Let $C$ be a circunference, $O$ is your circumcenter, $AB$ is your diameter and $R$ is any point in $C$ ($R$ is different of $A$ and $B$) Let $P$ be the foot of perpendicular by $O$ to $AR$, in the line $OP$ we match a point $Q$, where $QP$ is $\frac{OP}{2}$ and the point $Q$ isn't in the segment $OP$. In $Q$, we will do a parallel line to $AB$ that cut the line $AR$ in $T$. Denote $H$ the point of intersections of the line $AQ$ and $OT$. Show that $H$, $B$ and $R$ are collinears.

A table is covered by $15$ pieces of paper. Show that we can remove $7$ pieces so that the remaining $8$ cover at least $8/15$ of the table.

[b]p1.[/b] Let $a$ and $b$ be complex numbers such that $a^3 + b^3 = -17$ and $a + b = 1$. What is the value of $ab$? [b]p2.[/b] Let $AEFB$ be a right trapezoid, with $\angle AEF = \angle EAB = 90^o$. The two diagonals $EB$ and $AF$ intersect at point $D$, and $C$ is a point on $AE$ such that $AE \perp DC$. If $AB = 8$ and $EF = 17$, what is the lenght of $CD$? [b]p3.[/b] How many three-digit numbers $abc$ (where each of $a$, $b$, and $c$ represents a single digit, $a \ne 0$) are there such that the six-digit number $abcabc$ is divisible by $2$, $3$, $5$, $7$, $11$, or $13$? [b]p4.[/b] Let $S$ be the sum of all numbers of the form $\frac{1}{n}$ where $n$ is a postive integer and $\frac{1}{n}$ terminales in base $b$, a positive integer. If $S$ is $\frac{15}{8}$, what is the smallest such $b$? [b]p5.[/b] Sysyphus is having an birthday party and he has a square cake that is to be cut into $25$ square pieces. Zeus gets to make the first straight cut and messes up badly. What is the largest number of pieces Zeus can ruin (cut across)? Diagram? [b]p6.[/b] Given $(9x^2 - y^2)(9x^2 + 6xy + y^2) = 16$ and $3x + y = 2$. Find $x^y$. [b]p7.[/b] What is the prime factorization of the smallest integer $N$ such that $\frac{N}{2}$ is a perfect square, $\frac{N}{3}$ is a perfect cube, $\frac{N}{5}$ is a perfect fifth power? [b]p8.[/b] What is the maximum number of pieces that an spherical watermelon can be divided into with four straight planar cuts? [b]p9.[/b] How many ordered triples of integers $(x,y,z)$ are there such that $0 \le x, y, z \le 100$ and $$(x - y)^2 + (y - z)^2 + (z - x)^2 \ge (x + y - 2z) + (y + z - 2x)^2 + (z + x - 2y)^2.$$ [b]p10.[/b] Find all real solutions to $(2x - 4)^2 + (4x - 2)^3 = (4x + 2x - 6)^3$. [b]p11.[/b] Let $f$ be a function that takes integers to integers that also has $$f(x)=\begin{cases} x - 5\,\, if \,\, x \ge 50 \\ f (f (x + 12)) \,\, if \,\, x < 50 \end{cases}$$ Evaluate $f (2) + f (39) + f (58).$ [b]p12.[/b] If two real numbers are chosen at random (i.e. uniform distribution) from the interval $[0,1]$, what is the probability that theit difference will be less than $\frac35$? [b]p13.[/b] Let $a$, $b$, and $c$ be positive integers, not all even, such that $a < b$, $b = c - 2$, and $a^2 + b^2 = c^2$. What is the smallest possible value for $c$? [b]p14.[/b] Let $ABCD$ be a quadrilateral whose diagonals intersect at $O$. If $BO = 8$, $OD = 8$, $AO = 16$, $OC = 4$, and $AB = 16$, then find $AD$. [b]p15.[/b] Let $P_0$ be a regular icosahedron with an edge length of $17$ units. For each nonnegative integer $n$, recursively construct $P_{n+1}$ from Pn by performing the following procedure on each face of $P_n$: glue a regular tetrahedron to that face such that three of the vertices of the tetrahedron are the midpoints of the three adjacent edges of the face, and the last vertex extends outside of $P_n$. Express the number of square units in the surface area of $P_{17}$ in the form $$\frac{u^v\cdot w \sqrt{x}}{y^z}$$ , where $u, v, w, x, y$, and $z$ are integers, all greater than or equal to $2$, that satisfy the following conditions: the only perfect square that evenly divides $x$ is $1$, the GCD of $u$ and y is $1$, and neither $u$ nor $y$ divides $w$. Answers written in any other form will not be considered correct! PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The graph of the equation $x^3-2x^2y+xy^2-2y^3=0$ is the same as the graph of $\text{(A) }x^2+y^2=0\qquad\text{(B) }x=y\qquad\text{(C) }y=2x^2-x\qquad\text{(D) }x=y^3\qquad\text{(E) }x=2y$

Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$, a distance of $10$ km all uphill, then from town $B$ to town $C$, a distance of $15$ km all downhill, and then back to town $A$, a distance of $20$ km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the $45$-km ride than it takes Penny? $\textbf{(A)}\ 45\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 65\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 95$

Let $C$ be a right circular cone with apex $A$. Let $P_1, P_2, P_3, P_4$ and $P_5$ be points placed evenly along the circular base in that order, so that $P_1P_2P_3P_4P_5$ is a regular pentagon. Suppose that the shortest path from $P_1$ to $P_3$ along the curved surface of the cone passes through the midpoint of $AP_2$. Let $h$ be the height of $C$, and $r$ be the radius of the circular base of $C$. If $\left(\frac{h}{r}\right)^2$ can be written in simplest form as $\frac{a}{b}$ , find $a + b$.

