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 \[(x^2+y^2-2)^2+z^2=1\]

Find the sum of the elements of the set $$M = \left\{ \frac{n}{2}+\frac{m}{5} \,\, | m, n = 0, 1, 2,..., 100\right\}$$

There are $n$ boys $B_1, B_2, ... , B_n$ and $n$ girls $G_1, G_2, ... , G_n$. Each boy ranks the girls in order of preference, and each girl ranks the boys in order of preference. Show that we can arrange the boys and girls into n pairs so that we cannot find a boy and a girl who prefer each other to their partners. For example if $(B_1, G_3)$ and $(B_4, G_7)$ are two of the pairs, then it must not be the case that $B_4$ prefers $G_3$ to $G_7$ and $G_3$ prefers $B_4$ to $B_1$.

Find the greatest prime number $p$ for which there exists a prime number $q$ such that $p$ divides $4^q + 1$ and $q$ divides $4^p + 1$.

[b]2.[/b] Place 32 white and 32 black chessmen on the chessboard. Two chessmen of different colours will be said to form a "related pair" if they are placed either in the same row or in the same column. Determine the maximum and minimum number of related pairs (over all possible arrangements of the 64 chessmen considered. [b](C. 2)[/b]

For natural numbers $n$, let $r(n)$ be the number formed by reversing the digits of $n$, and take $f(n)$ to be the maximum value of $\frac{r(k)}k$ across all $n$-digit positive integers $k$. If we define $g(n)=\left\lfloor\frac1{10-f(n)}\right\rfloor$, what is the value of $g(20)$? [i]Proposed by Adam Bertelli[/i]

Show that the set $\{1,2,....,2^n\}$ can be partitioned in two classes, none of which contains an arithmetic progression of length $2n$.

Compute the $100^{\text{th}}$ smallest positive integer $n$ that satisfies the three congruences \[\begin{aligned} \left\lfloor \dfrac{n}{8} \right\rfloor &\equiv 3 \pmod{4}, \\ \left\lfloor \dfrac{n}{32} \right\rfloor &\equiv 2 \pmod{4}, \\ \left\lfloor \dfrac{n}{256} \right\rfloor &\equiv 1 \pmod{4}. \end{aligned}\] Here $\lfloor \cdot \rfloor$ denotes the greatest integer function. [i]Proposed by Michael Tang[/i]

A square with sides $1$ and four circles of radius $1$ considered each having a vertex of have the square as the center. Find area of the shaded part (see figure). [img]https://cdn.artofproblemsolving.com/attachments/b/6/6e28d94094d69bac13c2702853ac2c906a80a1.png[/img]

Given positive integers $n$ and $k$, $n > k^2 >4.$ In a $n \times n$ grid, a $k$[i]-group[/i] is a set of $k$ unit squares lying in different rows and different columns. Determine the maximal possible $N$, such that one can choose $N$ unit squares in the grid and color them, with the following condition holds: in any $k$[i]-group[/i] from the colored $N$ unit squares, there are two squares with the same color, and there are also two squares with different colors.

[b]p1.[/b] The case of Mr. Brown, Mr. Potter, and Mr. Smith is investigated. One of them has committed a crime. Everyone of them made two statements. Mr. Brown: I have not done it. Mr. Potter has not done it. Mr. Potter: Mr. Brown has not done it. Mr. Smith has done it. Mr. Smith: I have not done it. Mr. Brown has done it. It is known that one of them told the truth both times, one lied both times, and one told the truth one time and lied one time. Who has committed the crime? [b]p2.[/b] Is it possible to draw in a plane $1000001$ circles of the radius $1$ such that every circle touches exactly three other circles? [b]p3.[/b] Consider a circle of radius $R$ centered at the origin. A particle is “launched” from the $x$-axis at a distance, $d$, from the origin with $0 < d < R$, and at an angle, $\alpha$, with the $x$-axis. The particle is reflected from the boundary of the circle so that the [b]angle of incidence[/b] equals the [b]angle of reflection[/b]. Determine the angle $\alpha$ so that the path of the particle contacts the circle’s interior at exactly eight points. Please note that $\alpha$ should be determined in terms of the qunatities $R$ and $d$. [img]https://cdn.artofproblemsolving.com/attachments/e/3/b8ef9bb8d1b54c263bf2b68d3de60be5b41ad0.png[/img] [b]p4.[/b] Is it possible to find four different real numbers such that the cube of every number equals the square of the sum of the three others? [b]p5. [/b]The Fibonacci sequence $(1, 2, 3, 5, 8, 13, 21, . . .)$ is defined by the following formula: $f_n = f_{n-2} + f_{n-1}$, where $f_1 = 1$, $f_2 = 2$. Prove that any positive integer can be represented as a sum of different members of the Fibonacci sequence. [b]p6.[/b] $10,000$ points are arbitrary chosen inside a square of area $1$ m$^2$ . Does there exist a broken line connecting all these points, the length of which is less than $201$ m$^2? PS. You should use hide for answers.