Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set of rows and $C$ a set of columns, is called a [i]saddle pair[/i] if the following two conditions are satisfied: [list] [*] $(i)$ For each row $r^{\prime}$, there is $r \in R$ such that $a(r, c) \geqslant a\left(r^{\prime}, c\right)$ for all $c \in C$; [*] $(ii)$ For each column $c^{\prime}$, there is $c \in C$ such that $a(r, c) \leqslant a\left(r, c^{\prime}\right)$ for all $r \in R$. [/list] A saddle pair $(R, C)$ is called a [i]minimal pair[/i] if for each saddle pair $\left(R^{\prime}, C^{\prime}\right)$ with $R^{\prime} \subseteq R$ and $C^{\prime} \subseteq C$, we have $R^{\prime}=R$ and $C^{\prime}=C$. Prove that any two minimal pairs contain the same number of rows.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Matt has a twenty dollar bill and buys two items worth $\$7:99$ each. How much change does he receive, in dollars? [b]p2.[/b] The sum of two distinct numbers is equal to the positive difference of the two numbers. What is the product of the two numbers? [b]p3.[/b] Evaluate $$\frac{1 + 2 + 3 + 4 + 5 + 6 + 7}{8 + 9 + 10 + 11 + 12 + 13 + 14}.$$ [b]p4.[/b] A sphere with radius $r$ has volume $2\pi$. Find the volume of a sphere with diameter $r$. [b]p5.[/b] Yannick ran $100$ meters in $14.22$ seconds. Compute his average speed in meters per second, rounded to the nearest integer. [b]p6.[/b] The mean of the numbers $2, 0, 1, 5,$ and $x$ is an integer. Find the smallest possible positive integer value for $x$. [b]p7.[/b] Let $f(x) =\sqrt{2^2 - x^2}$. Find the value of $f(f(f(f(f(-1)))))$. [b]p8.[/b] Find the smallest positive integer $n$ such that $20$ divides $15n$ and $15$ divides $20n$. [b]p9.[/b] A circle is inscribed in equilateral triangle $ABC$. Let $M$ be the point where the circle touches side $AB$ and let $N$ be the second intersection of segment $CM$ and the circle. Compute the ratio $\frac{MN}{CN}$ . [b]p10.[/b] Four boys and four girls line up in a random order. What is the probability that both the first and last person in line is a girl? [b]p11.[/b] Let $k$ be a positive integer. After making $k$ consecutive shots successfully, Andy's overall shooting accuracy increased from $65\%$ to $70\%$. Determine the minimum possible value of $k$. [b]p12.[/b] In square $ABCD$, $M$ is the midpoint of side $CD$. Points $N$ and $P$ are on segments $BC$ and $AB$ respectively such that $ \angle AMN = \angle MNP = 90^o$. Compute the ratio $\frac{AP}{PB}$ . [b]p13.[/b] Meena writes the numbers $1, 2, 3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written? [b]p14.[/b] Find the smallest positive integer $N$ such that $2N$ is a perfect square and $3N$ is a perfect cube. [b]p15.[/b] A polyhedron has $60$ vertices, $150$ edges, and $92$ faces. If all of the faces are either regular pentagons or equilateral triangles, how many of the $92$ faces are pentagons? [b]p16.[/b] All positive integers relatively prime to $2015$ are written in increasing order. Let the twentieth number be $p$. The value of $\frac{2015}{p}-1$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute $a + b$. [b]p17.[/b] Five red lines and three blue lines are drawn on a plane. Given that $x$ pairs of lines of the same color intersect and $y$ pairs of lines of different colors intersect, find the maximum possible value of $y - x$. [b]p18.[/b] In triangle $ABC$, where $AC > AB$, $M$ is the midpoint of $BC$ and $D$ is on segment $AC$ such that $DM$ is perpendicular to $BC$. Given that the areas of $MAD$ and $MBD$ are $5$ and $6$, respectively, compute the area of triangle $ABC$. [b]p19.[/b] For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x, y \le 10$ is $(x + y)^2 + (xy - 1)^2$ a prime number? [b]p20.[/b] A solitaire game is played with $8$ red, $9$ green, and $10$ blue cards. Totoro plays each of the cards exactly once in some order, one at a time. When he plays a card of color $c$, he gains a number of points equal to the number of cards that are not of color $c$ in his hand. Find the maximum number of points that he can obtain by the end of the game. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A bar of chocolate is made of $10$ distinguishable triangles as shown below: [center][img]https://cdn.artofproblemsolving.com/attachments/3/d/f55b0af0ce320fbfcfdbfab6a5c9c9306bfd16.png[/img][/center] How many ways are there to divide the bar, along the edges of the triangles, into two or more contiguous pieces?

