An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants.

Let $ABC$ be a triangle with $\angle ABC = 60^{\circ}$. Find the angles of the triangle if $\angle BHI = 60^{\circ}$, where $H$ and $I$ are the orthocenter and incenter of $ABC$

Let $I$ be the incenter of the acute triangle $\triangle ABC$, and $BI$, $CI$ intersect the altitude of $\triangle ABC$ through $A$ at $U$, $V$, respectively. The circle with $AI$ as a diameter intersects $\odot(ABC)$ again at $T$, and $\odot(TUV)$ intersects the segment $BC$ and $\odot(ABC)$ at $P$, $Q$, respectively. Let $R$ be another intersection of $PQ$ and $\odot(ABC)$. Show that $AR\parallel BC$.

Let $ABCD$ be a square with the side of $20$ units. Amir divides this square into $400$ unit squares. Reza then picks $4$ of the vertices of these unit squares. These vertices lie inside the square $ABCD$ and define a rectangle with the sides parallel to the sides of the square $ABCD.$ There are exactly $24$ unit squares which have at least one point in common with the sides of this rectangle. Find all possible values for the area of a rectangle with these properties. [hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]

Prove that any identity that holds for every finite $ n$-distributive lattice also holds for the lattice of all convex subsets of the $ (n\minus{}1)$-dimensional Euclidean space. (For convex subsets, the lattice operations are the set-theoretic intersection and the convex hull of the set-theoretic union. We call a lattice $ n$-$ \textit{distributive}$ if \[ x \wedge (\bigvee_{i\equal{}0}^n y_i)\equal{}\bigvee_{j\equal{}0}^n(x \wedge (\bigvee_{0\leq i \leq n, \;i \not\equal{} j\ }y_i))\] holds for all elements of the lattice.) [i]A. Huhn[/i]

In some galaxy there are $1000000$ planets and on each of them there are at least $101$ portals. Each portal allows to teleport between some two planets, no two planets are connected by more than one portal. It is known that starting from any planet using portals you can get to any other planet, while it is impossible to return to that planet using once at most 5 different portals. Prove that starting from any planet you can get to any other planet within a year, using at most one portal daily (a year consists of 365 days). [i]M. Zorka[/i]

Let $ABC$ be an acute triangle with $AB<AC$ and incenter $I$. Let $D$ be the projection of $I$ onto $BC$. Let $H$ be the orthocenter of $ABC$ and suppose that $\angle IDH=\angle CBA-\angle ACB$. Prove that $AH=2ID$.

All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of$ k$ for which the same $k$-digit number can be found in at least two different places in this string?

The best parts of grandma’s $30$ cm $ \times 30$ cm square shaped pie are the edges. For this reason grandma’s three grandchildren would like to split the pie between each other so that everyone gets the same amount (of the area) of the pie, but also of the edges. Can they cut the pie into three connected pieces like that?

Determine, whether exists function $f$, which assigns each integer $k$, nonnegative integer $f(k)$ and meets the conditions: $f(0) > 0$, for each integer $k$ minimal number of the form $f(k - l) + f(l)$, where $l \in \mathbb{Z}$, equals $f(k)$.

Find all integers $a$ such that there are infinitely many positive integers $n$ such that $n$ divides $\phi(n)!+a$.

There is a sequence of integers $ a_1, a_2, \ldots, a_{2n+1} $ with the following property: after eliminating any term, the remaining ones can be divided into two groups of $ n $ terms such that the sum of the terms in the first group is equal to the sum words in the second. Prove that all terms of the sequence are equal.

There are three squares in the picture. Find the sum of angles $ADC$ and $BDC$. [img]https://cdn.artofproblemsolving.com/attachments/c/9/885a6c6253fca17e24528f8ba8a5d31a18c845.png[/img]

Find \[\lim_{n\to\infty}\int _0^{2n} e^{-2x}\left|x-2\lfloor\frac{x+1}{2}\rfloor\right|\ dx.\] [i]1985 Tohoku University entrance exam/Mathematics, Physics, Chemistry, Biology[/i]

Let $ABC$ be a triangle with side lengths $AB=13$, $BC=15$, $CA=14$. Let $\ell$ be the line passing through $A$ parallel to $BC$. Define $H$ as the orthocenter of $\triangle ABC$, and extend $BH$ to intersect $AC$ at $E$ and $\ell$ at $G$. Similarly, extend $CH$ to intersect $AB$ at $F$ and $\ell$ at $D$. Let $M$ be the midpoint of $BC$, and let $AM$ intersect the circumcircle of $AEF$ again at $P$. The ratio $\frac{PD}{PG}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Individual #15[/i]

Let $a$ and $b$ be positive integers such that $\text{gcd}(a, b) + \text{lcm}(a, b)$ is a multiple of $a+1$. If $b \le a$, show that $b$ is a perfect square.

