A point is picked uniformly at random inside of a square. Four segments are then drawn in connecting the point to each of the vertices of the square, cutting the square into four triangles. What is the probability that at least two of the resulting triangles are obtuse?

Let $MATH$ be a trapezoid with $MA=AT=TH=5$ and $MH=11$. Point $S$ is the orthocenter of $\triangle ATH$. Compute the area of quadrilateral $MASH$.

Assume the $n$ sets $A_1, A_2..., A_n$ are a partition of the set $A=\{1,2,...,29\}$, and the sum of any elements in $A_i$ , $(i=1,2,...,n)$ is not equal to $30$. Find the smallest possible value of $n$.

Let $ABC$ be an acute-angled triangle, and let $D, E$ and $K$ be the midpoints of its sides $AB, AC$ and $BC$ respectively. Let $O$ be the circumcentre of triangle $ABC$, and let $M$ be the foot of the perpendicular from $A$ on the line $BC$. From the midpoint $P$ of $OM$ we draw a line parallel to $AM$, which meets the lines $DE$ and $OA$ at the points $T$ and $Z$ respectively. Prove that: (a) the triangle $DZE$ is isosceles (b) the area of the triangle $DZE$ is given by the formula \[E_{DZE}=\frac{BC\cdot OK}{8}\]

Given is a $K_{2024}$ in which every edge has weight $1$ or $2$. If every cycle has even total weight, find the minimal value of the sum of all weights in the graph.

Let $ABC$ be a triangle with $AB = 16$, $AC = 10$, $BC = 18$. Let $D$ be a point on $AB$ such that $4AD = AB$ and let E be the foot of the angle bisector from $B$ onto $AC$. Let $P$ be the intersection of $CD$ and $BE$. Find the area of the quadrilateral $ADPE$.

Let $S \subset Z^+$ be a set of positive integers with the following property: for any $a, b \in S$, if $a \ne b$ then $a + b$ is a perfect square. Given that $2009 \in S$ and $2087 \in S$, what is the maximum number of elements in $S$?

Let $\omega$ be the circumcircle of triangle $ABC$. Tangent lines to $\omega$ at points $A$ and $C$ intersect at $K$. Line $BK$ intersects $\omega$ for the second time at $M$. On the line $BC$ point $N$ is chosen such that $\angle BAN = 90$. Line $MN$ intersects $\omega$ for the second time at $D$. Prove that $BD=BC$ [i]P. Chernikova[/i]

Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$. [i]Calvin Deng.[/i]

Point $P$ is taken on the extension of side $AB$ of an equilateral triangle $ABC$ so that $A$ is between $B$ and $P$. Denote by $a$ the side length of triangle $ABC$, by $r_1$ the inradius of triangle $PAC$, and by $r_2$ the exradius of triangle $PBC$ opposite $P$. Find the sum $r_1+r_2$ as a function in $a$.

How many diagonals does $11$-sided convex polygon have?

Let $I\subseteq \mathbb{R}$ be an interval and $f:I\rightarrow\mathbb{R}$ a function such that: \[|f(x)-f(y)|\le |x-y|,\quad\text{for all}\ x,y\in I. \] Show that $f$ is monotonic on $I$ if and only if, for any $x,y\in I$, either $f(x)\le f\left(\frac{x+y}{2}\right)\le f(y)$ or $f(y)\le f\left(\frac{x+y}{2}\right)\le f(x)$.

Positive integers $a,b,c$ satisfy that $a+b=b(a-c)$ and c+1 is a square of a prime. Prove that $a+b$ or $ab$ is a square.

