Let $\vartriangle ABC$ be a triangle. Let $Q$ be a point in the interior of $\vartriangle ABC$, and let $X, Y,Z$ denote the feet of the altitudes from $Q$ to sides $BC$, $CA$, $AB$, respectively. Suppose that $BC = 15$, $\angle ABC = 60^o$, $BZ = 8$, $ZQ = 6$, and $\angle QCA = 30^o$. Let line $QX$ intersect the circumcircle of $\vartriangle XY Z$ at the point $W\ne X$. If the ratio $\frac{ WY}{WZ}$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

Let $P$ be the probability that the product of $2020$ real numbers chosen independently and uniformly at random from the interval $[-1, 2]$ is positive. The value of $2P - 1$ can be written in the form $\left(\frac{m}{n} \right)^b$, where $m$, $n$ and $b$ are positive integers such that $m$ and $n$ are relatively prime and $b$ is as large as possible. Compute $m + n + b$.

Prove that there is no positive rational number $x$ such that \[x^{\lfloor x\rfloor }=\frac{9}{2}.\]

Ingrid and Erik are playing a game. For a given odd prime $p$, the numbers $1, 2, 3, ..., p-1$ are written on a blackboard. The players take turns making moves with Ingrid starting. A move consists of one of the players crossing out a number on the board that has not yet been crossed out. If the product of all currently crossed out numbers is $1 \pmod p$ after the move, the player whose move it was receives one point, otherwise, zero points are awarded. The game ends after all numbers have been crossed out. The player who has received the most points by the end of the game wins. If both players have the same score, the game ends in a draw. For each $p$, determine which player (if any) has a winning strategy

Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that \[\frac{a^p - a}{p}=b^2.\]

For each non-negative integer $n$, let $u_n = \left( 2+\sqrt{5} \right)^n + \left( 2-\sqrt{5} \right)^n$. a) Prove that $u_n$ is a positive integer for all $n \geq 0$. When $n$ changes, what is the largest possible remainder when $u_n$ is divided by $24$? b) Find all pairs of positive integers $(a, b)$ such that $a, b < 500$ and for all odd positive integers $n$, $u_n \equiv a^n - b^n \pmod {1111}$.

Suppose $50x$ is divisible by 100 and $kx$ is not divisible by 100 for all $k=1,2,\cdots, 49$ Find number of solutions for $x$ when $x$ takes values $1,2,\cdots 100$. [list=1] [*] 20 [*] 25 [*] 15 [*] 50 [/list]

Let $p(k)$ be the smallest prime not dividing $k$. Put $q(k) = 1$ if $p(k) = 2$, or the product of all primes $< p(k)$ if $p(k) > 2$. Define the sequence $x_0, x_1, x_2, ...$ by $x_0 = 1$, $x_{n+1} = \frac{x_np(x_n)}{q(x_n)}$. Find all $n$ such that $x_n = 111111$

Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.

Let $p$ be a prime number. Find all positive integers $a$ and $b$ such that: $\frac{4a + p}{b}+\frac{4b + p}{a}$ and $ \frac{a^2}{b}+\frac{b^2}{a}$ are integers.

$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$

Let $x_1,x_2,\dots,x_{2014}$ be integers among which no two are congurent modulo $2014$. Let $y_1,y_2,\dots,y_{2014}$ be integers among which no two are congurent modulo $2014$. Prove that one can rearrange $y_1,y_2,\dots,y_{2014}$ to $z_1,z_2,\dots,z_{2014}$, so that among \[x_1+z_1,x_2+z_2,\dots,x_{2014}+z_{2014}\] no two are congurent modulo $4028$.

Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$ has an integer solution.

Two positive integers have the sum $2002$. Can $2002$ divide their product?

Given a positive integer $\ell,$ define the sequence $\{a^{(\ell)}\}_{n=1}^{\infty}$ such that $a_n^{(\ell)}=\lfloor n + \sqrt[\ell]{n}+\tfrac{1}{2}\rfloor$ for all positive integers $n.$ Let $S$ denote the set of positive integers that appear in all three of the sequences $\{a_n^{(2)} \}_{n=1}^{\infty},$ $\{a_n^{(3)} \}_{n=1}^{\infty},$ and $\{a_n^{(4)} \}_{n=1}^{\infty}.$ Find the sum of the elements of $S$ that lie in the interval $[1,100].$

