Derek and Jacob have a cake in the shape a rectangle with dimensions $14$ inches by $9$ inches. They make a deal to split it: Derek takes home the portion of the cake that is less than one inch from the border, while Jacob takes home the remainder of the cake. Let $D : J$ be the ratio of the amount of cake Derek took to the amount of cake Jacob took, where $D$ and $J$ are relatively prime positive integers. Find $D + J$.

For integers $a$, $b$, $c$, and $d$, let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$. Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2)) = g(f(4)) = 0$.

Solve \[\left\{ \begin{array}{l} y(x+y)^2 = 9 \\ y(x^3-y^3) = 7 \\ \end{array} \right. \]

Determine all non-negative real numbers $a$, for which $f(a)=0$ for all functions $f: \mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0} $, who satisfy the equation $f(f(x) + f(y)) = yf(1 + yf(x))$ for all non-negative real numbers $x$ and $y$.

Let $a,b,c,m$ be integers, where $m>1$. Prove that if $$a^n+bn+c\equiv0\pmod m$$for each natural number $n$, then $b^2\equiv0\pmod m$. Must $b\equiv0\pmod m$ also hold?

Determine if there exists a strictly increasing sequence of positive integers $a_1$, $a_2$, ... such that $a_n \le n^3$ for every positive integer $n$ and that every positive integer can be written uniquely as the difference of two terms in the sequence.

Let $S$ denote the set of rational numbers in the interval $(0,1)$. Determine, with proof, if there exists a subset $T$ of $S$ such that every element in $S$ can be uniquely written as the sum of finitely many distinct elements in $T$.

$64$ non-negative numbers whose sum equals $1956$ are arranged in a square table, eight numbers in each row and each column. The sum of the numbers on the two longest diagonals is equal to $112$. The numbers situated symmetrically with respect to any of the longest diagonals are equal. (a) Prove that the sum of numbers in any column is less than $1035$. (b) Prove that the sum of numbers in any row is less than $518$.

Find all real numbers $x, y,z,t \in [0, \infty)$ so that $$x + y + z \le t, \,\,\, x^2 + y^2 + z^2 \ge t \,\,\, and \,\,\,x^3 + y^3 + z^3 \le t.$$

Determine all pairs $(a,b)$ of non-negative integers such that $$ \frac{a+b}{2}-\sqrt{ab}=1.$$

What necessary and sufficient condition should the coefficients $ a $, $ b $, $ c $, $ d $ satisfy so that the equation $$ax^3 + bx^2 + cx + d = 0$$ has two opposite roots?

Let $a, b, c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{1}{a}+\frac{3}{b}+\frac{5}{c} \ge 4a^2 + 3b^2 + 2c^2$$ When does the equality hold? Marius Stanean

Prove that for any real number $ x$ the following inequality is true: $ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$

Consider all positive multiples of $77$ less than $1,000,000.$ What is the sum of all the odd digits that show up?

For all positive numbers $a, b, c$ satisfying $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$, prove that: $$ \frac{a}{a+bc} + \frac{b}{b+ca} + \frac{c}{c+ab} \geq \frac{3}{4} .$$

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17$. What is the greatest possible perimeter of the triangle?

Three sequences ${a_n},{b_n},{c_n}$ satisfy the following conditions. [list] [*]$a_1=2,\,b_1=4,\,c_1=5$ [*]$\forall n,\; a_{n+1}=b_n+\frac{1}{c_n}, \, b_{n+1}=c_n+\frac{1}{a_n}, \, c_{n+1}=a_n+\frac{1}{b_n}$ [/list] Prove that for all positive integers $n$, $ $ $ $ $max(a_n,b_n,c_n)>\sqrt{2n+13}$.

On the first day of school, Ashley the teacher asked some of her students what their favorite color was and used those results to construct the pie chart pictured below. During this first day, $165$ students chose yellow as their favorite color. The next day, she polled $30$ additional students and was shocked when none of them chose yellow. After making a new pie chart based on the combined results of both days, Ashley noticed that the angle measure of the sector representing the students whose favorite color was yellow had decreased. Compute the difference, in degrees, between the old and the new angle measures. [img]https://cdn.artofproblemsolving.com/attachments/2/5/f605bf8d684075fe13fee9eb44f8f50b64c7d3.png[/img]

Solve the inequality $$3-2\left(3-2\left(3-...-2(3-2x)\right)...\right) >x$$. The total number of right parentheses is $100$.

Initially, a pair of numbers $(1,1)$ is written on the board. If for some $x$ and $y$ one of the pairs $(x, y-1)$ and $(x+y, y+1)$ is written on the board, then you can add the other one. Similarly for $(x, xy)$ and $(\frac {1} {x}, y)$. Prove that for each pair that appears on the board, its first number will be positive.

Consider the function $ f: \mathbb{N}_0\to\mathbb{N}_0$, where $ \mathbb{N}_0$ is the set of all non-negative integers, defined by the following conditions : $ (i)$ $ f(0) \equal{} 0$; $ (ii)$ $ f(2n) \equal{} 2f(n)$ and $ (iii)$ $ f(2n \plus{} 1) \equal{} n \plus{} 2f(n)$ for all $ n\geq 0$. $ (a)$ Determine the three sets $ L \equal{} \{ n | f(n) < f(n \plus{} 1) \}$, $ E \equal{} \{n | f(n) \equal{} f(n \plus{} 1) \}$, and $ G \equal{} \{n | f(n) > f(n \plus{} 1) \}$. $ (b)$ For each $ k \geq 0$, find a formula for $ a_k \equal{} \max\{f(n) : 0 \leq n \leq 2^k\}$ in terms of $ k$.

Compute the smallest positive integer $n$ for which $$\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}$$ is an integer.

[b]7.1.[/b] Prove that from the sides of an arbitrary quadrilateral you can fold a trapezoid. [b]7.2 / 6.2[/b] The numbers $A$ and $B$ are relatively prime. What common divisors can have the numbers $A+B$ and $A-B$? [b]7.3. / 6.4[/b] $15$ magazines lie on the table, completely covering it. Prove that it is possible to remove eight of them so that the remaining magz cover at least $7/15$ of the table area. [b]7.4[/b] In a six-digit number that is divisible by $7$, the last digit has been moved to the beginning. Prove that the resulting number is also divisible at $7$. [url=https://artofproblemsolving.com/community/c6h3391057p32066818]7.5*[/url] (asterisk problems in separate posts) [b]7.6 [/b] On sides $AB$ and $ BC$ of triangle $ABC$ , are constructed squares $ABDE$ and $BCKL$ with centers $O_1$ and $O_2$. $M_1$ and $M_2$ are midpoints of segments $DL$ and $AC$. Prove that $O_1M_1O_2M_2$ is a square. [img]https://cdn.artofproblemsolving.com/attachments/8/1/8aa816a84c5ac9de78b396096cf718063de390.png[/img] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].

Given real numbers $x,y,z,a$ satisfying: $$ x+y+z = a$$ $$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z} = \frac{1}{a} $$ Prove that at least one of the numbers $x,y,z$ is equal to $a$.

Let $f$ be a polynomial function with integer coefficients and $p$ be a prime number. Suppose there are at least four distinct integers satisfying $f(x) = p$. Show that $f$ does not have integer zeros.