Let $a$ and $b$ satisfy $a \ge b >0, a + b = 1$. i) Prove that if $m$ and $n$ are positive integers with $m < n$, then $a^m - a^n \ge b^m- b^n > 0$. ii) For each positive integer $n$, consider a quadratic function $f_n(x) = x^2 - b^nx- a^n$. Show that $f(x)$ has two roots that are in between $-1$ and $1$.

Solve the equation $mn =$ (gcd($m,n$))$^2$ + lcm($m, n$) in positive integers, where gcd($m, n$) – greatest common divisor of $m,n$, and lcm($m, n$) – least common multiple of $m,n$.

$a_1,a_2,...,a_{81}$ are nonzero, $a_i+a_{i+1}>0$ for $i=1,...,80$ and $a_1+a_2+...+a_{81}<0$. What is sign of $a_1*a_2*...*a_{81}$?

a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying: [b]i.[/b] each has $k$ elements; [b]ii.[/b] $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$) [b]iii.[/b] the union of the $5$ sets has exactly $f(k)$ elements. b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.

Can an arbitrary polygon be cut into isosceles trapezoids?

[b]p1.[/b] What is $20\times 20 - 19\times 19$? [b]p2.[/b] Andover has a total of $1440$ students and teachers as well as a $1 : 5$ teacher-to-student ratio (for every teacher, there are exactly $5$ students). In addition, every student is either a boarding student or a day student, and $70\%$ of the students are boarding students. How many day students does Andover have? [b]p3.[/b] The time is $2:20$. If the acute angle between the hour hand and the minute hand of the clock measures $x$ degrees, find $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/a/a18b089ae016b15580ec464c3e813d5cb57569.png[/img] [b]p4.[/b] Point $P$ is located on segment $AC$ of square $ABCD$ with side length $10$ such that $AP >CP$. If the area of quadrilateral $ABPD$ is $70$, what is the area of $\vartriangle PBD$? [b]p5.[/b] Andrew always sweetens his tea with sugar, and he likes a $1 : 7$ sugar-to-unsweetened tea ratio. One day, he makes a $100$ ml cup of unsweetened tea but realizes that he has run out of sugar. Andrew decides to borrow his sister's jug of pre-made SUPERSWEET tea, which has a $1 : 2$ sugar-to-unsweetened tea ratio. How much SUPERSWEET tea, in ml,does Andrew need to add to his unsweetened tea so that the resulting tea is his desired sweetness? [b]p6.[/b] Jeremy the architect has built a railroad track across the equator of his spherical home planet which has a radius of exactly $2020$ meters. He wants to raise the entire track $6$ meters off the ground, everywhere around the planet. In order to do this, he must buymore track, which comes from his supplier in bundles of $2$ meters. What is the minimum number of bundles he must purchase? Assume the railroad track was originally built on the ground. [b]p7.[/b] Mr. DoBa writes the numbers $1, 2, 3,..., 20$ on the board. Will then walks up to the board, chooses two of the numbers, and erases them from the board. Mr. DoBa remarks that the average of the remaining $18$ numbers is exactly $11$. What is the maximum possible value of the larger of the two numbers that Will erased? [b]p8.[/b] Nathan is thinking of a number. His number happens to be the smallest positive integer such that if Nathan doubles his number, the result is a perfect square, and if Nathan triples his number, the result is a perfect cube. What is Nathan's number? [b]p9.[/b] Let $S$ be the set of positive integers whose digits are in strictly increasing order when read from left to right. For example, $1$, $24$, and $369$ are all elements of $S$, while $20$ and $667$ are not. If the elements of $S$ are written in increasing order, what is the $100$th number written? [b]p10.[/b] Find the largest prime factor of the expression $2^{20} + 2^{16} + 2^{12} + 2^{8} + 2^{4} + 1$. [b]p11.[/b] Christina writes down all the numbers from $1$ to $2020$, inclusive, on a whiteboard. What is the sum of all the digits that she wrote down? [b]p12.[/b] Triangle $ABC$ has side lengths $AB = AC = 10$ and $BC = 16$. Let $M$ and $N$ be the midpoints of segments $BC$ and $CA$, respectively. There exists a point $P \ne A$ on segment $AM$ such that $2PN = PC$. What is the area of $\vartriangle PBC$? [b]p13.[/b] Consider the polynomial $$P(x) = x^4 + 3x^3 + 5x^2 + 7x + 9.$$ Let its four roots be $a, b, c, d$. Evaluate the expression $$(a + b + c)(a + b + d)(a + c + d)(b + c + d).$$ [b]p14.[/b] Consider the system of equations $$|y - 1| = 4 -|x - 1|$$ $$|y| =\sqrt{|k - x|}.$$ Find the largest $k$ for which this system has a solution for real values $x$ and $y$. [b]p16.[/b] Let $T_n = 1 + 2 + ... + n$ denote the $n$th triangular number. Find the number of positive integers $n$ less than $100$ such that $n$ and $T_n$ have the same number of positive integer factors. [b]p17.[/b] Let $ABCD$ be a square, and let $P$ be a point inside it such that $PA = 4$, $PB = 2$, and $PC = 2\sqrt2$. What is the area of $ABCD$? [b]p18.[/b] The Fibonacci sequence $\{F_n\}$ is defined as $F_0 = 0$, $F_1 = 1$, and $F_{n+2}= F_{n+1} + F_n$ for all integers $n \ge 0$. Let $$ S =\dfrac{1}{F_6 + \frac{1}{F_6}}+\dfrac{1}{F_8 + \frac{1}{F_8}}+\dfrac{1}{F_{10} +\frac{1}{F_{10}}}+\dfrac{1}{F_{12} + \frac{1}{F_{12}}}+ ... $$ Compute $420S$. [b]p19.[/b] Let $ABCD$ be a square with side length $5$. Point $P$ is located inside the square such that the distances from $P$ to $AB$ and $AD$ are $1$ and $2$ respectively. A point $T$ is selected uniformly at random inside $ABCD$. Let $p$ be the probability that quadrilaterals $APCT$ and $BPDT$ are both not self-intersecting and have areas that add to no more than $10$. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. Note: A quadrilateral is self-intersecting if any two of its edges cross. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For some integer n the equation $x^2 + y^2 + z^2 -xy -yz - zx = n$ has an integer solution $x, y, z$. Prove that the equation$ x^2 + y^2 - xy = n$ also has an integer solution $x, y$. Alexandr Yuran

