How many non-congruent isosceles triangles (including equilateral triangles) have positive integer side lengths and perimeter less than 20?

Do there exist distinct reals $x, y, z$, such that $\frac{1}{x^2+x+1}+\frac{1}{y^2+y+1}+\frac{1}{z^2+z+1}=4$?

A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$. [i]Proposed by Mongolia[/i]

Richard has a four infinitely large piles of coins: a pile of pennies (worth 1 cent each), a pile of nickels (5 cents), a pile of dimes (10 cents), and a pile of quarters (25 cents). He chooses one pile at random and takes one coin from that pile. Richard then repeats this process until the sum of the values of the coins he has taken is an integer number of dollars. (One dollar is 100 cents.) What is the expected value of this final sum of money, in cents? [i]Proposed by Lewis Chen[/i]

For what value of $ x$ does \[ \log_{\sqrt{2}} \sqrt{x} \plus{} \log_2 x \plus{} \log_4 (x^2) \plus{} \log_8 (x^3) \plus{} \log_{16} (x^4) \equal{} 40?\] $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 256 \qquad \textbf{(E)}\ 1024$

Circle $\omega_1$ has diameter $AB$, and circle $\omega_2$ has center $A$ and intersects $\omega_1$ at points $C$ and $D$. Let $E$ be the intersection of $AB$ and $CD$. Point $P$ is chosen on $\omega_2$ such that $P C = 8$, $P D = 14$, and $P E = 7$. Find the length of $P B$.

Let $X,Y$ be two points on the side $BC$ of triangle $ABC$ such that $2XY=BC$ ($X$ is between $B,Y$). Let $AA'$ be the diameter of the circumcirle of triangle $AXY$. Let $P$ be the point where $AX$ meets the perpendicular from $B$ to $BC$, and $Q$ be the point where $AY$ meets the perpendicular from $C$ to $BC$. Prove that the tangent line from $A'$ to the circumcircle of $AXY$ passes through the circumcenter of triangle $APQ$. [i]Proposed by Iman Maghsoudi[/i]

Starting at $(0, 0)$, Richard takes $2n+1$ steps, with each step being one unit either East, North, West, or South. For each step, the direction is chosen uniformly at random from the four possibilities. Determine the probability that Richard ends at $(1, 0)$.

Consider a triangle $\vartriangle ABC$ with $BC = 10$. An excircle is a circle that is tangent to one side of the triangle as well as the extensions of the other two sides; suppose that the excircle opposite vertex $B$ has center $I_2$ and exradius $r_2 = 11$, and suppose that the excircle opposite vertex $C$ has center $I_3$ and exradius $r_3 = 13$. Compute $I_2I_3$.

[u]Prottasha[/u] has a 10 sided dice. She throws the dice two times and sum the numbers she gets. Which number has the most probability to come out?

15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible to have equal numbers of apricots in all the boxes after $k$ moves.

Let $X = Q-\{-1,0,1\}$. We consider the function $f: X\to X$ given by $f (x) = x -\frac{1}{x} .$ Is there an $a \in X$ such that for every natural number n there is a $y \in X$ with $f (f (...( f (y)) ...)) = a$ where $f$ occurs exactly $n$ times on the left side?

Let $k\ge2$ be an integer. The function $f:\mathbb N\to\mathbb N$ is defined by $$f(n)=n+\left\lfloor\sqrt[k]{n+\sqrt[k]n}\right\rfloor.$$Determine the set of values taken by the function $f$.

For positive integers $x$, let$$f(x)=\begin{cases} \frac{f\left(\frac{x}{2}\right)}{2} &\mbox{if }x\mbox{ is even,} \\ 2^{-x} &\mbox{if }x\mbox{ is odd.} \end{cases}$$Find $f(1)+f(2)+f(3)+\dots$.

For each positive integer $n$, let $a_n = 0$ (or $1$) if the number of $1$’s in the binary representation of $n$ is even (or odd), respectively. Show that there do not exist positive integers $k$ and $m$ such that $$a_{k+j}=a_{k+m+j} =a_{k+2m+j}$$ for $0 \leq j \leq m-1.$

