Let $ ABCD$ be a convex quadrangle, $ P$ the intersection of lines $ AB$ and $ CD$, $ Q$ the intersection of lines $ AD$ and $ BC$ and $ O$ the intersection of diagonals $ AC$ and $ BD$. Show that if $ \angle POQ\equal{} 90^\circ$ then $ PO$ is the bisector of $ \angle AOD$ and $ OQ$ is the bisector of $ \angle AOB$.

$1$ or $-1$ is written in $50$ letters. These letters are put into $50$ envelopes. If you ask, you can learn the product of numbers written into any three letters. At least, how many questions are required to find the product of all of the $50$ numbers?

Let $ABCD$ be cyclic quadrilateral and $E$ the midpoint of $AC$. The circumcircle of $\triangle CDE$ intersect the side $BC$ at $F$, which is different from $C$. If $B'$ is the reflection of $B$ across $F$, prove that $EF$ is tangent to the circumcircle of $\triangle B'DF$. [i]Proposed by Nikola Velov[/i]

Prove that for all odd $ n > 1$, we have $ 8n \plus{} 4|C^{4n}_{2n}$.

A triangle $ABC$ is given. Find all the segments $XY$ that lie inside the triangle such that $XY$ and five of the segments $XA,XB, XC, YA,YB,YC$ divide the triangle $ABC$ into $5$ regions with equal areas. Furthermore, prove that all the segments $XY$ have a common point.

Determine the number of permutations $a_1, a_2, \dots, a_n$ of $1, 2, \dots, n$ such that for every positive integer $k$ with $1 \le k \le n$, there exists an integer $r$ with $0 \le r \le n - k$ which satisfies \[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \]

Find the greatest common divisor of all number in the set $( a^{41} - a | a \in \mathbb{N} and \geq 2 )$ . What is your guess if 41 is replaced by a natural number $n$

A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire? $\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{February 11}\qquad\textbf{(E) }\text{February 12}$

$C:(x-\arcsin a)(x-\arccos a)+(y-\arcsin a)(y+\arccos a)=0$. The length of string of $C$ cut by $l:x=\frac{\pi}{4}$ is $d$. When $a$ changes, the minumum value of $d$ is $\text{(A)}\frac{\pi}{4}\qquad\text{(B)}\frac{\pi}{3}\qquad\text{(C)}\frac{\pi}{2}\qquad\text{(D)}\pi$

[u]Round 1[/u] [b]p1.[/b] Solve the maze [img]https://cdn.artofproblemsolving.com/attachments/8/c/6439816a52b5f32c3cb415e2058556edb77c80.png[/img] [b]p2.[/b] Billiam can write a problem in $30$ minutes, Jerry can write a problem in $10$ minutes, and Evin can write a problem in $20$ minutes. Billiam begins writing problems alone at $3:00$ PM until Jerry joins himat $4:00$ PM, and Evin joins both of them at $4:30$ PM. Given that they write problems until the end of math team at $5:00$ PM, how many full problems have they written in total? [b]p3.[/b] How many pairs of positive integers $(n,k)$ are there such that ${n \choose k}= 6$? [u]Round 2 [/u] [b]p4.[/b] Find the sumof all integers $b > 1$ such that the expression of $143$ in base $b$ has an even number of digits and all digits are the same. [b]p5.[/b] Ιni thinks that $a \# b = a^2 - b$ and $a \& b = b^2 - a$, while Mimi thinks that $a \# b = b^2 - a$ and $a \& b = a^2 - b$. Both Ini and Mimi try to evaluate $6 \& (3 \# 4)$, each using what they think the operations $\&$ and $\#$ mean. What is the positive difference between their answers? [b]p6.[/b] A unit square sheet of paper lies on an infinite grid of unit squares. What is the maximum number of grid squares that the sheet of paper can partially cover at once? A grid square is partially covered if the area of the grid square under the sheet of paper is nonzero - i.e., lying on the edge only does not count. [u]Round 3[/u] [b]p7.[/b] Maya wants to buy lots of burgers. A burger without toppings costs $\$4$, and every added topping increases the price by 50 cents. There are 5 different toppings for Maya to choose from, and she can put any combination of toppings on each burger. How much would it cost forMaya to buy $1$ burger for each distinct set of toppings? Assume that the order in which the toppings are stacked onto the burger does not matter. [b]p8.[/b] Consider square $ABCD$ and right triangle $PQR$ in the plane. Given that both shapes have area $1$, $PQ =QR$, $PA = RB$, and $P$, $A$, $B$ and $R$ are collinear, find the area of the region inside both square $ABCD$ and $\vartriangle PQR$, given that it is not $0$. [b]p9.[/b] Find the sum of all $n$ such that $n$ is a $3$-digit perfect square that has the same tens digit as $\sqrt{n}$, but that has a different ones digit than $\sqrt{n}$. [u]Round 4[/u] [b]p10.[/b] Jeremy writes the string: $$LMTLMTLMTLMTLMTLMT$$ on a whiteboard (“$LMT$” written $6$ times). Find the number of ways to underline $3$ letters such that from left to right the underlined letters spell LMT. [b]p11.[/b] Compute the remainder when $12^{2022}$ is divided by $1331$. [b]p12.[/b] What is the greatest integer that cannot be expressed as the sum of $5$s, $23$s, and $29$s? [u]Round 5 [/u] [b]p13.[/b] Square $ABCD$ has point $E$ on side $BC$, and point $F$ on side $CD$, such that $\angle EAF = 45^o$. Let $BE = 3$, and $DF = 4$. Find the length of $FE$. [b]p14.[/b] Find the sum of all positive integers $k$ such that $\bullet$ $k$ is the power of some prime. $\bullet$ $k$ can be written as $5654_b$ for some $b > 6$. [b]p15.[/b] If $\sqrt[3]{x} + \sqrt[3]{y} = 2$ and $x + y = 20$, compute $\max \,(x, y)$. PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167372p28825861]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that the following inequality holds for all real $a$ and natural $n$: $$|a| \cdot |a-1|\cdot |a-2|\cdot ...\cdot |a-n| \ge \frac{n!F(a)}{2n}$$ $F(a)$ is the distance from $a$ to the closest integer.