Alex the Kat plays the following game. First, he writes the number $27000$ on a blackboard. Each minute, he erases the number on the blackboard and replaces it with a number chosen uniformly randomly from its positive divisors, including itself. Find the probability that, after $2019$ minutes, the number on the blackboard is $1$.

Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.

Let $a,b,c,d$ be non-negative integers. $(1)$ If $a^2+b^2-cd^2=2022 ,$ find the minimum of $a+b+c+d;$ $(1)$ If $a^2-b^2+cd^2=2022 ,$ find the minimum of $a+b+c+d .$

How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?

What is the value of $\frac{(2112-2021)^2}{169}$? $\textbf{(A) }7\qquad\textbf{(B) }21\qquad\textbf{(C) }49\qquad\textbf{(D) }64\qquad\textbf{(E) }91$

Define a sequence of polynomials $P_0,P_1,...$ by the recurrence $P_0(x)=1, P_1(x)=x, P_{n+1}(x) = 2xP_n(x)-P_{n-1}(x)$. Let $S=\left|P_{2017}'\left(\frac{i}{2}\right)\right|$ and $T=\left|P_{17}'\left(\frac{i}{2}\right)\right|$, where $i$ is the imaginary unit. Then $\frac{S}{T}$ is a rational number with fractional part $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m$. [i]Proposed by Tristan Shin[/i]

What is the largest integer that is a divisor of \[ (n\plus{}1)(n\plus{}3)(n\plus{}5)(n\plus{}7)(n\plus{}9) \]for all positive even integers $ n$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 165$

Prove that the sequence $ \{y_{n}\}_{n \ge 1}$ defined by \[ y_{0}=1, \; y_{n+1}= \frac{1}{2}\left( 3y_{n}+\sqrt{5y_{n}^{2}-4}\right) \] consists only of integers.