Prove that for any positive integer $k$ there exist infinitely many positive integers $m$ such that $3^k\mid m^3+10$.

Consider the sequence of numbers: $ 4, 7, 1, 8, 9, 7, 6, \ldots .$ For $ n > 2$, the $ n$th term of the sequence is the units digit of the sum of the two previous terms. Let $ S_n$ denote the sum of the first $ n$ terms of this sequence. The smallest value of $ n$ for which $ S_n > 10,000$ is: $ \textbf{(A)}\ 1992 \qquad \textbf{(B)}\ 1999 \qquad \textbf{(C)}\ 2001 \qquad \textbf{(D)}\ 2002 \qquad \textbf{(E)}\ 2004$

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

In an infinite sequence $a_1, a_2, a_3, \cdots$, the number $a_1$ equals $1$, and each $a_n, n > 1$, is obtained from $a_{n-1}$ as follows: [list]- if the greatest odd divisor of $n$ has residue $1$ modulo $4$, then $a_n = a_{n-1} + 1,$ - and if this residue equals $3$, then $a_n = a_{n-1} - 1.$[/list] Prove that in this sequence [b](a) [/b] the number $1$ occurs infinitely many times; [b](b)[/b] each positive integer occurs infinitely many times. (The initial terms of this sequence are $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots$ )

Given is a triangle $ABC$ with circumcentre $O$. The circumcircle of triangle $AOC$ intersects side $BC$ at $D$ and side $AB$ at $E$. Prove that the triangles $BDE$ and $AOC$ have circumradiuses of equal length.

Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have?

For a given positive integer $n$, it's known that there exist $2010$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$. Prove that there exist $2004$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$.

[b]p1.[/b] It’s currently $6:00$ on a $12$ hour clock. What time will be shown on the clock $100$ hours from now? Express your answer in the form hh : mm. [b]p2.[/b] A tub originally contains $10$ gallons of water. Alex adds some water, increasing the amount of water by 20%. Barbara, unhappy with Alex’s decision, decides to remove $20\%$ of the water currently in the tub. How much water, in gallons, is left in the tub? Express your answer as an exact decimal. [b]p3.[/b] There are $2000$ math students and $4000$ CS students at Berkeley. If $5580$ students are either math students or CS students, then how many of them are studying both math and CS? [b]p4.[/b] Determine the smallest integer $x$ greater than $1$ such that $x^2$ is one more than a multiple of $7$. [b]p5.[/b] Find two positive integers $x, y$ greater than $1$ whose product equals the following sum: $$9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29.$$ Express your answer as an ordered pair $(x, y)$ with $x \le y$. [b]p6.[/b] The average walking speed of a cow is $5$ meters per hour. If it takes the cow an entire day to walk around the edges of a perfect square, then determine the area (in square meters) of this square. [b]p7.[/b] Consider the cube below. If the length of the diagonal $AB$ is $3\sqrt3$, determine the volume of the cube. [img]https://cdn.artofproblemsolving.com/attachments/4/d/3a6fdf587c12f2e4637a029f38444914e161ac.png[/img] [b]p8.[/b] I have $18$ socks in my drawer, $6$ colored red, $8$ colored blue and $4$ colored green. If I close my eyes and grab a bunch of socks, how many socks must I grab to guarantee there will be two pairs of matching socks? [b]p9.[/b] Define the operation $a @ b$ to be $3 + ab + a + 2b$. There exists a number $x$ such that $x @ b = 1$ for all $b$. Find $x$. [b]p10.[/b] Compute the units digit of $2017^{(2017^2)}$. [b]p11.[/b] The distinct rational numbers $-\sqrt{-x}$, $x$, and $-x$ form an arithmetic sequence in that order. Determine the value of $x$. [b]p12.[/b] Let $y = x^2 + bx + c$ be a quadratic function that has only one root. If $b$ is positive, find $\frac{b+2}{\sqrt{c}+1}$. [b]p13.[/b] Alice, Bob, and four other people sit themselves around a circular table. What is the probability that Alice does not sit to the left or right of Bob? [b]p14.[/b] Let $f(x) = |x - 8|$. Let $p$ be the sum of all the values of $x$ such that $f(f(f(x))) = 2$ and $q$ be the minimum solution to $f(f(f(x))) = 2$. Compute $p \cdot q$. [b]p15.[/b] Determine the total number of rectangles ($1 \times 1$, $1 \times 2$, $2 \times 2$, etc.) formed by the lines in the figure below: $ \begin{tabular}{ | l | c | c | r| } \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular} $ [b]p16.[/b] Take a square $ABCD$ of side length $1$, and let $P$ be the midpoint of $AB$. Fold the square so that point $D$ touches $P$, and let the intersection of the bottom edge $DC$ with the right edge be $Q$. What is $BQ$? [img]https://cdn.artofproblemsolving.com/attachments/1/1/aeed2c501e34a40a8a786f6bb60922b614a36d.png[/img] [b]p17.[/b] Let $A$, $B$, and $k$ be integers, where $k$ is positive and the greatest common divisor of $A$, $B$, and $k$ is $1$. Define $x\# y$ by the formula $x\# y = \frac{Ax+By}{kxy}$ . If $8\# 4 = \frac12$ and $3\# 1 = \frac{13}{6}$ , determine the sum $A + B + k$. [b]p18.[/b] There are $20$ indistinguishable balls to be placed into bins $A$, $B$, $C$, $D$, and $E$. Each bin must have at least $2$ balls inside of it. How many ways can the balls be placed into the bins, if each ball must be placed in a bin? [b]p19.[/b] Let $T_i$ be a sequence of equilateral triangles such that (a) $T_1$ is an equilateral triangle with side length 1. (b) $T_{i+1}$ is inscribed in the circle inscribed in triangle $T_i$ for $i \ge 1$. Find $$\sum^{\infty}_{i=1} Area (T_i).$$ [b]p20.[/b] A [i]gorgeous [/i] sequence is a sequence of $1$’s and $0$’s such that there are no consecutive $1$’s. For instance, the set of all gorgeous sequences of length $3$ is $\{[1, 0, 0]$,$ [1, 0, 1]$, $[0, 1, 0]$, $[0, 0, 1]$, $[0, 0, 0]\}$. Determine the number of gorgeous sequences of length $7$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$. [i]Amol Aggarwal.[/i]

