Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.

Find with proof all non–zero real numbers $a$ and $b$ such that the three different polynomials $x^2 + ax + b, x^2 + x + ab$ and $ax^2 + x + b$ have exactly one common root.

In a plane, there are $n \geq 3$ circular beer mats $B_{1}, B_{2}, ..., B_{n}$ of equal size. $B_{k}$ touches $B_{k+1}$ ($k=1,2,...,n$); $B_{n+1}=B_{1}$. The beer mats are placed such that another beer mat $B$ of equal size touches all of them in the given order if rolling along the outside of the chain of beer mats. How many rotations $B$ makes untill it returns to it's starting position¿

Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red. Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$ [I]Netherlands[/i]

Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [i]2021 CCA Math Bonanza Individual Round #8[/i]

Prove that for each $x,y,z \in R$ it holds that $$\frac{x^2-y^2}{2x^2+1} +\frac{y^2-z^2}{2y^2+1}+\frac{z^2-x^2}{2z^2+1}\le 0$$

[b]p1.[/b] What is $20\times 19 + 20 \div (2 - 7)$? [b]p2.[/b] Will has three spinners. The first has three equally sized sections numbered $1$, $2$, $3$; the second has four equally sized sections numbered $1$, $2$, $3$, $4$; and the third has five equally sized sections numbered $1$, $2$, $3$, $4$, $5$. When Will spins all three spinners, the probability that the same number appears on all three spinners is $p$. Compute $\frac{1}{p}$. [b]p3.[/b] Three girls and five boys are seated randomly in a row of eight desks. Let $p$ be the probability that the students at the ends of the row are both boys. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$. [b]p4.[/b] Jaron either hits a home run or strikes out every time he bats. Last week, his batting average was $.300$. (Jaron's batting average is the number of home runs he has hit divided by the number of times he has batted.) After hitting $10$ home runs and striking out zero times in the last week, Jaron has now raised his batting average to $.310$. How many home runs has Jaron now hit? [b]p5.[/b] Suppose that the sum $$\frac{1}{1 \cdot 4} +\frac{1}{4 \cdot 7}+ ...+\frac{1}{97 \cdot 100}$$ is expressible as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p6.[/b] Let $ABCD$ be a unit square with center $O$, and $\vartriangle OEF$ be an equilateral triangle with center $A$. Suppose that $M$ is the area of the region inside the square but outside the triangle and $N$ is the area of the region inside the triangle but outside the square, and let $x = |M -N|$ be the positive difference between $M$ and $N$. If $$x =\frac1 8(p -\sqrt{q})$$ for positive integers $p$ and $q$, find $p + q$. [b]p7.[/b] Find the number of seven-digit numbers such that the sum of any two consecutive digits is divisible by $3$. For example, the number $1212121$ satisfies this property. [b]p8.[/b] There is a unique positive integer $x$ such that $x^x$ has $703$ positive factors. What is $x$? [b]p9.[/b] Let $x$ be the number of digits in $2^{2019}$ and let $y$ be the number of digits in $5^{2019}$. Compute $x + y$. [b]p10.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 13$ and $BC = 10$. Consider the set of all points $D$ in three-dimensional space such that $BCD$ is an equilateral triangle. This set of points forms a circle $\omega$. Let $E$ and $F$ be points on $\omega$ such that $AE$ and $AF$ are tangent to $\omega$. If $EF^2$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, determine $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The graph of $ y \equal{} x^6 \minus{} 10x^5 \plus{} 29x^4 \minus{} 4x^3 \plus{} ax^2$ lies above the line $ y \equal{} bx \plus{} c$ except at three values of $ x$, where the graph and the line intersect. What is the largest of those values? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

In a circle with center $M$, two chords $AC$ and $BD$ intersect perpendicularly. The circle of diameter $AM$ intersects the circle of diameter $BM$ besides $M$ also in point $P$. The circle of diameter $BM$ intersects the circle with diameter $CM$ besides $M$ also in point $Q$. The circle of diameter $CM$ intersects the circle of diameter $DM$ besides $M$ also in point $R$. The circle of diameter $DM$ intersects the circle of diameter $AM$ besides $M$ also in point $S$. Prove that quadrilateral $PQRS$ is a rectangle. [asy] unitsize (3 cm); pair A, B, C, D, M, P, Q, R, S; M = (0,0); A = dir(170); C = dir(10); B = dir(120); D = dir(240); draw(Circle(M,1)); draw(A--C); draw(B--D); draw(Circle(A/2,1/2)); draw(Circle(B/2,1/2)); draw(Circle(C/2,1/2)); draw(Circle(D/2,1/2)); P = (A + B)/2; Q = (B + C)/2; R = (C + D)/2; S = (D + A)/2; dot("$A$", A, A); dot("$B$", B, B); dot("$C$", C, C); dot("$D$", D, D); dot("$M$", M, E); dot("$P$", P, SE); dot("$Q$", Q, SE); dot("$R$", R, NE); dot("$S$", S, NE); [/asy]