A unit circle is centered at $(0, 0)$ on the $(x, y)$ plane. A regular hexagon passing through $(1, 0)$ is inscribed in the circle. Two points are randomly selected from the interior of the circle and horizontal lines are drawn through them, dividing the hexagon into at most three pieces. The probability that each piece contains exactly two of the hexagon's original vertices can be written as \[ \frac{2\left(\frac{m\pi}{n}+\frac{\sqrt{p}}{q}\right)^2}{\pi^2} \] for positive integers $m$, $n$, $p$, and $q$ such that $m$ and $n$ are relatively prime and $p$ is squarefree. Find $m+n+p+q$.

Let $ABC$ be a triangle with $\angle ABC=130$. Point $D$ on side $AC$ is the foot of the perpendicular from $B$. Points $E$ and $F$ are on sides $(AB)$ and $(BC)$ such that $DE=DF$ and $AEFC$ is cyclic. Find $\angle EDF$.

The positive real numbers $a,b,c$ are such that $21ab+2bc+8ca\leq 12$. Find the smallest value of $\frac{1}{a}+\frac{2}{b}+\frac{3}{c}$.

Prove that the number of ways of choosing $6$ among the first $49$ positive integers, at least two of which are consecutive, is equal to $\binom{49}6-\binom{44}6$.

How many positive integers have exactly three proper divisors, each of which is less than 50?

Let $a,b,c,d>0$ . Prove that $a^4+b^4+c^4+d^4 \geq 4abcd+4(a-b)^2 \sqrt{abcd}$

Find all continuous functions $f\colon \mathbb{R}_{>0}\to \mathbb{R}_{\geq 1}$ for which the following equation holds for all positive reals $x$, $y$: \[f\left(\frac{f(x)}{y}\right)-f\left(\frac{f(y)}{x}\right)=xy\left(f(x+1)-f(y+1)\right)\]

[b]p1.[/b] Find $$2^{\left(0^{\left(2^3\right)}\right)}$$ [b]p2.[/b] Amy likes to spin pencils. She has an $n\%$ probability of dropping the $n$th pencil. If she makes $100$ attempts, the expected number of pencils Amy will drop is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p3.[/b] Determine the units digit of $3 + 3^2 + 3^3 + 3^4 +....+ 3^{2022} + 3^{2023}$. [b]p4.[/b] Cyclic quadrilateral $ABCD$ is inscribed in circle $\omega$ with center $O$ and radius $20$. Let the intersection of $AC$ and $BD$ be $E$, and let the inradius of $\vartriangle AEB$ and $\vartriangle CED$ both be equal to $7$. Find $AE^2 - BE^2$. [b]p5.[/b] An isosceles right triangle is inscribed in a circle which is inscribed in an isosceles right triangle that is inscribed in another circle. This larger circle is inscribed in another isosceles right triangle. If the ratio of the area of the largest triangle to the area of the smallest triangle can be expressed as $a+b\sqrt{c}$, such that $a, b$ and $c$ are positive integers and no square divides $c$ except $1$, find $a + b + c$. [b]p6.[/b] Jonny has three days to solve as many ISL problems as he can. If the amount of problems he solves is equal to the maximum possible value of $gcd \left(f(x), f(x+1) \right)$ for $f(x) = x^3 +2$ over all positive integer values of $x$, then find the amount of problems Jonny solves. [b]p7.[/b] Three points $X$, $Y$, and $Z$ are randomly placed on the sides of a square such that $X$ and $Y$ are always on the same side of the square. The probability that non-degenerate triangle $\vartriangle XYZ$ contains the center of the square can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p8.[/b] Compute the largest integer less than $(\sqrt7 +\sqrt3)^6$. [b]p9.[/b] Find the minimum value of the expression $\frac{(x+y)^2}{x-y}$ given $x > y > 0$ are real numbers and $xy = 2209$. [b]p10.[/b] Find the number of nonnegative integers $n \le 6561$ such that the sum of the digits of $n$ in base $9$ is exactly $4$ greater than the sum of the digits of $n$ in base $3$. [b]p11.[/b] Estimation (Tiebreaker) Estimate the product of the number of people who took the December contest, the sum of all scores in the November contest, and the number of incorrect responses for Problem $1$ and Problem $2$ on the October Contest. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $AH_A$ and $BH_B$ be the altitudes of a triangle $ABC$. The line $H_AH_B$ meets the circumcircle of $ABC$ at points $P$ and $Q$. Let $A'$ be the reflection of $A$ about $BC$, and $B'$ be the reflection of $B$ about $CA$. Prove that $A',B', P,Q$ are concyclic.