A store increased the original price of a shirt by a certain percent and then decreased the new price by the same percent. Given that the resulting price was 84% of the original price, by what percent was the price increased and decreased? $\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40$

Consider all regular convex and star polygons inscribed in a given circle and having $n$ [i]sides[/i]. We call two such polygons to be equivalent if it is possible to obtain one from the other using a rotation about the center of the circle. How many classes of such polygons exist? [i]Mircea Becheanu[/i]

Given are arbitrary integers $a,b,p$. Prove that there always exist relatively prime integers $k$ and $\ell$ such that $ak+b\ell$ is divisible by $p$.

Find all triples of real numbers $(x, y,z)$ such that $$\begin{cases} x^4 + y^2 + 4 = 5yz \\ y^4 + z^2 + 4 = 5zx \\ z^4 + x^2 + 4 = 5xy\end{cases}$$

Let $ \alpha$ be the positive root of the equation $ x^2 \minus{} 1989x \minus{} 1 \equal{} 0.$ Prove that there exist infinitely many natural numbers $ n$ that satisfy the equation: \[ \lfloor \alpha n \plus{} 1989 \alpha \lfloor \alpha n \rfloor \rfloor \equal{} 1989n \plus{} \left( 1989^2 \plus{} 1 \right) \lfloor \alpha n \rfloor.\]

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 $x=5.1?$ $\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 41 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 57$

At Euclid High School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, 8 in Mr. Newton, and $9$ in Mrs. Young's class are taking the AMC $8$ this year. How many mathematics students at Euclid High School are taking the contest? $ \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30 $

A class collects $ \$50$ to buy flowers for a classmate who is in the hospital. Roses cost $ \$3$ each, and carnations cost $ \$2$ each. No other flowers are to be used. How many different bouquets could be purchased for exactly $ \$50$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$

Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]

In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$? [asy] import olympiad; size(180); real r = 3, s = 5, t = sqrt(r*r+s*s); defaultpen(linewidth(0.6) + fontsize(10)); pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0); draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D)); label("$A$",A,SW); label("$B$", B, NW); label("$C$",C,NE); label("$D$",D,SE); label("$P$",P,S); [/asy] $\textbf{(A) }3\sqrt 5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }6\sqrt 5 \qquad \textbf{(D) }20\qquad \textbf{(E) }25$

Determine all pairs of non-constant polynomials $p(x)$ and $q(x)$, each with leading coefficient $1$, degree $n$, and $n$ roots which are non-negative integers, that satisfy $p(x)-q(x)=1$.

Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different. [i]F. Petrov [/i]

Any two vertices $A,B$ of a regular $n$-gon are connected by an oriented segment (i.e. either $A\to B$ or $B\to A$). Find the maximum possible number of quadruples $(A,B,C,D)$ of vertices such that $A\to B\to C\to D\to A$.

A single player game has the following rules: initially, there are $10$ piles of stones with $1,2,...,10$ stones, respectively. A movement consists on making one of the following operations: i) to choose $2$ piles, both of them with at least $2$ stones, combine them and then add $2$ stones to the new pile; ii) to choose a pile with at least $4$ stones, remove $2$ stones from it, and then split it into two piles with amount of piles to be chosen by the player. The game continues until is not possible to make an operation. a) Give an example of a sequence of moves leading to the end of the game. b) Make a table with the total number of stones and the number of piles before and after the first 5 operations in your example above. c) Show that the number of piles with one stone in the end of the game is always the same, no matter how the movements are made.

Consider sequences $a_0$,$a_1$,$a_2$,$\cdots$ of non-negative integers defined by selecting any $a_0$,$a_1$,$a_2$ (not all 0) and for each $n$ $\geq$ 3 letting $a_n$ = |$a_n-1$ - $a_n-3$| 1-In the particular case that $a_0$ = 1,$a_1$ = 3 and $a_2$ = 2, calculate the beginning of the sequence, listing $a_0$,$a_1$,$\cdots$,$a_{19}$,$a_{20}$. 2-Prove that for each sequence, there is a constant $c$ such that $a_i$ $\leq$ $c$ for all $i$ $\geq$ 0. Note that the constant $c$ my depend on the numbers $a_0$,$a_1$ and $a_2$ 3-Prove that, for each choice of $a_0$,$a_1$ and $a_2$, the resulting sequence is eventually periodic. 4-Prove that, the minimum length p of the period described in (3) is the same for all permitted starting values $a_0$,$a_1$,$a_2$ of the sequence

Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for any $x,y\in \mathbb{R}$, there holds \[f(x+f(y))+f(xy)=yf(x)+f(y)+f(f(x)).\]

In the figure below, each side of the rhombus has length 5 centimeters. [asy] import graph; unitsize(2.0cm); real w = sqrt(3); draw((w, 0) -- (0, 1) -- (-w, 0) -- (0, -1) -- cycle); filldraw(Circle((-0.5, 0), 0.8 / sqrt(pi)), gray); label("$60^\circ$", (w - 0.1, 0), W); [/asy] The circle lies entirely within the rhombus. The area of the circle is $n$ square centimeters, where $n$ is a positive integer. Compute the number of possible values of $n$.

The symbols $ (a,b,\ldots,g)$ and $ [a,b,\ldots,g]$ denote the greatest common divisor and least common multiple, respectively, of the positive integers $ a,b,\ldots,g$. For example, $ (3,6,18)\equal{}3$ and $ [6,15]\equal{}30$. Prove that \[ \frac{[a,b,c]^2}{[a,b][b,c][c,a]}\equal{}\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}.\]

Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that - The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$, - $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$, - The perimeter of $A_1A_2\dots A_{11}$ is $20$. If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive integers $m,n,p,q$ with $n$ squarefree and $\gcd(m,p,q)=1$, find $m+n+p+q$.

There is a rectangular sheet of paper on an infinite blackboard. Marvin secretly chooses a convex $2024$-gon $P$ that lies fully on the piece of paper. Tigerin wants to find the vertices of $P$. In each step, Tigerin can draw a line $g$ on the blackboard that is fully outside the piece of paper, then Marvin replies with the line $h$ parallel to $g$ that is the closest to $g$ which passes through at least one vertex of $P$. Prove that there exists a positive integer $n$, independent of the choice of the polygon, such that Tigerin can always determine the vertices of $P$ in at most $n$ steps.

If $x$ and $y$ are positive real numbers such that $(x + \sqrt{x^2 + 1})(y +\sqrt{y^2 + 1}) = 2018$: The minimum possible value of $x + y$ is A. $\frac{2017}{\sqrt{2018}}$ B. $\frac{2018}{\sqrt{2019}}$ C. $\frac{2017}{2\sqrt{2018}}$ D. $\frac{2019}{\sqrt{2018}}$ E. $\sqrt{3}$

If $0 <x_1,x_2,...,x_n\le 1$, where $n \ge 1$, show that $$\frac{x_1}{1+(n-1)x_1}+\frac{x_2}{1+(n-1)x_2}+...+\frac{x_n}{1+(n-1)x_n}\le 1$$

On each square of an $n$x$n$ board sleeps a dragon. Two dragons are called neighbors if their squares have a side in common. Each turn, Minnie wakes up a dragon which has a living neighbor and Max directs it towards one of its living neighbors. The dragon than breathes fire on that neighbor and destroys it, and then goes back to sleep. Minnie's goal is to minimize the snoring of the dragons and leave as few living dragons as possible. Max is a member of PETD (People for the Ethical Treatment of Dragons), and he wants to save as many dragons as he can. How many dragons will stay alive at the end if 1. $n=4$? 2. $n=5$?

There is point $D$ on edge $AC$ isosceles triangle $ABC$ with base $BC$. There is point $K$ on the smallest arc $CD$ of circumcircle of triangle $BCD$. Ray $CK$ intersects line parallel to line $BC$ through $A$ at point $T$. Let $M$ be midpoint of segment $DT$. Prove that $\angle AKT=\angle CAM$.

Given is a triangle $ABC$ and the points $M, P$ lie on the segments $AB, BC$, respectively, such that $AM=BC$ and $CP=BM$. If $AP$ and $CM$ meet at $O$ and $2\angle AOM=\angle ABC$, find the measure of $\angle ABC$.