Found problems: 85335
2003 Bundeswettbewerb Mathematik, 3
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$.
Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$.
[i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$.
Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.
2012 India National Olympiad, 6
Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \ne 0$, $f(1) = 0$ and
$(i) f(xy) + f(x)f(y) = f(x) + f(y)$
$(ii)\left(f(x-y) - f(0)\right ) f(x)f(y) = 0 $
for all $x,y \in \mathbb{Z}$, simultaneously.
$(a)$ Find the set of all possible values of the function $f$.
$(b)$ If $f(10) \ne 0$ and $f(2) = 0$, find the set of all integers $n$ such that $f(n) \ne 0$.
2012 Grigore Moisil Intercounty, 3
Find for which natural numbers $ n\ge 2 $ there exist two real matrices $ A,B $ of order $ n $ that satisy the property:
$$ (AB)^2=0\neq (BA)^2 $$
[i]Dan Bărbosu[/i]
2012 Princeton University Math Competition, A6
Let an be a sequence such that $a_0 = 0$ and:
$a_{3n+1} = a_{3n} + 1 = a_n + 1$
$a_{3n+2} = a_{3n} + 2 = a_n + 2$
for all natural numbers $n$. How many $n$ less than $2012$ have the property that $a_n = 7$?
2018 CMIMC Team, 9-1/9-2
Andy rolls a fair 4-sided dice, numbered 1 to 4, until he rolls a number that is less than his last roll. If the expected number of times that Andy will roll the dice can be expressed as a reduced fraction $\frac{p}{q}$, find $p + q$.
Let $T = TNYWR$. The solutions in $z$ to the equation \[\left(z + \frac Tz\right)^2 = 1\] form the vertices of a quadrilateral in the complex plane. Compute the area of this quadrilateral.
2016 ASDAN Math Tournament, 7
Heesu, Xingyou, and Bill are in a class with $9$ other children. The teacher randomly arranges the children in a circle for story time. However, Heesu, Xingyou, and Bill want to sit near each other. Compute the probability that all $3$ children are seated within a consecutive group of $5$ seats.
2021 European Mathematical Cup, 2
Let $ABC$ be a triangle and let $D, E$ and $F$ be the midpoints of sides $BC, CA$ and $AB$, respectively.
Let $X\ne A$ be the intersection of $AD$ with the circumcircle of $ABC$. Let $\Omega$ be the circle through $D$ and $X$,
tangent to the circumcircle of $ABC$. Let $Y$ and $Z$ be the intersections of the tangent to $\Omega$ at $D$ with the
perpendicular bisectors of segments $DE$ and $DF$, respectively. Let $P$ be the intersection of $YE$ and $ZF$ and
let $G$ be the centroid of $ABC$. Show that the tangents at $B$ and $C$ to the circumcircle of $ABC$ and the line $PG$ are concurrent.
2010 LMT, 15
Al is bored of Rock Paper Scissors, and wants to invent a new game: $Z-Y-X-W-V.$ Two players, each choose to play either $Z, Y, X, W,$ or $V.$ If they play the same thing, the result is a tie. However, Al must come up with a ’pecking order’, that is, he must decide which plays beat which. For each of the $10$ pairs of distinct plays that the two players can make, Al randomly decides a winner. For example, he could decide that $W$ beats $Y$ and that $Z$ beats $X,$ etc. What is the probability that after Al makes all of these $10$ choices, the game is balanced, that is, playing each letter results in an equal probability of winning?
1965 Spain Mathematical Olympiad, 6
We have an empty equilateral triangle with length of a side $l$. We put the triangle, horizontally, over a sphere of radius $r$. Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $r$)?
2018 Indonesia MO, 3
Alzim and Badril are playing a game on a hexagonal lattice grid with 37 points (4 points a side), all of them uncolored. On his turn, Alzim colors one uncolored point with the color red, and Badril colors [b]two[/b] uncolored points with the color blue. The game ends either when there is an equilateral triangle whose vertices are all red, or all points are colored. If the former happens, then Alzim wins, otherwise Badril wins. If Alzim starts the game, does Alzim have a strategy to guarantee victory?
