
Find all monotonic functions $u:\mathbb{R}\rightarrow\mathbb{R}$ which have the property that there exists a strictly monotonic function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x+y)=f(x)u(x)+f(y) \]
for all $x,y\in\mathbb{R}$.
[i]Vasile Pop[/i]
2009 Moldova Team Selection Test, 3
[color=darkblue]The sequence $ (a_n)_{n \in \mathbb{N}}$ is defined as follows:
\[ a_n \equal{} \dfrac{2}{3 \plus{} 1} \plus{} \dfrac{2^2}{3^2 \plus{} 1} \plus{} \dfrac{2^3}{3^4 \plus{} 1} \plus{} \ldots \plus{} \dfrac{2^{n \plus{} 1}}{3^{2^n} \plus{} 1}
\]
Prove that $ a_n < 1$ for any $ n \in \mathbb{N}$[/color]
2012 Middle European Mathematical Olympiad, 1
Let $ \mathbb{R} ^{+} $ denote the set of all positive real numbers. Find all functions $ \mathbb{R} ^{+} \to \mathbb{R} ^{+} $ such that
\[ f(x+f(y)) = yf(xy+1)\]
holds for all $ x, y \in \mathbb{R} ^{+} $.
1987 IMO Longlists, 11
Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$. The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$.
We want to compute, using a computer, the extreme values of $g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)}$ for $x - t, x + t \in [0, 1]$. At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?
2014 District Olympiad, 3
Let $p$ and $n$ be positive integers, with $p\geq2$, and let $a$ be a real number such that $1\leq a<a+n\leq p$. Prove that the set
\[ \mathcal {S}=\left\{\left\lfloor \log_{2}x\right\rfloor +\left\lfloor \log_{3}x\right\rfloor +\cdots+\left\lfloor \log_{p}x\right\rfloor\mid x\in\mathbb{R},a\leq x\leq a+n\right\} \]
has exactly $n+1$ elements.
2011 All-Russian Olympiad Regional Round, 10.6
2011 numbers are written on a board. For any three numbers, their sum is also among numbers written on the board. What is the smallest number of zeros among all 2011 numbers?
(Author: I. Bogdanov)
1993 Brazil National Olympiad, 2
A real number with absolute value less than $1$ is written in each cell of an $n\times n$ array, so that the sum of the numbers in each $2\times 2$ square is zero. Show that for odd $n$ the sum of all the numbers is less than $n$.
1987 Balkan MO, 2
Find all real numbers $x,y$ greater than $1$, satisfying the condition that the numbers $\sqrt{x-1}+\sqrt{y-1}$ and $\sqrt{x+1}+\sqrt{y+1}$ are nonconsecutive integers.
1991 Iran MO (2nd round), 1
Prove that there exist at least six points with rational coordinates on the curve of the equation
\[y^3=x^3+x+1370^{1370}\]
1981 Canada National Olympiad, 1
For any real number $t$, denote by $[t]$ the greatest integer which is less than or equal to $t$. For example: $[8] = 8$, $[\pi] = 3$, and $[-5/2] = -3$. Show that the equation
\[[x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345\]
has no real solution.
1998 Greece National Olympiad, 4
Let a function $g:\mathbb{N}_0\to\mathbb{N}_0$ satisfy $g(0)=0$ and $g(n)=n-g(g(n-1))$ for all $n\ge 1$. Prove that:
a) $g(k)\ge g(k-1)$ for any positive integer $k$.
b) There is no $k$ such that $g(k-1)=g(k)=g(k+1)$.
2010 India IMO Training Camp, 11
Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$
2009 China Western Mathematical Olympiad, 1
Let $M$ be the set of the real numbers except for finitely many elements. Prove that for every positive integer $n$ there exists a polynomial $f(x)$ with $\deg f = n$, such that all the coefficients and the $n$ real roots of $f$ are all in $M$.
1971 IMO Longlists, 12
A system of n numbers $x_1, x_2, \ldots, x_n$ is given such that
\[x_1 = \log_{x_{n-1}} x_n, x_2 = \log_{x_{n}} x_1, \ldots, x_n = \log_{x_{n-2}} x_{n-1}.\]
Prove that $\prod_{k=1}^n x_k =1.$
2004 Hong kong National Olympiad, 1
Let $a_{1},a_{2},...,a_{n+1}(n>1)$ are positive real numbers such that $a_{2}-a_{1}=a_{3}-a_{2}=...=a_{n+1}-a_{n}$. Prove that $\sum_{k=2}^{n}\frac{1}{a_{k}^{2}}\leq\frac{n-1}{2}.\frac{a_{1}a_{n}+a_{2}a_{n+1}}{a_{1}a_{2}a_{n}a_{n+1}}$
1989 IberoAmerican, 1
Determine all triples of real numbers that satisfy the following system of equations:
\[x+y-z=-1\\ x^2-y^2+z^2=1\\ -x^3+y^3+z^3=-1\]