Petya digs the garden bed alone for $a$ minutes longer than he does with Vasya. Vasya digs up the same bed for $b$ minutes longer than he would have done with Petya. How many minutes does it take Vasya and Petya to dig up the same bed together? orthogonal).

Show that there is no integer-valued function on the integers such that $f(m+f(n))=f(m)-n$ for all $m,n$.

(a) Given $2 + 4 + 6 + \dots + p = 6480$, find $p$. (b) Given $7 + 11 + 15 + \dots + q = 5250$, find $q$. (c) Given $2^2 + 4^2 + 6^2 + \dots + r^2 - 1^2 - 3^2 - 5^2 - \dots - (r-1)^2 = 2485$, compute $r$.

[b]p17.[/b] Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ? [b]p18.[/b] Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play? [b]p19.[/b] Compute the sum of all primes $p$ such that $2^p + p^2$ is also prime. [b]p20.[/b] In how many ways can one color the $8$ vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be the product of $n$ real numbers $x_1$, $x_2$,$...$, $x_n$. Prove that if $p - x_k$ is an odd integer for $k = 1, 2,..., n$, then each of the numbers $x_1$, $x_2$,$...$, $x_n$is irrational. (G Galperin)

Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that [i](a)[/i] $ f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n$ for all integers $ m$ and $ n;$ [i](b)[/i] $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \mathbb{Z}.$

We say that a rectangle and a triangle are [i]similar[/i], if they have the same area and the same perimeter. Let $P$ be a rectangle for which the ratio of the longer to the shorter side is at least $\lambda -1+\sqrt{\lambda (\lambda -2)}$ where $\lambda =\frac{3\sqrt{3}}{2}$. Prove that there exists a tringle that is [i]similar[/i] to $P$.

If $0<a\leq b\leq c$ prove that $$\frac{(c-a)^2}{6c}\leq \frac{a+b+c}{3}-\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$

$$Problem3;$$If distinct real numbers x,y satisfy $\{x\} = \{y\}$ and $\{x^3\}=\{y^3\}$ prove that $x$ is a root of a quadratic equation with integer coefficients.

Let $P(x)$ be a quadratic polynomial with nonnegative coeficients. Show that for any real numbers $x$ and $y$, we have the inequality $P(xy)^2 \leqslant P(x^2)P(y^2)$. [i]E. Malinnikova[/i]

Does \[\left\{ \begin{array}{l} x^y = z \\ y^z = x \\ z^x = y \\ \end{array} \right. \] have any solutions in positive reals apart from $x = y = z= 1$?

Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$. Prove the inequality $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$

[u]Round 1[/u] [b]p1.[/b] Is it possible to draw some number of diagonals in a convex hexagon so that every diagonal crosses EXACTLY three others in the interior of the hexagon? (Diagonals that touch at one of the corners of the hexagon DO NOT count as crossing.) [b]p2.[/b] A $ 3\times 3$ square grid is filled with positive numbers so that (a) the product of the numbers in every row is $1$, (b) the product of the numbers in every column is $1$, (c) the product of the numbers in any of the four $2\times 2$ squares is $2$. What is the middle number in the grid? Find all possible answers and show that there are no others. [b]p3.[/b] Each letter in $HAGRID$'s name represents a distinct digit between $0$ and $9$. Show that $$HAGRID \times H \times A\times G\times R\times I\times D$$ is divisible by $3$. (For example, if $H=1$, $A=2$, $G=3$, $R = 4$, $I = 5$, $D = 64$, then $HAGRID \times H \times A\times G\times R\times I\times D= 123456\times 1\times2\times3\times4\times5\times 6$). [b]p4.[/b] You walk into a room and find five boxes sitting on a table. Each box contains some number of coins, and you can see how many coins are in each box. In the corner of the room, there is a large pile of coins. You can take two coins at a time from the pile and place them in different boxes. If you can add coins to boxes in this way as many times as you like, can you guarantee that each box on the table will eventually contain the same number of coins? [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2[/u] [b]p6.[/b] After going for a swim in his vault of gold coins, Scrooge McDuck decides he wants to try to arrange some of his gold coins on a table so that every coin he places on the table touches exactly three others. Can he possibly do this? You need to justify your answer. (Assume the gold coins are circular, and that they all have the same size. Coins must be laid at on the table, and no two of them can overlap.) [b]p7.[/b] You have a deck of $50$ cards, each of which is labeled with a number between $1$ and $25$. In the deck, there are exactly two cards with each label. The cards are shuffled and dealt to $25$ students who are sitting at a round table, and each student receives two cards. The students will now play a game. On every move of the game, each student takes the card with the smaller number out of his or her hand and passes it to the person on his/her right. Each student makes this move at the same time so that everyone always has exactly two cards. The game continues until some student has a pair of cards with the same number. Show that this game will eventually end. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_0,a_1,\dots$ be a sequence of positive reals such that $$ a_{n+2} \leq \frac{2023a_n}{a_na_{n+1}+2023}$$ for all integers $n\geq 0$. Prove that either $a_{2023}<1$ or $a_{2024}<1$.

