Find all five-digit numbers that satisfy the following conditions: 1. The number is a palindrome. 2. The middle digit is twice the value of the first digit. 3. The number is a perfect square. [i]Proposed by Tamar Turashvili, Georgia [/i]

Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation \[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\] For all $x,y\in\mathbb R.$

[b]p1.[/b] What is the slope of the line perpendicular to the the graph $\frac{x}{4}+\frac{y}{9}= 1$ at $(0, 9)$? [b]p2.[/b] A boy is standing in the middle of a very very long staircase and he has two pogo sticks. One pogo stick allows him to jump $220$ steps up the staircase. The second pogo stick allows him to jump $125$ steps down the staircase. What is the smallest positive number of steps that he can reach from his original position by a series of jumps? [b]p3.[/b] If you roll three fair six-sided dice, what is the probability that the product of the results will be a multiple of $3$? [b]p4.[/b] Right triangle $ABC$ has squares $ABXY$ and $ACWZ$ drawn externally to its legs and a semicircle drawn externally to its hypotenuse $BC$. If the area of the semicircle is $18\pi$ and the area of triangle $ABC$ is $30$, what is the sum of the areas of squares $ABXY$ and $ACWZ$? [img]https://cdn.artofproblemsolving.com/attachments/5/1/c9717e7731af84e5286335420b73b974da2643.png[/img] [b]p5.[/b] You have a bag containing $3$ types of pens: red, green, and blue. $30\%$ of the pens are red pens, and $20\%$ are green pens. If, after you add $10$ blue pens, $60\%$ of the pens are blue pens, how many green pens did you start with? [b]p6.[/b] Canada gained partial independence from the United Kingdom in $1867$, beginning its long role as the headgear of the United States. It gained its full independence in $1982$. What is the last digit of $1867^{1982}$? [b]p7.[/b] Bacon, Meat, and Tomato are dealing with paperwork. Bacon can fill out $5$ forms in $3$ minutes, Meat can fill out $7$ forms in $5$ minutes, and Tomato can staple $3$ forms in $1$ minute. A form must be filled out and stapled together (in either order) to complete it. How long will it take them to complete $105$ forms? [b]p8.[/b] Nice numbers are defined to be $7$-digit palindromes that have no $3$ identical digits (e.g., $1234321$ or $5610165$ but not $7427247$). A pretty number is a nice number with a $7$ in its decimal representation (e.g., $3781873$). What is the $7^{th}$ pretty number? [b]p9.[/b] Let $O$ be the center of a semicircle with diameter $AD$ and area $2\pi$. Given square $ABCD$ drawn externally to the semicircle, construct a new circle with center $B$ and radius $BO$. If we extend $BC$, this new circle intersects $BC$ at $P$. What is the length of $CP$? [img]https://cdn.artofproblemsolving.com/attachments/b/1/be15e45cd6465c7d9b45529b6547c0010b8029.png[/img] [b]p10.[/b] Derek has $10$ American coins in his pocket, summing to a total of $53$ cents. If he randomly grabs $3$ coins from his pocket, what is the probability that they're all different? [b]p11.[/b] What is the sum of the whole numbers between $6\sqrt{10}$ and $7\pi$ ? [b]p12.[/b] What is the volume of a cylinder whose radius is equal to its height and whose surface area is numerically equal to its volume? [b]p13.[/b] $15$ people, including Luke and Matt, attend a Berkeley Math meeting. If Luke and Matt sit next to each other, a fight will break out. If they sit around a circular table, all positions equally likely, what is the probability that a fight doesn't break out? [b]p14.[/b] A non-degenerate square has sides of length $s$, and a circle has radius $r$. Let the area of the square be equal to that of the circle. If we have a rectangle with sides of lengths $r$, $s$, and its area has an integer value, what is the smallest possible value for $s$? [b]p15.[/b] How many ways can you arrange the letters of the word "$BERKELEY$" such that no two $E$'s are next to each other? [b]p16.[/b] Kim, who has a tragic allergy to cake, is having a birthday party. She invites $12$ people but isn't sure if $11$ or $12$ will show up. However, she needs to cut the cake before the party starts. What is the least number of slices (not necessarily of equal size) that she will need to cut to ensure that the cake can be equally split among either $11$ or $12$ guests with no excess? [b]p17.[/b] Tom has $2012$ blue cards, $2012$ red cards, and $2012$ boxes. He distributes the cards in such a way such that each box has at least $1$ card. Sam chooses a box randomly, then chooses a card randomly. Suppose that Tom arranges the cards such that the probability of Sam choosing a blue card is maximized. What is this maximum probability? [b]p18.[/b] Allison wants to bake a pie, so she goes to the discount market with a handful of dollar bills. The discount market sells different fruit each for some integer number of cents and does not add tax to purchases. She buys $22$ apples and $7$ boxes of blueberries, using all of her dollar bills. When she arrives back home, she realizes she needs more fruit, though, so she grabs her checkbook and heads back to the market. This time, she buys $31$ apples and $4$ boxes of blueberries, for a total of $60$ cents more than her last visit. Given she spent less than $100$ dollars over the two trips, how much (in dollars) did she spend on her first trip to the market? [b]p19.[/b] Consider a parallelogram $ABCD$. Let $k$ be the line passing through A and parallel to the bisector of $\angle ABC$, and let $\ell$ be the bisector of $\angle BAD$. Let $k$ intersect line $CD$ at $E$ and $\ell$ intersect line $CD$ at $F$. If $AB = 13$ and $BC = 37$, find the length $EF$. [b]p20.[/b] Given for some real $a, b, c, d,$ $$P(x) = ax^4 + bx^3 + cx^2 + dx$$ $$P(-5) = P(-2) = P(2) = P(5) = 1$$ Find $P(10).$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose $a$ and $b$ are two positive real numbers such that the roots of the cubic equation $x^3 - ax + b = 0$ are all real. If $\alpha$ is a root of this cubic with minimal absolute value, prove that \[ \dfrac{b}{a} < \alpha < \dfrac{3b}{2a}. \]

