The quadrilateral $ABCD$ is such that $AB=AD=1$ and $\angle A=90^\circ$. If $CB=c$, $CA=b$, and $CD=a$, then prove that $(2-a^2-c^2 )^2+(2b^2-a^2-c^2 )^2=4a^2 c^2$ and $(a-c)^2\leq 2b^2\leq (a+c)^2$.

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the lateral sides of all such trapezoids share a common point.

Some lottery is played as follows. A lottery participant buys a card with $10$ numbered cells. He has the right to cross out any $4$ of these $10$ cells. Then a drawing occurs, during which some $7$ out of $10$ cells become winning. The player wins when all $4$ squares he crosses out are winning. The question arises, what is the smallest number of cards that can be used so that, if filled out correctly, at least one of these cards will win in any case? We do not suggest that you answer this question (we ourselves do not know the answer), although, of course, we will be very glad if you do and will evaluate this achievement accordingly. The task is; to indicate a certain number $n$ and a method of filling n cards that guarantees at least one win. The smaller $n$, the higher the rating of the work.

Several straight lines such that no two are parallel, cut the plane into several regions. A point $A$ is marked inside of one region. Prove that a point, separated from $A$ by each of these lines, exists if and only if $A$ belongs to an unbounded region.

In this diagram, not drawn to scale, figures $\text{I}$ and $\text{III}$ are equilateral triangular regions with respective areas of $32\sqrt{3}$ and $8\sqrt{3}$ square inches. Figure $\text{II}$ is a square region with area $32$ sq. in. Let the length of segment $AD$ be decreased by $12\frac{1}{2} \%$ of itself, while the lengths of $AB$ and $CD$ remain unchanged. The percent decrease in the area of the square is: [asy] draw((0,0)--(22.6,0)); draw((0,0)--(5.66,9.8)--(11.3,0)--(11.3,5.66)--(16.96,5.66)--(16.96,0)--(19.45,4.9)--(22.6,0)); label("A", (0,0), S); label("B", (11.3,0), S); label("C", (16.96,0), S); label("D", (22.6,0), S); label("I", (5.66, 3.9)); label("II", (14.15,2.83)); label("III", (19.7,2)); [/asy] $\textbf{(A)}\ 12\frac{1}{2} \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 75 \qquad\textbf{(E)}\ 87\frac{1}{2}$

Juan must pay $4$ bills. He goes to an ATM, but doesn't remember the amount of the bills. Just know that a) Each account is a multiple of $1,000$ and is at least $4,000$. b) The accounts total 2$00, 000$. What is the least number of times Juan must use the ATM to make sure he can pay the bills with exact change without any excess money? The cashier has banknotes of $2, 000$, $5, 000$, $10, 000$, and $20,000$. Juan can decide how much money he asks the cashier each time, but you cannot decide how many bills of each type to give to the cashier.

If $R$ and $S$ are two rectangles with integer sides such that the perimeter of $R$ equals the area of $S$ and the perimeter of $S$ equals the area of $R$, then we call $R$ and $S$ a friendly pair of rectangles. Find all friendly pairs of rectangles.

Let $A$ be a real $n \times n$ matrix and suppose that for every positive integer $m$ there exists a real symmetric matrix $B$ such that $$2021B = A^m+B^2.$$ Prove that $|\text{det} A| \leq 1$.

Let $a_0,a_1,a_2,\ldots$ be a sequence of integers and $b_0,b_1,b_2,\ldots$ be a sequence of [i]positive[/i] integers such that $a_0=0,a_1=1$, and \[ a_{n+1} = \begin{cases} a_nb_n+a_{n-1} & \text{if $b_{n-1}=1$} \\ a_nb_n-a_{n-1} & \text{if $b_{n-1}>1$} \end{cases}\qquad\text{for }n=1,2,\ldots. \] for $n=1,2,\ldots.$ Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to $2017$.

There are five research labs on Mars. Is it always possible to divide Mars to five connected congruent regions such that each region contains exactly on research lab. [img]http://i37.tinypic.com/f2iq8g.png[/img]

The circle $\Omega$ with center $I_A$, is the $A$-excircle of triangle $ABC$. Which is tangent to $AB,AC$ at $F,E$ respectivly. Point $D$ is the reflection of $A$ through $I_AB$. Lines $DI_A$ and $EF$ meet at $K$. Prove that ,circumcenter of $DKE$ , midpoint of $BC$ and $I_A$ are collinear.

1) The numbers $1,2,3,\ldots,2010$ are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers $a$ and $b$ with $ab-a-b$ while the second player replaces them with $ab+a+b.$ The game ends when a single number remains on the blackboard. If this number is smaller than $1\cdot2\cdot3\cdot\ldots\cdot2010$ then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy? 2) The numbers $1,2,3,\ldots,2010$ are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers $a$ and $b$ with $ab-a-b+2$ while the second player replaces them with $ab+a+b.$ The game ends when a single number remains on the blackboard. If this number is smaller than $1\cdot2\cdot3\cdot\ldots\cdot2010$ then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy?

Find the number of ordered triples of positive integers $(a, b, c)$ such that $abc$ divides $(ab + 1)(bc + 1)(ca + 1)$.

The points $D, E, F$ lie respectively on the sides $BC$, $CA$, $AB$ of the triangle ABC such that $F B = BD$, $DC = CE$, and the lines $EF$ and $BC$ are parallel. Tangent to the circumscribed circle of triangle $DEF$ at point $F$ intersects line $AD$ at point $P$. Perpendicular bisector of segment $EF$ intersects the segment $AC$ at $Q$. Prove that the lines $P Q$ and $BC$ are parallel.

