[u]Round 1 [/u] [b]p1.[/b] In the convex quadrilateral $ABCD$ with diagonals $AC$ and $BD$, you know that angle $BAC$ is congruent to angle $CBD$, and that angle $ACD$ is congruent to angle $ADB$. Show that angle $ABC$ is congruent to angle $ADC$. [img]https://cdn.artofproblemsolving.com/attachments/5/d/41cd120813d5541dc73c5d4a6c86cc82747fcc.png[/img] [b]p2.[/b] In how many different ways can you place $12$ chips in the squares of a $4 \times 4$ chessboard so that (a) there is at most one chip in each square, and (b) every row and every column contains exactly three chips. [b]p3.[/b] Students from Hufflepuff and Ravenclaw were split into pairs consisting of one student from each house. The pairs of students were sent to Honeydukes to get candy for Father's Day. For each pair of students, either the Hufflepuff student brought back twice as many pieces of candy as the Ravenclaw student or the Ravenclaw student brought back twice as many pieces of candy as the Hufflepuff student. When they returned, Professor Trelawney determined that the students had brought back a total of $1000$ pieces of candy. Could she have possibly been right? Why or why not? Assume that candy only comes in whole pieces (cannot be divided into parts). [b]p4.[/b] While you are on a hike across Deception Pass, you encounter an evil troll, who will not let you across the bridge until you solve the following puzzle. There are six stones, two colored red, two colored yellow, and two colored green. Aside from their colors, all six stones look and feel exactly the same. Unfortunately, in each colored pair, one stone is slightly heavier than the other. Each of the lighter stones has the same weight, and each of the heavier stones has the same weight. Using a balance scale to make TWO measurements, decide which stone of each color is the lighter one. [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2 [/u] [b]p6.[/b] Consider a set of finitely many points on the plane such that if we choose any three points $A,B,C$ from the set, then the area of the triangle $ABC$ is less than $1$. Show that all of these points can be covered by a triangle whose area is less than $4$. [b]p7.[/b] A palindrome is a number that is the same when read forward and backward. For example, $1771$ and $23903030932$ are palindromes. Can the number obtained by writing the numbers from $1$ to $n$ in order be a palindrome for some $n > 1$ ? (For example, if $n = 11$, the number obtained is $1234567891011$, which is not a palindrome.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a given prime. For each prime $r$, we defind the function as following $F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}$. 1. Show that $F(r)$ is a positive integer for any prime $r \ne p$. 2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \ne p, s \ne p$ and $r \ne s$. 3. Fix a prime $r \ne p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p| q - 1$ but $p^2 \nmid q - 1$.

Determine the number of positive integer solutions $(x,y, z)$ to the equation $xyz = 2(x + y + z)$.

Let $d$ be a real number such that $d^2=r^2+s^2$, where $r$ and $s$ are rational numbers. Prove that we can color all points of the plane with rational coordinates with two different colors such that the points with distance $d$ have different colors.

Let $a$ be [i]good[/i] if the number of prime divisors of $a$ is equal to $2$. Do there exist $18$ consecutive good natural numbers?

Let $p$ be a prime number. Prove that there exists $n \in Z$ such that $p | n^2 -n+3$ if and only if there exists $m \in Z$ such that $p | m^2 -m+25$.

(a) Find all triples of integers $(a,b,c)$ for which $\frac14=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}$. (b) Determine all positive integers $n$ for which there exist positive integers $x_1,x_2,\ldots,x_n$ such that $1=\frac1{x_1^2}+\frac1{x_2^2}+\ldots+\frac1{x_n^2}$.

Let $n\geq 2021$. Let $a_1<a_2<\cdots<a_n$ an arithmetic sequence such that $a_1>2021$ and $a_i$ is a prime number for all $1\leq i\leq n$. Prove that for all $p$ prime with $p<2021, p$ divides the diference of the arithmetic sequence.

Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers. (a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter. (b) Let K be an arbitrary integral disguisable triangle for which no smaller integral disguisable triangle similar to it exists. Prove that at least two side lengths of K are perfect squares.

Find all triples \( (x, y, z) \) of positive integers that satisfy the equation \[ x + xy + xyz = 31. \]

Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$

a) Determine all 4-tuples $(x_0,x_1,x_2,x_3)$ of pairwise distinct intergers such that each $x_k$ is coprime to $x_{k+1}$(indices reduces modulo 4) and the cyclic sum $\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_1}$ is an interger. b)Show that there are infinitely many 5-tuples $(x_0,x_1,x_2,x_3,x_4)$ of pairwise distinct intergers such that each $x_k$ is coprime to $x_{k+1}$(indices reduces modulo 5) and the cyclic sum $\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_4}+\frac{x_4}{x_0}$ is an interger.

Prove that it is possible to find $2^{n+1}$ of $2^n$ digit numbers containing only "$1$" and "$2$" as digits, such that every two of them distinguish at least in $2^{n-1}$ digits.

Determine all 6-digit numbers $(abcdef)$ so that $(abcdef) = (def)^2$ where $\left( x_1x_2...x_n \right)$ is no multiplication but an n-digit number.

Denote by $P(n)$ the product of the digits of a positive integer $n$. For example, $P(1948)=1\cdot9\cdot4\cdot8=288$. [list=a] [*] Evaluate the sum $P(1)+P(2)+\dots+P(2017)$. [*] Determine the maximum value of $\frac{P(n)}{n}$ where $2017\leq n\leq5777$. [/list]

Prove that the number $A=\frac{(4n)!}{(2n)!n!}$ is an integer and divisible by $2^{n+1}$, where $n$ is a positive integer.

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$. [i]Proposed by Stephan Wagner, South Africa[/i]

Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$. [i]Evan O' Dorney.[/i]

[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].

There are three bags of marbles. Bag two has twice as many marbles as bag one. Bag three has three times as many marbles as bag one. Half the marbles in bag one, one third the marbles in bag two, and one fourth the marbles in bag three are green. If all three bags of marbles are dumped into a single pile, $\frac{m}{n}$ of the marbles in the pile would be green where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Given $n$ a positive integer, define $T_n$ the number of quadruples of positive integers $(a,b,x,y)$ such that $a>b$ and $n= ax+by$. Prove that $T_{2023}$ is odd.

Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.

A list of numbers $a_1,a_2,\ldots,a_m$ contains an arithmetic trio $a_i, a_j, a_k$ if $i < j < k$ and $2a_j = a_i + a_k$. Let $n$ be a positive integer. Show that the numbers $1, 2, 3, \ldots, n$ can be reordered in a list that does not contain arithmetic trios.

Let $(a_{n})_{n \ge 1}$ be a sequence of integers satisfying the inequality \[ 0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1 \] for all $n \ge 2$. Prove that the sequence $(a_{n})$ is periodic. Any Hints or Sols for this hard problem?? :help:

A natural number $k>1$ is given. The sum of some divisor of $k$ and some divisor of $k - 1$ is equal to $a$,where $a>k + 1$. Prove that at least one of the numbers $a - 1$ or $a + 1$ composite.