A set S consists of triangles whose sides have integer lengths less than $5$, and no two elements of S are congruent or similar. What is the largest number of elements that $S$ can have? ${\textbf{(A)}\ \ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}}\ 11\qquad\textbf{(E)}\ 12 $

Yang the Spinning Square Sheep is a square in the plane such that his four legs are his four vertices. Yang can do two different types of [i]tricks[/i]: (a) Yang can choose one of his sides, then reflect himself over the side. (b) Yang can choose one of his legs, then rotate $90^\circ$ counterclockwise around the leg. Yang notices that after $2016$ tricks, each leg ends up in exactly the same place the leg started out in! Let there be $N$ ways for Yang to perform his $2016$ tricks. What is the remainder when $N$ is divided by $100000$? [i] Proposed by James Lin [/i]

Three sequences ${a_n},{b_n},{c_n}$ satisfy the following conditions. [list] [*]$a_1=2,\,b_1=4,\,c_1=5$ [*]$\forall n,\; a_{n+1}=b_n+\frac{1}{c_n}, \, b_{n+1}=c_n+\frac{1}{a_n}, \, c_{n+1}=a_n+\frac{1}{b_n}$ [/list] Prove that for all positive integers $n$, $ $ $ $ $max(a_n,b_n,c_n)>\sqrt{2n+13}$.

Show that for all $n\ge 3$ there are $n$ different positive integers $x_1,x_2, ...,x_n$ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= 1.$$

Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

Find functions $ f: \mathbb{N} \rightarrow \mathbb{N}$, such that $ f(x^2 \plus{} f(y)) \equal{} xf(x) \plus{} y$, for $ x,y \in \mathbb{N}$.

Let us call an integer sequence $\{ a_1,a_2, \dots \}$ nice if there exist a function $f: \mathbb{Z^+} \to \mathbb{Z^+} $ such that $$a_i \equiv a_j \pmod{n} \iff i\equiv j \pmod{f(n)}$$ for all $i,j,n \in \mathbb{Z^+}$. Find all nice sequences.

Determine the smallest natural number $n =>2$ with the property: For every positive integers $a_1, a_2,. . . , a_n$ the product of all differences $a_j-a_i$, $1 <=i <j <=n$, is divisible by 2001.

A [i]kite[/i] is a quadrilateral whose diagonals are perpendicular. Let kite $ABCD$ be such that $\angle B = \angle D = 90^\circ$. Let $M$ and $N$ be the points of tangency of the incircle of $ABCD$ to $AB$ and $BC$ respectively. Let $\omega$ be the circle centered at $C$ and tangent to $AB$ and $AD$. Construct another kite $AB^\prime C^\prime D^\prime$ that is similar to $ABCD$ and whose incircle is $\omega$. Let $N^\prime$ be the point of tangency of $B^\prime C^\prime$ to $\omega$. If $MN^\prime \parallel AC$, then what is the ratio of $AB:BC$?

On the first day of school, Ashley the teacher asked some of her students what their favorite color was and used those results to construct the pie chart pictured below. During this first day, $165$ students chose yellow as their favorite color. The next day, she polled $30$ additional students and was shocked when none of them chose yellow. After making a new pie chart based on the combined results of both days, Ashley noticed that the angle measure of the sector representing the students whose favorite color was yellow had decreased. Compute the difference, in degrees, between the old and the new angle measures. [img]https://cdn.artofproblemsolving.com/attachments/2/5/f605bf8d684075fe13fee9eb44f8f50b64c7d3.png[/img]

Let \(d(n)\) denote the number of positive divisors of \(n\). The sequence \(a_0\), \(a_1\), \(a_2\), \(\ldots\) is defined as follows: \(a_0=1\), and for all integers \(n\ge1\), \[a_n=d(a_{n-1})+d(d(a_{n-2}))+\cdots+ {\underbrace{d(d(\ldots d(a_0)\ldots))}_{n\text{ times}}}.\] Show that for all integers \(n\ge1\), we have \(a_n\le3n\). [i]Proposed by Karthik Vedula[/i]

Acute-angled triangle ${ABC}$ with ${AB<AC<BC}$ and let be ${c(O,R)}$ it’s circumcircle. Diameters ${BD}$ and ${CE}$ are drawn. Circle ${c_1(A,AE)}$ interescts ${AC}$ at ${K}$. Circle ${{c}_{2}(A,AD)}$ intersects ${BA}$ at ${L}$ .(${A}$ lies between ${B}$ and ${L}$). Prove that lines ${EK}$ and ${DL}$ intersect at circle $c$ . by Evangelos Psychas (Greece)

Show that there is a square with side length $14$ whose floor may be covered (exact coverage of the square area) by $21$ squares so that between them there is exactly $6$ squares with side length $1$, $5$ squares with side length $2$, $4$ squares with side length $3$, $3$ squares with side length $4$, $2$ squares with side length $5$ and a square with side length $6$ .

