Found problems: 85335
2017 Irish Math Olympiad, 3
A line segment $B_0B_n$ is divided into $n$ equal parts at points $B_1,B_2,...,B_{n-1} $ and $A$ is a point such that $\angle B_0AB_n$ is a right angle. Prove that :
$$\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2$$
2006 Spain Mathematical Olympiad, 1
Find all the functions $f:(0,+\infty) \to R $ that satisfy the equation
$$f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy)$$
for all pairs of $x,y$ real and positive numbers, where $\lambda$ is a positive real number such that $f(\lambda )=1$
2021 Caucasus Mathematical Olympiad, 6
A row of 2021 balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins, if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?
1968 All Soviet Union Mathematical Olympiad, 111
The city is a rectangle divided onto squares by $m$ streets coming from the West to the East and $n$ streets coming from the North to the South. There are militioners (policemen) on the streets but not on the crossroads. They watch the certain automobile, moving along the closed route, marking the time and the direction of its movement. Its trace is not known in advance, but they know, that it will not pass over the same segment of the way twice. What is the minimal number of the militioners providing the unique determination of the route according to their reports?
2005 Croatia National Olympiad, 3
Find the locus of points inside a trihedral angle such that the sum of their distances from the faces of the trihedral angle has a fixed positive value $a$.
1965 Polish MO Finals, 4
Prove that if the integers $ a $ and $ b $ satisfy the equation
$$ 2a^2 + a = 3b^2 + b,$$
then the numbers $ a - b $ and $ 2a + 2b + 1 $ are squares of integers.
2015 Belarus Team Selection Test, 3
The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $P$ and $Q$ respectively. $N$ and $M$ are the midpoints of $AC$ and $BC$ respectively. Let $X=AM\cap BP, Y=BN\cap AQ$. Given $C,X,Y$ are collinear, prove that $CX$ is the angle bisector of the angle $ACB$.
I. Gorodnin
1956 Polish MO Finals, 1
Solve the system of equations
$$
\begin{array}{l}<br />
x^2y^2 + x^2z^2 = axyz\\<br />
y^2z^2 + y^2x^2 = bxyz\\<br />
z^2x^2 + z^2y^2 = cxyz.<br />
\end{array}$$
2008 AMC 10, 13
Doug can paint a room in $ 5$ hours. Dave can paint the same room in $ 7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $ t$ be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by $ t$?
$ \textbf{(A)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\plus{}1)\equal{}1 \qquad
\textbf{(B)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\plus{}1\equal{}1 \qquad
\textbf{(C)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\equal{}1 \\
\textbf{(D)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\minus{}1)\equal{}1 \qquad
\textbf{(E)}\ (5\plus{}7)t\equal{}1$
1997 Brazil National Olympiad, 3
a) Show that there are no functions $f, g: \mathbb R \to \mathbb R$ such that $g(f(x)) = x^3$ and $f(g(x)) = x^2$ for all $x \in \mathbb R$.
b) Let $S$ be the set of all real numbers greater than 1. Show that there are functions $f, g : S \to S$ satsfying the condition above.
1955 Putnam, B4
Do there exist $1,000,000$ consecutive integers each of which contains a repeated prime factor?
2023 Junior Balkan Team Selection Tests - Moldova, 4
On the board there are three real numbers $(a,b,c)$. During a $procedure$ the numbers are erased and in their place another three numbers a written, either $(c,b,a)$ or every time a nonzero real number $ d $ is chosen and the numbers $(a, 2ad+b, ad^2+bd+c)$ are written.
1) If we start with $(1,-2,-1)$ written on the board, can we have the numbers $(2,0,-1)$ on the board after a finite number of procedures?
2) If we start with $(1,-2,-1)$ written on the board, can we have the numbers $(2,-1,-1)$ on the board after a finite number of procedures?
2012 Centers of Excellency of Suceava, 4
Let $ O $ be the circumcenter of a triangle $ ABC $ with $ \angle BAC=60^{\circ } $ whose incenter is denoted by $ I. $ Let $ B_1,C_1 $ be the intersection of $ BI,CI $ with the circumcircle of $ ABC, $ respectively. Denote by $ O_1,O_2 $ the circumcenters of $ BIC_1,CIB_1, $ respectively. Show that $ O_1,I,O,O_2 $ are collinear.
