In a quadrilateral $ABCD$ the intersection of the diagonals is called $P$. Point $X$ is the orthocentre of triangle $PAB$. (The orthocentre of a triangle is the point where the three altitudes of the triangle intersect.) Point $Y$ is the orthocentre of triangle $PCD$. Suppose that $X$ lies inside triangle $PAB$ and $Y$ lies inside triangle $PCD$. Moreover, suppose that $P$ is the midpoint of line segment $XY$ . Prove that $ABCD$ is a parallelogram. [asy] import geometry; unitsize (1.5 cm); pair A, B, C, D, P, X, Y; A = (0,0); B = (2,-0.5); C = (3.5,2.2); D = A + C - B; P = (A + C)/2; X = orthocentercenter(A,B,P); Y = orthocentercenter(C,D,P); draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(A--extension(A,X,B,P), dotted); draw(B--extension(B,X,A,P), dotted); draw(P--extension(P,X,A,B), dotted); draw(C--extension(C,Y,D,P), dotted); draw(D--extension(D,Y,C,P), dotted); draw(P--extension(P,Y,C,D), dotted); dot("$A$", A, W); dot("$B$", B, S); dot("$C$", C, E); dot("$D$", D, N); dot("$P$", P, E); dot("$X$", X, NW); dot("$Y$", Y, SE); [/asy]

What is the value of $2-(-2)^{-2}$? $ \textbf{(A) } -2 \qquad\textbf{(B) } \dfrac{1}{16} \qquad\textbf{(C) } \dfrac{7}{4} \qquad\textbf{(D) } \dfrac{9}{4} \qquad\textbf{(E) } 6 $

Let $a_1, a_2, ..., a_{12}$ be the vertices of a regular dodecagon $D_1$ ($12$-gon). The four vertices $a_1$, $a_4$, $a_7$, $a_{10}$ form a square, as do the four vertices $a_2$, $a_5$, $a_8$, $a_{11}$ and $a_3$, $a_6$, $a_9$, $a_{12}$. Let $D_2$ be the polygon formed by the intersection of these three squares. If we let$ [A]$ denotes the area of polygon $A$, compute $\frac{[D_2]}{[D_1]}$.

Let $ABCD$ be a cyclic quadrilateral such that $AB \cdot BC=2 \cdot AD \cdot DC$. Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \le 9AC^2$. [color=#FF0000]Moderator says: Use the search before posting contest problems [url]http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=530783[/url][/color]

Let $\eta \in [0, 1]$ be a relative measure of material absorbence. $\eta$ values for materials combined together are additive. $\eta$ for a napkin is $10$ times that of a sheet of paper, and a cardboard roll has $\eta = 0.75$. Justin can create a makeshift cup with $\eta = 1$ using $50$ napkins and nothing else. How many sheets of paper would he need to add to a cardboard roll to create a makeshift cup with $\eta = 1$?

If $a_1, ... ,a_{12}$ are twelve nonzero integers such that $a^6_1+...·+a^6_{12} = 450697$, what is the value of $a^2_1+...+a^2_{12}$?

A permutation of the integers \(2020, 2021,...,2118, 2119\) is a list \(a_1,a_2,a_3,...,a_{100}\) where each one of the numbers appears exactly once. For each permutation we define the partial sums. $s_1=a_1$ $s_2=a_1+a_2$ $s_3=a_1+a_2+a_3$ $...$ $s_{100}=a_1+a_2+...+a_{100}$ How many of these permutations satisfy that none of the numbers \(s_1,...,s_{100}\) is divisible by $3$?

Let $a<b<c<d$ be integers. If one of the roots of the equation $(x-a)(x-b)(x-c)(x-d)-9$ is $x=7$, what is $a+b+c+d$? $ \textbf{(A)}\ 14 \qquad\textbf{(B)}\ 21 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 42 \qquad\textbf{(E)}\ 63 $

Problem 4. Let$\left( A,+,\cdot \right)$ be a ring with the property that $x=0$ is the only solution of the ${{x}^{2}}=0,x\in A$ecuation. Let $B=\left\{ a\in A|{{a}^{2}}=1 \right\}$. Prove that: (a) $ab-ba=bab-a$, whatever would be $a\in A$ and $b\in B$. (b) $\left( B,\cdot \right)$ is a group

Let $N=10^{2024}$. $S$ is a square in the Cartesian plane with side length $N$ and the sides parallel to the coordinate axes. Inside there are $N$ points $P_1$, $P_2$, $\dots$, $P_N$ all of which have different $x$ coordinates, and the absolute value of the slope of any connected line between these points is at most $1$. Prove that there exists a line $l$ such that at least $2024$ of these points is at most distance $1$ away from $l$.

Prove that $$\sum_{k=0}^\infty x^k\frac{1+x^{2k+2}}{(1-x^{2k+2})^2}=\sum_{k=0}^\infty(-1)^k\frac{x^k}{(1-x^{k+1})^2}$$for all $x\in(-1,1)$.

