If $x\neq y$ and the sequences $x,a_1,a_2,y$ and $x,b_1,b_2,b_3,y$ each are in arithmetic progression, then $(a_2-a_1)/(b_2-b_1)$ equals $\textbf{(A) }\frac{2}{3}\qquad\textbf{(B) }\frac{3}{4}\qquad\textbf{(C) }1\qquad\textbf{(D) }\frac{4}{3}\qquad \textbf{(E) }\frac{3}{2}$

Let $\triangle ABC$ and $A'$,$B'$,$C'$ the symmetrics of vertex over opposite sides.The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$.$B_1$ and $C_1$ are defined similarly.Prove that lines $AA_1$,$BB_1$ and $CC_1$ are concurent.

In $\triangle PQR$, $PQ=8$, $QR=13$, and $RP=15$. Prove that there is a point $S$ on line segment $\overline{PR}$, but not at its endpoints, such that $PS$ and $QS$ are also integers. [asy] size(200); defaultpen(linewidth(0.8)); pair P=origin,Q=(8,0),R=(7,10),S=(3/2,15/7); draw(P--Q--R--cycle); label("$P$",P,W); label("$Q$",Q,E); label("$R$",R,NE); draw(Q--S,linetype("4 4")); label("$S$",S,NW); [/asy]

On the chessboard, $ 32$ black pawns and $ 32$ white pawns are arranged. In every move, a pawn can capture another pawn of the opposite color, moving diagonally to an adjacent square where the captured one stands. White pawns move only in upper-left or upper-right directions, while black ones can move in down-left or in down-right directions only; the captured pawn is removed from the board. A pawn cannot move without capturing an opposite pawn. Find the least possible number of pawns which can stay on the chessboard.

Let $S$ be the set of integers modulo $2020$. Suppose that $a_1,a_2,...,a_{2020},b_1,b_2,...,b_{2020}, c$ are arbitrary elements of $S$. For any $x_1,x_2,...,x_{2020}\in S$, define $f(x_1,x_2,...,x_{2020})$ to be the $2020$-tuple whose $i$th coordinate is $x_{i-2} + a_i x_{2019} + b_ix_{2020} + cx_i$, where we set $x_{-1}=x_0=0$. Let $m$ be the smallest positive integer such that, for some values of $a_1,a_2,...,a_{2020},b_1,b_2,...,b_{2020}, c$, we have, for all $x_1,x_2,...,x_{2020}\in S$, that $f^m (x_1, x_2, ..., x_{2020} ) = (0,0,...,0)$ . For this value of $m$, there are exactly $n$ choices of the tuple $(a_1,a_2,...,a_{2020},b_1,b_2,...,b_{2020},c)$ such that, for all $x_1,x_2,...,x_{2020}\in S$, $f^m (x_1, x_2, ..., x_{2020} ) = (0,0,...,0)$. Compute $100m+n$. [i]Proposed by Vincent Huang[/i]

Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j=1}^n a_j \geq \sqrt{n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$ \[ \sum_{j=1}^n a_j^2 > \frac{1}{4} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right). \]

Let $a,b,c,x,y,z$ be positive real numbers such that $ay+bz+cx \le az+bx+cy$. Prove that $$ \frac{xy}{ax+bx+cy}+\frac{yz}{by+cy+az}+\frac{zx}{cz+az+bx} \le \frac{x+y+z}{a+b+c}$$

In the plane $xOy$, a lot of points are considered $$X = \{P (a, b) | (a, b) \in \{1, 2,..., 10\} \times \{1, 2,..., 10 \}\}$$ Determine the number of different lines that can be obtained by joining two of them between the points of the set $X$; so that any two lines are not parallel.

Prove that in any triangle $ABC$, \[ 0 < \cot { \left( \frac{A}{4} \right)} - \tan{ \left( \frac{B}{4} \right) } - \tan{ \left( \frac{C}{4} \right) } - 1 < 2 \cot { \left( \frac{A}{2} \right) }. \]

We say that a natural number of at least two digits $E$ is [i]special [/i] if each time two adjacent digits of $E$ are added, a divisor of $E$ is obtained. For example, $2124$ is special, since the numbers $2 + 1$, $1 + 2$ and $2 + 4$ are all divisors of $2124$. Find the largest value of $n$ for which there exist $n$ consecutive natural numbers such that they are all special.

In a cycling competition with $14$ stages, one each day, and $100$ participants, a competitor was characterized by finishing $93^{\text{rd}}$ each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.)