[i]Cătălin Țigăeru[/i]
1991 IMTS, 2
Note that 1990 can be "turned into a square" by adding a digit on its right, and some digits on its left; i.e., $419904 = 648^2$. Prove that 1991 cannot be turned into a square by the same procedure; i.e., there are no digits $d,x,y,..$ such that $...yx1991d$ is a perfect square.
1970 Miklós Schweitzer, 7
Let us use the word $ N$-measure for nonnegative, finitely additive set functions defined on all subsets of the positive integers, equal to $ 0$ on finite sets, and equal to $ 1$ on the whole set. We say that the system $ \Upsilon$ of sets determines the $ N$-measure $ \mu$ if any $ N$-measure coinciding with $ \mu$ on all elements of $ \Upsilon$ is necessarily identical with $ \mu$.
Prove the existence of an $ N$-measure $ \mu$ that cannot be determined by a system of cardinality less than continuum.
[i]I. Juhasz[/i]
1992 IMO Longlists, 80
Given a graph with $n$ vertices and a positive integer $m$ that is less than $ n$, prove that the graph contains a set of $m+1$ vertices in which the difference between the largest degree of any vertex in the set and the smallest degree of any vertex in the set is at most $m-1.$
2021 Mexico National Olympiad, 2
Let $ABC$ be a triangle with $\angle ACB > 90^{\circ}$, and let $D$ be a point on $BC$ such that $AD$ is perpendicular to $BC$. Consider the circumference $\Gamma$ with with diameter $BC$. A line $\ell$ passes through $D$ and is tangent to $\Gamma$ at $P$, cuts $AC$ at $M$ (such that $M$ is in between $A$ and $C$), and cuts the side $AB$ at $N$. Prove that $M$ is the midpoint of $DP$ if and only if $N$ is the midpoint of $AB$.
2012 Germany Team Selection Test, 1
Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.
[i]Proposed by Luxembourg[/i]
2020-2021 OMMC, 7
An infinitely large grid is filled such that each grid square contains exactly one of the digits $\{ 1,2,3,4\},$ each digit appears at least once, and the digit in each grid square equals the digit located $5$ squares above it as well as the digit located $5$ squares to the right. A group of $4$ horizontally adjacent digits or $4$ vertically adjacent digits is chosen randomly, and depending on its orientation is read left to right or top to bottom to form an $4$-digit integer. The expected value of this integer is also a $4$-digit integer $N$. Given this, find the last three digits of the sum of all possible values of $N$.
2019 Slovenia Team Selection Test, 1
Let $ABC$ be a non-right isosceles triangle such that $AC = BC$. Let $D$ be such a point on the perpendicular bisector of $AB$, that $AD$ is tangent on the $ABC$ circumcircle. Let $E$ be such a point on $AB$, that $CE$ and $AD$ are perpendicular and let $F$ be the second intersection of line $AC$ and the circle $CDE$. Prove that $DF$ and $AB$ are parallel.
2023 Math Prize for Girls Problems, 2
In the $xy$-coordinate plane, the horizontal line $y = k$ intersects the graph of the cubic $2x^3 + 6x^2 - 4x + 5$ in three points $P$, $Q$, and $R$. Given that $Q$ is the midpoint of $P$ and $R$, what is $k$?
2019 Brazil Team Selection Test, 1
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
2007 Romania National Olympiad, 1
Let $A,B\in\mathcal{M}_{2}(\mathbb{R})$ (real $2\times 2$ matrices), that satisfy $A^{2}+B^{2}=AB$. Prove that $(AB-BA)^{2}=O_{2}$.
1985 Traian Lălescu, 1.1
Consider the function $ f:\mathbb{R}\longrightarrow\mathbb{R} ,\quad f(x)=\max (x-3,2) . $ Find the perimeter and the area of the figure delimited by the lines $ x=-3,x=1, $ the $ Ox $ axis, and the graph of $ f. $
2008 Tournament Of Towns, 7
A test consists of $30$ true or false questions. After the test (answering all $30$ questions), Victor gets his score: the number of correct answers. Victor is allowed to take the test (the same questions ) several times. Can Victor work out a strategy that insure him to get a perfect score after
[b](a) [/b] $30$th attempt?
[b](b)[/b] $25$th attempt?
(Initially, Victor does not know any answer)