Given two distinct circles touching each other internally, show how to construct a triangle with the inner circle as its incircle and the outer circle as its nine point circle.

Circle $\omega_1$ with center $K$ of radius $4$ and circle $\omega_2$ of radius $6$ intersect at points $W$ and $U$. If the incenter of $\triangle KWU$ lies on circle $\omega_2$, find the length of $\overline{WU}$. (Note: The incenter of a triangle is the intersection of the angle bisectors of the angles of the triangle) [i]Proposed by Bradley Guo[/i]

Consider a circle tangent to sides $AB$ and $AC$ (these sides are not equal) of triangle $ABC$ and the circumscribed circle around it. Let $K$, $M$ and $P$ be the touchpoints of this circle with the sides of the triangle and with the circle circumscribed around it, respectively, and let $L$ be the midpoint of the arc $BC$ (not containing $A$). Prove that the lines $KM$, $PL$ and $BC$ intersect at one point.

A competition involving $n\ge 2$ players was held over $k$ days. In each day, the players received scores of $1,2,3,\ldots , n$ points with no players receiving the same score. At the end of the $k$ days, it was found that each player had exactly $26$ points in total. Determine all pairs $(n,k)$ for which this is possible.

Let $A,B$ be two $n\times n$ complex matrices of the same rank, and let $k$ be a positive integer. Prove that $A^{k+1}B^k = A$ if and only if $B^{k+1}A^k = B$.

A circle $\omega$ with center $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of this arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\omega$ in points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of quadrilateral $ABCD$ meet in point $X$, and the diagonals of $ABCD$ meet in point $Y$. Prove that points $X, Y, I$ and $M$ are collinear.

Let $p(x)$ be a nonconstant polynomial with real coefficients. For every positive integer~$n$, let $$q_n(x) = (x+1)^np(x)+x^n p(x+1) .$$ Prove that there are only finitely many numbers $n$ such that all roots of $q_n(x)$ are real.

A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)

Let $k$ be a real number. Show that the polynomial $p (x) = x^3-24x + k$ has at most an integer root.

In a chess tournament, each pair of players play one game. A lost game yields 0 points, a won game yields 1 point and a tied game yields $\frac12$ point. After the tournament, it turns out that in each group of three players, at least one got $1 \frac12$ points in the games against the two others. What is the largest number of players that may have participated?

Josh writes the numbers $1,2,3,\dots,99,100$. He marks out $1$, skips the next number $(2)$, marks out $3$, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number $(2)$, skips the next number $(4)$, marks out $6$, skips $8$, marks out $10$, and so on to the end. Josh continues in this manner until only one number remains. What is that number? $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 56 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 96$

For $f_n(x)=x^n$ and $a\neq 1$ consider I. $(f_{11}(a)f_{13}(a))^{14}$ II. $f_{11}(a)f_{13}(a)f_{14}(a)$ III. $(f_{11}(f_{13}(a)))^{14}$ IV. $f_{11}(f_{13}(f_{14}(a)))$ Which of these equal $f_{2002}(a)$? $\textbf{(A) }\text{I and II only}\qquad\textbf{(B) }\text{II and III only}\qquad\textbf{(C) }\text{III and IV only}\qquad\textbf{(D) }\text{II, III, and IV only}\qquad\textbf{(E) }\text{all of them}$

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for every $x,y \in \mathbb{R}$ $$f(x^{4}-y^{4})+4f(xy)^{2}=f(x^{4}+y^{4})$$

Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students. [asy] size(8cm); void drawbar(real x, real h) { fill((x-0.15,0.5)--(x+0.15,0.5)--(x+0.15,h)--(x-0.15,h)--cycle,gray); } draw((0.5,0.5)--(7.5,0.5)--(7.5,5)--(0.5,5)--cycle); for (real i=1; i<5; i=i+0.5) { draw((0.5,i)--(7.5,i),gray); } drawbar(1.0,1.0); drawbar(2.0,2.0); drawbar(3.0,1.5); drawbar(4.0,3.5); drawbar(5.0,4.5); drawbar(6.0,2.0); drawbar(7.0,1.5); for (int i=1; i<8; ++i) { label("$"+string(i)+"$",(i,0.25)); } for (int i=1; i<9; ++i) { label("$"+string(i)+"$",(0.5,0.5*(i+1)),W); } label("Number of Days of Exercise",(4,-0.1)); label(rotate(90)*"Number of Students",(-0.1,2.75)); [/asy]What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class? $\textbf{(A) } 3.50 \qquad \textbf{(B) } 3.57 \qquad \textbf{(C) } 4.36 \qquad \textbf{(D) } 4.50 \qquad \textbf{(E) } 5.00$

