The circle $x^2+ y^2 = r^2$ meets the coordinate axis at $A = (r,0), B = (-r,0), C = (0,r)$ and $D = (0,-r).$ Let $P = (u,v)$ and $Q = (-u,v)$ be two points on the circumference of the circle. Let $N$ be the point of intersection of $PQ$ and the $y$-axis, and $M$ be the foot of the perpendicular drawn from $P$ to the $x$-axis. If $r^2$ is odd, $u = p^m > q^n = v,$ where $p$ and $q$ are prime numbers and $m$ and $n$ are natural numbers, show that \[ |AM| = 1, |BM| = 9, |DN| = 8, |PQ| = 8. \]

Let $p\geq5$ be a prime and $S_k=1^k+2^k+...+(p-1)^k,\forall k\in\mathbb{N}.$ Prove that there is an infinity of numbers $n\in\mathbb{N}$ such that $p^3$ divides $S_n$ and $ p $ divides $S_{n-1}$ and $S_{n-2}.$

Solve following system equations: \[\left\{ \begin{array}{c} 3x+4y=26\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \sqrt{x^2+y^2-4x+2y+5}+\sqrt{x^2+y^2-20x-10y+125}=10\ \end{array} \right.\ \ \]

For some integer $m$, the polynomial $x^3-2011x+m$ has the three integer roots $a$, $b$, and $c$. Find $|a|+|b|+|c|$.

A subset $S$ of $\mathbb N$ is [i]eventually linear[/i] iff there are $k,N\in\mathbb N$ that for $n>N,n\in S\Longleftrightarrow k|n$. Let $S$ be a subset of $\mathbb N$ that is closed under addition. Prove that $S$ is eventually linear.

Let $m \ge 2$ and $n \ge 1$ be integers and $A=(a_{ij})$ a square matrix of order $n$ with integer entries. Prove that for any permutation $\sigma \in S_n$ there is a function $\varepsilon : \{1,2,\ldots,n\} \to \{0,1\}$ such that replacing the entries $a_{\sigma(1)1},$ $a_{\sigma(2)2}, $ $\ldots,$ $a_{\sigma(n)n}$ of $A$ respectively by $$a_{\sigma(1)1}+\varepsilon(1), ~a_{\sigma(2)2}+\varepsilon(2), ~\ldots, ~a_{\sigma(n)n}+\varepsilon(n),$$ the determinant of the matrix $A_{\varepsilon}$ thus obtained is not divisible by $m.$

Natural numbers $a$, $b$ are written in decimal using the same digits (i.e. every digit from 0 to 9 appears the same number of times in $a$ and in $b$). Prove that if $a+b=10^{1000}$ then both numbers $a$ and $b$ are divisible by $10$.

A rectangle $3 \times 5$ is divided into $15$ $1 \times 1$ cells. The middle $3$ cells that have no common points with the border of the rectangle are deleted. Is it possible to put in the remaining $12$ cells numbers $1, 2, \ldots, 12$ in some order, so that the sums of the numbers in the cells along each of the four sides of the rectangle are equal? [i]Proposed by Mariia Rozhkova[/i]

Given integer $n\geq 2$, two invisible [color=#eee]rabbits[/color] (rabbits) discussed their strategy and was sent to two points $A, B$ with distance $n$ units on a plane, respectively. However, they do not know whether they are on the same or different side of the plane (when facing each other, the might view the left/right direction as the same or different). They both can see points $A,B$, and they need to hop to each other's starting point. Each hop would measure $1$ unit in distance, and they would jump and land at the same time for each round. However, if at any time they landed no more than $1$ unit away, they'll both turn into deer. Find the minimum number of round they need to reach their destiny while ensuring they won't turn into deer. [i]Proposed by redshrimp[/i]

Let $a, b, c$ be pairwise coprime natural numbers. A positive integer $n$ is said to be [i]stubborn[/i] if it cannot be written in the form $n = bcx+cay+abz$, for some $x, y, z \in\mathbb{ N}.$ Determine the number of stubborn numbers.

A jacket and a shirt originally sold for $ \$80$ and $ \$40$, respectively. During a sale Chris bought the $ \$80$ jacket at a $40\%$ discount and the $ \$40$ shirt at a $55\%$ discount. The total amount saved was what percent of the total of the original prices? $\text{(A)}\ 45\% \qquad \text{(B)}\ 47\dfrac{1}{2}\% \qquad \text{(C)}\ 50\% \qquad \text{(D)}\ 79\dfrac{1}{6}\% \qquad \text{(E)}\ 95\%$.

One needs to ffll the cells of an $n\times n$ table ($n > 1$) with distinct integers from $1$ to $n^2$ so that every two consecutive integers are placed in cells that share a side, while every two integers with the same remainder if divided by $n$ are placed in distinct rows and distinct columns. For which $n$ is this possible? (Alexandr Gribalko)

Compute \[\sum_{a_1=0}^\infty\sum_{a_2=0}^\infty\cdots\sum_{a_7=0}^\infty\dfrac{a_1+a_2+\cdots+a_7}{3^{a_1+a_2+\cdots+a_7}}.\]

