Found problems: 85335
2024 Harvard-MIT Mathematics Tournament, 2
Jerry and Neil have a $3$-sided die that rolls the numbers $1,2,$ and $3,$ each with probability $\tfrac{1}{3}.$ Jerry rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry's. Compute the probability that Neil's final number is $3.$
2007 Junior Balkan Team Selection Tests - Romania, 1
Let us consider $a,b$ two integers. Prove that there exists and it is unique a pair of integers $(x,y)$ such that: \[(x+2y-a)^{2}+(2x-y-b)^{2}\leq 1.\]
2011 AMC 8, 19
How many rectangles are in this figure?
[asy]
pair A,B,C,D,E,F,G,H,I,J,K,L;
A=(0,0);
B=(20,0);
C=(20,20);
D=(0,20);
draw(A--B--C--D--cycle);
E=(-10,-5);
F=(13,-5);
G=(13,5);
H=(-10,5);
draw(E--F--G--H--cycle);
I=(10,-20);
J=(18,-20);
K=(18,13);
L=(10,13);
draw(I--J--K--L--cycle);[/asy]
$ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 $
2006 All-Russian Olympiad, 5
Two sequences of positive reals, $ \left(x_n\right)$ and $ \left(y_n\right)$, satisfy the relations $ x_{n \plus{} 2} \equal{} x_n \plus{} x_{n \plus{} 1}^2$ and $ y_{n \plus{} 2} \equal{} y_n^2 \plus{} y_{n \plus{} 1}$ for all natural numbers $ n$. Prove that, if the numbers $ x_1$, $ x_2$, $ y_1$, $ y_2$ are all greater than $ 1$, then there exists a natural number $ k$ such that $ x_k > y_k$.
Kyiv City MO Juniors 2003+ geometry, 2012.8.3
On the circle $\gamma$ the points $A$ and $B$ are selected. The circle $\omega$ touches the segment $AB$ at the point $K$ and intersects the circle $\gamma$ at the points $M$ and $N$. The points lie on the circle $\gamma$ in the following order: $A, \, \, M, \, \, N, \, \, B$. Prove that $\angle AMK = \angle KNB$.
(Yuri Biletsky)
2013 Moldova Team Selection Test, 4
Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as:
$x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$.
$a)$ Prove that the sequence is converging.
$b)$ Find $\lim_{n\rightarrow \infty}{x_n}$.
2021 Princeton University Math Competition, A4 / B6
There are n lilypads in a row labeled $1, 2, \dots, n$ from left to right. Fareniss the Frog picks a lilypad at random to start on, and every second she jumps to an adjacent lilypad; if there are two such lilypads, she is twice as likely to jump to the right as to the left. After some finite number of seconds, there exists two lilypads $A$ and $B$ such that Fareniss is more than $1000$ times as likely to be on $A$ as she is to be on $B$. What is the minimal number of lilypads $n$ such that this situation must occur?
2025 JBMO TST - Turkey, 1
Let $ABCD$ be a cyclic quadrilateral and let the intersection point of lines $AB$ and $CD$ be $E$. Let the points $K$ and $L$ be arbitrary points on sides $CD$ and $AB$ respectively, which satisfy the conditions
$$\angle KAD = \angle KBC \quad \text{and} \quad \angle LDA = \angle LCB.$$
Prove that $EK = EL$.
1995 Chile National Olympiad, 5
A tamer wants to line up five lions and four tigers. We know that a tiger cannot go after another. How many ways can the beasts be distributed? The tamer cannot distinguish two animals of the same species.
2015 All-Russian Olympiad, 6
A field has a shape of checkboard $\text{41x41}$ square. A tank concealed in one of the cells of the field. By one shot, a fighter airplane fires one of the cells. If a shot hits the tank, then the tank moves to a neighboring cell of the field, otherwise it stays in its cell (the cells are neighbours if they share a side). A pilot has no information about the tank , one needs to hit it twice. Find the least number of shots sufficient to destroy the tank for sure. [i](S.Berlov,A.Magazinov)[/i]
1995 AIME Problems, 5
For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i=\sqrt{-1}.$ Find $b.$
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 $n$,
$$(a+b)^n-a^n-b^n \ge \frac{2^n-2}{2^{n-2}} \cdot ab(a+b)^{n-2}.$$