The figure shows a line intersecting a square lattice. The area of some arising quadrilaterals are also indicated. What is the area of the region with the question mark? [img]https://cdn.artofproblemsolving.com/attachments/0/d/4d5741a63d052e3f6971f87e60ca7df7302fb0.png[/img]

Let $ MN$ be a line parallel to the side $ BC$ of a triangle $ ABC$, with $ M$ on the side $ AB$ and $ N$ on the side $ AC$. The lines $ BN$ and $ CM$ meet at point $ P$. The circumcircles of triangles $ BMP$ and $ CNP$ meet at two distinct points $ P$ and $ Q$. Prove that $ \angle BAQ = \angle CAP$. [i]Liubomir Chiriac, Moldova[/i]

Let $n$ be a given positive integer. Determine all positive divisors $d$ of $3n^2$ such that $n^2 + d$ is the square of an integer.

Let $p$ be an odd prime. Prove that \[1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.\]

Let $\triangle ABC$ such that $AB=AC$, $D$ and $E$ points on $AB$ and $BC$, respectively, with $DE\parallel AC$. Let $F$ on line $DE$ such that $CADF$ it´s a parallelogram. If $O$ is the circumcenter of $\triangle BDE$, prove that $O,F,A$ and $D$ lie on a circle.

Let $P$ be a polynomial with integer coefficients. There exist integers $a$ and $b$ such that $P(a) \cdot P(b)=-(a-b)^2$. Prove that $P(a)+P(b)=0$.

Different points $A_1, A_2,..., A_n$ in the plane ($n> 3$) are such that the triangle $A_iA_jA_k$ is obtuse for all the different $i,j,k \in\{1,2,...,n\}$. Prove that there is a point $A_{n + 1}$ in the plane, such that the triangle $A_iA_jA_{n + 1}$ is obtuse for all different $i,j \in\{1,2,...,n\}$

Find all triangles such that its angles form an arithmetic sequence and the corresponding sides form a geometric sequence.

12 points are located on a clock with the sme distance , numbers $1,2,3 , ... 12$ are marked on each point in clockwise order . Use 4 kinds of colors (red,yellow,blue,green) to colour the the points , each kind of color has 3 points . N ow , use these 12 points as the vertex of convex quadrilateral to construct $n$ convex quadrilaterals . They satisfies the following conditions: (1). the colours of vertex of every convex quadrilateral are different from each other . (2). for every 3 quadrilaterals among them , there exists a colour such that : the numbers on the 3 points painted into this colour are different from each other . Find the maximum $n$ .

Let $a,b,c$ be three different positive integers. Show that the sequence \[a+b+c,ab+bc+ca,3abc\] could be neither an arithmetic nor geometric progression. [i]Fajar Yuliawan, Bandung[/i]

[u]Round 5[/u] [b]p13.[/b] Initially, the three numbers $20$, $201$, and $2016$ are written on a blackboard. Each minute, Zhuo selects two of the numbers on the board and adds $1$ to each. Find the minimum $n$ for which Zhuo can make all three numbers equal to $n$. [b]p14.[/b] Call a three-letter string rearrangeable if, when the first letter is moved to the end, the resulting string comes later alphabetically than the original string. For example, $AAA$ and $BAA$ are not rearrangeable, while $ABB$ is rearrangeable. How many three-letters strings with (not necessarily distinct) uppercase letters are rearrangeable? [b]p15.[/b] Triangle $ABC$ is an isosceles right triangle with $\angle C = 90^o$ and $AC = 1$. Points $D$, $E$ and $F$ are chosen on sides $BC$,$CA$ and $AB$, respectively, such that $AEF$, $BFD$, $CDE$, and $DEF$ are isosceles right triangles. Find the sum of all distinct possible lengths of segment $DE$. [u]Round 6[/u] [b]p16.[/b] Let $p, q$, and $r$ be prime numbers such that $pqr = 17(p + q + r)$. Find the value of the product $pqr$. [b]p17.[/b] A cylindrical cup containing some water is tilted $45$ degrees from the vertical. The point on the surface of the water closest to the bottom of the cup is $6$ units away. The point on the surface of the water farthest from the bottom of the cup is $10$ units away. Compute the volume of the water in the cup. [b]p18.[/b] Each dot in an equilateral triangular grid with $63$ rows and $2016 = \frac12 \cdot 63 \cdot 64$ dots is colored black or white. Every unit equilateral triangle with three dots has the property that exactly one of its vertices is colored black. Find all possible values of the number of black dots in the grid. [u]Round 7[/u] [b]p19.[/b] Tomasz starts with the number $2$. Each minute, he either adds $2$ to his number, subtracts $2$ from his number, multiplies his number by $2$, or divides his number by $2$. Find the minimum number of minutes he will need in order to make his number equal $2016$. [b]p20.[/b] The edges of a regular octahedron $ABCDEF$ are painted with $3$ distinct colors such that no two edges with the same color lie on the same face. In how many ways can the octahedron be painted? Colorings are considered different under rotation or reflection. [b]p21.[/b] Jacob is trapped inside an equilateral triangle $ABC$ and must visit each edge of triangle $ABC$ at least once. (Visiting an edge means reaching a point on the edge.) His distances to sides $AB$, $BC$, and $CA$ are currently $3$, $4$, and $5$, respectively. If he does not need to return to his starting point, compute the least possible distance that Jacob must travel. [u]Round 8[/u] [b]p22.[/b] Four integers $a, b, c$, and $d$ with a $\le b \le c \le d$ satisfy the property that the product of any two of them is equal to the sum of the other two. Given that the four numbers are not all equal, determine the $4$-tuple $(a, b, c, d)$. [b]p23.[/b] In equilateral triangle $ABC$, points $D$,$E$, and $F$ lie on sides $BC$,$CA$ and $AB$, respectively, such that $BD = 4$ and $CD = 5$. If $DEF$ is an isosceles right triangle with right angle at $D$, compute $EA + FA$. [b]p24.[/b] On each edge of a regular tetrahedron, four points that separate the edge into five equal segments are marked. There are sixteen planes that are parallel to a face of the tetrahedron and pass through exactly three of the marked points. When the tetrahedron is cut along each of these sixteen planes, how many new tetrahedrons are produced? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2934049p26256220]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the greatest positive integer $k$ such that the following inequality holds for all $a,b,c\in\mathbb{R}^+$ satisfying $abc=1$ \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \]

