In a class of $26$ students, everyone is being graded on five subjects with one of three possible marks. Prove that if $25$ of these students have received their marks, then we can grade the last one in such a way that their marks differ from these of any other student on at least two subjects.

Eight points are chosen on the circumference of a circle, labelled $P_1$, $P_2$, ..., $P_8$ in clockwise order. A route is a sequence of at least two points $P_{a_1}$, $P_{a_2}$, $...$, $P_{a_n}$ such that if an ant were to visit these points in their given order, starting at $P_{a_1}$ and ending at $P_{a_n}$, by following $n-1$ straight line segments (each connecting each $P_{a_i}$ and $P_{a_{i+1}}$), it would never visit a point twice or cross its own path. Find the number of routes.

Prove that from every set of $n+1$ natural numbers, whose prime factors are in a given set of $n$ prime numbers, one can select several distinct numbers whose product is a perfect square.

[u]Round 1[/u] [b]p1.[/b] In order to make good salad dressing, Bob needs a $0.9\%$ salt solution. If soy sauce is $15\%$ salt, how much water, in mL, does Bob need to add to $3$ mL of pure soy sauce in order to have a good salad dressing? [b]p2.[/b] Alex the Geologist is buying a canteen before he ventures into the desert. The original cost of a canteen is $\$20$, but Alex has two coupons. One coupon is $\$3$ off and the other is $10\%$ off the entire remaining cost. Alex can use the coupons in any order. What is the least amount of money he could pay for the canteen? [b]p3.[/b] Steve and Yooni have six distinct teddy bears to split between them, including exactly $1$ blue teddy bear and $1$ green teddy bear. How many ways are there for the two to divide the teddy bears, if Steve gets the blue teddy bear and Yooni gets the green teddy bear? (The two do not necessarily have to get the same number of teddy bears, but each teddy bear must go to a person.) [u]Round 2[/u] [b]p4.[/b] In the currency of Mathamania, $5$ wampas are equal to $3$ kabobs and $10$ kabobs are equal to $2$ jambas. How many jambas are equal to twenty-five wampas? [b]p5.[/b] A sphere has a volume of $81\pi$. A new sphere with the same center is constructed with a radius that is $\frac13$ the radius of the original sphere. Find the volume, in terms of $\pi$, of the region between the two spheres. [b]p6.[/b] A frog is located at the origin. It makes four hops, each of which moves it either $1$ unit to the right or $1$ unit to the left. If it also ends at the origin, how many $4$-hop paths can it take? [u]Round 3[/u] [b]p7.[/b] Nick multiplies two consecutive positive integers to get $4^5 - 2^5$ . What is the smaller of the two numbers? [b]p8.[/b] In rectangle $ABCD$, $E$ is a point on segment $CD$ such that $\angle EBC = 30^o$ and $\angle AEB = 80^o$. Find $\angle EAB$, in degrees. [b]p9.[/b] Mary’s secret garden contains clones of Homer Simpson and WALL-E. A WALL-E clone has $4$ legs. Meanwhile, Homer Simpson clones are human and therefore have $2$ legs each. A Homer Simpson clone always has $5$ donuts, while a WALL-E clone has $2$. In Mary’s secret garden, there are $184$ donuts and $128$ legs. How many WALL-E clones are there? [u]Round 4[/u] [b]p10.[/b] Including Richie, there are $6$ students in a math club. Each day, Richie hangs out with a different group of club mates, each of whom gives him a dollar when he hangs out with them. How many dollars will Richie have by the time he has hung out with every possible group of club mates? [b]p11.[/b] There are seven boxes in a line: three empty, three holding $\$10$ each, and one holding the jackpot of $\$1, 000, 000$. From the left to the right, the boxes are numbered $1, 2, 3, 4, 5, 6$ and $7$, in that order. You are told the following: $\bullet$ No two adjacent boxes hold the same contents. $\bullet$ Box $4$ is empty. $\bullet$ There is one more $\$10$ prize to the right of the jackpot than there is to the left. Which box holds the jackpot? [b]p12.[/b] Let $a$ and $b$ be real numbers such that $a + b = 8$. Let $c$ be the minimum possible value of $x^2 + ax + b$ over all real numbers $x$. Find the maximum possible value of $c$ over all such $a$ and $b$. [u]Round 5[/u] [b]p13.[/b] Let $ABCD$ be a rectangle with $AB = 10$ and $BC = 12$. Let M be the midpoint of $CD$, and $P$ be a point on $BM$ such that $BP = BC$. Find the area of $ABPD$. [b]p14.[/b] The number $19$ has the following properties: $\bullet$ It is a $2$-digit positive integer. $\bullet$ It is the two leading digits of a $4$-digit perfect square, because $1936 = 44^2$. How many numbers, including $19$, satisfy these two conditions? [b]p15.[/b] In a $3 \times 3$ grid, each unit square is colored either black or white. A coloring is considered “nice” if there is at most one white square in each row or column. What is the total number of nice colorings? Rotations and reflections of a coloring are considered distinct. (For example, in the three squares shown below, only the rightmost one has a nice coloring. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e6932c822bec77aa0b07c98d1789e58416b912.png[/img] PS. You should use hide for answers. Rest rounds have been posted [url=https://artofproblemsolving.com/community/c4h2786958p24498425]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For positive integers $a$ and $b$, an $(a,b)$-shuffle of a deck of $a+b$ cards is any shuffle that preserves the relative order of the top $a$ cards and the relative order of the bottom $b$ cards. Let $n$, $k$, $a_1$, $a_2$, $\dots$, $a_k$, $b_1$, $b_2$, $\dots$, $b_k$ be fixed positive integers such that $a_i+b_i=n$ for all $1\leq i\leq k$. Big Bird has a deck of $n$ cards and will perform an $(a_i,b_i)$-shuffle for each $1\leq i\leq k$, in ascending order of $i$. Suppose that Big Bird can reverse the order of the deck. Prove that Big Bird can also achieve any of the $n!$ permutations of the cards. [i]Linus Tang[/i]