[b]p1.[/b] In rectangle $ABMC$, $AB= 5$ and $BM= 8$. If point $X$ is the midpoint of side $AC$, what is the area of triangle $XCM$? [b]p2.[/b] Find the sum of all possible values of $a+b+c+d$ such that $(a, b, c, d)$ are quadruplets of (not necessarily distinct) prime numbers satisfying $a \cdot b \cdot c \cdot d = 4792$. [b]p3.[/b] How many integers from $1$ to $2022$ inclusive are divisible by $6$ or $24$, but not by both? [b]p4.[/b] Jerry begins his English homework at $07:39$ a.m. At $07:44$ a.m., he has finished $2.5\%$ of his homework. Subsequently, for every five minutes that pass, he completes three times as much homework as he did in the previous five minute interval. If Jerry finishes his homework at $AB : CD$ a.m., what is $A + B + C + D$? For example, if he finishes at $03:14$ a.m., $A + B + C + D = 0 + 3 + 1 + 4$. [b]p5.[/b] Advay the frog jumps $10$ times on Mondays, Wednesdays and Fridays. He jumps $7$ times on Tuesdays and Saturdays. He jumps $5$ times on Thursdays and Sundays. How many times in total did Advay jump in November if November $17$th falls on a Thursday? (There are $30$ days in November). [b]p6.[/b] In the following diagram, $\angle BAD\cong \angle DAC$, $\overline{CD} = 2\overline{BD}$, and $ \angle AEC$ and $\angle ACE$ are complementary. Given that $\overline{BA} = 210$ and $\overline{EC} = 525$, find $\overline{AE}$. [img]https://cdn.artofproblemsolving.com/attachments/5/3/8e11caf2d7dbb143a296573f265e696b4ab27e.png[/img] [b]p7.[/b] How many trailing zeros are there when $2021!$ is expressed in base $2021$? [b]p8.[/b] When two circular rings of diameter $12$ on the Olympic Games Logo intersect, they meet at two points, creating a $60^o$ arc on each circle. If four such intersections exist on the logo, and no region is in $3$ circles, the area of the regions of the logo that exist in exactly two circles is $a\pi - b\sqrt{c}$ where $a$, $b$, $c$ are positive integers and $\sqrt{c}$ is fully simplified find $a + b + c$. [b]p9.[/b] If $x^2 + ax - 3$ is a factor of $x^4 - x^3 + bx^2 - 5x - 3$, then what is $|a + b|$? [b]p10.[/b] Let $(x, y, z)$ be the point on the graph of $x^4 +2x^2y^2 +y^4 -2x^2 -2y^2 +z^2 +1 = 0$ such that $x+y +z$ is maximized. Find $a+b$ if $xy +xz +yz$ can be expressed as $\frac{a}{b}$ where $a$, $b$ are relatively prime positive integers. [b]p11.[/b] Andy starts driving from Pittsburgh to Columbus and back at a random time from $12$ pm to $3$ pm. Brendan starts driving from Pittsburgh to Columbus and back at a random time from $1$ pm to $4$ pm. Both Andy and Brendan take $3$ hours for the round trip, and they travel at constant speeds. The probability that they pass each other closer to Pittsburgh than Columbus is$ m/n$, for relatively prime positive integers $m$ and $n$. What is $m + n$? [b]p12.[/b] Consider trapezoid $ABCD$ with $AB$ parallel to $CD$ and $AB < CD$. Let $AD \cap BC = O$, $BO = 5$, and $BC = 11$. Drop perpendicular $AH$ and $BI$ onto $CD$. Given that $AH : AD = \frac23$ and $BI : BC = \frac56$ , calculate $a + b + c + d - e$ if $AB + CD$ can be expressed as $\frac{a\sqrt{b} + c\sqrt{d}}{e}$ where $a$, $b$, $c$, $d$, $e$ are integers with $gcd(a, c, e) = 1$ and $\sqrt{b}$, $\sqrt{d}$ are fully simplified. [b]p13.[/b] The polynomials $p(x)$ and $q(x)$ are of the same degree and have the same set of integer coefficients but the order of the coefficients is different. What is the smallest possible positive difference between $p(2021)$ and $q(2021)$? [b]p14.[/b] Let $ABCD$ be a square with side length $12$, and $P$ be a point inside $ABCD$. Let line $AP$ intersect $DC$ at $E$. Let line $DE$ intersect the circumcircle of $ADP$ at $F \ne D$. Given that line $EB$ is tangent to the circumcircle of $ABP$ at $B$, and $FD = 8$, find $m + n$ if $AP$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. [b]p15.[/b] A three digit number $m$ is chosen such that its hundreds digit is the sum of the tens and units digits. What is the smallest positive integer $n$ such that $n$ cannot divide $m$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine what day of the week day was: June $6$, $1944$. Note: Leap years are those that are multiples of $4$ and do not end in $00$ or that are multiples of $400$, for example $1812$, $1816$, $1820$, $1600$, $2000$, but $1800$, $1810$, $2100$ are not leaps. Giving the answer without any mathematical justification will not award points.

Find all positive integers $k$, for which the product of some consecutive $k$ positive integers ends with $k$. [i]Proposed by Oleksiy Masalitin[/i]

Let $0 < a_1 < a_2 <... < a_{16} < 122$ be $16$ integers. Prove that there exist integers $(p, q, r, s)$, with $1 \le p < r \le s < q \le 16$, such that $a_p + a_q = a_r + a_s$. An additional $2$ points will be awarded for this problem, if you can find a larger bound than $122$ (with proof).

On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.

Let $m,n$ and $N$ be positive integers and $\mathbb{Z}_{N}=\{0,1,\dots,N-1\}$ a set of residues modulo $N$. Consider a table $m\times n$ such that each one of the $mn$ cells has an element of $\mathbb{Z}_{N}$. A [i]move[/i] is choose an element $g\in \mathbb{Z}_{N}$, a cell in the table and add $+g$ to the elements in the same row/column of the chosen cell(the sum is modulo $N$). Prove that if $N$ is coprime with $m-1,n-1,m+n-1$ then any initial arrangement of your elements in the table cells can become any other arrangement using an finite quantity of moves.

Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows: \[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\] Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ . [i]Proposed by Marcin Kuczma, Poland[/i]

Does there exist a positive integer such that its last digit is nonzero and that it becomes exactly two times bigger when the order of its digits is reversed?

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50)=12$ because the prime factorization of $50$ is $ 2 \cdot 5^2 $, and $ 2 + 5 + 5 = 12 $. What is the range of the function $r$, $ \{ r(n) : n \ \text{is a composite positive integer} \} $? [b](A)[/b] the set of positive integers [b](B)[/b] the set of composite positive integers [b](C)[/b] the set of even positive integers [b](D)[/b] the set of integers greater than 3 [b](E)[/b] the set of integers greater than 4