$ABCD$ is a cyclic quadrilateral. The lines $AD$ and $BC$ meet at X, and the lines $AB$ and $CD$ meet at $Y$ . The line joining the midpoints $M$ and $N$ of the diagonals $AC$ and $BD$, respectively, meets the internal bisector of angle $AXB$ at $P$ and the external bisector of angle $BYC$ at $Q$. Prove that $PXQY$ is a rectangle

Consider a convex solid $ K$ in space and two parallel planes $ \epsilon _1$ and $ \epsilon _2$ on the distance $ 1$ tangent to $ K$. A plane $ \epsilon$ between $ \epsilon _1$ and $ \epsilon _2$ is on the distance $ d_1$ from $ \epsilon _1$. Find all $ d_1$ such that the part of $ K$ between $ \epsilon _1$ and $ \epsilon$ always has a volume not exceeding half the volume of $ K$.

Let $S$ be the set of $n$-uples $\left( x_{1}, x_{2}, \ldots, x_{n}\right)$ such that $x_{i}\in \{ 0, 1 \}$ for all $i \in \overline{1,n}$, where $n \geq 3$. Let $M(n)$ be the smallest integer with the property that any subset of $S$ with at least $M(n)$ elements contains at least three $n$-uples \[\left( x_{1}, \ldots, x_{n}\right), \, \left( y_{1}, \ldots, y_{n}\right), \, \left( z_{1}, \ldots, z_{n}\right) \] such that \[\sum_{i=1}^{n}\left( x_{i}-y_{i}\right)^{2}= \sum_{i=1}^{n}\left( y_{i}-z_{i}\right)^{2}= \sum_{i=1}^{n}\left( z_{i}-x_{i}\right)^{2}. \] (a) Prove that $M(n) \leq \left\lfloor \frac{2^{n+1}}{n}\right\rfloor+1$. (b) Compute $M(3)$ and $M(4)$.

Find the locus of points $M$ inside the rhombus $ABCD$ such that the sum of angles $AMB$ and $CMD$ equals $180^o$ .

Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$. (Slovakia)

[b]p1[/b]. Prove that there is no interger $n$ such that $n^2 +1$ is divisible by $7$. [b]p2.[/b] Find all solutions of the system $$x^2-yz=1$$ $$y^2-xz=2$$ $$z^2-xy=3$$ [b]p3.[/b] A triangle with long legs is an isoceles triangle in which the length of the two equal sides is greater than or equal to the length of the remaining side. What is the maximum number, $n$ , of points in the plane with the property that every three of them form the vertices of a triangle with long legs? Prove all assertions. [b]p4.[/b] Prove that the area of a quadrilateral of sides $a, b, c, d$ which can be inscribed in a circle and circumscribed about another circle is given by $A=\sqrt{abcd}$ [b]p5.[/b] Prove that all of the squares of side length $$\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},...,\frac{1}{n},...$$ can fit inside a square of side length $1$ without overlap. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In $\vartriangle ABC, \angle A= 2 \angle B$. Let $a,b,c$ be the lengths of its sides $BC,CA,AB$, respectively. Prove that $a^2 = b(b + c)$.

[b]p1.[/b] (a) Suppose $101$ Dalmatians chase $2012$ squirrels. Each squirrel gets chased by at most one Dalmatian, and each Dalmatian chases at least one squirrel. Show that two Dalmatians chase the same number of squirrels. (b) What is the largest number of Dalmatians that can chase $2012$ squirrels in a way that each Dalmatian chases at least one squirrel and no two Dalmatians chase the same number of squirrels? [b]p2.[/b] Lucy and Linus play the following game. They start by putting the integers $1, 2, 3, ..., 2012$ in a hat. In each round of the game, Lucy and Linus each draw a number from the hat. If the two numbers are $a$ and $b$, they throw away these numbers and put the number $|a - b|$ back into the hat. After $2011$ rounds, there is only one number in the hat. If it is even, Lucy wins. If it is odd, Linus wins. (a) Prove that there is a sequence of drawings that makes Lucy win. (b) Prove that Lucy always wins. [b]p3.[/b] Suppose $x$ is a positive real number and $x^{1990}$, $x^{2001}$, and $x^{2012}$ differ by integers. Prove that $x$ is an integer. [b]p4.[/b] Suppose that each point in three-dimensional space is colored with one of five colors and suppose that each color is used at least once. Prove that there is some plane that contains at least four of the colors. [b]p5.[/b] Two circles, $C_1$ and $C_2$, are tangent at point $A$, with $C_1$ lying inside $C_2$ (and $C_1 \ne C_2$). The line through their centers intersects $C_1$ at $B_1$ and $C_2$ at $B_2$. A line $L$ is drawn through $A$ and it intersects $C_1$ at $P_1$ (with $P_1 \ne A$) and intersects $C_2$ at $P_2$ (with $P_2 \ne A$). The perpendicular from $P_2$ to the line $B_1B_2$ intersects the line $B_1B_2$ at $F$. Prove that if the line $P_1F$ is tangent to $C_1$ then $F$ is the midpoint of the line segment $B_1B_2$. [img]https://cdn.artofproblemsolving.com/attachments/9/e/4db59be9fa764d3e910a828ed3296907ca5657.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An arbitrary triangle $ABC$ is given and a point $P$ lies on the side $AB$. It is requested to draw through $P$ a line that divides the triangle into two figures of the same area.

A circle touches the extensions of sides $CA$ and $CB$ of triangle $ABC$, and also touches side $AB$ of this triangle at point $P$. Prove that the radius of the circle tangent to segments $AP$, $CP$ and the circumscribed circle of this triangle is equal to the radius of the inscribed circle in this triangle.

The median $AM$ is drawn in the acute-angled triangle $ABC$ with different sides. Its extension intersects the circumscribed circle $w$ of this triangle at the point $P$. Let $A {{H} _ {1}}$ be the altitude $\Delta ABC$, $H$ be the point of intersection of its altitudes. The rays $MH$ and $P {{H} _ {1}}$ intersect the circle $w$ at the points $K$ and $T$, respectively. Prove that the circumscribed circle of $\Delta KT {{H} _ {1}}$ touches the segment $BC$. (Hilko Danilo)

[b]p1.[/b] Find all triples of positive integers such that the sum of their reciprocals is equal to one. [b]p2.[/b] Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$. [b]p3.[/b] There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that (a) for each chip positioned on the circle the antipodal position is occupied by a chip of different color; (b) there are no two neighboring blue chips. [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle for which $AB+BC = 3AC$. Let $D$ and $E$ be the points of tangency of the incircle with the sides $AB$ and $BC$ respectively, and let $K$ and $L$ be the other endpoints of the diameters originating from $D$ and $E.$ Prove that $C , A, L$, and $K$ lie on a circle.

Let two circles $C_1,C_2$ with different radius be externally tangent at a point $T$. Let $A$ be on $C_1$ and $B$ be on $C_2$, with $A,B \ne T$ such that $\angle ATB = 90^o$. (a) Prove that all such lines $AB$ are concurrent. (b) Find the locus of the midpoints of all such segments $AB$.

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of \[a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}.\]

Altitudes $AA_1$, $BB_1$, $CC_1$ of an acute triangle $ABC$ meet at $H$. $A_0$, $B_0$, $C_0$ are the midpoints of $BC$, $CA$, $AB$ respectively. Points $A_2$, $B_2$, $C_2$ on the segments $AH$, $BH$, $HC_1$ respectively are such that $\angle A_0B_2A_2 = \angle B_0C_2B_2 = \angle C_0A_2C_2 =90^\circ$. Prove that the lines $AC_2$, $BA_2$, $CB_2$ are concurrent.

Let $ABC$ ne a right triangle with $\angle ACB=90^o$. Let $E, F$ be respecitvely the midpoints of the $BC, AC$ and $CD$ be it's altitude. Next, let $P$ be the intersection of the internal angle bisector from $A$ and the line $EF$. Prove that $P$ is the center of the circle inscribed in the triangle $CDE$ .

[u]Set 1[/u] [b]p1.[/b] Farmer John has $4000$ gallons of milk in a bucket. On the first day, he withdraws $10\%$ of the milk in the bucket for his cows. On each following day, he withdraws a percentage of the remaining milk that is $10\%$ more than the percentage he withdrew on the previous day. For example, he withdraws $20\%$ of the remaining milk on the second day. How much milk, in gallons, is left after the tenth day? [b]p2.[/b] Will multiplies the first four positive composite numbers to get an answer of $w$. Jeremy multiplies the first four positive prime numbers to get an answer of $j$. What is the positive difference between $w$ and $j$? [b]p3.[/b] In Nathan’s math class of $60$ students, $75\%$ of the students like dogs and $60\%$ of the students like cats. What is the positive difference between the maximum possible and minimum possible number of students who like both dogs and cats? [u]Set 2[/u] [b]p4.[/b] For how many integers $x$ is $x^4 - 1$ prime? [b]p5.[/b] Right triangle $\vartriangle ABC$ satisfies $\angle BAC = 90^o$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 60$ and $AB = 65$, find the area of $\vartriangle ABC$. [b]p6.[/b] Define $n! = n \times (n - 1) \times ... \times 1$. Given that $3! + 4! + 5! = a^2 + b^2 + c^2$ for distinct positive integers $a, b, c$, find $a + b + c$. [u]Set 3[/u] [b]p7.[/b] Max nails a unit square to the plane. Let M be the number of ways to place a regular hexagon (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Vincent, on the other hand, nails a regular unit hexagon to the plane. Let $V$ be the number of ways to place a square (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Find the nonnegative difference between $M$ and $V$ . [b]p8.[/b] Let a be the answer to this question, and suppose $a > 0$. Find $\sqrt{a +\sqrt{a +\sqrt{a +...}}}$ . [b]p9.[/b] How many ordered pairs of integers $(x, y)$ are there such that $x^2 - y^2 = 2019$? [u]Set 4[/u] [b]p10.[/b] Compute $\frac{p^3 + q^3 + r^3 - 3pqr}{p + q + r}$ where $p = 17$, $q = 7$, and $r = 8$. [b]p11.[/b] The unit squares of a $3 \times 3$ grid are colored black and white. Call a coloring good if in each of the four $2 \times 2$ squares in the $3 \times 3$ grid, there is either exactly one black square or exactly one white square. How many good colorings are there? Consider rotations and reflections of the same pattern distinct colorings. [b]p12.[/b] Define a $k$-[i]respecting [/i]string as a sequence of $k$ consecutive positive integers $a_1$, $a_2$, $...$ , $a_k$ such that $a_i$ is divisible by $i$ for each $1 \le i \le k$. For example, $7$, $8$, $9$ is a $3$-respecting string because $7$ is divisible by $1$, $8$ is divisible by $2$, and $9$ is divisible by $3$. Let $S_7$ be the set of the first terms of all $7$-respecting strings. Find the sum of the three smallest elements in $S_7$. [u]Set 5[/u] [b]p13.[/b] A triangle and a quadrilateral are situated in the plane such that they have a finite number of intersection points $I$. Find the sum of all possible values of $I$. [b]p14.[/b] Mr. DoBa continuously chooses a positive integer at random such that he picks the positive integer $N$ with probability $2^{-N}$ , and he wins when he picks a multiple of 10. What is the expected number of times Mr. DoBa will pick a number in this game until he wins? [b]p15.[/b] If $a, b, c, d$ are all positive integers less than $5$, not necessarily distinct, find the number of ordered quadruples $(a, b, c, d)$ such that $a^b - c^d$ is divisible by $5$. PS. You had better use hide for answers. Last 4 sets have been posted [url=https://artofproblemsolving.com/community/c4h2777362p24370554]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ l_a,l_b,l_c$ be three parallel lines passing through $ A,B,C$ respectively. Let $ l_a'$ be reflection of $ l_a$ into $ BC$. $ l_b'$ and $ l_c'$ are defined similarly. Prove that $ l_a',l_b',l_c'$ are concurrent if and only if $ l_a$ is parallel to Euler line of triangle $ ABC$.

Given triangle $ ABC$. Point $ O$ is the center of the excircle touching the side $ BC$. Point $ O_1$ is the reflection of $ O$ in $ BC$. Determine angle $ A$ if $ O_1$ lies on the circumcircle of $ ABC$.

Which of the following has the greatest number of line of symmetry? $ \textbf{(A)}\ \text{ Equilateral Triangle}$ $\textbf{(B)}\ \text{Non-square rhombus} $ $\textbf{(C)}\ \text{Non-square rectangle}$ $\textbf{(D)}\ \text{Isosceles Triangle}$ $\textbf{(E)}\ \text{Square} $

Let $ f: [0,1]\to\mathbb R$ be a derivable function, with a continuous derivative $ f'$ on $ [0,1]$. Prove that if $ f\left( \frac 12\right) \equal{} 0$, then \[ \int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.\]

Let $AA_1$ and $BB_1$ be the altitudes of an acute-angled, non-isosceles triangle $ABC$. Also, let $A_0$ and $B_0$ be the midpoints of its sides $BC$ and $CA$, respectively. The line $A_1B_1$ intersects the line $A_0B_0$ at a point $C'$. Prove that the line $CC'$ is perpendicular to the Euler line of the triangle $ABC$ (this is the line that joins the orthocenter and the circumcenter of the triangle $ABC$).

Let $ABC$ be an isosceles triangle, and point $D$ in its interior such that $$D \hat{B} C=30^\circ, D \hat{B}A=50^\circ, D \hat{C}B=55^\circ$$ (a) Prove that $\hat B=\hat C=80^\circ$. (b) Find the measure of the angle $D \hat{A} C$.