Let $n$ to be a positive integer. A family $\wp$ of intervals $[i, j]$ with $0 \le i < j \le n$ and $i$, $j$ integers is considered [i]happy[/i] if, for any $I_1 = [i_1, j_1] \in \wp$ and $I_2 = [i_2, j_2] \in \wp$ such that $I_1 \subset I_2$, we have $i_1 = i_2$ [b]or[/b] $j_1 = j_2$. Determine the maximum number of elements of a [i]happy[/i] family.

[b]p1.[/b] Find the largest possible number of consecutive $9$’s in which an integer between $10,000,000$ and $13,371,337$ can end. For example, $199$ ends in two $9$’s, while $92,999$ ends in three $9$’s. [b]p2.[/b] Let $ABCD$ be a square of side length $2$. Equilateral triangles $ABP$, $BCQ$, $CDR$, and $DAS$ are constructed inside the square. Compute the area of quadrilateral $PQRS$. [b]p3.[/b] Evaluate the expression $7 \cdot 11 \cdot 13 \cdot 1003 - 3 \cdot 17 \cdot 59 \cdot 331$. [b]p4.[/b] Compute the number of positive integers $c$ such that there is a non-degenerate obtuse triangle with side lengths $21$, $29$, and $c$. [b]p5.[/b] Consider a $5\times 5$ board, colored like a chessboard, such that the four corners are black. Determine the number of ways to place $5$ rooks on black squares such that no two of the rooks attack one another, given that the rooks are indistinguishable and the board cannot be rotated. (Two rooks attack each other if they are in the same row or column.) [b]p6.[/b] Let $ABCD$ be a trapezoid of height $6$ with bases $AB$ and $CD$. Suppose that $AB = 2$ and $CD = 3$, and let $F$ and $G$ be the midpoints of segments $AD$ and $BC$, respectively. If diagonals $AC$ and $BD$ intersect at point $E$, compute the area of triangle $FGE$. [b]p7.[/b] A regular octahedron is a solid with eight faces that are congruent equilateral triangles. Suppose that an ant is at the center of one face of a regular octahedron of edge length $10$. The ant wants to walk along the surface of the octahedron to reach the center of the opposite face. (Two faces of an octahedron are said to be opposite if they do not share a vertex.) Determine the minimum possible distance that the ant must walk. [b]p8.[/b] Let $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, and $D_1D_2D_3$ be triangles in the plane. All the sides of the four triangles are extended into lines. Determine the maximum number of pairs of these lines that can meet at $60^o$ angles. [b]p9.[/b] For an integer $n$, let $f_n(x)$ denote the function $f_n(x) =\sqrt{x^2 - 2012x + n}+1006$. Determine all positive integers $a$ such that $f_a(f_{2012}(x)) = x$ for all $x \ge 2012$. [b]p10.[/b] Determine the number of ordered triples of integers $(a, b, c)$ such that $(a + b)(b + c)(c + a) = 1800$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Twenty players participated in a chess tournament. Each player faced every other player exactly once and each match ended with either player winning or a draw. In this tournament, it was noticed that for every match that ended in a draw, each of the other $18$ players won at least one of the two players involved in it. We also know that at least two games ended in a draw. Show that it is possible to name the players as $P_1, P_2, ...., P_{20}$ so that player $P_k$ beat player $P_{k+1}$, to each $k \in \{ 1, 2, 3,... , 19\}$.