1978 Vietnam National Olympiad, 4
Find three rational numbers $\frac{a}{d}, \frac{b}{d}, \frac{c}{d}$ in their lowest terms such that they form an arithmetic progression and $\frac{b}{a} =\frac{a + 1}{d + 1}, \frac{c}{b} = \frac{b + 1}{d + 1}$.
2023 Math Prize for Girls Problems, 14
Five points are chosen uniformly and independently at random on the surface of a sphere. Next, 2 of these 5 points are randomly picked, with every pair equally likely. What is the probability that the 2 points are separated by the plane containing the other 3 points?
2008 Princeton University Math Competition, A6/B8
$xxxx$
$xx$
$x$
$x$
In how many ways can you fill in the $x$s with the numbers $1-8$ so that for each $x$, the numbers below and to the right are higher.
2021 Polish MO Finals, 4
Prove that for every pair of positive real numbers $a, b$ and for every positive integer $n$,
$$(a+b)^n-a^n-b^n \ge \frac{2^n-2}{2^{n-2}} \cdot ab(a+b)^{n-2}.$$
2017 Tuymaada Olympiad, 4
There are 25 masks of different colours. k sages play the following game. They are shown all the masks. Then the sages agree on their strategy. After that the masks are put on them so that each sage sees the masks on the others but can not see who wears each mask and does not see his own mask. No communication is allowed. Then each of them simultaneously names one colour trying to guess the colour of his mask. Find the minimum k for which the sages can agree so that at least one of them surely guesses the colour of his mask.
( S. Berlov )
1995 All-Russian Olympiad, 3
Does there exist a sequence of natural numbers in which every natural number occurs exactly once, such that for each $k = 1, 2, 3, \dots$ the sum of the first $k$ terms of the sequence is divisible by $k$?
[i]A. Shapovalov[/i]
1999 Harvard-MIT Mathematics Tournament, 10
If $5$ points are placed in the plane at lattice points (i.e. points $(x,y)$ where $x $and $y$ are both integers) such that no three are collinear, then there are $10$ triangles whose vertices are among these points. What is the minimum possible number of these triangles that have area greater than $1/2$?
2012 Morocco TST, 2
Find all positive integer $n$ and prime number $p$ such that $p^2+7^n$ is a perfect square
2016 Purple Comet Problems, 1
Two integers have a sum of 2016 and a difference of 500. Find the larger of the two integers.
2025 Harvard-MIT Mathematics Tournament, 7
There exists a unique triple $(a,b,c)$ of positive real numbers that satisfies the equations $$2(a^2+1)=3(b^2+1)=4(c^2+1) \quad \text{and} \quad ab+bc+ca=1.$$ Compute $a+b+c.$
1992 China National Olympiad, 1
A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$, $BD$ of $ABCD$ meet at $P$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ meet at $P$ and $Q$ ($O,P,Q$ are pairwise distinct). Show that $\angle OQP=90^{\circ}$.
2002 Moldova Team Selection Test, 4
The sequence Pn (x), n ∈ N of polynomials is defined as follows:
P0 (x) = x, P1 (x) = 4x³ + 3x
Pn+1 (x) = (4x² + 2)Pn (x) − Pn−1 (x), for all n ≥ 1
For every positive integer m, we consider the set A(m) = { Pn (m) | n ∈ N }. Show that the sets A(m) and A(m+4) have no common elements.
2023 AIME, 7
Each vertex of a regular dodecagon (12-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.
2015 Saudi Arabia JBMO TST, 4
Let $a,b$ and $c$ be positive numbers with $a^2+b^2+c^2=3$. Prove that $a+b+c\ge 3\sqrt[5]{abc}$.
2023 CMIMC Geometry, 2
Two circles have radius $2$ and $3$, and the distance between their centers is $10$. Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$.
(A [i]common external tangent[/i] is a tangent line to two circles such that the circles are on the same side of the line, while a [i]common internal tangent[/i] is a tangent line to two circles such that the circles are on opposite sides of the line).
[i]Proposed by Connor Gordon)[/i]