Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality.

Let $ABC$ be a triangle, and let $E$ and $F$ be two arbitrary points on the sides $AB$ and $AC$, respectively. The circumcircle of triangle $AEF$ meets the circumcircle of triangle $ABC$ again at point $M$. Let $D$ be the reflection of point $M$ across the line $EF$ and let $O$ be the circumcenter of triangle $ABC$. Prove that $D$ is on $BC$ if and only if $O$ belongs to the circumcircle of triangle $AEF$.

Let $ABC$ be a right-angle triangle with $\angle B=90^{\circ}$. Let $D$ be a point on $AC$ such that the inradii of the triangles $ABD$ and $CBD$ are equal. If this common value is $r^{\prime}$ and if $r$ is the inradius of triangle $ABC$, prove that \[ \cfrac{1}{r'}=\cfrac{1}{r}+\cfrac{1}{BD}. \]

The remainder when the product $1492\cdot 1776\cdot 1812\cdot 1996$ is divided by $5$ is $\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4$

Let $a,b,c,d$ be positive real numbers such that $a^2+b^2+c^2+d^2=4$. Prove that exist two of $a,b,c,d$ with sum less or equal to $2$.

Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$. Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$. Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$. (Hungary)

The circle contains a closed $100$-part broken line, such that no three segments pass through one point. All its corners are obtuse, and their sum in degrees is divided by $720$. Prove that this broken line has an odd number of self-intersection points.

Let $\triangle{ABC}$ has circumcircle $\Gamma$, drop the perpendicular line from $A$ to $BC$ and meet $\Gamma$ at point $D$, similarly, altitude from $B$ to $AC$ meets $\Gamma$ at $E$. Prove that if $AB=DE, \angle{ACB}=60^{\circ}$ (sorry it is from my memory I can't remember the exact problem, but it means the same)

Determine all real constants $t$ such that whenever $a$, $b$ and $c$ are the lengths of sides of a triangle, then so are $a^2+bct$, $b^2+cat$, $c^2+abt$.

[u]Round 9[/u] [b]p25. [/b]Define a sequence $\{a_n\}_{n \ge 1}$ of positive real numbers by $a_1 = 2$ and $a^2_n -2a_n +5 =4a_{n-1}$ for $n \ge 2$. Suppose $k$ is a positive real number such that $a_n <k$ for all positive integers $n$. Find the minimum possible value of $k$. [b]p26.[/b] Let $\vartriangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $C A = 15$. Suppose the incenter of $\vartriangle ABC$ is $I$ and the incircle is tangent to $BC$ and $AB$ at $D$ and $E$, respectively. Line $\ell$ passes through the midpoints of $BD$ and $BE$ and point $X$ is on $\ell$ such that $AX \parallel BC$. Find $X I$ . [b]p27.[/b] Let $x, y, z$ be positive real numbers such that $x y + yz +zx = 20$ and $x^2yz +x y^2z +x yz^2 = 100$. Additionally, let $s = \max (x y, yz,xz)$ and $m = \min(x, y, z)$. If $s$ is maximal, find $m$. [u]Round 10[/u] [b]p28.[/b] Let $\omega_1$ be a circle with center $O$ and radius $1$ that is internally tangent to a circle $\omega_2$ with radius $2$ at $T$ . Let $R$ be a point on $\omega_1$ and let $N$ be the projection of $R$ onto line $TO$. Suppose that $O$ lies on segment $NT$ and $\frac{RN}{NO} = \frac4 3$ . Additionally, let $S$ be a point on $\omega_2$ such that $T,R,S$ are collinear. Tangents are drawn from $S$ to $\omega_1$ and touch $\omega_1$ at $P$ and $Q$. The tangent to $\omega_1$ at $R$ intersects $PQ$ at $Z$. Find the area of triangle $\vartriangle ZRS$. [b]p29.[/b] Let $m$ and $n$ be positive integers such that $k =\frac{ m^2+n^2}{mn-1}$ is also a positive integer. Find the sum of all possible values of $k$. [b]p30.[/b] Let $f_k (x) = k \cdot \ min (x,1-x)$. Find the maximum value of $k \le 2$ for which the equation $f_k ( f_k ( f_k (x))) = x$ has fewer than $8$ solutions for $x$ with $0 \le x \le 1$. [u]Round 11[/u] In the following problems, $A$ is the answer to Problem $31$, $B$ is the answer to Problem $32$, and $C$ is the answer to Problem $33$. For this set, you should find the values of $A$,$B$, and $C$ and submit them as answers to problems $31$, $32$, and $33$, respectively. Although these answers depend on each other, each problem will be scored separately. [b]p31.[/b] Find $$A \cdot B \cdot C + \dfrac{1}{B+ \dfrac{1}{C +\dfrac{1}{B+\dfrac{1}{...}}}}$$ [b]p32.[/b] Let $D = 7 \cdot B \cdot C$. An ant begins at the bottom of a unit circle. Every turn, the ant moves a distance of $r$ units clockwise along the circle, where $r$ is picked uniformly at random from the interval $\left[ \frac{\pi}{2D} , \frac{\pi}{D} \right]$. Then, the entire unit circle is rotated $\frac{\pi}{4}$ radians counterclockwise. The ant wins the game if it doesn’t get crushed between the circle and the $x$-axis for the first two turns. Find the probability that the ant wins the game. [b]p33.[/b] Let $m$ and $n$ be the two-digit numbers consisting of the products of the digits and the sum of the digits of the integer $2016 \cdot B$, respectively. Find $\frac{n^2}{m^2 - mn}$. [u]Round 12[/u] [b]p34.[/b] There are five regular platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For each of these solids, define its adjacency angle to be the dihedral angle formed between two adjacent faces. Estimate the sum of the adjacency angles of all five solids, in degrees. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).$ [b]p35.[/b] Estimate the value of $$\log_{10} \left(\prod_{k|2016} k!\right), $$ where the product is taken over all positive divisors $k$ of $2016$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lceil 15 \cdot \min \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).$ [b]p36.[/b] Estimate the value of $\sqrt{2016}^{\sqrt[4]{2016}}$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lceil 15 \cdot \min \left(\frac{\ln E}{\ln A}, 2- \frac{\ln E}{\ln A}\right) \rceil \right).$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3158474p28715078]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