Let $m$ and $n$ be positive integers with $m\le 2000$ and $k=3-\frac{m}{n}$. Find the smallest positive value of $k$.

1.1 Let $f(A)$ denote the difference between the maximum value and the minimum value of a set $A$. Find the sum of $f(A)$ as $A$ ranges over the subsets of $\{1, 2, \dots, n\}$. 1.2 All cells of an $8 × 8$ board are initially white. A move consists of flipping the color (white to black or vice versa) of cells in a $1\times 3$ or $3\times 1$ rectangle. Determine whether there is a finite sequence of moves resulting in the state where all $64$ cells are black. 1.3 Prove that for all positive integers $m$, there exists a positive integer $n$ such that the set $\{n, n + 1, n + 2, \dots , 3n\}$ contains exactly $m$ perfect squares.

Let $a,b,c > 0$ and $ab+bc+ca = 1$. Prove the inequality $3\sqrt[3]{\frac{1}{abc} +6(a+b+c) }\le \frac{\sqrt[3]3}{abc}$

Does there exist a solution in real numbers of the system of equations \[\left\{ \begin{array}{rcl} (x - y)(z - t)(z - x)(z - t)^2 = A, \\ (y - z)(t - x)(t - y)(x - z)^2 = B,\\ (x - z)(y - t)(z - t)(y - z)^2 = C,\\ \end{array} \right.\] when a) $A=2, B=8, C=6;$ b) $A=2, B=6, C=8.$?

Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote \[ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. \] Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$

Let $x, y, z$ be positive real numbers whose sum is $2012$. Find the maximum value of $$ \frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{(x^4 + y^4 + z^4)}$$

Find all polynomials $Q(x)= ax^2+bx+c$ with integer coefficients for which there exist three different prime numbers $p_1, p_2, p_3$ such that $|Q(p_1)| = |Q(p_2)| = |Q(p_3)| = 11$.

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

Let $n$ and $k$ be positive integers. Consider $n$ infinite arithmetic progressions of nonnegative integers with the property that among any $k$ consecutive nonnegative integers, at least one of $k$ integers belongs to one of the $n$ arithmetic progressions. Let $d_1,d_2,\ldots,d_n$ denote the differences of the arithmetic progressions, and let $d=\min\{d_1,d_2,\ldots,d_n\}$. In terms of $n$ and $k$, what is the maximum possible value of $d$?

Solve the system of equations: $ \begin{matrix} x^2 + x - 1 = y \\ y^2 + y - 1 = z \\ z^2 + z - 1 = x. \end{matrix} $

Let $k,n$ and $r$ be positive integers. (a) Let $Q(x)=x^k+a_1x^{k+1}+\cdots+a_nx^{k+n}$ be a polynomial with real coefficients. Show that the function $\frac{Q(x)}{x^k}$ is strictly positive for all real $x$ satisfying $$0<|x|<\frac1{1+\sum\limits_{i=1}^n |a_i|}$$ (b) Let $P(x)=b_0+b_1x+\cdots+b_rx^r$ be a non zero polynomial with real coefficients. Let $m$ be the smallest number such that $b_m \neq 0$. Prove that the graph of $y=P(x)$ cuts the $x$-axis at the origin (i.e., $P$ changes signs at $x=0$) if and only if $m$ is an odd integer.