Find the greatest number of depicted pieces composed of $4$ unit squares that can be placed without overlapping on an $n \times n$ grid (where n is a positive integer) in such a way that it is possible to move from some corner to the opposite corner via uncovered squares (moving between squares requires a common edge). The shapes can be rotated and reflected. [img]https://cdn.artofproblemsolving.com/attachments/b/d/f2978a24fdd737edfafa5927a8d2129eb586ee.png[/img]

A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled? $ \textbf{(A)}\ 1404 \qquad \textbf{(B)}\ 1462 \qquad \textbf{(C)}\ 1604 \qquad \textbf{(D)}\ 1605 \qquad \textbf{(E)}\ 1804$

When the natural numbers $P$ and $P'$, with $P>P'$, are divided by the natural number $D$, the remainders are $R$ and $R'$, respectively. When $PP'$ and $RR'$ are divided by $D$, the remainders are $r$ and $r'$, respectively. Then: $\textbf{(A) }r>r'\text{ always}\qquad \textbf{(B) }r<r'\text{ always}\qquad$ $\textbf{(C) }r>r'\text{ sometimes, and }r<r'\text{ sometimes}$ $\textbf{(D) }r>r'\text{ sometimes, and }r=r'\text{ sometimes}$ $\textbf{(E) }r=r'\text{ always}$

It is given $2015$ numbers such that every one of them when gets replaced with sum of the rest, we get same $2015$ same numbers. Prove that product of all numbers is $0$

Consider three sides of an $n \times n \times n$ cube that meet at one of the corners of the cube. For which $n$ is it possible to use this completely and without overlapping to cover strips of paper of size $3 \times 1$? The paper strips can also do this glued over the edges between these cube faces.

Which of the following can be used to distinguish a solid ball from a hollow sphere of the same radius and mass? $\textbf{(A)}$ Measurements of the orbit of a test mass around the object. $\textbf{(B)}$ Measurements of the time it takes the object to roll down an inclined plane. $\textbf{(C)}$ Measurements of the tidal forces applied by the object to a liquid body. $\textbf{(D)}$ Measurements of the behavior of the object as it oats in water. $\textbf{(E)}$ Measurements of the force applied to the object by a uniform gravitational field.

Solve the equations $$ \frac{dy}{dx}=z(y+z)^n, \;\; \; \frac{dz}{dx} = y(y+z)^n,$$ given the initial conditions $y=1$ and $z=0$ when $x=0.$

Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq 2$. Prove that \[ \max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j \]

Let $p$ be a prime number of the form $4k+3$. Prove that there are no integers $w,x,y,z$ whose product is not divisible by $p$, such that: \[ w^{2p}+x^{2p}+y^{2p}=z^{2p}. \]

If $x$ and $\log_{10} x$ are real numbers and $\log_{10} x<0$, then: $\textbf{(A)}\ x<0 \qquad \textbf{(B)}\ -1<x<1 \qquad \textbf{(C)}\ 0<x\le 1 $ $ \textbf{(D)}\ -1<x<0 \qquad \textbf{(E)}\ 0<x<1$

Are there integers $x,y$ and $z$ which satisfy the equation $$z^2=(x+1)(y^2-1)+n$$ when (a) $n=2006$ (b) $n=2007$?

In a group of interpreters each one speaks one of several foreign languages, 24 of them speak Japanese, 24 Malaysian, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malaysian and exactly 12 speak Farsi.

What is the square root of the sum of the first 2006 positive odd integers?

A positive integer $m$ is called a [i]beautiful [/i] integer if that there exists a positive integer $n$ such that $m = n^2+ n + 1$. Prove that there are infinitely many [i]beautiful [/i] integers with square factors, and the square factors of different beautiful integers are relative prime.

A billiard table. (see picture) A white ball is on $p_1$ and a red ball is on $p_2$. The white ball is shot towards the red ball as shown on the pic, hitting 3 sides first. Find the minimal distance the ball must travel. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=280[/img]

Let $a_1, a_2, \ldots , a_{11}$ be 11 pairwise distinct positive integer with sum less than 2007. Let S be the sequence of $1,2, \ldots ,2007$. Define an [b]operation[/b] to be 22 consecutive applications of the following steps on the sequence $S$: on $i$-th step, choose a number from the sequense $S$ at random, say $x$. If $1 \leq i \leq 11$, replace $x$ with $x+a_i$ ; if $12 \leq i \leq 22$, replace $x$ with $x-a_{i-11}$ . If the result of [b]operation[/b] on the sequence $S$ is an odd permutation of $\{1, 2, \ldots , 2007\}$, it is an [b]odd operation[/b]; if the result of [b]operation[/b] on the sequence $S$ is an even permutation of $\{1, 2, \ldots , 2007\}$, it is an [b]even operation[/b]. Which is larger, the number of odd operation or the number of even permutation? And by how many? Here $\{x_1, x_2, \ldots , x_{2007}\}$ is an even permutation of $\{1, 2, \ldots ,2007\}$ if the product $\prod_{i > j} (x_i - x_j)$ is positive, and an odd one otherwise.