One hundred points labeled $1$ to $100$ are arranged in a $10\times 10$ grid such that adjacent points are one unit apart. The labels are increasing left to right, top to bottom (so the first row has labels $1$ to $10,$ the second row has labels $11$ to $20,$ and so on). Convex polygon $\mathcal{P}$ has the property that every point with a label divisible by $7$ is either on the boundary or in the interior of $\mathcal{P}.$ Compute the smallest possible area of $\mathcal{P}.$

Let $a$ and $n$ be two integers greater than $1$. Prove that if $n$ divides $(a-1)^k$ for some integer $k\geq 2$, then $n$ also divides $a^{n-1}+a^{n-2}+\cdots+a+1$.

A box contains red, green, blue, and white balls, 111 balls in all. If you take out 100 balls without looking, then there will always be 4 balls of different colors among them. What is the smallest number of balls you must take out without looking to guarantee that among them there will always be balls of at least 3 different colors?

Let $a_1,a_2, a_3,...$ be a sequence of real numbers which satisfy the relation $a_{n+1} =\sqrt{a_n^2 + 1}$ Suppose that there exists a positive integer $n_0$ such that $a_{2n_0} = 3a_{n_0}$ . Find the value of $a_{46}$.

Let $ U$ be an $ n \times n$ orthogonal matrix. Prove that for any $ n \times n$ matrix $ A$, the matrices \[ A_m=\frac{1}{m+1} \sum_{j=0}^m U^{-j}AU^j\] converge entrywise as $ m \rightarrow \infty.$ [i]L. Kovacs[/i]

A line $l$ passes through the center $O$ of an equilateral triangle $\Delta ABC$, which intersects $CA$ in $N$ and $BC$ in $M$. Prove that we can construct a triangle with $AM$,$BN$, and $MN$ such that the altitude to $MN$ (in this triangle) is constant when $l$ changes.

A rectangle is divided into $n^2$ smaller rectangle by $n - 1$ horizontal lines and $n - 1$ vertical lines. Between those rectangles there are exactly $5660$ which are not congruent. For what minimum value of $n$ is this possible?

[color=darkred]The altitude $[BH]$ dropped onto the hypotenuse of a triangle $ABC$ intersects the bisectors $[AD]$ and $[CE]$ at $Q$ and $P$ respectively. Prove that the line passing through the midpoints of the segments $[QD]$ and $[PE]$ is parallel to the line $AC$ .[/color]

Let $n$ be an integer with $n\ge 3$. Consider all dissections of a convex $n$-gon into triangles by $n-3$ non-intersecting diagonals, and all colourings of the triangles with black and white so that triangles with a common side are always of a different colour. Find the least possible number of black triangles.

Given triangle $ ABC$ and $ AB\equal{}c$, $ AC\equal{}b$ and $ BC\equal{}a$ satisfying $ a \geq b \geq c$, $ BE$ and $ CF$ are two interior angle bisectors. $ P$ is a point inside triangle $ AEF$. $ R$ and $ Q$ are the projections of $ P$ on sides $ AB$ and $ AC$. Prove that $ PR \plus{} PQ \plus{} RQ < b$.

Find the minimum positive value of $ 1*2*3*4*...*2020*2021*2022$ where you can replace $*$ as $+$ or $-$

Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] [*]$m > 1$ and $\gcd(m,n) = 1$; [*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ are not divisible by $n$; and [*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc. [i]Carl Schildkraut[/i]

The mass of metal deposited by the electrolysis of an aqueous solution of metal ions increases in direct proportion to which property? $\text{I. electrolysis current}$ $\text{II. electrolysis time}$ $\text{III. metal ion charge}$ $ \textbf{(A)}\ \text{I only} \qquad\textbf{(B)}\ \text{III only} \qquad\textbf{(C)}\ \text{I and II only} \qquad\textbf{(D)}\ \text{I, II, and III}\qquad$

Solve the equation $\sqrt{x^2 - 4x + 13} + 1 = 2x$

a) Is it possible to place $2024$ checkers on a board $70 \times 70$ so that any square $2 \times 2$ contains even number of checkers? b) Is it possible to place $2023$ checkers on a board $70 \times 70$ so that any square $2 \times 2$ contains odd number of checkers?