The number of real triples $(x , y , z )$ that satisfy the equation $x^4 + 4y^4 + z^4 + 4 = 8xyz$ is (A): $0$, (B): $1$, (C): $2$, (D): $8$, (E): None of the above.

Show that there are infinitely many tuples $(a,b,c,d)$ of natural numbers such that $a^3 + b^4 + c^5 = d^7$.

Let $P_1(x, y)$ and $P_2(x, y)$ be two relatively prime polynomials with complex coefficients. Let $Q(x, y)$ and $R(x, y)$ be polynomials with complex coefficients and each of degree not exceeding $d$. Prove that there exist two integers $A_1, A_2$ not simultaneously zero with $|A_i| \leq d + 1 \ (i = 1, 2)$ and such that the polynomial $A_1P_1(x, y) + A_2P_2(x, y)$ is coprime to $Q(x, y)$ and $R(x, y).$

A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are positive integers, $a$ and $e$ are relatively prime, and neither $c$ nor $f$ is divisible by the square of any prime. Find the remainder when the product $abcdef$ is divided by 1000.

Fix a positive integer $n\geq 3$. Does there exist infinitely many sets $S$ of positive integers $\lbrace a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n\rbrace$, such that $\gcd (a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n)=1$, $\lbrace a_i\rbrace _{i=1}^n$, $\lbrace b_i\rbrace _{i=1}^n$ are arithmetic progressions, and $\prod_{i=1}^n a_i = \prod_{i=1}^n b_i$?

Let $p$ and $q$ be odd prime numbers and $a$ a positive integer so that $p|a^q+1$ and $q|a^p+1$. Show that $p|a+1$ or $q|a+1$. [i]Authored by Nikola Velov[/i]

Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.

A student wrote a correct addition operation $A/B+C/D = E/F$ on the blackboard, where both summands are irreducible and $F$ is the least common multiple of $B$ and $D$. After that, the student reduced the sum $E/F$ correctly by an integer $d$. Prove that $d$ is a common divisor of $B$ and $D$.

Given quadrilateral $ ABCD$. The circumcircle of $ ABC$ is tangent to side $ CD$, and the circumcircle of $ ACD$ is tangent to side $ AB$. Prove that the length of diagonal $ AC$ is less than the distance between the midpoints of $ AB$ and $ CD$.

Determine all finite increasing arithmetic progressions in which each term is the reciprocal of a positive integer and the sum of all the terms is $1$.

For any integers $m,n$, we have the set $A(m,n) = \{ x^2+mx+n \mid x \in \mathbb{Z} \}$, where $\mathbb{Z}$ is the integer set. Does there exist three distinct elements $a,b,c$ which belong to $A(m,n)$ and satisfy the equality $a=bc$?

Find all positive reals $x,y,z $ such that \[2x-2y+\dfrac1z = \dfrac1{2014},\hspace{0.5em} 2y-2z +\dfrac1x = \dfrac1{2014},\hspace{0.5em}\text{and}\hspace{0.5em} 2z-2x+ \dfrac1y = \dfrac1{2014}.\]

If $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}$, where $a,b,c$ are positive integers with no common factor, prove that $(a +b)$ is a square.

Find all natural solutions of $3^{x}= 2^{x}y+1.$

Let $a$ and $n$ be positive integers such that: 1. $a^{2^n}-a$ is divisible by $n$, 2. $\sum\limits_{k=1}^{n} k^{2024}a^{2^k}$ is [i]not[/i] divisible by $n$. Prove that $n$ has a prime factor [i]smaller[/i] than $2024$. [i]Proposed by Shantanu Nene[/i]

Find all quadruplets $(a, b, c, d)$ of positive integers such that \[ \left( 1 + \frac{1}{a} \right) \left( 1 + \frac{1}{b} \right) \left( 1 + \frac{1}{c} \right) \left( 1 + \frac{1}{d} \right) = 4. \]

A positive integer $b$ is called good if there exist positive integers $1=a_1, a_2, \ldots, a_{2023}=b$ such that $|a_{i+1}-a_i|=2^i$. Find the number of the good integers.

Prove that the following inequality holds with the exception of finitely many positive integers $n$: \[\sum_{i=1}^n\sum_{j=1}^n gcd(i,j)>4n^2.\]

The representation of real number $a$ as a decimal infinite fraction contain all $10$ digits. For a positive integer $n$ let $v_n$ be the number of all segments of length $n$ that occur. Prove that, if $v_n \leq n + 8$ for some positive integer $n$, then the number $a$ is rational.