Solve the inequality $$3-2\left(3-2\left(3-...-2(3-2x)\right)...\right) >x$$. The total number of right parentheses is $100$.

Let $A_1, A_2, \ldots , A_m$ be subsets of the set $\{ 1,2, \ldots , n \}$, such that the cardinal of each subset $A_i$, such $1 \le i \le m$ is not divisible by $30$, while the cardinal of each of the subsets $A_i \cap A_j$ for $1 \le i,j \le m$, $i \neq j$ is divisible by $30$. Prove \begin{align*} 2m - \left \lfloor \frac{m}{30} \right \rfloor \le 3n \end{align*}

Let $\{X_n\}$ and $\{Y_n\}$ be sequences defined as follows: \[X_0=Y_0=X_1=Y_1=1,\] \begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*} Let $k$ be the largest integer that satisfies all of the following conditions: [list=i][*] $|X_i-k|\leq 2007$, for some positive integer $i$; [*] $|Y_j-k|\leq 2007$, for some positive integer $j$; and [*] $k<10^{2007}.$[/list] Find the remainder when $k$ is divided by $2007$.

Define a figure which is constructed by unit squares "cross star" if it satisfies the following conditions: $(1)$Square bar $AB$ is bisected by square bar $CD$ $(2)$At least one square of $AB$ lay on both sides of $CD$ $(3)$At least one square of $CD$ lay on both sides of $AB$ There is a rectangular grid sheet composed of $38\times 53=2014$ squares,find the number of such cross star in this rectangle sheet

Prove that in any set of $17$ distinct natural numbers one can either find five numbers so that four of them are divisible into the other or five numbers none of which is divisible into any other. (An established theorem)

A square paper of side $n$ is divided into $n^2$ unit square cells. A maze is drawn on the paper with unit walls between some cells in such a way that one can reach every cell from every other cell not crossing any wall. Find, in terms of $n$, the largest possible total length of the walls.

Adam has a single stack of $3 \cdot 2^n$ rocks, where $n$ is a nonnegative integer. Each move, Adam can either split an existing stack into two new stacks whose sizes differ by $0$ or $1$, or he can combine two existing stacks into one new stack. Adam keeps performing such moves until he eventually gets at least one stack with $2^n$ rocks. Find, with proof, the minimum possible number of times Adam could have combined two stacks. [i]Proposed by Anthony Wang[/i]

The real numbers $a_1,a_2,a_3$ and $b{}$ are given. The equation \[(x-a_1)(x-a_2)(x-a_3)=b\]has three distinct real roots, $c_1,c_2,c_3.$ Determine the roots of the equation \[(x+c_1)(x+c_2)(x+c_3)=b.\][i]Proposed by A. Antropov and K. Sukhov[/i]

Let $\triangle ABC$ be a triangle with side‐lengths $a,b,c$, incenter $I$, and circumradius $R$. Denote by $P$ the area of $\triangle ABC$, and let $P_1,\;P_2,\;P_3$ be the areas of triangles $\triangle ABI$, $\triangle BCI$, and $\triangle CAI$, respectively. Prove that \[ \frac{abc}{12R} \;\le\; \frac{P_1^2 + P_2^2 + P_3^2}{P} \;\le\; \frac{3R^3}{4\sqrt[3]{abc}}. \]

Fourteen friends met at a party. One of them, Fredek, wanted to go to bed early. He said goodbye to 10 of his friends, forgot about the remaining 3, and went to bed. After a while he returned to the party, said goodbye to 10 of his friends (not necessarily the same as before), and went to bed. Later Fredek came back a number of times, each time saying goodbye to exactly 10 of his friends, and then went back to bed. As soon as he had said goodbye to each of his friends at least once, he did not come back again. In the morning Fredek realized that he had said goodbye a different number of times to each of his thirteen friends! What is the smallest possible number of times that Fredek returned to the party?

The polynomial $ P(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}d$ has three distinct real roots. The polynomial $ P(Q(x))$, where $ Q(x)\equal{}x^2\plus{}x\plus{}2001$, has no real roots. Prove that $ P(2001)>\frac{1}{64}$.

Prove that if $ M $, $ N $, $ P $ are the feet of the altitudes of acute-angled triangle $ ABC $, then the ratio of the perimeter of triangle $ MNP $ to the perimeter of triangle $ ABC $ is equal to the ratio of the radius of the circle inscribed in triangle $ ABC $ to the radius of the circle circumscribed about triangle $ ABC $.