[u]Round 1[/u] [b]p1.[/b] The alphabet of the Aau-Bau language consists of two letters: A and B. Two words have the same meaning if one of them can be constructed from the other by replacing any AA with A, replacing any BB with B, or by replacing any ABA with BAB. For example, the word AABA means the same thing as ABA, and AABA also means the same thing as ABAB. In this language, is it possible to name all seven days of the week? [b]p2.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken. [img]https://cdn.artofproblemsolving.com/attachments/7/6/0fd93a0deaa71a5bb1599d2488f8b4eac5d0eb.jpg[/img] [b]p3.[/b] A playground has a swing-set with exactly three swings. When 3rd and 4th graders from Dr. Anna’s math class play during recess, she has a rule that if a $3^{rd}$ grader is in the middle swing there must be $4^{th}$ graders on that person’s left and right. And if there is a $4^{th}$ grader in the middle, there must be $3^{rd}$ graders on that person’s left and right. Dr. Anna calculates that there are $350$ different ways her students can arrange themselves on the three swings with no empty seats. How many students are in her class? [img]https://cdn.artofproblemsolving.com/attachments/5/9/4c402d143646582376d09ebbe54816b8799311.jpg[/img] [b]p4.[/b] The archipelago Artinagos has $19$ islands. Each island has toll bridges to at least $3$ other islands. An unsuspecting driver used a bad mapping app to plan a route from North Noether Island to South Noether Island, which involved crossing $12$ bridges. Show that there must be a route with fewer bridges. [img]https://cdn.artofproblemsolving.com/attachments/e/3/4eea2c16b201ff2ac732788fe9b78025004853.jpg[/img] [b]p5.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9, ... , 121)$ are in one column? [u]Round 2[/u] [b]p6.[/b] Hungry and Sneaky have opened a rectangular box of chocolates and are going to take turns eating them. The chocolates are arranged in a $2m \times 2n$ grid. Hungry can take any two chocolates that are side-by-side, but Sneaky can take only one at a time. If there are no more chocolates located side-by-side, all remaining chocolates go to Sneaky. Hungry goes first. Each player wants to eat as many chocolates as possible. What is the maximum number of chocolates Sneaky can get, no matter how Hungry picks his? [img]https://cdn.artofproblemsolving.com/attachments/b/4/26d7156ca6248385cb46c6e8054773592b24a3.jpg[/img] [b]p7.[/b] There is a thief hiding in the sultan’s palace. The palace contains $2019$ rooms connected by doors. One can walk from any room to any other room, possibly through other rooms, and there is only one way to do this. That is, one cannot walk in a loop in the palace. To catch the thief, a guard must be in the same room as the thief at the same time. Prove that $11$ guards can always find and catch the thief, no matter how the thief moves around during the search. [img]https://cdn.artofproblemsolving.com/attachments/a/b/9728ac271e84c4954935553c4d58b3ff4b194d.jpg[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the largest number of obtuse triangles that can be composed of $16$ different segments (each triangle is composed of three segments), if the largest of these segments does not exceed twice the smallest?

Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?

Let $\triangle ABC$ be an acute triangle. Let us denote the foot of the altitudes from the vertices $A, B$ and $C$ to the opposite sides by $D, E$ and $F,$ respectively, and the intersection point of the altitudes of the triangle $ABC$ by $H.$ Let $P$ be the intersection of the line $BE$ and the segment $DF.$ A straight line passing through $P$ and perpendicular to $BC$ intersects $AB$ at $Q.$ Let $N$ be the intersection of the segment $EQ$ with the perpendicular drawn from $A.$ Prove that $N$ is the midpoint of segment $AH.$

Let $c_1,c_2,c_3,\ldots, c_{2008}$ be complex numbers such that \[|c_1|=|c_2|=|c_3|=\cdots=|c_{2008}|=1492,\] and let $S(2008,t)$ be the sum of all products of these $2008$ complex numbers taken $t$ at a time. Let $Q$ be the maximum possible value of \[\left|\dfrac{S(2008,1492)}{S(2008,516)}\right|.\] Find the remainder when $Q$ is divided by $2008$.

Estimate the number of pairs of integers $1\leq a,b\leq1000$ satisfying $\gcd(a,b)=\gcd(a+1,b+1)$. An estimate of $E$ earns $2^{1-0.00002|E-A|}$ points, where $A$ is the actual answer. [i]2020 CCA Math Bonanza Lightning Round #5.3[/i]

All integer numbers are colored in 3 colors in arbitrary way. Prove that there are two distinct numbers whose difference is a perfect square and the numbers are colored in the same color.

Call a permutation $a_1,a_2,\ldots,a_n$ [i]quasi-increasing[/i] if $a_k\le a_{k+1}+2$ for each $1\le k\le n-1$. For example, $54321$ and $14253$ are quasi-increasing permutations of the integers $1,2,3,4,5$, but $45123$ is not. Find the number of quasi-increasing permutations of the integers $1,2,\ldots,7$.

On the trip home from the meeting where this AMC$ 10$ was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $ bbcac$, where $ 0 \le a < b < c \le 9$, and $ b$ was the average of $ a$ and $ c$. How many different five-digit numbers satisfy all these properties? $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 20\qquad \textbf{(E)}\ 24$

For every positive integer $n$ we denote by $d(n)$ the sum of its digits in the decimal representation. Prove that for each positive integer $k$ there exists a positive integer $m$ such that the equation $x+d(x)=m$ has exactly $k$ solutions in the set of positive integers.

Three circles of equal radius have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point $O$.

Sketch the geodesics on the surface \[(x^2+y^2-2)^2+z^2=1\]