[b]p1.[/b] Replace $*$’s by an arithmetic operations (addition, subtraction, multiplication or division) to obtain true equality $$2*0*1*6*7=1.$$ [b]p2.[/b] The interval of length $88$ cm is divided into three unequal parts. The distance between middle points of the left and right parts is $46$ cm. Find the length of the middle part. [b]p3.[/b] A $5\times 6$ rectangle is drawn on a square grid. Paint some cells of the rectangle in such a way that every $3\times 2$ sub‐rectangle has exactly two cells painted. [b]p4.[/b] There are $8$ similar coins. $5$ of them are counterfeit. A detector can analyze any set of coins and show if there are counterfeit coins in this set. The detector neither determines which coins nare counterfeit nor how many counterfeit coins are there. How to run the detector twice to find for sure at least one counterfeit coin? [b]p5.[/b] There is a set of $20$ weights of masses $1, 2, 3,...$ and $20$ grams. Can one divide this set into three groups of equal total masses? [b]p6.[/b] Replace letters $A,B,C,D,E,F,G$ by the digits $0,1,...,9$ to get true equality $AB+CD=EF * EG$ (different letters correspond to different digits, same letter means the same digit, $AB$, $CD$, $EF$, and $EG$ are two‐digit numbers). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let the Sierpinski triangle graph $S_n$ be defined as follows. $S_0$ consists of three vertices and three edges in a triangle. $S_1$ consists of $6$ vertices and $9$ edges. To make $S_n+1,$ we take three copies of $S_n$ and merge vertices at the corners, where the bottom-left corner of the top copy merges with the top corner of the bottom-left copy, etc. Then the number of cycles on $S_4,$ which visit each vertex exactly once and traverse each edge at most once, can be expressed as $p_1^{e_1}p_2^{e_2}$ for some primes $p_1, p_2$ and positive integers $e_1, e_2.$ Find $p_1 + p_2 + e_1 + e_2.$ [center] [img]https://cdn.artofproblemsolving.com/attachments/6/2/51d83da65910cd32ce0b235a9615ec467870e1.png[/img] [/center]

In each of the different grades of a high school there are an odd number of pupils. Each pupil has a best friend (who possibly is in a different grade). Everyone is the best friend of their best friend. In the upcoming school trip, every pupil goes to either Rome or Paris. Show that the pupils can be distributed over the two destinations in such a way that (i) every student goes to the same destination as their best friend; (ii) for each grade the absolute difference between the number of pupils that are going to Rome and that of those who are going to Paris is equal to $1$.

Let $a_1, a_2, \dots, a_n$ be a sequence of real numbers, and let $m$ be a fixed positive integer less than $n$. We say an index $k$ with $1\le k\le n$ is good if there exists some $\ell$ with $1\le \ell \le m$ such that $a_k+a_{k+1}+...+a_{k+\ell-1}\ge0$, where the indices are taken modulo $n$. Let $T$ be the set of all good indices. Prove that $\sum\limits_{k \in T}a_k \ge 0$. [i]Proposed by Mark Sellke[/i]

Let \(n\) be a positive integer and consider an \(n\times n\) square grid. For \(1\le k\le n\), a [i]python[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single row, and no other cells. Similarly, an [i]anaconda[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single column, and no other cells. The grid contains at least one python or anaconda, and it satisfies the following properties: [list] [*]No cell is occupied by multiple snakes. [*]If a cell in the grid is immediately to the left or immediately to the right of a python, then that cell must be occupied by an anaconda. [*]If a cell in the grid is immediately to above or immediately below an anaconda, then that cell must be occupied by a python. [/list] Prove that the sum of the squares of the lengths of the snakes is at least \(n^2\). [i]Proposed by Linus Tang[/i]

Let $n \ge 3$ be an integer, and consider a circle with $n + 1$ equally spaced points marked on it. Consider all labellings of these points with the numbers $0, 1, ... , n$ such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called [i]beautiful[/i] if, for any four labels $a < b < c < d$ with $a + d = b + c$, the chord joining the points labelled $a$ and $d$ does not intersect the chord joining the points labelled $b$ and $c$. Let $M$ be the number of beautiful labelings, and let N be the number of ordered pairs $(x, y)$ of positive integers such that $x + y \le n$ and $\gcd(x, y) = 1$. Prove that $$M = N + 1.$$

Consider the following array: \[ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots \] Find the 5-th number on the $ n$-th row with $ n>5$.

