We say that a subset \( T \) of \(\{1, 2, \dots, 2024\}\) is [b]kawaii[/b] if \( T \) has the following properties: 1. \( T \) has at least two distinct elements; 2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \). For example, the subset \( T = \{31, 71, 2024\} \) is [b]kawaii[/b], but \( T = \{5, 15, 75\} \) is not [b]kawaii[/b] because \( 15 - 5 = 10 \) divides \( 15 + 5 = 20 \). What is the largest possible number of elements that a [b]kawaii [/b]subset can have?

Find, with proof, all positive integers $n$ such that the equation $x^{3}+y^{3}+z^{3}=nx^{2}y^{2}z^{2}$ has a solution in positive integers.

Let $p$ be an odd prime, and let $(p^p)!=mp^k$ for some positive integers $m$ and $k$. Find in terms of $p$ the number of ordered pairs $(m,k)$ satisfying $m+k\equiv0\pmod p$.

Prove that for all integers $ a > 1$ and $ b > 1$ there exists a function $ f$ from the positive integers to the positive integers such that $ f(a\cdot f(n)) \equal{} b\cdot n$ for all $ n$ positive integer.

Find the value of $$(2^{40}+12^{41}+23^{42}+67^{43}+87^{44})^{45!+46}\mod11$$ (variation but same answer) [hide=Answer]3[/hide]

Let $p=ab+bc+ac$ be a prime number where $a,b,c$ are different two by two, show that $a^3,b^3,c^3$ gives different residues modulo $p$

Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying: (1)$\tau (n)=a$ (2)$n|\phi (n)+\sigma (n)$ Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$, $\sigma (n)$ is the sum of positive divisors of $n$.

[b]p16.[/b] Let $P(x)$ be a quadratic such that $P(-2) = 10$, $P(0) = 5$, $P(3) = 0$. Then, find the sum of the coefficients of the polynomial equal to $P(x)P(-x)$. [b]p17.[/b] Suppose that $a < b < c < d$ are positive integers such that the pairwise differences of $a, b, c, d$ are all distinct, and $a + b + c + d$ is divisible by $2023$. Find the least possible value of $d$. [b]p18.[/b] Consider a right rectangular prism with bases $ABCD$ and $A'B'C'D'$ and other edges $AA'$, $BB'$, $CC'$ and $DD'$. Suppose $AB = 1$, $AD = 2$, and $AA' = 1$. $\bullet$ Let $X$ be the plane passing through $A$, $C'$, and the midpoint of $BB'$. $\bullet$ Let $Y$ be the plane passing through $D$, $B'$, and the midpoint of $CC'$. Then the intersection of $X$, $Y$ , and the prism is a line segment of length $\ell$. Find $\ell$. PS. You should use hide for answers.

Find all positive intger solutions of $3^x+29=2^y$.

For a prime $q$, let $\Phi_q(x)=x^{q-1}+x^{q-2}+\cdots+x+1$. Find the sum of all primes $p$ such that $3 \le p \le 100$ and there exists an odd prime $q$ and a positive integer $N$ satisfying \[\dbinom{N}{\Phi_q(p)}\equiv \dbinom{2\Phi_q(p)}{N} \not \equiv 0 \pmod p. \][i]Proposed by Sammy Luo[/i]

Find all positive integers $a,b,c$ satisfying $(a,b)=(b,c)=(c,a)=1$ and \[ \begin{cases} a^2+b\mid b^2+c\\ b^2+c\mid c^2+a \end{cases} \] and none of prime divisors of $a^2+b$ are congruent to $1$ modulo $7$

For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set $$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$ where $i = 1, 2$. Determine the smallest positive integer $m$ such that $2f_1(m) - f_2(m) = 2017$.

Solve the following system \[ \left\{ \begin{array}{l} xyzu-x^3=9 \\ x+yz=\dfrac{3}{2}u \\ \end{array} \right. \] in positive integers $x$, $y$, $z$ and $u$.

Find all integer solutions of the equation $$y^2+2y=x^4+20x^3+104x^2+40x+2003.$$ Note: has appeared many times before, see [url=https://artofproblemsolving.com/community/q1_%22x%5E4%2B20x%5E3%2B104x%5E2%22]here[/url]

We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets. For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$. [i]Proposed by Victor Wang[/i]

Define sequence $(a_n):a_n=2^n+3^n+6^n+1(n\in\mathbb{Z}_+)$. Are there intenger $k\geq2$, satisfying that $\gcd(k,a_i)=1$ for all $k\in\mathbb{Z}_+$? If yes, find the smallest $k$. If not, prove this.