For odd numbers $a_1 , ..., a_k$ and even numbers $b_1 , ..., b_k$ , we know that $\sum_ {j = 1}^k a_j^n = \sum_{j = 1}^k b_j^n$ is satisfied for n = 1,2, ..., N. Prove that $k\geq 2^N$ and that for $k = 2^N$ there exists a solution $(a_1,...,b_1,...)$ with the above properties.

Find the greatest value of $C$ for which, for any $x, y, z,u$, and such that for $0\le x\le y \le z\le u$, holds the inequality $$(x + y +z + u)^2 \ge Cyz .$$

Do there exist polynomials $a(x)$, $b(x)$, $c(y)$, $d(y)$ such that \[1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)\] holds identically?

There is no sequence $x_n$ strictly increasing with terms natural numbers such that : $$ x_n+x_{k}=x_{nk}, \ \ for \, any \,\,\, n, k \in \mathbb{N}^*$$

Let $n$ be a natural number and let $P (t) = 1 + t + t^2 + ... + t^{2n}$. If $x \in R$ such that $P (x)$ and $P (x^2)$ are rational numbers, prove that $x$ is rational number.

A polynomial $P(x)$ is called [i]nice[/i] if $P(0) = 1$ and the nonzero coefficients of $P(x)$ alternate between $1$ and $-1$ when written in order. Suppose that $P(x)$ is nice, and let $m$ and $n$ be two relatively prime positive integers. Show that \[Q(x) = P(x^n) \cdot \frac{(x^{mn} - 1)(x-1)}{(x^m-1)(x^n-1)}\] is nice as well.

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

Find all vectors of $n$ real numbers $(x_1,x_2,\ldots,x_n)$ such that \[ \left\{ \begin{array}{ccc} x_1 & = & \dfrac 1{x_2} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n } \\ x_2 & = & \dfrac 1{x_1} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n} \\ \ & \cdots & \ \\ x_n & = & \dfrac 1{x_1} + \dfrac 1{x_2} + \cdots + \dfrac 1{x_{n-1}} \end{array} \right. \] [i]Mircea Becheanu[/i]

Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x \plus{} y)(f(x) \minus{} y) \equal{} xf(x) \minus{} yf(y) \] for all $ x,y \in \mathbb{R}$.

