Points $M$ and $K$ are chosen on lateral sides $AB,AC$ of an isosceles triangle $ABC$ and point $D$ is chosen on $BC$ such that $AMDK$ is a parallelogram. Let the lines $MK$ and $BC$ meet at point $L$, and let $X,Y$ be the intersection points of $AB,AC$ with the perpendicular line from $D$ to $BC$. Prove that the circle with center $L$ and radius $LD$ and the circumcircle of triangle $AXY$ are tangent.

The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$? $\textbf{(A)}\ \frac{8a^2}{(a+1)^2}\qquad\textbf{(B)}\ \frac{4a}{a+1}\qquad\textbf{(C)}\ \frac{8a}{a+1}\qquad\textbf{(D)}\ \frac{8a^2}{a^2+1}\qquad\textbf{(E)}\ \frac{8a}{a^2+1}$

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

Find all positive integer solutions $(a, b, c)$ to the function $a^{2} + b^{2} + c^{2} = 2005$, where $a \leq b \leq c$.

In the accompanying figure, the outer square has side length 40. A second square S' of side length 15 is constructed inside S with the same center as S and with sides parallel to those of S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S'. The result is a four-pointed starlike figure inscribed in S. The star figure is cut out and then folded to form a pyramid with base S'. Find the volume of this pyramid. [asy] draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0)); draw((2.5,2.5)--(4,0)--(5.5,2.5)--(8,4)--(5.5,5.5)--(4,8)--(2.5,5.5)--(0,4)--(2.5,2.5)--(5.5,2.5)--(5.5,5.5)--(2.5,5.5)--(2.5,2.5)); [/asy]

Al and Bob are playing Rock Paper Scissors. Al plays rock. What is the probability that Al wins, given that Bob plays randomly and has an equal probability of playing rock, paper, and scissors?

If the point $(x,-4)$ lies on the straight line joining the points $(0,8)$ and $(-4,0)$ in the xy-plane, then $x$ is equal to $\textbf{(A) }-2\qquad\textbf{(B) }2\qquad\textbf{(C) }-8\qquad\textbf{(D) }6\qquad \textbf{(E) }-6$

The number of points equidistant from a circle and two parallel tangents to the circle is: $\textbf{(A) }0\qquad \textbf{(B) }2\qquad \textbf{(C) }3\qquad \textbf{(D) }4\qquad \textbf{(E) }\text{infinite}$

a) Prove that the fraction $\frac{3n+5}{2n+3}$ is irreducible for every $n \in N$ b) Let $x,y$ be digits of decimal representation system with $x>0$, and $\frac{\overline{xy}+12}{\overline{xy}-3}\in N$, prove that $x+y=9$. Is the converse true?

$ ABCD$ is a quadrilateral with $ BC$ parallel to $ AD$. $ M$ is the midpoint of $ CD$, $ P$ is the midpoint of $ MA$ and $ Q$ is the midpoint of $ MB$. The lines $ DP$ and $ CQ$ meet at $ N$. Prove that $ N$ is inside the quadrilateral $ ABCD$.

Find all integer solutions of the equation the equation $2x^2-y^{14}=1$.

The water city of Platense consists of many platforms and bridges between them. Each bridge connects two platforms and there is not two bridges connecting the same two platforms. The mayor wants to switch some bridges by a series of moves in the following way: if there are three platforms $A,B,C$ and bridges $AB$ and $AC$ ([b]no[/b] bridge $BC$), he can switch bridge $AB$ to a bridge $BC$. A configuration of bridges is [i]good[/i] if it is possible to go to any platfom from any platform using only bridges. Starting in a good configuration, prove that the mayor can reach any other good configuration, whose the quantity of bridges is the same, using the allowed moves.

Let $m$ be a positive integers. A square room with corners at $(0,0), (2m,0), (0,2m),$ $(2m,2m)$ has mirrors as walls. At each integer lattice point $(i,j)$ with $0 < i, j < 2m$ a single small double sided mirror is oriented parallel to either the $x$ or $y$ axis. A beam of light is shone from a corner making a $45^\circ$ angle with each of the walls. Prove that the opposite corner is not lit.

Let $ABC$ be an isosceles right triangle at $A$ with $AB=AC$. Let $M$ and $N$ be on side $BC$, with $M$ between $B$ and $N,$ such that $$BM^2+ NC^2= MN^2.$$ Determine the measure of the angle $\angle MAN.$

Let $\alpha, c > 0$, define $x_1 = c$ and let $x_{n + 1} = x_n e^{-x_n^\alpha}$ for $n \geq 1$. For which values of $\beta$ does $\sum_{i = 1}^{\infty} x_n^\beta$ converge?