Determine all triples $(a, b, c)$ of positive integers such that $$a! + b! = 2^{c!}.$$ [i](Walther Janous)[/i]

The largest of the following integers which divides each of the numbers of the sequence $ 1^5 \minus{} 1,\, 2^5 \minus{} 2,\, 3^5 \minus{} 3,\, \cdots, n^5 \minus{} n, \cdots$ is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 120\qquad \textbf{(E)}\ 30$

Show that there are infinitely many positive integers $a$ such that the number $n^4+a$ is composite for every positive integer $n.$ Give 5 (different) numbers $a$ with the mentioned property.

Let $(a_n)_{n = 0}^{\infty} $ be a sequence of positive integers such that for every $n \geq 0$ it is true that $$a_{n+2} = a_0 a_1 + a_1 a_2 + ... + a_n a_{n+1} - 1 $$ a) Prove that there exist a prime number which divides infinitely many $a_n$ b) Prove that there exist infinitely many such prime numbers

Assume the $m$ is a given integer greater than $ 1$. Find the largest number $C$ such that for all $n \in N$ we have $$\sum_{1\le k \le m ,\,\, (k,m)=1}\frac{1}{k}\ge C \sum_{k=1}^{m}\frac{1}{k}$$

Given is the equation $x^2+y^2-axy+2=0$ where $a$ is a positive integral parameter. $i.$Show that,for $a\neq 4$ there exist no pairs $(x,y)$ of positive integers satisfying the equation. $ii.$ Show that,for $a=4$ there exist infinite pairs $(x,y)$ of positive integers satisfying the equation,and determine those pairs.