A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \ge 3$, determine the largest integer $m$ such that no planar $n$-gon splits into less than $m$ triangles.

There are 47 students in a classroom with seats arranged in 6 rows $ \times$ 8 columns, and the seat in the $ i$-th row and $ j$-th column is denoted by $ (i,j).$ Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $ (i,j),$ if his/her new seat is $ (m,n),$ we say that the student is moved by $ [a, b] \equal{} [i \minus{} m, j \minus{} n]$ and define the position value of the student as $ a\plus{}b.$ Let $ S$ denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values of $ S.$

Let $G$ be a graph, not containing $K_4$ as a subgraph and $|V(G)|=3k$ (I interpret this to be the number of vertices is divisible by 3). What is the maximum number of triangles in $G$?

At the Berkeley Math Tournament, teams are composed of $6$ students, each of whom pick two distinct subject tests out of $5$ choices. How many different distributions across subjects are possible for a team?

On a right triangle of paper, two points $A$ and $B$ have been painted. You have scissors and you have the right to make cuts (on paper) as follows: cut through a height of the given triangle. In doing so, remove, without the respective altitude, one of the two triangles and continue the process. Prove that after a finite number of cuts you can separate points $A$ and $B$ leaving one of them outside the remaining triangles.

In a class, there are $n{}$ children of different heights. Denote by $A{}$ the number of ways to arrange them all in a row, numbered $1,2,\ldots,n$ from left to right, so that each person with an odd number is shorter than each of his neighbors. Let $B{}$ be the number of ways to organize $n-1$ badminton games between these children so that everyone plays at most two games with children shorter than himself and at most one game with children taller than himself (the order of the games is not important). Prove that $A = B$.

Justine has a coin which will come up the same as the last flip $\frac23$ of the time and the other side $\frac13$ of the time. She flips it and it comes up heads. She then flips it $2010$ more times. What is the probability that the last flip is heads?

Let $k$ and $n$ be positive integers with $k < n$. Find the number of subsets of $\{1, 2, . . . , n\}$ such that the difference between the largest and smallest elements in the subset is $k$.

Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.

Some $1\times 2$ dominoes, each covering two adjacent unit squares, are placed on a board of size $n\times n$ such that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is $2008$, find the least possible value of $n$.