Let $ 9 < n_{1} < n_{2} < \ldots < n_{s} < 1992$ be positive integers and \[ P(x) \equal{} 1 \plus{} x^{2} \plus{} x^{9} \plus{} x^{n_{1}} \plus{} \cdots \plus{} x^{n_{s}} \plus{} x^{1992}.\] Prove that if $ x_{0}$ is real root of $ P(x)$ then $ x_{0}\leq\frac {1 \minus{} \sqrt {5}}{2}$.

There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let $p_{a,b}$ be the sum of the zeroes of $P(x)$. Find the sum of the $p_{a,b}$'s for all possible combinations of $a$ and $b$.

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

Let the vertex set \( V \) of a graph be partitioned into \( h \) parts \( (V = V_1 \cup V_2 \cup \cdots \cup V_h) \), with \(|V_1| = n_1, |V_2| = n_2, \ldots, |V_h| = n_h \). If there is an edge between any two vertices only when they belong to different parts, the graph is called a complete \( h \)-partite graph, denoted as \( k(n_1, n_2, \ldots, n_h) \). Let \( n \) and \( r \) be positive integers, \( n \geq 6 \), \( r \leq \frac{2}{3}n \). Consider the complete \( r + 1 \)-partite graph \( k\left(\underbrace{1, 1, \ldots, 1}_{r}, n - r\right) \). Answer the following questions: 1. Find the maximum number of disjoint circles (i.e., circles with no common vertices) in this complete \( r + 1 \)-partite graph. 2. Given \( n \), for all \( r \leq \frac{2}{3}n \), find the maximum number of edges in a complete \( r + 1 \)-partite graph \( k(1, 1, \ldots, 1, n - r) \) where no more than one circle is disjoint.

[b]p1.[/b] What is $20\times 19 + 20 \div (2 - 7)$? [b]p2.[/b] Will has three spinners. The first has three equally sized sections numbered $1$, $2$, $3$; the second has four equally sized sections numbered $1$, $2$, $3$, $4$; and the third has five equally sized sections numbered $1$, $2$, $3$, $4$, $5$. When Will spins all three spinners, the probability that the same number appears on all three spinners is $p$. Compute $\frac{1}{p}$. [b]p3.[/b] Three girls and five boys are seated randomly in a row of eight desks. Let $p$ be the probability that the students at the ends of the row are both boys. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$. [b]p4.[/b] Jaron either hits a home run or strikes out every time he bats. Last week, his batting average was $.300$. (Jaron's batting average is the number of home runs he has hit divided by the number of times he has batted.) After hitting $10$ home runs and striking out zero times in the last week, Jaron has now raised his batting average to $.310$. How many home runs has Jaron now hit? [b]p5.[/b] Suppose that the sum $$\frac{1}{1 \cdot 4} +\frac{1}{4 \cdot 7}+ ...+\frac{1}{97 \cdot 100}$$ is expressible as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p6.[/b] Let $ABCD$ be a unit square with center $O$, and $\vartriangle OEF$ be an equilateral triangle with center $A$. Suppose that $M$ is the area of the region inside the square but outside the triangle and $N$ is the area of the region inside the triangle but outside the square, and let $x = |M -N|$ be the positive difference between $M$ and $N$. If $$x =\frac1 8(p -\sqrt{q})$$ for positive integers $p$ and $q$, find $p + q$. [b]p7.[/b] Find the number of seven-digit numbers such that the sum of any two consecutive digits is divisible by $3$. For example, the number $1212121$ satisfies this property. [b]p8.[/b] There is a unique positive integer $x$ such that $x^x$ has $703$ positive factors. What is $x$? [b]p9.[/b] Let $x$ be the number of digits in $2^{2019}$ and let $y$ be the number of digits in $5^{2019}$. Compute $x + y$. [b]p10.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 13$ and $BC = 10$. Consider the set of all points $D$ in three-dimensional space such that $BCD$ is an equilateral triangle. This set of points forms a circle $\omega$. Let $E$ and $F$ be points on $\omega$ such that $AE$ and $AF$ are tangent to $\omega$. If $EF^2$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, determine $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a square $ABCD$, let us consider an equilateral triangle $KLM$, whose vertices $K$, $L$ and $M$ belong to the sides $AB$, $BC$ and $CD$ respectively. Find the locus of the midpoints of the sides $KL$ for all possible equilateral triangles $KLM$. Note: The set of points that satisfy a property is called a locus.

Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips? $ \textbf{(A)} 4 \qquad\textbf{(B)} 5 \qquad\textbf{(C)} 6 \qquad\textbf{(D)} 7 \qquad\textbf{(E)} 9 $

Find the number of three-digit numbers such that its first two digits are each divisible by its third digit.

Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying \[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\ 1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\ + f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\ + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases} \] for all nonnegative integers $ p$, $ q$, $ r$.

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that \[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\] for all $x,y>0.$