[i]20 problems for 20 minutes.[/i] [b]p1.[/b] Euclid eats $\frac17$ of a pie in $7$ seconds. Euler eats $\frac15$ of an identical pie in $10$ seconds. Who eats faster? [b]p2.[/b] Given that $\pi = 3.1415926...$ , compute the circumference of a circle of radius 1. Express your answer as a decimal rounded to the nearest hundred thousandth (i.e. $1.234562$ and $1.234567$ would be rounded to $1.23456$ and $1.23457$, respectively). [b]p3.[/b] Alice bikes to Wonderland, which is $6$ miles from her house. Her bicycle has two wheels, and she also keeps a spare tire with her. If each of the three tires must be used for the same number of miles, for how many miles will each tire be used? [b]p4.[/b] Simplify $\frac{2010 \cdot 2010}{2011}$ to a mixed number. (For example, $2\frac12$ is a mixed number while $\frac52$ and $2.5$ are not.) [b]p5.[/b] There are currently $175$ problems submitted for $EMC^2$. Chris has submitted $51$ of them. If nobody else submits any more problems, how many more problems must Chris submit so that he has submitted $\frac13$ of the problems? [b]p6.[/b] As shown in the diagram below, points $D$ and $L$ are located on segment $AK$, with $D$ between $A$ and $L$, such that $\frac{AD}{DK}=\frac{1}{3}$ and $\frac{DL}{LK}=\frac{5}{9}$. What is $\frac{DL}{AK}$? [img]https://cdn.artofproblemsolving.com/attachments/9/a/3f92bd33ffbe52a735158f7ebca79c4c360d30.png[/img] [b]p7.[/b] Find the number of possible ways to order the letters $G, G, e, e, e$ such that two neighboring letters are never $G$ and $e$ in that order. [b]p8.[/b] Find the number of odd composite integers between $0$ and $50$. [b]p9.[/b] Bob tries to remember his $2$-digit extension number. He knows that the number is divisible by $5$ and that the first digit is odd. How many possibilities are there for this number? [b]p10.[/b] Al walks $1$ mile due north, then $2$ miles due east, then $3$ miles due south, and then $4$ miles due west. How far, in miles, is he from his starting position? (Assume that the Earth is flat.) [b]p11.[/b] When n is a positive integer, $n!$ denotes the product of the first $n$ positive integers; that is, $n! = 1 \cdot 2 \cdot 3 \cdot ... \cdot n$. Given that $7! = 5040$, compute $8! + 9! + 10!$. [b]p12.[/b] Sam's phone company charges him a per-minute charge as well as a connection fee (which is the same for every call) every time he makes a phone call. If Sam was charged $\$4.88$ for an $11$-minute call and $\$6.00$ for a $19$-minute call, how much would he be charged for a $15$-minute call? [b]p13.[/b] For a positive integer $n$, let $s_n$ be the sum of the n smallest primes. Find the least $n$ such that $s_n$ is a perfect square (the square of an integer). [b]p14.[/b] Find the remainder when $2011^{2011}$ is divided by $7$. [b]p15.[/b] Let $a, b, c$, and $d$ be $4$ positive integers, each of which is less than $10$, and let $e$ be their least common multiple. Find the maximum possible value of $e$. [b]p16.[/b] Evaluate $100 - 1 + 99 - 2 + 98 - 3 + ... + 52 - 49 + 51 - 50$. [b]p17.[/b] There are $30$ basketball teams in the Phillips Exeter Dorm Basketball League. In how ways can $4$ teams be chosen for a tournament if the two teams Soule Internationals and Abbot United cannot be chosen at the same time? [b]p18.[/b] The numbers $1, 2, 3, 4, 5, 6$ are randomly written around a circle. What is the probability that there are four neighboring numbers such that the sum of the middle two numbers is less than the sum of the other two? [b]p19.[/b] What is the largest positive $2$-digit factor of $3^{2^{2011}} - 2^{2^{2011}}$? [b]p20.[/b] Rhombus $ABCD$ has vertices $A = (-12,-4)$, $B = (6, b)$, $C = (c,-4)$ and $D = (d,-28)$, where $b$, $c$, and $d$ are integers. Find a constant $m$ such that the line y = $mx$ divides the rhombus into two regions of equal area. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Today is Saturday, April $25$, $2020$. What is the value of $6 + 4 + 25 + 2020$? [b]p2.[/b] The figure below consists of a $2$ by $3$ grid of squares. How many squares of any size are in the grid? $\begin{tabular}{|l|l|l|} \hline & & \\ \hline & & \\ \hline \end{tabular}$ [b]p3.[/b] James is playing a game. He first rolls a six-sided dice which contains a different number on each side, then randomly picks one of twelve dierent colors, and finally ips a quarter. How many different possible combinations of a number, a color and a flip are there in this game? [b]p4.[/b] What is the sum of the number of diagonals and sides in a regular hexagon? [b]p5.[/b] Mickey Mouse and Minnie Mouse are best friends but they often fight. Each of their fights take up exactly one hour, and they always fight on prime days. For example, they fight on January $2$nd, $3$rd, but not the $4$th. Knowing this, how many total times do Mickey and Minnie fight in the months of April, May and June? [b]p6.[/b] Apple always loved eating watermelons. Normal watermelons have around $13$ black seeds and $25$ brown seeds, whereas strange watermelons had $45$ black seeds and $2$ brown seeds. If Apple bought $14$ normal watermelons and $7$ strange watermelons, then let $a$ be the total number of black seeds and $b$ be the total number of brown seeds. What is $a - b$? [b]p7.