A sequence $(a_1,a_2,...,a_{675})$ is given so that each term is an alphabet in the English language (no distinction is made between lower and upper case letters). It is known that in the sequence $a$ is never followed by $b$ and $c$ is never followed by $d$. Show that there are integers $m$ and $n$ with $1 \le m < n \le 674$ such that $a_m = a_n$ and $a_{m+1} = a_{n+1}$·

Let $f : Z_{>0} \to Z_{>0}$ be a function which satisfies $k|f^k(x)-x$ for all $k, x \in Z_{>0}$ and $f(x)-x \le 2023$. If $f(1) = 2000$, what can $f$ be? [i]Remark[/i]: Here, $f^k (x)$ denotes the $k$-fold application of $f$ to $x$.

The trapezoid ABCD has bases with lengths 7 and 17 and area 120. Find the difference of the areas of $\triangle$ACD and $\triangle$CDB. [asy] pair A, B, C, D; A = (0, 0); B = (17, 0); C = (1, 10); D = (10, 10); draw(A--B--D--C--cycle); label("$A$", A, W); label("$B$", B, E); label("$C$", C, W); label("$D$", D, E); draw(B--C); draw(A--D); [/asy]

Two circles meeting at points $A, B$ and a point $O$ lying outside them are given. Using a compass and a ruler construct a ray with origin $O$ meeting the first circle at point $C$ and the second one at point $D$ in such a way that the ratio $OC : OD$ be maximal.

Let $n\ge 4$ be a positive integer and let $M$ be a set of $n$ points in the plane, where no three points are collinear and not all of the $n$ points being concyclic. Find all real functions $f:M\to\mathbb{R}$ such that for any circle $\mathcal{C}$ containing at least three points from $M$, the following equality holds: \[\sum_{P\in\mathcal{C}\cap M} f(P)=0\] [i]Dorel Mihet[/i]

Find $x+y+z$ when $$a_1x+a_2y+a_3z= a$$$$b_1x+b_2y+b_3z=b$$$$c_1x+c_2y+c_3z=c$$ Given that $$a_1\left(b_2c_3-b_3c_2\right)-a_2\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c_2-b_2c_1\right)=9$$$$a\left(b_2c_3-b_3c_2\right)-a_2\left(bc_3-b_3c\right)+a_3\left(bc_2-b_2c\right)=17$$$$a_1\left(bc_3-b_3c\right)-a\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c-bc_1\right)=-8$$$$a_1\left(b_2c-bc_2\right)-a_2\left(b_1c-bc_1\right)+a\left(b_1c_2-b_2c_1\right)=7.$$ [i]2017 CCA Math Bonanza Lightning Round #5.1[/i]