In triangle $\vartriangle ABC$, $M$ is the midpoint of $\overline{AB}$ and $N$ is the midpoint of $\overline{AC}$. Let $P$ be the midpoint of $\overline{BN}$ and let $Q$ be the midpoint of $\overline{CM}$. If $AM = 6$, $BC = 8$ and $BN = 7$, compute the perimeter of triangle $\vartriangle NP Q$.

Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$. Prove the inequality $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$

How many zeros does $100!$ have at its end in the usual decimal representation?

Let $AC>BC$ in $\triangle ABC$ and $M$ and $N$ be the midpoints of $AC$ and $BC$ respectively. The angle bisector of $\angle B$ intersects $\overline{MN}$ at $P$. The incircle of $\triangle ABC$ has center $I$ and touches $BC$ at $Q$. The perpendiculars from $P$ and $Q$ to $MN$ and $BC$ respectively intersect at $R$. Let $S=AB\cap RN$. a) Prove that $PCQI$ is cyclic b) Express the length of the segment $BS$ with $a$, $b$, $c$ - the side lengths of $\triangle ABC$ .

In how many ways can the following figure be tiled with $2 \times 1$ dominos? [asy] defaultpen(linewidth(.8)); size(5.5cm); int i; for(i = 1; i<6; i = i+1) { draw((.5 + i,6-i)--(.5 + i,i-6)--(-(.5 + i),i-6)--(-(.5 + i),6-i)--cycle);} draw((.5,5)--(.5,-5)^^(-.5,5)--(-.5,-5)^^(5.5,0)--(-5.5,0)); [/asy] [i]Proposed by Lewis Chen[/i]

Given a triangle $ABC$, and let $ E $ and $ F $ be the feet of altitudes from vertices $ B $ and $ C $ to the opposite sides. Denote $ O $ and $ H $ be the circumcenter and orthocenter of triangle $ ABC $. Given that $ FA=FC $, prove that $ OEHF $ is a parallelogram.

Let $m$ and $n$ be positive integers. Fuming Zeng gives James a rectangle, such that $m-1$ lines are drawn parallel to one pair of sides and $n-1$ lines are drawn parallel to the other pair of sides (with each line distinct and intersecting the interior of the rectangle), thus dividing the rectangle into an $m\times n$ grid of smaller rectangles. Fuming Zeng chooses $m+n-1$ of the $mn$ smaller rectangles and then tells James the area of each of the smaller rectangles. Of the $\dbinom{mn}{m+n-1}$ possible combinations of rectangles and their areas Fuming Zeng could have given, let $C_{m,n}$ be the number of combinations which would allow James to determine the area of the whole rectangle. Given that \[A=\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{C_{m,n}\binom{m+n}{m}}{(m+n)^{m+n}},\] then find the greatest integer less than $1000A$. [i]Proposed by James Lin

Find all four-digit numbers in which the thousands digit is equal to the hundreds digit and the tens digit is equal to the units digit and which are squares of integers.

Consider the set \[M = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in\mathcal{M}_2(\mathbb{C})\ |\ ab = cd \right\}.\] a) Give an example of matrix $A\in M$ such that $A^{2017}\in M$ and $A^{2019}\in M$, but $A^{2018}\notin M$. b) Show that if $A\in M$ and there exists the integer number $k\ge 1$ such that $A^k \in M$, $A^{k + 1}\in M$ si $A^{k + 2} \in M$, then $A^n\in M$, for any integer number $n\ge 1$.

Let $X$ be a bounded, nonempty set of points in the Cartesian plane. Let $f(X)$ be the set of all points that are at a distance of at most $1$ from some point in $X$. Let $f_n(X) = f(f(\cdots(f(X))\cdots))$ ($n$ times). Show that $f_n(X)$ becomes “more circular” as $n$ gets larger. In other words, if $r_n = \sup\{\text{radii of circles contained in } f_n(X) \}$ and $R_n = \inf \{\text{radii of circles containing } f_n(X)\}$, then show that $R_n/r_n$ gets arbitrarily close to $1$ as $n$ becomes arbitrarily large. [hide]I'm not sure that I'm posting this in a right forum. If it's in a wrong forum, please mods move it.[/hide]

Alice has got a circular key ring with $n$ keys, $n\ge3$. When she takes it out of her pocket, she does not know whether it got rotated and/or flipped. The only way she can distinguish the keys is by coloring them (a color is assigned to each key). What is the minimum number of colors needed?

The sum of the numerical coefficients in the complete expansion of $ (x^2 \minus{} 2xy \plus{} y^2)^7$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 128 \qquad \textbf{(E)}\ 128^2$

Show that there exists infinite triples $(x,y,z) \in N^3$ such that $x^2+y^2+z^2=3xyz$.

Prove that for a positive number $ r>1, $ there is a nondecreasing sequence of positive numbers $ \left( a_v\right)_{v\ge 1} $ such that $$ r=\lim_{n\to\infty }\sum_{i=1}^n \frac{a_i}{a_{i+1}} . $$