Find such real $p, q, a, b$, that for all $x$ an equality is held: $$(2x-1)^{20} - (ax+b)^{20} = (x^2+px+q)^{10}$$

Integers between $1$ and $7$ are written on a blackboard. It is possible that not all the numbers from 1 to 7 are present, and it is also possible that one, some or all of the numbers are repeated, one or more times. A move consists of choosing one or more numbers on the blackboard, where all distinct, delete them and write different numbers in their place, such that the written numbers together with those deleted form the whole set $\{1, 2, 3, 4, 5, 6 , 7\}$ For example, moves allowed are: • delete a $4$ and a $5$, and write in their place the numbers $1, 2, 3, 6$ and $7$; • deleting a $1$, a $2$, a $3$, a $4$, a $5$, a $6$ and a $7$ and write nothing in their place. Prove that, if it is possible to find a sequence of moves, starting from the initial situation, leading to have on board a single number (written once), then this number does not depend on the sequence of moves used.

Let $A$ and $B$ two countries which inlude exactly $2011$ cities.There is exactly one flight from a city of $A$ to a city of $B$ and there is no domestic flights (flights are bi-directional).For every city $X$ (doesn't matter from $A$ or from $B$), there exist at most $19$ different airline such that airline have a flight from $X$ to the another city.For an integer $k$, (it doesn't matter how flights arranged ) we can say that there exists at least $k$ cities such that it is possible to trip from one of these $k$ cities to another with same airline.So find the maximum value of $k$.

Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions: (a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$; (b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.

Triangle $ABC$ is such that $\angle BAC>\angle ABC>60^o$. The perpendicular bisector of $\overline{AB}$ intersects the segment $\overline {BC}$ at $O$. Suppose there exists a point $D$ on the segment $\overline{AC}$ such that $OD=AB$ and $\angle ODA=30^o$. Find $\angle BAC$. [i](Vlad Robu)[/i]

Let $d(n)$ be the number of positive divisors of a positive integer $n$. Let $\mathbb{N}$ be the set of all positive integers. Say that a function $F$ from $\mathbb{N}$ to $\mathbb{N}$ is [i]divisor-respecting[/i] if $d(F(mn)) = d(F(m)) d(F(n))$ for all positive integers $m$ and $n$, and $d(F(n)) \le d(n)$ for all positive integers $n$. Find all divisor-respecting functions. Justify your answer.

All the species of plants existing in Russia are catalogued (numbered by integers from $2$ to $2000$ ; one after another, without omissions or repetitions). For any pair of species the gcd of their catalogue numbers was calculated and recorded but the catalogue numbers themselves were lost. Is it possible to restore the catalogue numbers from the data in hand?

A natural number is written on the blackboard. Whenever number $ x$ is written, one can write any of the numbers $ 2x \plus{} 1$ and $ \frac {x}{x \plus{} 2}$. At some moment the number $ 2008$ appears on the blackboard. Show that it was there from the very beginning.

Let $E(n)$ denote the largest integer $k$ such that $5^k$ divides $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$ Calculate $$\lim_{n\to \infty} \frac{E(n)}{n^2 }.$$