Let $ \mathcal{F}$ be a nonempty family of sets with the following properties: (a) If $ X \in \mathcal{F}$, then there are some $ Y \in \mathcal{F}$ and $ Z \in \mathcal{F}$ such that $ Y \cap Z =\emptyset$ and $ Y \cup Z=X$. (b) If $ X \in \mathcal{F}$, and $ Y \cup Z =X , Y \cap Z=\emptyset$, then either $ Y \in \mathcal{F}$ or $ Z \in \mathcal{F}$. Show that there is a decreasing sequence $ X_0 \supseteq X_1 \supseteq X_2 \supseteq ...$ of sets $ X_n \in \mathcal{F}$ such that \[ \bigcap_{n=0}^{\infty} X_n= \emptyset.\] [i]F. Galvin[/i]

Determine all pairs of functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any $x,y\in \mathbb{R}$, \[f(x)f(y)=g(x)g(y)+g(x)+g(y).\]

The incircle $I$ of $\triangle ABC$ is tangent to sides $AB,BC,CA$ at $D,E,F$ respectively. Line $EF$ intersects lines $AI,BI,DI$ at $M,N,K$ respectively. Prove that $DM\cdot KE=DN\cdot KF$.

Determine for which integers $n \geqslant 4$ the cells of a $1 \times (2n+1)$ table can be filled with the numbers $1, 2, 3, \dots, 2n + 1$ such that the following conditions are satisfied: [list=i] [*]Each of the numbers $1, 2, 3, \dots, 2n + 1$ appears exactly once. [*]In any $1 \times 3$ rectangle, one of the numbers is the arithmetic mean of the other two. [*]The number $1$ is located in the middle cell of the table. [/list]

Let the four real solutions to the equation $x^2 + \frac{144}{x^2} = 25$ be $r_1, r_2, r_3$, and $r_4$. Find $|r_1| +|r_2| +|r_3| +|r_4|$.

Are there any triples $(a,b,c)$ of positive integers such that $(a-2)(b-2)(c-2)+12$ is a prime number that properly divides the positive number $a^2+b^2+c^2+abc-2017$?

Let $A$ be a complex $n \times n$ Matrix for $n >1$. Let $A^{H}$ be the conjugate transpose of $A$. Prove that $A\cdot A^{H} =I_{n}$ if and only if $A=S\cdot (S^{H})^{-1}$ for some complex Matrix $S$.

In a board, the positive integer $N$ is written. In each round, Olive can realize any one of the following operations: I - Switch the current number by a positive multiple of the current number. II - Switch the current number by a number with the same digits of the current number, but the digits are written in another order(leading zeros are allowed). For instance, if the current number is $2022$, Olive can write any of the following numbers $222,2202,2220$. Determine all the positive integers $N$, such that, Olive can write the number $1$ after a finite quantity of rounds.

Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i\geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.

Triangle $\vartriangle ABC$ is inscribed in circle $O$ and has sides $AB = 47$, $BC = 69$, and $CA = 34$. Let $E$ be the point on $O$ such that $\overline{AE}$ and $\overline{BC}$ intersect inside $O$, $8$ units away from $B$. Let $P$ and $Q$ be the points on $\overleftrightarrow{BE}$ and $\overleftrightarrow{CE}$, respectively, such that $\angle EPA$ and $\angle EQA$ are right angles. Suppose lines $\overleftrightarrow{AP}$ and $\overleftrightarrow{AQ}$ respectively intersect $O$ again at $X$ and $Y$ . Compute the distance $XY$.

Let $\vartriangle ABC$ be equilateral. Two points $D$ and $E$ are on side $BC$ (with order $B, D, E, C$), and satisfy $\angle DAE = 30^o$ . If $BD = 2$ and $CE = 3$, what is $BC$? [img]https://cdn.artofproblemsolving.com/attachments/c/8/27b756f84e086fe31b5ea695f51fb6c78b63d0.png[/img]

Let $p$ be a prime number. At a school of $p^{2020}$ students it is required that each club consist of exactly $p$ students. Is it possible for each pair of students to have exactly one club in common?

Francisco wrote a sequence of numbers starting with $25$. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was $8$, what is the fifth term of the sequence?

Find all functions $f: \mathbb{N}_{0}\rightarrow \mathbb{N}_{0}$ such that for all $n\in \mathbb{N}_{0}$: \[f(f(n))+f(n)=2n+6.\]

Let $m,n$ be integers greater than 1,$m \geq n$,$a_1,a_2,\dots,a_n$ are $n$ distinct numbers not exceed $m$,which are relatively primitive.Show that for any real $x$,there exists $i$ for which $||a_ix|| \geq \frac{2}{m(m+1)} ||x||$,where $||x||$ denotes the distance between $x$ and the nearest integer to $x$ .

Nair and Yuli play the following game: $1.$ There is a coin to be moved along a horizontal array with $203$ cells. $2.$ At the beginning, the coin is at the first cell, counting from left to right. $3.$ Nair plays first. $4.$ Each of the players, in their turns, can move the coin $1$, $2$, or $3$ cells to the right. $5.$ The winner is the one who reaches the last cell first. What strategy does Nair need to use in order to always win the game?