[b]p1.[/b] a) Given four points in the plane, no three of which lie on the same line, each subset of three points determines the vertices of a triangle. Can all these triangles have equal areas? If so, give an example of four points (in the plane) with this property, and then describe all arrangements of four joints (in the plane) which permit this. If no such arrangement exists, prove this. b) Repeat part a) with "five" replacing "four" throughout. [b]p2.[/b] Three people at the base of a long stairway begin a race up the stairs. Person A leaps five steps with each stride (landing on steps $5$, $10$, $15$, etc.). Person B leaps a little more slowly but covers six steps with each stride. Person C leaps seven steps with each stride. A picture taken near the end of the race shows all three landing simultaneously, with Person A twenty-one steps from the top, person B seven steps from the top, and Person C one step from the top. How many steps are there in the stairway? If you can find more than one answer, do so. Justify your answer. [b]p3. [/b]Let $S$ denote the sum of an infinite geometric series. Suppose the sum of the squares of the terms is $2S$, and that df the cubes is $64S/13$. Find the first three terms of the original series. [b]p4.[/b] $A$, $B$ and $C$ are three equally spaced points on a circular hoop. Prove that as the hoop rolls along the horizontal line $\ell$, the sum of the distances of the points $A, B$, and $C$ above line $\ell$ is constant. [img]https://cdn.artofproblemsolving.com/attachments/3/e/a1efd0975cf8ff3cf6acb1da56da1dce35d81e.png[/img] [b]p5.[/b] A set of $n$ numbers $x_1,x_2,x_3,...,x_n$ (where $n>1$) has the property that the $k^{th}$ number (that is, $x_k$ ) is removed from the set, the remaining $(n-1)$ numbers have a sum equal to $k$ (the subscript o $x_k$ ), and this is true for each $k = 1,2,3,...,n$. a) SoIve for these $n$ numbers b) Find whether at least one of these $n$ numbers can be an integer. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The distances from a point $ P$ inside an equilateral triangle to the vertices of the triangle are $ 3,4$, and $ 5$. Find the area of the triangle.

A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds $T_{max}$. One end of the cord is attached to a fixed point, the other is attached to an object of mass $3m$. If a second, smaller object of mass m moving at an initial speed $v_0$ strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of $v_f$. All motion occurs on a horizontal, frictionless surface. Find the ratio of the total kinetic energy of the system of two masses after the perfectly elastic collision and the cord has broken to the initial kinetic energy of the smaller mass prior to the collision. $ \textbf{(A)}\ 1/4 \qquad\textbf{(B)}\ 1/3 \qquad\textbf{(C)}\ 1/2 \qquad\textbf{(D)}\ 3/4 \qquad\textbf{(E)}\ \text{none of the above} $

In space, we are given the points $A,B,C$ and a sphere with center $O$ and radius $1$. Find the point $X$ from the sphere for which the sum $f(X)=|XA|^2+|XB|^2+|XC|^2$ attains its maximal and minimal value. Prove that if the segments $OA,OB,OC$ are pairwise perpendicular and $d$ is the distance from the center $O$ to the centroid of the triangle $ABC$ then: (a) the maximum of $f(X)$ is equal to $9d^2+3+6d$; (b) the minimum of $f(X)$ is equal to $9d^2+3-6d$. [i]K. Dochev and I. Dimovski[/i]

Find the least integer $k$ such that for any $2011 \times 2011$ table filled with integers Kain chooses, Abel be able to change at most $k$ cells to achieve a new table in which $4022$ sums of rows and columns are pairwise different.

Let $A$, $B$, $C$, $D$ be four points such that no three are collinear and $D$ is not the orthocenter of $ABC$. Let $P$, $Q$, $R$ be the orthocenters of $\triangle BCD$, $\triangle CAD$, $\triangle ABD$, respectively. Suppose that the lines $AP$, $BQ$, $CR$ are pairwise distinct and are concurrent. Show that the four points $A$, $B$, $C$, $D$ lie on a circle. [i]Andrew Gu[/i]

Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as $\frac{a}{b}$ for positive integers $a,b$ with $\gcd (a,b) = 1$, then find $a + b$. [i]Proposed by Vismay Sharan[/i]

Hugo places a chess piece on the top left square of a $20 \times 20$ chessboard and makes $10$ moves with it. On each of these $10$ moves, he moves the piece either one square horizontally (left or right) or one square vertically (up or down). After the last move, he draws an $X$ on the square that the piece occupies. When Hugo plays the game over and over again, what is the largest possible number of squares that could eventually be marked with an $X$? Prove that your answer is correct.

(A.Zaslavsky, 8) A triangle can be dissected into three equal triangles. Prove that some its angle is equal to $ 60^{\circ}$.

In $\vartriangle ABC$, the points $D, E$ and $F$ lie on $AB, BC$ and $CA$ respectively. The line segments $AE, BF$ and $CD$ meet at the point $G$. Suppose that the area of each of $\vartriangle BGD, \vartriangle ECG$ and $\vartriangle GFA$ is $1$ cm$^2$. Prove that the area of each of $\vartriangle BEG, \vartriangle GCF$ and $\vartriangle ADG$ is also $1$ cm$^2$. [img]https://cdn.artofproblemsolving.com/attachments/e/7/ec090135bd2e47a9681d767bb984797d87218c.png[/img]

Prove that we can put in any arbitrary triangle with sidelengths $a,b,c$ such that $0\le a,b,c \le \sqrt2$ into a unit cube.