[u]Round 1[/u] [b]p1.[/b] Ravi has a bag with $100$ slips of paper in it. Each slip has one of the numbers $3, 5$, or $7$ written on it. Given that half of the slips have the number $3$ written on them, and the average of the values on all the slips is $4.4$, how many slips have $7$ written on them? [b]p2.[/b] In triangle $ABC$, point $D$ lies on side $AB$ such that $AB \perp CD$. It is given that $\frac{CD}{BD}=\frac12$, $AC = 29$, and $AD = 20$. Find the area of triangle $BCD$. [b]p3.[/b] Compute $(123 + 4)(123 + 5) - 123\cdot 132$. [u]Round 2[/u] [b]p4. [/b] David is evaluating the terms in the sequence $a_n = (n + 1)^3 - n^3$ for $n = 1, 2, 3,....$ (that is, $a_1 = 2^3 - 1^3$ , $a_2 = 3^3 - 2^3$, $a_3 = 4^3 - 3^3$, and so on). Find the first composite number in the sequence. (An positive integer is composite if it has a divisor other than 1 and itself.) [b]p5.[/b] Find the sum of all positive integers strictly less than $100$ that are not divisible by $3$. [b]p6.[/b] In how many ways can Alex draw the diagram below without lifting his pencil or retracing a line? (Two drawings are different if the order in which he draws the edges is different, or the direction in which he draws an edge is different). [img]https://cdn.artofproblemsolving.com/attachments/9/6/9d29c23b3ca64e787e717ceff22d45851ae503.png[/img] [u]Round 3[/u] [b]p7.[/b] Fresh Mann is a $9$th grader at Euclid High School. Fresh Mann thinks that the word vertices is the plural of the word vertice. Indeed, vertices is the plural of the word vertex. Using all the letters in the word vertice, he can make $m$ $7$-letter sequences. Using all the letters in the word vertex, he can make $n$ $6$-letter sequences. Find $m - n$. [b]p8.[/b] Fresh Mann is given the following expression in his Algebra $1$ class: $101 - 102 = 1$. Fresh Mann is allowed to move some of the digits in this (incorrect) equation to make it into a correct equation. What is the minimal number of digits Fresh Mann needs to move? [b]p9.[/b] Fresh Mann said, “The function $f(x) = ax^2+bx+c$ passes through $6$ points. Their $x$-coordinates are consecutive positive integers, and their y-coordinates are $34$, $55$, $84$, $119$, $160$, and $207$, respectively.” Sophy Moore replied, “You’ve made an error in your list,” and replaced one of Fresh Mann’s numbers with the correct y-coordinate. Find the corrected value. [u]Round 4[/u] [b]p10.[/b] An assassin is trying to find his target’s hotel room number, which is a three-digit positive integer. He knows the following clues about the number: (a) The sum of any two digits of the number is divisible by the remaining digit. (b) The number is divisible by $3$, but if the first digit is removed, the remaining two-digit number is not. (c) The middle digit is the only digit that is a perfect square. Given these clues, what is a possible value for the room number? [b]p11.[/b] Find a positive real number $r$ that satisfies $$\frac{4 + r^3}{9 + r^6}=\frac{1}{5 - r^3}- \frac{1}{9 + r^6}.$$ [b]p12.[/b] Find the largest integer $n$ such that there exist integers $x$ and $y$ between $1$ and $20$ inclusive with $$\left|\frac{21}{19} -\frac{x}{y} \right|<\frac{1}{n}.$$ PS. You had better use hide for answers. Last rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784267p24464980]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Do there exist postive integers $a_1<a_2<\cdots<a_{100}$ such that for $2\le k\le100$ the greatest common divisor of $a_{k-1}$ and $a_k$ is greater than the greatest common divisor of $a_k$ and $a_{k+1}$?

Show that the only solutions of te equation \[ p^{k} + 1 = q^{m} \], in positive integers $k,q,m > 1$ and prime $p$ are (i) $(p,k,q,m) = (2,3,3,2)$ (ii) $k=1 , q=2,$and $p$ is a prime of the form $2^{m} -1$, $m > 1 \in \mathbb{N}$

Find all primes $p$ and $q$, with $p \le q$, so that $$p (2q + 1) + q (2p + 1) = 2 (p^2 + q^2).$$

a) Denote by $S(n)$ the sum of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $S(1),S(2),...$. Show that the number obtained is irrational. b) Denote by $P(n)$ the product of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $P(1),P(2),...$. Show that the number obtained is irrational.

Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Compute $\displaystyle \sum_{i=1}^{\phi(2023)} \dfrac{\gcd(i,\phi(2023))}{\phi(2023)}$. [i]Proposed by Giacomo Rizzo[/i]