For every integer $n \ge 2$ let $B_n$ denote the set of all binary $n$-nuples of zeroes and ones, and split $B_n$ into equivalence classes by letting two $n$-nuples be equivalent if one is obtained from the another by a cyclic permutation.(for example 110, 011 and 101 are equivalent). Determine the integers $n \ge 2$ for which $B_n$ splits into an odd number of equivalence classes.

A number is called [i]flippy[/i] if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15$? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

Five lilypads lie in a line on a pond. At first, a frog sits on the third lilypad. Then, each minute there is a $\tfrac{1}{2}$ probability that the frog jumps to the lilypad to its left and $\tfrac{1}{2}$ probability that it jumps to its right. If the frog jumps to the left from the leftmost lilypad or right from the rightmost lilypad, it will fall in the pond and stay there forever. Compute the probability that the frog is not in the pond after $14$ minutes have passed.

Find all functions $ f:Z\to Z$ with the following property: if $x+y+z=0$, then $f(x)+f(y)+f(z)=xyz.$

A non-zero polynomial $ S\in\mathbb{R}[X,Y]$ is called homogeneous of degree $ d$ if there is a positive integer $ d$ so that $ S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y)$ for any $ \lambda\in\mathbb{R}$. Let $ P,Q\in\mathbb{R}[X,Y]$ so that $ Q$ is homogeneous and $ P$ divides $ Q$ (that is, $ P|Q$). Prove that $ P$ is homogeneous too.

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

Evan has a simple graph with $v$ vertices and $e$ edges. Show that he can delete at least $\frac{e-v+1}{2}$ edges so that each vertex still has at least half of its original degree.

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. A group of haikus Some have one syllable less Sixteen in total. The group of haikus Some have one syllable more Eighteen in total. What is the largest Total count of syllables That the group can’t have? (For instance, a group Sixteen, seventeen, eighteen Fifty-one total.) (Also, you can have No sixteen, no eighteen Syllable haikus) [i]Proposed by Jeff Lin[/i]

Let $P(x)$ be a polynomial such that, when divided by $x - 2$, the remainder is $3$ and, when divided by $x - 3$, the remainder is $2$. If, when divided by $(x - 2)(x - 3)$, the remainder is $ax + b$, find $a^2 + b^2$.

Give an acute-angled triangle $ABC$ with area $S$, let points $A',B',C'$ be located as follows: $A'$ is the point where altitude from $A$ on $BC$ meets the outwards facing semicirle drawn on $BX$ as diameter.Points $B',C'$ are located similarly. Evaluate the sum $T=($area $\vartriangle BCA')^2+($area $\vartriangle CAB')^2+($area $\vartriangle ABC')^2$.

3. Plane is divided with horizontal and vertical lines into unit squares. Into each square we write a positive integer so that each positive integer appears exactly once. Determine whether it is possible to write numbers in such a way, that each written number is a divisor of a sum of its four neighbours.

Let be the sequence $ \left( a_n \right)_{n\ge 0} $ of positive real numbers defined by $$ a_n=1+\frac{a_{n-1}}{n} ,\quad\forall n\ge 1. $$ Calculate $ \lim_{n\to\infty } a_n ^n . $ [i]Florian Dumitrel[/i]

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]