Found problems: 85335
1997 South africa National Olympiad, 6
Six points are connected in pairs by lines, each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. (A path is closed if it begins and ends at the same point.)
2015 Azerbaijan JBMO TST, 1
Let $x,y$ and $z$ be non-negative real numbers satisfying the equation $x+y+z=xyz$. Prove that $2(x^2+y^2+z^2)\geq3(x+y+z)$.
2023 Thailand October Camp, 1
Let $C$ be a finite set of chords in a circle such that each chord passes through the midpoint of some other chord. Prove that any two of these chords intersect inside the circle.
2019 Yasinsky Geometry Olympiad, p1
The sports ground has the shape of a rectangle $ABCD$, with the angle between the diagonals $AC$ and $BD$ is equal to $60^o$ and $AB >BC$. The trainer instructed Andriyka to go first $10$ times on the route $A-C-B-D-A$, and then $15$ more times along the route $A-D-A$. Andriyka performed the task, moving a total of $4.5$ km. What is the distance $AC$?
2020 Princeton University Math Competition, B1
Find all pairs of natural numbers $(n, k)$ with the following property:
Given a $k\times k$ array of cells, such that every cell contains one integer, there always exists a path from the left to the right edges such that the sum of the numbers on the path is a multiple of $n$.
Note: A path from the left to the right edge is a sequence of cells of the array $a_1, a_2, ... , a_m$ so that $a_1$ is a cell of the leftmost column, $a_m$ is the cell of the rightmost column, and $a_{i}$, $a_{i+1}$ share an edge for all $i = 1, 2, ... , m -1$.
2006 Purple Comet Problems, 9
Moving horizontally and vertically from point to point along the lines in the diagram below, how many routes are there from point $A$ to point $B$ which consist of six horizontal moves and six vertical moves?
[asy]
for(int i=0; i<=6;++i)
{
draw((i,i)--(6,i),black);
draw((i,i)--(i,0),black);
for(int a=i; a<=6;++a)
{
dot((a,i));
}
}
label("A",(0,0),W);
label("B",(6,6),N);
[/asy]
2007 China National Olympiad, 1
Let $O, I$ be the circumcenter and incenter of triangle $ABC$. The incircle of $\triangle ABC$ touches $BC, CA, AB$ at points $D, E, F$ repsectively. $FD$ meets $CA$ at $P$, $ED$ meets $AB$ at $Q$. $M$ and $N$ are midpoints of $PE$ and $QF$ respectively. Show that $OI \perp MN$.
2015 JHMT, 3
Consider a triangular pyramid $ABCD$ with equilateral base $ABC$ of side length $1$. $AD = BD =CD$ and $\angle ADB = \angle BDC = \angle ADC = 90^o$ . Find the volume of $ABCD$.
2024 USAMTS Problems, 4
During a lecture, each of $26$ mathematicians falls asleep exactly once, and stays asleep for a nonzero amount of time. Each mathematician is awake at the moment the lecture starts, and the moment the lecture finishes. Prove that there are either $6$ mathematicians such that no two are asleep at the same time, or $6$ mathematicians such that there is some point in time during which all $6$ are asleep.
1970 IMO Longlists, 34
In connection with a convex pentagon $ABCDE$ we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.
1988 AMC 12/AHSME, 12
Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the [b]sum[/b] of Jack's integer and Jill's integer?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ \text{each digit is equally likely} $
1995 Turkey MO (2nd round), 5
Let $t(A)$ denote the sum of elements of a nonempty set $A$ of integers, and define $t(\emptyset)=0$. Find a set $X$ of positive integers such that for every integers $k$ there is a unique ordered pair of disjoint subsets $(A_{k},B_{k})$ of $X$ such that $t(A_{k})-t(B_{k}) = k$.
2004 Bundeswettbewerb Mathematik, 2
Consider a triangle whose sidelengths $a$, $b$, $c$ are integers, and which has the property that one of its altitudes equals the sum of the two others.
Then, prove that $a^2+b^2+c^2$ is a perfect square.
2005 ISI B.Stat Entrance Exam, 3
Let $f$ be a function defined on $\{(i,j): i,j \in \mathbb{N}\}$ satisfying
(i) $f(i,i+1)=\frac{1}{3}$ for all $i$
(ii) $f(i,j)=f(i,k)+f(k,j)-2f(i,k)f(k,j)$ for all $k$ such that $i <k<j$.
Find the value of $f(1,100)$.
2023 Bulgaria JBMO TST, 1
Let $ABCDE$ be a cyclic pentagon such that $BC = DE$ and $AB$ is parallel to $DE$. Let $X, Y,$ and $Z$ be the midpoints of $BD, CE,$ and $AE$ respectively. Show that $AE$ is tangent to the circumcircle of the triangle $XYZ$.
Proposed by [i]Nikola Velov, Macedonia[/i]
2023 Greece National Olympiad, 4
A class consists of 26 students with two students sitting on each desk. Suddenly, the students decide to change seats, such that every two students that were previously sitting together are now apart. Find the maximum value of positive integer $N$ such that, regardless of the students' sitting positions, at the end there is a set $S$ consisting of $N$ students satisfying the following property: every two of them have never been sitting together.
2013 Tuymaada Olympiad, 6
Solve the equation $p^2-pq-q^3=1$ in prime numbers.
[i]A. Golovanov[/i]
2001 Iran MO (2nd round), 1
Find all polynomials $P$ with real coefficients such that $\forall{x\in\mathbb{R}}$ we have:
\[ P(2P(x))=2P(P(x))+2(P(x))^2. \]
2001 Estonia National Olympiad, 1
A convex $n$-gon has exactly three obtuse interior angles. Find all possible values of $n$.
1992 USAMO, 3
For a nonempty set $\, S \,$ of integers, let $\, \sigma(S) \,$ be the sum of the elements of $\, S$. Suppose that $\, A = \{a_1, a_2, \ldots, a_{11} \} \,$ is a set of positive integers with $\, a_1 < a_2 < \cdots < a_{11} \,$ and that, for each positive integer $\, n\leq 1500, \,$ there is a subset $\, S \,$ of $\, A \,$ for which $\, \sigma(S) = n$. What is the smallest possible value of $\, a_{10}$?
2009 IMO Shortlist, 5
Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\]
[i]Proposed by Igor Voronovich, Belarus[/i]
2014 Argentina Cono Sur TST, 5
In an acute triangle $ABC$, let $D$ be a point in $BC$ such that $AD$ is the angle bisector of $\angle{BAC}$. Let $E \neq B$ be the point of intersection of the circumcircle of triangle $ABD$ with the line perpendicular to $AD$ drawn through $B$. Let $O$ be the circumcenter of triangle $ABC$. Prove that $E$, $O$, and $A$ are collinear.
2000 Swedish Mathematical Competition, 2
$p(x)$ is a polynomial such that $p(y^2+1) = 6y^4 - y^2 + 5$. Find $p(y^2-1)$.
2004 China Team Selection Test, 1
Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}\equal{}\angle{PBC}\equal{}\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}\plus{} S_{B_1CA}\plus{} S_{C_1AB} \geq \lambda S_{ABC}$.
TNO 2024 Junior, 4
Tomás is an avid domino player. One day, while playing with the tiles, he realized he could arrange all the tiles in a single row following the rules, meaning that the number on the right side of each tile matches the number on the left side of the next tile. If the number on the left side of the first tile is 5, what is the number on the right side of the last tile?