[b]p1.[/b] The remainder of a number when divided by $7$ is $5$. If I multiply the number by $32$ and add $18$ to the product, what is the new remainder when divided by $7$? [b]p2.[/b] If a fair coin is flipped $15$ times, what is the probability that there are more heads than tails? [b]p3.[/b] Let $-\frac{\sqrt{p}}{q}$ be the smallest nonzero real number such that the reciprocal of the number is equal to the number minus the square root of the square of the number, where $p$ and $q$ are positive integers and $p$ is not divisible the square of any prime. Find $p + q$. [b]p4.[/b] Rachel likes to put fertilizers on her grass to help her grass grow. However, she has cows there as well, and they eat $3$ little fertilizer balls on average. If each ball is spherical with a radius of $4$, then the total volume that each cow consumes can be expressed in the form $a\pi$ where $a$ is an integer. What is $a$? [b]p5.[/b] One day, all $30$ students in Precalc class are bored, so they decide to play a game. Everyone enters into their calculators the expression $9 \diamondsuit 9 \diamondsuit 9 ... \diamondsuit 9$, where $9$ appears $2020$ times, and each $\diamondsuit$ is either a multiplication or division sign. Each student chooses the signs randomly, but they each choose one more multiplication sign than division sign. Then all $30$ students calculate their expression and take the class average. Find the expected value of the class average. [b]p6.[/b] NaNoWriMo, or National Novel Writing Month, is an event in November during which aspiring writers attempt to produce novel-length work - formally defined as $50,000$ words or more - within the span of $30$ days. Justin wants to participate in NaNoWriMo, but he's a busy high school student: after accounting for school, meals, showering, and other necessities, Justin only has six hours to do his homework and perhaps participate in NaNoWriMo on weekdays. On weekends, he has twelve hours on Saturday and only nine hours on Sunday, because he goes to church. Suppose Justin spends two hours on homework every single day, including the weekends. On Wednesdays, he has science team, which takes up another hour and a half of his time. On Fridays, he spends three hours in orchestra rehearsal. Assume that he spends all other time on writing. Then, if November $1$st is a Friday, let $w$ be the minimum number of words per minute that Justin must type to finish the novel. Round $w$ to the nearest whole number. [b]p7.[/b] Let positive reals $a$, $b$, $c$ be the side lengths of a triangle with area $2030$. Given $ab + bc + ca = 15000$ and $abc = 350000$, find the sum of the lengths of the altitudes of the triangle. [b]p8.[/b] Find the minimum possible area of a rectangle with integer sides such that a triangle with side lengths $3$, $4$, $5$, a triangle with side lengths $4$, $5$, $6$, and a triangle with side lengths $\frac94$, $4$, $4$ all fit inside the rectangle without overlapping. [b]p9.[/b] The base $16$ number $10111213...99_{16}$, which is a concatenation of all of the (base $10$) $2$-digit numbers, is written on the board. Then, the last $2n$ digits are erased such that the base $10$ value of remaining number is divisible by $51$. Find the smallest possible integer value of $n$. [b]p10.[/b] Consider sequences that consist entirely of $X$'s, $Y$ 's and $Z$'s where runs of consecutive $X$'s, $Y$ 's, and $Z$'s are at most length $3$. How many sequences with these properties of length $8$ are there? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there is a polynomial $P(x)$ with integer coefficients such that for all $x$ in the interval $\left[ \frac{1}{10} , \frac{9}{10}\right]$ we have $$\left|P(x) -\frac12 \right| < \frac{ 1}{1000 }.$$

$\textbf{Problem C.1}$ There are two piles of coins, each containing $2010$ pieces. Two players $A$ and $B$ play a game taking turns ($A$ plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?

If $x_i$ is an integer greater than 1, let $f(x_i)$ be the greatest prime factor of $x_i,x_{i+1} =x_i-f(x_i)$ ($i\ge 0$ and i is an integer). (1) Prove that for any integer $x_0$ greater than 1, there exists a natural number$k(x_0)$, such that $x_{k(x_0)+1}=0$ Grade 10: (2) Let $V_{(x_0)}$ be the number of different numbers in $f(x_0),f(x_1),\cdots,f(x_{k(x_0)})$. Find the largest number in $V(2),V(3),\cdots,V(781)$ and give reasons. Note: Bai Lu Zhou Academy was founded in 1241 and has a history of 781 years. Grade 11: (2) Let $V_{(x_0)}$ be the number of different numbers in $f(x_0),f(x_1),\cdots,f(x_{k(x_0)})$. Find the largest number in $V(2),V(3),\cdots,V(2022)$ and give reasons.

A positive integer $n$ between $1$ and $N=2007^{2007}$ inclusive is selected at random. If $a$ and $b$ are natural numbers such that $a/b$ is the probability that $N$ and $n^3-36n$ are relatively prime, find the value of $a+b$.

Gary has $2$ children. We know one is a boy born on a Friday. Assume birthing boys and girls are equally likely, being born on any day of the week is equally likely, and that these properties are independent of each other, as well as independent from child to child. The probability that both of Gary's children are boys is $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime integers. Find $a + b.$

The real numbers $a_1,a_2,\dots,a_n$ are given such that $|a_i|\leq 1$ for all $i=1,2,\dots,n$ and $a_1+a_2+\cdots+a_n=0$. a) Prove that there exists $k\in\{1,2,\dots,n\}$ such that \[ |a_1+2a_2+\cdots+ka_k|\leq\frac{2k+1}{4}. \] b) Prove that for $n > 2$ the bound above is the best possible. [i]Radu Gologan, Dan Schwarz[/i]

Find the largest positive integer $n$ that cannot be written as $n = 20a + 28b + 35c$ for nonnegative integers $a, b$, and $c$.