[/b] Jerry and Justin both roll a die once. The probability that Jerry's roll is greater than Justin's can be expressed as a fraction in the form $\frac{m}{n}$ in simplified terms. What is $m + n$? [b]p8.[/b] Taylor wants to color the sides of an octagon. What is the minimum number of colors Taylor will need so that no adjacent sides of the octagon will be filled in with the same color? [b]p9.[/b] The point $\frac23$ of the way from ($-6, 8$) to ($-3, 5$) can be expressed as an ordered pair $(a, b)$. What is $|a - b|$? [b]p10.[/b] Mary Price Maddox laughs $7$ times per class. If she teaches $4$ classes a day for the $5$ weekdays every week but doesn't laugh on Wednesdays, then how many times does she laugh after $5$ weeks of teaching? [b]p11.[/b] Let $ABCD$ be a unit square. If $E$ is the midpoint of $AB$ and $F$ lies inside $ABCD$ such that $CFD$ is an equilateral triangle, the positive difference between the area of $CED$ and $CFD$ can be expressed in the form $\frac{a-\sqrt{b}}{c}$ , where $a$, $b$, $c$ are in lowest simplified terms. What is $a + b + c$? [b]p12.[/b] Eddie has musician's syndrome. Whenever a song is a $C$, $A$, or $F$ minor, he begins to cry and his body becomes very stiff. On the other hand, if the song is in $G$ minor, $A$ at major, or $E$ at major, his eyes open wide and he feels like the happiest human being ever alive. There are a total of $24$ keys. How many different possibilities are there in which he cries while playing one song with two distinct keys? [b]p13.[/b] What positive integer must be added to both the numerator and denominator of $\frac{12}{40}$ to make a fraction that is equivalent to $\frac{4}{11}$ ? [b]p14.[/b] The number $0$ is written on the board. Each minute, Gene the genie either multiplies the number on the board by $3$ or $9$, each with equal probability, and then adds either $1$,$2$, or $3$, each with equal probability. Find the expected value of the number after $3$ minutes. [b]p15.[/b] $x$ satisfies $\dfrac{1}{x+ \dfrac{1}{1+\frac{1}{2}}}=\dfrac{1}{2+ \dfrac{1}{1- \dfrac{1}{2+\frac{1}{2}}}}$ Find $x$. [b]p16.[/b] How many different points in a coordinate plane can a bug end up on if the bug starts at the origin and moves one unit to the right, left, up or down every minute for $8$ minutes? [b]p17.[/b] The triplets Addie, Allie, and Annie, are racing against the triplets Bobby, Billy, and Bonnie in a relay race on a track that is $100$ feet long. The first person of each team must run around the entire track twice and tag the second person for the second person to start running. Then, the second person must run once around the entire track and tag the third person, and finally, the third person would only have to run around half the track. Addie and Bob run first, Allie and Billy second, Annie and Bonnie third. Addie, Allie, and Annie run at $50$ feet per minute (ft/m), $25$ ft/m, and $20$ ft/m, respectively. If Bob, Billy, and Bonnie run half as fast as Addie, Allie, and Annie, respectively, then how many minutes will it take Bob, Billy, and Bonnie to finish the race. Assume that everyone runs at a constant rate. [b]p18.[/b] James likes to play with Jane and Jason. If the probability that Jason and Jane play together is $\frac13$, while the probability that James and Jason is $\frac14$ and the probability that James and Jane play together is $\frac15$, then the probability that they all play together is $\frac{\sqrt{p}}{q}$ for positive integers $p$, $q$ where $p$ is not divisible by the square of any prime. Find $p + q$. [b]p19.[/b] Call an integer a near-prime if it is one more than a prime number. Find the sum of all near-primes less than$ 1000$ that are perfect powers. (Note: a perfect power is an integer of the form $n^k$ where $n, k \ge 2$ are integers.) [b]p20.[/b] What is the integer solution to $\sqrt{\frac{2x-6}{x-11}} = \frac{3x-7}{x+6}$ ? [b]p21.[/b] Consider rectangle $ABCD$ with $AB = 12$ and $BC = 4$ with $F$,$G$ trisecting $DC$ so that $F$ is closer to $D$. Then $E$ is on $AB$. We call the intersection of $EF$ and $DB$ $X$, and the intersection of $EG$ and $DB$ is $Y$. If the area of $\vartriangle XY E$ is \frac{8}{15} , then what is the length of $EB$? [b]p22.[/b] The sum $$\sum^{\infty}_{n=2} \frac{1}{4n^2-1}$$ can be expressed as a common fraction $\frac{a}{b}$ in lowest terms. Find $a + b$. [b]p23.[/b] In square $ABCD$, $M$, $N$, $O$, $P$ are points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$ and $\overline{DA}$, respectively. If $AB = 4$, $AM = BM$ and $DP = 3AP$, the least possible value of $MN + NO + OP$ can be expressed as $\sqrt{x}$ forsome integer x. Find x: [b]p24.[/b] Grand-Ovich the ant is at a vertex of a regular hexagon and he moves to one of the adjacent vertices every minute with equal probability. Let the probability that after $8$ minutes he will have returned to the starting vertex at least once be the common fraction $\frac{a}{b}$ in lowest terms. What is $a + b$? [b]p25.[/b] Find the last two non-zero digits at the end of $2020!$ written as a two digit number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.