[b]p1.[/b] Fun facts! We know that $1008^2-1007^2 = 1008+1007$ and $1009^2-1008^2 = 1009+1008$. Now compute the following: $$1010^2 - 1009^2 - 1.$$ [b]p2.[/b] Let $m$ be the smallest positive multiple of $2018$ such that the fraction $m/2019$ can be simplified. What is the number $m$? [b]p3.[/b] Given that $n$ satisfies the following equation $$n + 3n + 5n + 7n + 9n = 200,$$ find $n$. [b]p4.[/b] Grace and Somya each have a collection of coins worth a dollar. Both Grace and Somya have quarters, dimes, nickels and pennies. Serena then observes that Grace has the least number of coins possible to make one dollar and Somya has the most number of coins possible. If Grace has $G$ coins and Somya has $S$ coins, what is $G + S$? [b]p5.[/b] What is the ones digit of $2018^{2018}$? [b]p6.[/b] Kaitlyn plays a number game. Each time when Kaitlyn has a number, if it is even, she divides it by $2$, and if it is odd, she multiplies it by $5$ and adds $1$. Kaitlyn then takes the resulting number and continues the process until she reaches $1$. For example, if she begins with $3$, she finds the sequence of $6$ numbers to be $$3, 3 \cdot 5 + 1 = 16, 16/2 = 8, 8/2 = 4, 4/2 = 2, 2/2 = 1.$$ If Kaitlyn's starting number is $51$, how many numbers are in her sequence, including the starting number and the number $1$? [b]p7.[/b] Andrew likes both geometry and piano. His piano has $88$ keys, $x$ of which are white and $y$ of which are black. Each white key has area $3$ and each black key has area $11$. If the keys of his piano have combined area $880$, how many black keys does he have? [b]p8.[/b] A six-sided die contains the numbers $1$, $2$, $3$, $4$, $5$, and $6$ on its faces. If numbers on opposite faces of a die always sum to $7$, how many distinct dice are possible? (Two dice are considered the same if one can be rotated to obtain the other.) [b]p9.[/b] In $\vartriangle ABC$, $AB$ is $12$ and $AC$ is $15$. Alex draws the angle bisector of $BAC$, $AD$, such that $D$ is on $BC$. If $CD$ is $10$, then the area of $\vartriangle ABC$ can be expressed in the form $\frac{m \sqrt{n}}{p}$ where $m, p$ are relatively prime and $n$ is not divisible by the square of any prime. Find $m + n + p$. [b]p10.[/b] Find the smallest positive integer that leaves a remainder of $2$ when divided by $5$, a remainder of $3$ when divided by $6$, a remainder of $4$ when divided by $7$, and a remainder of $5$ when divided by $8$. [b]p11.[/b] Chris has a bag with $4$ marbles. Each minute, Chris randomly selects a marble out of the bag and flips a coin. If the coin comes up heads, Chris puts the marble back in the bag, while if the coin comes up tails, Chris sets the marble aside. What is the expected number of seconds it will take Chris to empty the bag? [b]p12.[/b] A real fixed point $x$ of a function $f(x)$ is a real number such that $f(x) = x$. Find the absolute value of the product of the real fixed points of the function $f(x) = x^4 + x - 16$. [b]p13.[/b] A triangle with angles $30^o$, $75^o$, $75^o$ is inscribed in a circle with radius $1$. The area of the triangle can be expressed as $\frac{a+\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p14.[/b] Dora and Charlotte are playing a game involving flipping coins. On a player's turn, she first chooses a probability of the coin landing heads between $\frac14$ and $\frac34$ , and the coin magically flips heads with that probability. The player then flips this coin until the coin lands heads, at which point her turn ends. The game ends the first time someone flips heads on an odd-numbered flip. The last player to flip the coin wins. If both players are playing optimally and Dora goes first, let the probability that Charlotte win the game be $\frac{a}{b}$ . Find $a \cdot b$. [b]p15.[/b] Jonny is trying to sort a list of numbers in ascending order by swapping pairs of numbers. For example, if he has the list $1$, $4$, $3$, $2$, Jonny would swap $2$ and $4$ to obtain $1$, $2$, $3$, $4$. If Jonny is given a random list of $400$ distinct numbers, let $x$ be the expected minimum number of swaps he needs. Compute $\left \lfloor \frac{x}{20} \right \rfloor$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all continuous functions $f:\left[0,1\right]\rightarrow[0,\infty)$ such that: $\int_{0}^{1}f\left(x\right)dx\cdotp\int_{0}^{1}f^{2}\left(x\right)dx\cdotp...\cdotp\int_{0}^{1}f^{2020}\left(x\right)dx=\left(\int_{0}^{1}f^{2021}\left(x\right)dx\right)^{1010}$