Let $n$ a positive integer. We call a pair $(\pi ,C)$ composed by a permutation $\pi$$:$ {$1,2,...n$}$\rightarrow${$1,2,...,n$} and a binary function $C:$ {$1,2,...,n$}$\rightarrow${$0,1$} "revengeful" if it satisfies the two following conditions: $1)$For every $i$ $\in$ {$1,2,...,n$}, there exist $j$ $\in$ $S_{i}=${$i, \pi(i),\pi(\pi(i)),...$} such that $C(j)=1$. $2)$ If $C(k)=1$, then $k$ is one of the $v_{2}(|S_{k}|)+1$ highest elements of $S_{k}$, where $v_{2}(t)$ is the highest nonnegative integer such that $2^{v_{2}(t)}$ divides $t$, for every positive integer $t$. Let $V$ the number of revengeful pairs and $P$ the number of partitions of $n$ with all parts powers of $2$. Determine $\frac{V}{P}$.

Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all $2$-digit primes. Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of you two uniform numbers is today's date. What number does Caitlin wear? $\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$

There are $12$ monsters in a plane. Each monster is capable of spraying fire in a $30$-degree cone. Prove that monsters can destroy the plane.

Find all $x$ and $y$ which are rational multiples of $\pi$ with $0<x<y<\frac{\pi}{2}$ and $\tan x+\tan y =2$.

$p$ passengers get on a train with $n$ wagons. Find the probability of being at least one passenger at each wagon.

A triangle $OAB$ with $\angle A=90^{\circ}$ lies inside another triangle with vertex $O$. The altitude of $OAB$ from $A$ until it meets the side of angle $O$ at $M$. The distances from $M$ and $B$ to the second side of angle $O$ are $2$ and $1$ respectively. Find the length of $OA$.

Let $\displaystyle{A}$ and $\displaystyle{B}$ be real symmetric matrixes with all eigenvalues strictly greater than $\displaystyle{1}$. Let $\displaystyle{\lambda }$ be a real eigenvalue of matrix $\displaystyle{{\rm A}{\rm B}}$. Prove that $\displaystyle{\left| \lambda \right| > 1}$. [i]Proposed by Pavel Kozhevnikov, MIPT, Moscow.[/i]

We call a permutation of the numbers $1$, $2$, $3$, $\dots$ , $n$ 'kawaii' if there is exactly one number that is greater than its position. For example: $1$, $4$, $3$, $2$ is a kawaii permutation (when $n=4$) because only the number $4$ is greater than its position $2$. How many kawaii permutations are there if $n=14$?

There are 1000 towns $A_{1},A_{2},\ldots ,A_{1000}$ with airports in a country and some of them are connected via flights. It's known that the $i$-th town is connected with $d_{i}$ other towns where $d_{1}\leq d_{2}\leq \ldots \leq d_{1000}$ and $d_{j}\geq j+1$ for every $j=1,2,\ldots 999-d_{999}$. Prove that if the airport of any town $A_{k}$ is closed, then we'd still be able to get from any town $A_{i}$ to any $A_{j}$ for $i,j\neq k$ (possibly by more than one flight).

Georg is putting his $250$ stamps in a new album. On the first page he places one stamp and then on every page just as many or twice as many stamps as on the preceding page. In this way he ends up precisely having put all $250$ stamps in the album. How few pages are sufficient for him?

Three circles with centers $V_0$, $V_1$, $V_2$ and radii $33$, $30$, $25$ respectively and mutually externally tangent: $P_i$ is the tangency point between circles $V_{i+1}$ and $V_{i+2}$, where indeces are taken modulo $3$. For $i=0,1,2$, line $P_{i+1}P_{i+2}$ intersects circle $V_{i+1}$ at $P_{i+2}$ and $Q_i$, and the same line intersects circle $V_{i+2}$ at $P_{i+1}$ and $R_i$. If $Q_0R_1$ intersects $Q_2R_0$ at $X$, then the distance from $X$ to line $R_1Q_2$ can be expressed as $\tfrac{a\sqrt b}c$, where the integer $b$ is not divisible by the square of any prime, and positive integers $a$ and $c$ are relatively prime. Find the value of $b+c$.

Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use? $\textbf{(A) }48\qquad\textbf{(B) }87\qquad\textbf{(C) }91\qquad\textbf{(D) }96\qquad \textbf{(E) }120$