Let $\triangle ABC$ be a triangle with $BC = 4, CA= 5, AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Sreejato Bhattacharya[/i]

A circle touches the lateral sides of a trapezoid $ABCD$ at points $B$ and $C$, and its center lies on $AD$. Prove that the diameter of the circle is less than the medial line of the trapezoid.

Let $x, y$ be positive integers such that $\gcd(x,y)=1$ and $x+3y^2$ is a perfect square. Prove that $x^2+9y^4$ can't be a perfect square.

There are 25 ants on a number line; five at each of the coordinates $1$, $2$, $3$, $4$, and $5$. Each minute, one ant moves from its current position to a position one unit away. What is the minimum number of minutes that must pass before it is possible for no two ants to be on the same coordinate? [i]Ray Li[/i]

Let $ABC$ be an isosceles triangle with $AC=BC$. Let $P,Q,R$ be points on the sides $AB, BC, CA$ of the triangle such that $CQPR$ is a parallelogram. Show that the reflection of $P$ over $QR$ lies on the circumcircle of $ABC$.

Given a line $ m $ and a segment $ AB $ parallel to it. Divide the segment $ AB $ into three equal parts using only a ruler, i.e. drawing only the lines.

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

Let $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minimum possible value of $\left\lvert d - \sqrt{n} \right\rvert$ can be written in the form $a\sqrt{b} + c$, where $b$ is a positive integer not divisible by the square of any prime, and $a$ and $c$ are nonzero integers (not necessarily positive). Compute $a+b+c$. [i]Proposed by Matthew Lerner-Brecher[/i]

Let $a, b, c$, be real numbers such that $$a^2b^2 + 18 abc > 4b^3+4a^3c+27c^2 .$$ Prove that $a^2>3b$.

While driving his car, Ken pulled off the road to get gasoline when he was of $\frac{7}{12}$ the way through his trip. After driving another eleven miles, he noticed that he was $\frac{13}{20}$ of the way through his trip. How many miles long was his entire trip?

The diagonals of the convex quadrilateral $ABCD$ intersect at the point $O$. The points $X$ and $Y$ are symmetrical to the point $O$ with respect to the midpoints of the sides $BC$ and $AD$, respectively. It is known that $AB = BC = CD$. Prove that the point of intersection of the perpendicular bisectors of the diagonals of the quadrilateral lies on the line $XY$.

Let $ABCDEFG$ be half of a regular dodecagon . Let $P$ be the intersection of the lines $AB$ and $GF$, and let $Q$ be the intersection of the lines $AC$ and $GE$. Prove that $Q$ is the circumcenter of the triangle $AGP$.

Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.

[b]1.[/b] Let $a_{1}, a_{2}, \dots , a_{n}$ and $b_{1}, b_{2}, \dots , b_{m}$ be $n+m$ unit vectors in the $r$-dimensional Euclidean space $E_{r} (n,m \leq r)$; let $a_{1}, a_{2}, \dots , a_{n}$ as well as $b_{1}, b_{2}, \dots , b_{m}$ be mutually orthogonal. For any vector $x \in E_{r}$, consider $Tx= \sum_{i=1}^{n}\sum_{k=1}^{m}(x,a_{i})(a_{i},b_{k})b_{k}$ ($(a,b)$ denotes the scalar product of $a$ and $b$). Show that the sequence $(T^{k}x)^{\infty}_{ k =0}$, where $T^{0} x= x$ and $T^{k} x = T(T^{k-1}x)$, is convergent and give a geometrical characterization of how the limit depends on $x$. [b](S. 14)[/b]

Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly? $\text{(A)}\ \$3.63 \qquad\text{(B)}\ \$5.13\qquad\text{(C)}\ \$6.30 \qquad\text{(D)}\ \$7.45 \qquad\text{(E)}\ \$9.07$

Given $2020*2020$ chessboard, what is the maximum number of warriors you can put on its cells such that no two warriors attack each other. Warrior is a special chess piece which can move either $3$ steps forward and one step sideward and $2$ step forward and $2$ step sideward in any direction.

Each lottery ticket has a 9-digit numbers, which uses only the digits $1$, $2$, $3$. Each ticket is colored [color=red]red[/color],[color=blue] blue [/color]or [color=green]green[/color]. If two tickets have numbers which differ in all nine places, then the tickets have different colors. Ticket $122222222$ is red, and ticket $222222222$ is [color=green]green.[/color] What color is ticket $123123123$ ?

In an acute triangle $ABC$, the points $H$, $G$, and $M$ are located on $BC$ in such a way that $AH$, $AG$, and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$, $AB=10$, and $AC=14$. Find the area of triangle $ABC$.

Find any two consecutive natural numbers, each of which is divisible by the square of the sum of its digits.