Cookies should be placed on a $7 \times 7$ chess board, so that never four cookies can form a rectangle whose sides are parallel to those of the chessboard. Find the maximum number of biscuits that can be positioned in this way. Note: If you do the same job for a $13 \times 13$ chessboard, you get a biscuit. If you solve it for an infinite number of squares of chessboards, you get two biscuits. If you solve them for all sidelengths, , you even get two three biscuits (We cannot distribute more cookies, otherwise we run the risk of them to form a rectangle).

We removed the middle square of $2 \times 2$ from the $8 \times 8$ board. a) How many checkers can be placed on the remaining $60$ boxes so that there are no two not jeopardize? b) How many at least checkers can be placed on the board so that they are at risk all $60$ squares? (A lady is threatening the box she stands on, as well as any box she can get to in one move without going over any of the four removed boxes.)

First $20$ positive integers are written on a board. It is known that, after you erase a number from the board, there exists a number that is equal to the arithmetic mean of the rest of the numbers left on the board. Find all the numbers that could've been erased.

Fred the Four-Dimensional Fluffy Sheep is walking in 4-dimensional space. He starts at the origin. Each minute, he walks from his current position $(a_1, a_2, a_3, a_4)$ to some position $(x_1, x_2, x_3, x_4)$ with integer coordinates satisfying \[(x_1-a_1)^2 + (x_2-a_2)^2 + (x_3-a_3)^2 + (x_4-a_4)^2 = 4 \quad \text{and} \quad |(x_1 + x_2 + x_3 + x_4) - (a_1 + a_2 + a_3 + a_4)| = 2.\] In how many ways can Fred reach $(10, 10, 10, 10)$ after exactly 40 minutes, if he is allowed to pass through this point during his walk?

Three different faces of a regular dodecahedron are selected at random and painted. What is the probability that there is at least one pair of painted faces that share an edge?

Let $S=\{1,2,\cdots, n\}$ and let $T$ be the set of all ordered triples of subsets of $S$, say $(A_1, A_2, A_3)$, such that $A_1\cup A_2\cup A_3=S$. Determine, in terms of $n$, \[ \sum_{(A_1,A_2,A_3)\in T}|A_1\cap A_2\cap A_3|\]

At first all $2^n$ rows of a $2^n \times n$ table were filled with all different $n$-tuples of numbers $+1$ and $-1$. Then some of the numbers were replaced by Os. Prove that one can choose a (non-empty) set of rows such that: (a) the sum of all the numbers in all the chosen rows is $0$; (b) the sum of all the chosen rows equals the zero row, that is, the sum of numbers in each column of the chosen rows equals $0$. (G Kondakov, V Chernorutskii)

A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle: Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members. Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.

Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.

Let $ A$ be a set of $ 2n$ points on the plane such that no three points are collinear. Prove that for any distinct two points $ a,b\in A$ there exists a line that partitions $ A$ into two subsets each containing $ n$ points and such that $ a,b$ lie on different sides of the line.

The set $\{1,2,\dots\>,n\}$ is called $P$. The function $f: P \to \{1,2,\dots\>,m\}$ satisfies \[f(A\cap B)=\min (f(A), f(B)).\] What is the relationship between the number of possible functions $f$ with the sum $\displaystyle \sum_{j=1}^m j^n$? There is a nice and easy solution to this. Too bad I did not think of it...

Consider two finite sequences of real numbers \( a_1, a_2, \dots, a_n \) and \( b_1, b_2, \dots, b_n \). Let \( \alpha(x) = \#\{i | a_i = x \} \) and \( \beta(x) = \#\{i | b_i = -x \} \). Prove that there exists a permutation \( \sigma \in S_n \) (the symmetric group of \( n \) elements) such that \( a_{\sigma(i)} + b_i \neq 0 \) for all \( i = 1, \dots, n \) if and only if \( \alpha(x) + \beta(x) \leq n \) for all \( x \in \mathbb{R} \).