[u]Round 1[/u] [b]p1.[/b] Three snails – Alice, Bobby, and Cindy – were racing down a road. Whenever one snail passed another, it waved at the snail it passed. During the race, Alice waved $3$ times and was waved at twice. Bobby waved $4$ times and was waved at $3$ times. Cindy waved $5$ times. How many times was she waved at? [b]p2.[/b] Sherlock and Mycroft are playing Battleship on a $4\times 4$ grid. Mycroft hides a single $3\times 1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser? [b]p3.[/b] Thirty girls – $13$ of them in red dresses and $17$ in blue dresses – were dancing in a circle, hand-in-hand. Afterwards, each girl was asked if the girl to her right was in a blue dress. Only the girls who had both neighbors in red dresses or both in blue dresses told the truth. How many girls could have answered “Yes”? [b]p4.[/b] Herman and Alex play a game on a $5\times 5$ board. On his turn, a player can claim any open square as his territory. Once all the squares are claimed, the winner is the player whose territory has the longer border. Herman goes first. If both play their best, who will win, or will the game end in a draw? [img]https://cdn.artofproblemsolving.com/attachments/5/7/113d54f2217a39bac622899d3d3eb51ec34f1f.png[/img] [b]p5.[/b] Is it possible to find $2014$ distinct positive integers whose sum is divisible by each of them? [u]Round 2[/u] [b]p6.[/b] Hermione and Ron play a game that starts with 129 hats arranged in a circle. They take turns magically transforming the hats into animals. On each turn, a player picks a hat and chooses whether to change it into a badger or into a raven. A player loses if after his or her turn there are two animals of the same species right next to each other. Hermione goes first. Who loses? [b]p7.[/b] Three warring states control the corner provinces of the island whose map is shown below. [img]https://cdn.artofproblemsolving.com/attachments/e/a/4e2f436be1dcd3f899aa34145356f8c66cda82.png[/img] As a result of war, each of the remaining $18$ provinces was occupied by one of the states. None of the states was able to occupy any province on the coast opposite their corner. The states would like to sign a peace treaty. To do this, they each must send ambassadors to a place where three provinces, one controlled by each state, come together. Prove that they can always find such a place to meet. For example, if the provinces are occupied as shown here, the squares mark possible meeting spots. [img]https://cdn.artofproblemsolving.com/attachments/e/b/81de9187951822120fc26024c1c1fbe2138737.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\theta$ be a real number such that $1 + \sin 2\theta -\left(\frac12 \sin 2\theta\right)^2= 0$. Compute the maximum value of $(1 + \sin \theta )(1 + \cos \theta)$.

Let $N$ be an integer greater than $1$ and let $T_n$ be the number of non empty subsets $S$ of $\{1,2,.....,n\}$ with the property that the average of the elements of $S$ is an integer.Prove that $T_n - n$ is always even.

Find the number of positive integers $n$ less than $10000$ such that there are more $4$’s in the digits of $n + 1$ than in the digits of $n$.

Let $P, Q \in \mathbb{R}[x]$ be relatively prime nonconstant polynomials. Show that there can be at most three real numbers $\lambda$ such that $P + \lambda Q$ is the square of a polynomial. [i]Alison Miller[/i]

Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. (See figure.) Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993? [asy] int x=101, y=3*floor(x/4); draw(Arc(origin, 1, 360*(y-3)/x, 360*(y+4)/x)); int i; for(i=y-2; i<y+4; i=i+1) { dot(dir(360*i/x)); } label("3", dir(360*(y-2)/x), dir(360*(y-2)/x)); label("2", dir(360*(y+1)/x), dir(360*(y+1)/x)); label("1", dir(360*(y+3)/x), dir(360*(y+3)/x));[/asy]

Let $S_n = n^2 + 20n + 12$, $n$ a positive integer. What is the sum of all possible values of $n$ for which $S_n$ is a perfect square?

Anna and Boris have the same copy of $5\times5$ table filled with $25$ distinct numbers. After choosing the maximal number in the table, Anna erases the row and the column that contain this number. Then she continue the same operations with a smaller table till it is possible.Boris basically does the same; however, each time choosing the minimal number in a table. Can it happen that the total sum of the numbers chosen by Boris a) is greater than the total sum of the numbers chosen by Anna? [i](6 points)[/i] b) is greater than the total sum of any $5$ numbers of initial table given that no two of the numbers are in the same row or in the same column? [i](2 points)[/i]

Triangle $ ABC$ has $ AB \equal{} 2 \cdot AC$. Let $ D$ and $ E$ be on $ \overline{AB}$ and $ \overline{BC}$, respectively, such that $ \angle{BAE} \equal{} \angle{ACD}.$ Let $ F$ be the intersection of segments $ AE$ and $ CD$, and suppose that $ \triangle{CFE}$ is equilateral. What is $ \angle{ACB}$? $ \textbf{(A)}\ 60^{\circ}\qquad \textbf{(B)}\ 75^{\circ}\qquad \textbf{(C)}\ 90^{\circ}\qquad \textbf{(D)}\ 105^{\circ}\qquad \textbf{(E)}\ 120^{\circ}$

$ABC$ is a triangle, and $E$ and $F$ are points on the segments $BC$ and $CA$ respectively, such that $\frac{CE}{CB}+\frac{CF}{CA}=1$ and $\angle CEF=\angle CAB$. Suppose that $M$ is the midpoint of $EF$ and $G$ is the point of intersection between $CM$ and $AB$. Prove that triangle $FEG$ is similar to triangle $ABC$.