Compute the positive difference between the two real solutions to the equation $$(x-1)(x-4)(x-2)(x-8)(x-5)(x-7)+48\sqrt 3 = 0.$$

A company sells detergent in three different sized boxes: small (S), medium (M) and large (L). The medium size costs $50\%$ more than the small size and contains $20\%$ less detergent than the large size. The large size contains twice as much detergent as the small size and costs $30\%$ more than the medium size. Rank the three sizes from best to worst buy. $ \textbf{(A)}\ \text{SML}\qquad\textbf{(B)}\ \text{LMS}\qquad\textbf{(C)}\ \text{MSL}\qquad\textbf{(D)}\ \text{LSM}\qquad\textbf{(E)}\ \text{MLS} $

A [i]T-tetromino[/i] is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square. Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes.

[b]p1[/b]. Your Halloween took a bad turn, and you are trapped on a small rock above a sea of lava. You are on rock $1$, and rocks $2$ through $12$ are arranged in a straight line in front of you. You want to get to rock $12$. You must jump from rock to rock, and you can either (1) jump from rock $n$ to $n + 1$ or (2) jump from rock $n$ to $n + 2$. Unfortunately, you are weak from eating too much candy, and you cannot do (2) twice in a row. How many different sequences of jumps will take you to your destination? [b]p2.[/b] Find the number of ordered triples $(p; q; r)$ such that $p, q, r$ are prime, $pq + pr$ is a perfect square and $p + q + r \le 100$. [b]p3.[/b] Let $x, y, z$ be nonzero complex numbers such that $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} \ne 0$ and $$x^2(y + z) + y^2(z + x) + z^2(x + y) = 4(xy + yz + zx) = -3xyz.$$ Find $\frac{x^3 + y^3 + z^3}{x^2 + y^2 + z^2}$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a >1$ be a positive integer. Prove that the set $\{a^2+a-1,a^3+a-1,\cdots\}$ have a subset $S$ with infinite members and for any two members of $S$ like $x,y$ we have $\gcd(x,y)=1$. Then prove that the set of primes has infinite members.

Two polynomials $ f(x)=a_{100}x^{100}+a_{99}x^{99}+\dots+a_{1}x+a_{0}$ and $ g(x)=b_{100}x^{100}+b_{99}x^{99}+\dots+b_{1}x+b_{0}$ of degree $ 100$ differ from each other by a permutation of coefficients. It is known that $ a_{i}\ne b_{i}$ for $ i=0, 1, 2, \dots, 100$. Is it possible that $ f(x)\geq g(x)$ for all real $ x$?

Consider the sequence $$1,1,2,1,2,4,1,2,4,8,1,2,4,8,16,1, . . .$$ formed by writing the first power of two, followed by the first two powers of two, followed by the first three powers of two, and so on. Find the smallest positive integer $N$ such that $N > 100$ and the sum of the first $N$ terms of this sequence is a power of two.

A circle with radius $4$ and $251$ distinct points inside the circle are given. Show that it is possible to draw a circle with radius $1$ and containing at least $11$ of these points.

Let b be a given real number. The sequence of integers $a_1, a_2,a_3, ...$ is such that $a_1 =(b]$ and $a_{n+1}=(a_n+b]$ for all $n\ge 1$ Prove that the sum $a_1+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{n}$ is an integer number for any natural $n$ . (In the condition of the problem, $(x]$ denotes the smallest integer that is greater than or equal to $x$)

In Brawl Stars, Rico can shoot his opponent directly or make his bullet bounce off the wall at the same angle. His opponent is $15$ feet in front of him and there are infinitely long walls $1$ feet to the left and right of Rico. If Rico's bullet travels $d$ feet before hitting the opponent, find the sum of all possible integer values of $d$.

An ant is at vertex of a cube. Every $10$ minutes it moves to an adjacent vertex along an edge. If $N$ is the number of one hour journeys that end at the starting vertex, find the sum of the squares of the digits of $N$.

Find the best rational approximation $x$ to $\sqrt[3]{2016}$ such that $|x-\sqrt[3]{2016}|$ is as small as possible. You may either find an $x=\tfrac{a}{b}$ where $a,b$ are coprime integers or find a decimal approximation. Let $C$ be the actual answer and $A$ be the answer you submit. Your score will be given by $\lceil10+\tfrac{16.5}{0.1+e^{30|A-C|}}\rceil$, where $\lceil x\rceil$ denote the smallest integer which is $\geq x$.

Given a rectangle $ ABCD,$ let $ AB \equal{} b, AD \equal{} a ( a\geq b),$ three points $ X,Y,Z$ are put inside or on the boundary of the rectangle, arbitrarily. Find the maximum of the minimum of the distances between any two points among the three points. (Denote it by $ a,b$)

Let $f(n)$ be the sum of the digits of $n$. Find $\displaystyle{\sum_{n=1}^{99}f(n)}$.