Found problems: 85335
1992 China National Olympiad, 2
Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)
2018 Serbia Team Selection Test, 2
Let $n$ be a fixed positive integer and let $x_1,\ldots,x_n$ be positive real numbers. Prove that
$$x_1\left(1-x_1^2\right)+x_2\left(1-(x_1+x_2)^2\right)+\cdots+x_n\left(1-(x_1+...+x_n)^2\right)<\frac{2}{3}.$$
2001 239 Open Mathematical Olympiad, 2
For any positive numbers $ a_1 , a_2 , \dots, a_n $ prove the inequality $$\!
\left(\!1\!+\!\frac{1}{a_1(1+a_1)} \!\right)\!
\left(\!1\!+\!\frac{1}{a_2(1+a_2)} \! \right) \! \dots \!
\left(\!1\!+\!\frac{1}{a_n(1+a_n)} \! \right) \geq
\left(\!1\!+\!\frac{1}{p(1+p)} \! \right)^{\! n} \! ,$$
where $p=\sqrt[n]{a_1 a_2 \dots a_n}$.
2004 Thailand Mathematical Olympiad, 2
Let $f : Q \to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$. If $f(2004) = 2547$, find $f(2547)$.
2011 Sharygin Geometry Olympiad, 8
Using only the ruler, divide the side of a square table into $n$ equal parts.
All lines drawn must lie on the surface of the table.
2018 Abels Math Contest (Norwegian MO) Final, 2
The circumcentre of a triangle $ABC$ is called $O$. The points $A',B'$ and $C'$ are the reflections of $O$ in $BC, CA$, and $AB$, respectively. Show that the three lines $AA' , BB'$, and $CC'$ meet in a common point.
2022 BMT, 5
Theo and Wendy are commuting to school from their houses. Theo travels at $x$ miles per hour, while Wendy travels at $x + 5$ miles per hour. The school is $4$ miles from Theo’s house and $10$ miles from Wendy’s house. If Wendy’s commute takes double the amount of time that Theo’s commute takes, how many minutes does it take Wendy to get to school?
2014 Taiwan TST Round 2, 2
Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.
2013 China Western Mathematical Olympiad, 6
Let $PA, PB$ be tangents to a circle centered at $O$, and $C$ a point on the minor arc $AB$. The perpendicular from $C$ to $PC$ intersects internal angle bisectors of $AOC,BOC$ at $D,E$. Show that $CD=CE$
1997 Italy TST, 3
Determine all triples $(x,y, p)$ with $x$, $y$ positive integers and $p$ a prime number verifying the equation $p^x -y^p = 1$.
1954 Czech and Slovak Olympiad III A, 4
Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$.
(a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s.
(b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.
2009 Philippine MO, 2
[b](a)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n + 1 = x^2$.
[b](b)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n = x^2 + 1$.
2023 Indonesia TST, A
Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation:
\[Q(a+b) = \frac{P(a) - P(b)}{a - b}\]
$\forall a, b \in \mathbb{Z}^+$ and $a>b$
2006 Lithuania National Olympiad, 4
Find the maximal cardinality $|S|$ of the subset $S \subset A=\{1, 2, 3, \dots, 9\}$ given that no two sums $a+b | a, b \in S, a \neq b$ are equal.
2017 Dutch IMO TST, 1
Let $a, b,c$ be distinct positive integers, and suppose that $p = ab+bc+ca$ is a prime number.
$(a)$ Show that $a^2,b^,c^2$ give distinct remainders after division by $p$.
(b) Show that $a^3,b^3,c^3$ give distinct remainders after division by $p$.
LMT Team Rounds 2021+, A10
Pieck the Frog hops on Pascal's Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
[i]Proposed by Steven Yu[/i]
LMT Guts Rounds, 12
$a,b,c,d,e$ are equal to $1,2,3,4,5$ in some order, such that no two of $a,b,c,d,e$ are equal to the same integer. Given that $b \leq d, c \geq a,a \leq e,b \geq e,$ and that $d\neq5,$ determine the value of $a^b+c^d+e.$
1987 IMO Longlists, 17
Consider the number $\alpha$ obtained by writing one after another the decimal representations of $1, 1987, 1987^2, \dots$ to the right the decimal point. Show that $\alpha$ is irrational.
2025 District Olympiad, P4
Let $ABCDEF$ be a convex hexagon with $\angle A = \angle C=\angle E$ and $\angle B = \angle D=\angle F$.
[list=a]
[*] Prove that there is a unique point $P$ which is equidistant from sides $AB,CD$ and $EF$.
[*] If $G_1$ and $G_2$ are the centers of mass of $\triangle ACE$ and $\triangle BDF$, show that $\angle G_1PG_2=60^{\circ}$.
2021 Turkey MO (2nd round), 6
In a school, there are 2021 students, each having exactly $k$ friends. There aren't three students such that all three are friends with each other. What is the maximum possible value of $k$?
1980 All Soviet Union Mathematical Olympiad, 292
Find real solutions of the system :
$$\begin{cases} \sin x + 2 \sin (x+y+z) = 0 \\
\sin y + 3 \sin (x+y+z) = 0\\
\sin z + 4 \sin (x+y+z) = 0\end{cases}$$
2004 All-Russian Olympiad Regional Round, 8.3
In an acute triangle, the distance from the midpoint of any side to the opposite vertex is equal to the sum of the distances from it to sides of the triangle. Prove that this triangle is equilateral.
2001 South africa National Olympiad, 3
For a certain real number $x$, the differences between $x^{1919}$, $x^{1960}$ and $x^{2001}$ are all integers. Prove that $x$ is an integer.
1997 Tournament Of Towns, (555) 5
Each face of a cube is of the same size as each square of a chessboard. The cube is coloured black and white, placed on one of the squares of the chessboard and rolled so that each square of the chessboard is visited exactly once. Can this be done in such a way that the colour of the visited square and the colour of the bottom face of the cube are always the same?
(A Shapovalov)
1974 Kurschak Competition, 2
$S_n$ is a square side $\frac{1}{n}$. Find the smallest $k$ such that the squares $S_1, S_2,S_3, ...$ can be put into a square side $k$ without overlapping.