Prove that for any $n$ ($n \geq 2$) pairwise distinct fractions in the interval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.

Some positive integers are initially written on a board, where each $2$ of them are different. Each time we can do the following moves: (1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 numbers (written in the board) in the form $n, n+4$ we can erase them and write down $n-1$ After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that: Independetly of the starting numbers, each number which appears in any move is greater or equal to $c$

For any two points $A (x_1 , y_1)$ and $B (x_2, y_2)$, the distance $r (A, B)$ between them is determined by the equality $r(A, B) = | x_1- x_2 | + | y_1 - y_2 |$. Prove that the triangle inequality $r(A, C) + r(C, B) \ge r(A, B)$. holds for the distance introduced in this way . Let $A$ and $B$ be two points of the plane (you can take $A(1, 3)$, $B(3, 7)$). Find the locus of points $C$ for which a) $r(A, C) + r(C, B) = r(A, B)$ b) $r(A, C) = r(C, B).$

Prove that the polynomial $x^4+\lambda x^3+\mu x^2+\nu x+1$ has no real roots if $\lambda, \mu , \nu $ are real numbers satisfying \[|\lambda |+|\mu |+|\nu |\le \sqrt{2} \]

Provided that the roots of the polynom $ X^n+a_1X^{n-1} +a_2X^{n-2} +\cdots +a_{n-1}X +a_n:\in\mathbb{R}[X] , $ of degree $ n\ge 2, $ are all real and pairwise distinct, prove that there exists is a neighbourhood $ \mathcal{V} $ of $ \left( a_1,a_2,\ldots ,a_n \right) $ in $ \mathbb{R}^n $ and $ n $ functions $ x_1,x_2,\ldots ,x_n\in\mathcal{C}^{\infty } \left( \mathcal{V} \right) $ whose values at $ \left( a_1,a_2,\ldots ,a_n \right) $ are roots of the mentioned polynom.

Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$. Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right|  < 1$ for all real numbers $x$.

If $\theta$ is the unique solution in $(0,\pi)$ to the equation $2\sin(x)+3\sin(\tfrac{3x}{2})+\sin(2x)+3\sin(\tfrac{5x}{2})=0,$ then $\cos(\theta)=\tfrac{a-\sqrt{b}}{c}$ for positive integers $a,b,c$ such that $a$ and $c$ are relatively prime. Find $a+b+c.$

$a_1,a_2,\dots ,a_n>0$ are positive real numbers such that $\sum_{i=1}^{n} \frac{1}{a_i}=n$ prove that: $\sum_{i<j} \left(\frac{a_i-a_j}{a_i+a_j}\right)^2\le\frac{n^2}{2}\left(1-\frac{n}{\sum_{i=1}^{n}a_i}\right)$

Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ such that for all integers $x$, $y$, $$f(x-f(y))=f(f(y))+f(x-2y)$$ [i]Proposed by Ivan Chan Kai Chin[/i]

The polynomial of seven variables $$ Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2) $$ is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients: $$ Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2. $$ Find all possible values of $P_1(1,1,\ldots,1)$. [i](A. Yuran)[/i]

Let ${a_n}$ be a sequence defined by $a_1=2$, $a_{n+1}=\left[ \frac {3a_n}{2}\right]$ $\forall n \in \mathbb N$ $0.a_1a_2...$ rational or irrational?

Let $\delta$ be a symbol such that $\delta \neq 0$ and $\delta^2 = 0$. Define $\mathbb R[\delta] = \{a + b \delta | a, b \in \mathbb R\}$, where $a+ b \delta = c+ d \delta$ if and only if $a = c$ and $b = d$, and define \[(a + b \delta) + (c + d \delta) = (a + c) + (b + d) \delta,\]\[(a + b \delta) \cdot (c + d \delta) = ac + (ad + bc) \delta.\] Let $P(x)$ be a polynomial with real coefficients. Show that $P(x)$ has a multiple real root if and only if $P(x)$ has a non-real root in $\mathbb R[\delta].$