Points $A, B, C, D$ lie on a line $l$, in that order. Find the locus of points $P$ in the plane for which $\angle{APB}=\angle{CPD}$.

Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids". Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".

Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$. [list=a] [*] Determine its units digit. [*] Determine its tens digit. [/list]

For how many positive integers $n\le100$ is it true that $10n$ has exactly three times as many positive divisors as $n$ has?

Find the ratio between the sum of the areas of the circles and the area of the fourth circle that are shown in the figure Each circle passes through the center of the previous one and they are internally tangent. [img]https://cdn.artofproblemsolving.com/attachments/d/2/29d2be270f7bcf9aee793b0b01c2ef10131e06.jpg[/img]

Prove that the number $19^{1976} + 76^{1976}$: $(a)$ is divisible by the (Fermat) prime number $F_4 = 2^{2^4} + 1$; $(b)$ is divisible by at least four distinct primes other than $F_4$.

Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1,3,4,5,6,$ and $9$. What is the sum of the possible values for $w$? $ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93 $

Suppose $r$, $s$, and $t$ are nonzero reals such that the polynomial $x^2 + rx + s$ has $s$ and $t$ as roots, and the polynomial $x^2 + tx + r$ has $5$ as a root. Compute $s$.

Let $x_1$ be a given positive integer. A sequence $\{x_n\}_ {n\geq 1}$ of positive integers is such that $x_n$, for $n \geq 2$, is obtained from $x_ {n-1}$ by adding some nonzero digit of $x_ {n-1}$. Prove that a) the sequence contains an even term; b) the sequence contains infinitely many even terms.

The four sets A, B, C, and D each have $400$ elements. The intersection of any two of the sets has $115$ elements. The intersection of any three of the sets has $53$ elements. The intersection of all four sets has $28$ elements. How many elements are there in the union of the four sets?

Find the least possible value of the following expression $\lfloor \frac{a+b}{c+d}\rfloor +\lfloor \frac{a+c}{b+d}\rfloor +\lfloor \frac{a+d}{b+c}\rfloor + \lfloor \frac{c+d}{a+b}\rfloor +\lfloor \frac{b+d}{a+c}\rfloor +\lfloor \frac{b+c}{a+d}\rfloor$ where $a$, $b$, $c$ and $d$ are positive real numbers.

Suppose $f(x)$ is a polynomial with integer coefficients such that $f(0) = 11$ and $f(x_1) = f(x_2) = ... = f(x_n) = 2002$ for some distinct integers $x_1, x_2, . . . , x_n$. Find the largest possible value of $n$.

Let $x,y\in \mathbb{N}$ and $k=\frac{x^2+y^2}{2xy+1}$. Determine all natural values of $k$.

Let $ABC$ be an acute-angled triangle with circumscribed circle $k$ and centre of the circumscribed circle $O$. A line through $O$ intersects the sides $AB$ and $AC$ at $D$ and $E$.Denote by $B'$ and $C'$ the reflections of $B$ and $C$ over $O$, respectively. Prove that the circumscribed circles of $ODC'$ and $OEB'$ concur on $k$.

A set of $n$ lights, numbered $1$ to $n$, are initially off. At every moment, it is possible to perform one of the following operations: $\bullet$ change the state of lamp $1$, $\bullet$ change the state of lamp $2$, as long as lamp $1$ is on, $\bullet$ change the state of lamp $k > 2$, as long as lamp $k - 1$ is on and all lamps $1, . . . , k - 2$ are off. It shows that it is possible, after a certain number of operations, to have only the lamp left on.

Boy A and $2016$ flags are on a circumference whose length is $1$ of a circle. He wants to get all flags by moving on the circumference. He can get all flags by moving distance $l$ regardless of the positions of boy A and flags. Find the possible minimum value as $l$ like this. Note that boy A doesn’t have to return to the starting point to leave gotten flags.

There are $100$ points on the circumference of a circle, arbitrarily labelled by $1,2,\ldots,100$. For each three points, call their triangle [b]clockwise[/b] if the increasing order of them is in clockwise order. Prove that it is impossible to have exactly $2017$ [b]clockwise[/b] triangles.

Prove that there exist infinitely many pairs $\left(x;\;y\right)$ of different positive rational numbers, such that the numbers $\sqrt{x^2+y^3}$ and $\sqrt{x^3+y^2}$ are both rational.

Let $k>1$ be a positive integer. On a $\text{k} \times \text{k}$ square grid, Tom and Jerry are on opposite corners, with Tom at the top right corner. Both can move to an adjacent square every move, where two squares are adjacent if they share a side. Tom and Jerry alternate moves, with Jerry going first. Tom [i]catches[/i] Jerry if they are on the same square. We aim to answer to the following question: What is the smallest number of moves that Tom needs to guarantee catching Jerry? (a) Without proof, find the answer in the cases of $k=2,3,4$, and (correctly) guess what the answer is in terms of $k$. We'll refer to this answer as $A(k)$. (b) Find a strategy that Jerry can use to guarantee that Tom takes at least $A(k)$ moves to catch Jerry. Now, you will find a strategy for Tom to catch Jerry in at most $A(k)$ moves, no matter what Jerry does. (c) Find, with proof, a working strategy for $k=5$. (d) Find, with proof, a working strategy for all $k \geq 2$.

If $(\log_2 3)^x-(\log_5 3)^x\geq (\log_2 3)^{-y}-(\log_5 3)^{-y}$, then $\text{(A)}x-y\geq0\qquad\text{(B)}x+y\geq0\qquad\text{(C)}x-y\leq0\qquad\text{(D)}x+y\leq0$

A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)

Find all non-integers $x$ such that $x+\frac{13}{x}=[x]+\frac{13}{[x]} . $where$[x]$ mean the greatest integer $n$ , where $n\leq x.$

Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$ $\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 41 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 57$

Find all primes $p$ such that $p+2$ and $p^2+2p-8$ are also primes.