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

[u]Round 1[/u] [b]p1.[/b] Goldilocks enters the home of the three bears – Papa Bear, Mama Bear, and Baby Bear. Each bear is wearing a different-colored shirt – red, green, or blue. All the bears look the same to Goldilocks, so she cannot otherwise tell them apart. The bears in the red and blue shirts each make one true statement and one false statement. The bear in the red shirt says: “I'm Blue's dad. I'm Green's daughter.” The bear in the blue shirt says: “Red and Green are of opposite gender. Red and Green are my parents.” Help Goldilocks find out which bear is wearing which shirt. [b]p2.[/b] The University of Washington is holding a talent competition. The competition has five contests: math, physics, chemistry, biology, and ballroom dancing. Any student can enter into any number of the contests but only once for each one. For example, a student may participate in math, biology, and ballroom. It turned out that each student participated in an odd number of contests. Also, each contest had an odd number of participants. Was the total number of contestants odd or even? [b]p3.[/b] The $99$ greatest scientists of Mars and Venus are seated evenly around a circular table. If any scientist sees two colleagues from her own planet sitting an equal number of seats to her left and right, she waves to them. For example, if you are from Mars and the scientists sitting two seats to your left and right are also from Mars, you will wave to them. Prove that at least one of the $99$ scientists will be waving, no matter how they are seated around the table. [b]p4.[/b] One hundred boys participated in a tennis tournament in which every player played each other player exactly once and there were no ties. Prove that after the tournament, it is possible for the boys to line up for pizza so that each boy defeated the boy standing right behind him in line. [b]p5.[/b] To celebrate space exploration, the Science Fiction Museum is going to read Star Wars and Star Trek stories for $24$ hours straight. A different story will be read each hour for a total of $12$ Star Wars stories and $12$ Star Trek stories. George and Gene want to listen to exactly $6$ Star Wars and $6$ Star Trek stories. Show that no matter how the readings are scheduled, the friends can find a block of $12$ consecutive hours to listen to the stories together. [u]Round 2[/u] [b]p6.[/b] $2013$ people attended Cinderella's ball. Some of the guests were friends with each other. At midnight, the guests started turning into mice. After the first minute, everyone who had no friends at the ball turned into a mouse. After the second minute, everyone who had exactly one friend among the remaining people turned into a mouse. After the third minute, everyone who had two human friends left in the room turned into a mouse, and so on. What is the maximal number of people that could have been left at the ball after $2013$ minutes? [b]p7.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $12$ points that are vertices of a regular polygon with $12$ sides. Rafael must draw segments that have their two ends at two of the points drawn. He is allowed to have each point be an endpoint of more than one segment and for the segments to intersect, but he is prohibited from drawing three segments that are the three sides of a triangle in which each vertex is one of the $12$ starting points. Find the maximum number of segments Rafael can draw and justify why he cannot draw a greater number of segments.

In a country, there are some two-way roads between the cities. There are $2010$ roads connected to the capital city. For all cities different from the capital city, there are less than $2010$ roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number is even. $k$ roads connected to the capital city will be deleted. It is wanted that whatever the road network is, if we can reach from one city to another at the beginning, then we can reach after the deleting process also. Find the maximum value of $k.$

A trawler is about to fish in territorial waters of a neighboring country, for what he has no licence. Whenever he throws the net, the coast-guard may stop him with the probability $1/k$, where $k$ is a fixed positive integer. Each throw brings him a fish landing of a fixed weight. However, if the coast-guard stops him, they will confiscate his entire fish landing and demand him to leave the country. The trawler plans to throw the net $n$ times before he returns to territorial waters in his country. Find $n$ for which his expected profit is maximal.

Let $\mathcal{F}$ be a family of $4$-element subsets of a set of size $5^m$, where $m$ is a fixed positive integer. If the intersection of any two sets in $\mathcal{F}$ does not have size exactly $2$, find the maximal value of $|\mathcal{F}|$.

Given a set of $n$ positive numbers. For each its nonempty subset consider the sum of all the subset's numbers. Prove that you can divide those sums onto $n$ groups in such a way, that the least sum in every group is not less than a half of the greatest sum in the same group.

There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors. P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.

$f,g$ are two permutations of set $X=\{1,\dots,n\}$. We say $f,g$ have common points iff there is a $k\in X$ that $f(k)=g(k)$. a) If $m>\frac{n}{2}$, prove that there are $m$ permutations $f_{1},f_{2},\dots,f_{m}$ from $X$ that for each permutation $f\in X$, there is an index $i$ that $f,f_{i}$ have common points. b) Prove that if $m\leq\frac{n}{2}$, we can not find permutations $f_{1},f_{2},\dots,f_{m}$ satisfying the above condition.

What is the maximal number of checkers that can be placed on an $8\times 8$ checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker?