Let $m,n$ be positive integers with $m \geq n$, and let $S$ be the set of all $n$-term sequences of positive integers $(a_1, a_2, \ldots a_n)$ such that $a_1 + a_2 + \cdots + a_n = m$. Show that \[\sum_S 1^{a_1} 2^{a_2} \cdots n^{a_n} = {n \choose n} n^m - {n \choose n-1} (n-1)^m + \cdots + (-1)^{n-2} {n \choose 2} 2^m + (-1)^{n-1} {n \choose 1}.\]

Zeroes are written in all cells of a $5\times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and the cells having a common side with it. Is it possible to obtain the number 2012 in all cells simultaneously?

A $8\times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.

There are $2000$ components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts two or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?

Prove that the number permutations $ \alpha$ of $ \{1,2,\dots,n\}$ s.t. there does not exist $ i<j<n$ s.t. $ \alpha(i)<\alpha(j\plus{}1)<\alpha(j)$ is equal to the number of partitions of that set.

In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.

Five guys join five girls for a night of bridge. Bridge games are always played by a team of two guys against a team of two girls. The guys and girls want to make sure that every guy and girl play against each other an equal number of times. Given that at least one game is played, what is the least number of games necessary to accomplish this?

Color every vertex of $2008$-gon with two colors, such that adjacent vertices have different color. If sum of angles of vertices of first color is same as sum of angles of vertices of second color, than we call $2008$-gon as interesting. Convex $2009$-gon one vertex is marked. It is known, that if remove any unmarked vertex, then we get interesting $2008$-gon. Prove, that if we remove marked vertex, then we get interesting $2008$-gon too.

For a given integer $n\ge 2$, let $a_0,a_1,\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\{ a_i+a_j \mid 0\le i \le j \le n \}$.

In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. For what is the smallest amount of rubles you can rearrange chips in reverse order?

Determine the smallest possible value of $| A_{1} \cup A_{2} \cup A_{3} \cup A_{4} \cup A_{5} |$, where $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ sets simultaneously satisfying the following conditions: $(i)$ $| A_{i}\cap A_{j} | = 1$ for all $1\leq i < j\leq 5$, i.e. any two distinct sets contain exactly one element in common; $(ii)$ $A_{i}\cap A_{j} \cap A_{k}\cap A_{l} =\varnothing$ for all $1\leq i<j<k<l\leq 5$, i.e. any four different sets contain no common element. Where $| S |$ means the number of elements of $S$.

A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies \[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \] for all $k=1,2,\ldots 9$. Find the number of interesting numbers.

Let $n \ge 5$ be an integer. Prove that $n$ is prime if and only if for any representation of $n$ as a sum of four positive integers $n = a + b + c + d$, it is true that $ab \ne cd$.

In an ${8}$×${8}$ squares chart , we dig out $n$ squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart . Then find the mininum value of $n$ .

Let $ n $ be a positive integer. Consider words of length $n$ composed of letters from the set $ \{ M, E, O \} $. Let $ a $ be the number of such words containing an even number (possibly 0) of blocks $ ME $ and an even number (possibly 0) blocks of $ MO $ . Similarly let $ b $ the number of such words containing an odd number of blocks $ ME $ and an odd number of blocks $ MO $. Prove that $ a>b $.

There is a stone on each square of $n\times n$ chessboard. We gather $n^2$ stones and distribute them to the squares (again each square contains one stone) such that any two adjacent stones are again adjacent. Find all distributions such that at least one stone at the corners remains at its initial square. (Two squares are adjacent if they share a common edge.)

Given are the real line and the two unique marked points $0$ and $1$. We can perform as many times as we want the following operation: we take two already marked points $a$ and $b$ and mark the reflection of $a$ over $b$. Let $f(n)$ be the minimum number of operations needed to mark on the real line the number $n$ (which is the number at a distance $\left| n\right|$ from $0$ and it is on the right of $0$ if $n>0$ and on the left of $0$ if $n<0$). For example, $f(0)=f(1)=0$ and $f(-1)=f(2)=1$. Find $f(n)$.

Let $n$ be a positive integer and suppose that $\phi(n)=\frac{n}{k}$, where $k$ is the greatest perfect square such that $k \mid n$. Let $a_1,a_2,\ldots,a_n$ be $n$ positive integers such that $a_i=p_1^{a_1i} \cdot p_2^{a_2i} \cdots p_n^{a_ni}$, where $p_i$ are prime numbers and $a_{ji}$ are non-negative integers, $1 \leq i \leq n, 1 \leq j \leq n$. We know that $p_i\mid \phi(a_i)$, and if $p_i\mid \phi(a_j)$, then $p_j\mid \phi(a_i)$. Prove that there exist integers $k_1,k_2,\ldots,k_m$ with $1 \leq k_1 \leq k_2 \leq \cdots \leq k_m \leq n$ such that \[\phi(a_{k_{1}} \cdot a_{k_{2}} \cdots a_{k_{m}})=p_1 \cdot p_2 \cdots p_n.\]

Let $n\geq 3$ a positive integer. In each cell of a $n\times n$ chessboard one must write $1$ or $2$ in such a way the sum of all written numbers in each $2\times 3$ and $3\times 2$ sub-chessboard is even. How many different ways can the chessboard be completed?

A convex solid is formed in four-dimensional Euclidean space with vertices at the $24$ possible permutations of $\{1, 2, 3, 4\}$ (so $(1, 2, 3, 4)$, $(1, 2, 4, 3)$, etc.). What is the product of the number of faces and edges of this solid?

In CMI, each person has atmost $3$ friends. A disease has infected exactly $2023$ peoplein CMI . Each day, a person gets infected if and only if atleast two of their friends were infected on the previous day. Once someone is infected, they can neither die nor be cured. Given that everyone in CMI eventually got infected, what is the maximum possible number of people in CMI?

Carlos and Yue play the following game: First Carlos writes a $+$ sign or a $-$ sign in front of each of the $50$ numbers $1,2,\cdots,50$. Then, in turns, each one chooses a number from the sequence obtained; Start by choosing Yue. If the absolute value of the sum of the $25$ numbers that Carlos chose is greater than or equal to the absolute value of the sum of the $25$ numbers that Yue chose, Carlos wins. In the other case, Yue wins. Determine which of the two players can develop a strategy that will ensure victory, no matter how well their opponent plays, and describe said strategy.

[b]p1.[/b] Farmer James goes to Kristy’s Krispy Chicken to order a crispy chicken sandwich. He can choose from $3$ types of buns, $2$ types of sauces, $4$ types of vegetables, and $4$ types of cheese. He can only choose one type of bun and cheese, but can choose any nonzero number of sauces, and the same with vegetables. How many different chicken sandwiches can Farmer James order? [b]p2.[/b] A line with slope $2$ and a line with slope $3$ intersect at the point $(m, n)$, where $m, n > 0$. These lines intersect the $x$ axis at points $A$ and $B$, and they intersect the y axis at points $C$ and $D$. If $AB = CD$, find $m/n$. [b]p3.[/b] A multi-set of $11$ positive integers has a median of $10$, a unique mode of $11$, and a mean of $ 12$. What is the largest possible number that can be in this multi-set? (A multi-set is a set that allows repeated elements.) [b]p4.[/b] Farmer James is swimming in the Eggs-Eater River, which flows at a constant rate of $5$ miles per hour, and is recording his time. He swims $ 1$ mile upstream, against the current, and then swims $1$ mile back to his starting point, along with the current. The time he recorded was double the time that he would have recorded if he had swum in still water the entire trip. To the nearest integer, how fast can Farmer James swim in still water, in miles per hour? [b]p5.[/b] $ABCD$ is a square with side length $60$. Point $E$ is on $AD$ and $F$ is on $CD$ such that $\angle BEF = 90^o$. Find the minimum possible length of $CF$. [b]p6.[/b] Farmer James makes a trianglomino by gluing together $5$ equilateral triangles of side length $ 1$, with adjacent triangles sharing an entire edge. Two trianglominoes are considered the same if they can be matched using only translations and rotations (but not reflections). How many distinct trianglominoes can Farmer James make? [b]p7.[/b] Two real numbers $x$ and $y$ satisfy $x^2 - y^2 = 2y - 2x$ , and $x + 6 = y^2 + 2y$. What is the sum of all possible values of$ y$? [b]p8.[/b] Let $N$ be a positive multiple of $840$. When $N$ is written in base $6$, it is of the form $\overline{abcdef}_6$ where $a, b, c, d, e, f$ are distinct base $6$ digits. What is the smallest possible value of $N$, when written in base $6$? [b]p9.[/b] For $S = \{1, 2,..., 12\}$, find the number of functions $f : S \to S$ that satisfy the following $3$ conditions: (a) If $n$ is divisible by $3$, $f(n)$ is not divisible by $3$, (b) If $n$ is not divisible by $3$, $f(n)$ is divisible by $3$, and (c) $f(f(n)) = n$ holds for exactly $8$ distinct values of $n$ in $S$. [b]p10.[/b] Regular pentagon $JAMES$ has area $ 1$. Let $O$ lie on line $EM$ and $N$ lie on line $MA$ so that $E, M, O$ and $M, A, N$ lie on their respective lines in that order. Given that $MO = AN$ and $NO = 11 \cdot ME$, find the area of $NOM$. [b]p11.[/b] Hen Hao is flipping a special coin, which lands on its sunny side and its rainy side each with probability $1/2$. Hen Hao flips her coin ten times. Given that the coin never landed with its rainy side up twice in a row, find the probability that Hen Hao’s last flip had its sunny side up. [b]p12.[/b] Find the product of all integer values of a such that the polynomial $x^4 + 8x^3 + ax^2 + 2x - 1$ can be factored into two non-constant polynomials with integer coefficients. [b]p13.[/b] Isosceles trapezoid $ABCD$ has $AB = CD$ and $AD = 6BC$. Point $X$ is the intersection of the diagonals $AC$ and $BD$. There exist a positive real number $k$ and a point $P$ inside $ABCD$ which satisfy $$[PBC] : [PCD] : [PDA] = 1 : k : 3,$$ where $[XYZ]$ denotes the area of triangle $XYZ$. If $PX \parallel AB$, find the value of $k$. [b]p14.[/b] How many positive integers $n < 1000$ are there such that in base $10$, every digit in $3n$ (that isn’t a leading zero) is greater than the corresponding place value digit (possibly a leading zero) in $n$? For example, $n = 56$, $3n = 168$ satisfies this property as $1 > 0$, $6 > 5$, and $8 > 6$. On the other hand, $n = 506$, $3n = 1518$ does not work because of the hundreds place. [b]p15.[/b] Find the greatest integer that is smaller than $$\frac{2018}{37^2}+\frac{2018}{39^2}+ ... +\frac{2018}{ 107^2}.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [u]Set 3[/u] [b]P3.11[/b] Find all possible values of $c$ in the following system of equations: $$a^2 + ab + c^2 = 31$$ $$b^2 + ab - c^2 = 18$$ $$a^2 - b^2 = 7$$ [b]P3.12 / R5.25[/b] In square $ABCD$ with side length $13$, point $E$ lies on segment $CD$. Segment $AE$ divides $ABCD$ into triangle $ADE$ and quadrilateral $ABCE$. If the ratio of the area of $ADE$ to the area of $ABCE$ is $4 : 11$, what is the ratio of the perimeter of $ADE$ to the perimeter of$ ABCE$? [b]P3.13[/b] Thomas has two distinct chocolate bars. One of them is $1$ by $5$ and the other one is $1$ by $3$. If he can only eat a single $1$ by $1$ piece off of either the leftmost side or the rightmost side of either bar at a time, how many different ways can he eat the two bars? [b]P3.14[/b] In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. The entire triangle is revolved about side $BC$. What is the volume of the swept out region? [b]P3.15[/b] Find the number of ordered pairs of positive integers $(a, b)$ that satisfy the equation $a(a -1) + 2ab + b(b - 1) = 600$. [u]Set 4[/u] [b]P4.16[/b] Compute the sum of the digits of $(10^{2017} - 1)^2$ . [b]P4.17[/b] A right triangle with area $210$ is inscribed within a semicircle, with its hypotenuse coinciding with the diameter of the semicircle. $2$ semicircles are constructed (facing outwards) with the legs of the triangle as their diameters. What is the area inside the $2$ semicircles but outside the first semicircle? [b]P4.18[/b] Find the smallest positive integer $n$ such that exactly $\frac{1}{10}$ of its positive divisors are perfect squares. [b]P4.19[/b] One day, Sambuddha and Jamie decide to have a tower building competition using oranges of radius $1$ inch. Each player begins with $14$ oranges. Jamie builds his tower by making a $3$ by $3$ base, placing a $2$ by $2$ square on top, and placing the last orange at the very top. However, Sambuddha is very hungry and eats $4$ of his oranges. With his remaining $10$ oranges, he builds a similar tower, forming an equilateral triangle with $3$ oranges on each side, placing another equilateral triangle with $2$ oranges on each side on top, and placing the last orange at the very top. What is the positive difference between the heights of these two towers? [b]P4.20[/b] Let $r, s$, and $t$ be the roots of the polynomial $x^3 - 9x + 42$. Compute the value of $(rs)^3 + (st)^3 + (tr)^3$. [u]Set 5[/u] [b]P5.21[/b] For all integers $k > 1$, $\sum_{n=0}^{\infty}k^{-n} =\frac{k}{k -1}$. There exists a sequence of integers $j_0, j_1, ...$ such that $\sum_{n=0}^{\infty}j_n k^{-n} =\left(\frac{k}{k -1}\right)^3$ for all integers $k > 1$. Find $j_{10}$. [b]P5.22[/b] Nimi is a triangle with vertices located at $(-1, 6)$, $(6, 3)$, and $(7, 9)$. His center of mass is tied to his owner, who is asleep at $(0, 0)$, using a rod. Nimi is capable of spinning around his center of mass and revolving about his owner. What is the maximum area that Nimi can sweep through? [b]P5.23[/b] The polynomial $x^{19} - x - 2$ has $19$ distinct roots. Let these roots be $a_1, a_2, ..., a_{19}$. Find $a^{37}_1 + a^{37}_2+...+a^{37}_{19}$. [b]P5.24[/b] I start with a positive integer $n$. Every turn, if $n$ is even, I replace $n$ with $\frac{n}{2}$, otherwise I replace $n$ with $n-1$. Let $k$ be the most turns required for a number $n < 500$ to be reduced to $1$. How many values of $n < 500$ require k turns to be reduced to $1$? [b]P5.25[/b] In triangle $ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Let $I$ and $O$ be the incircle and circumcircle of $ABC$, respectively. The altitude from $A$ intersects $I$ at points $P$ and $Q$, and $O$ at point $R$, such that $Q$ lies between $P$ and $R$. Find $PR$. PS. You should use hide for answers. R1-15 /P1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here[/url], and R16-30 /P6-10/ P26-30 [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].