Let $a,b,c$ be positive integers such that $a|b^2, b|c^2$ and $c|a^2$. Prove that $abc|(a+b+c)^{7}$

[u]Round 5[/u] [b]p13.[/b] You have a $26 \times 26$ grid of squares. Color each randomly with red, yellow, or blue. What is the expected number (to the nearest integer) of $2 \times 2$ squares that are entirely red? [b]p14.[/b] Four snakes are boarding a plane with four seats. Each snake has been assigned to a different seat. The first snake sits in the wrong seat. Any subsequent snake will sit in their assigned seat if vacant, if not, they will choose a random seat that is available. What is the expected number of snakes who sit in their correct seats? [b]p15.[/b] Let $n \ge 1$ be an integer and $a > 0$ a real number. In terms of n, find the number of solutions $(x_1, ..., x_n)$ of the equation $\sum^n_{i=1}(x^2_i + (a - x_i)^2) = na^2$ such that $x_i$ belongs to the interval $[0, a]$ , for $i = 1, 2, . . . , n$. [u]Round 6 [/u] [b]p16.[/b] All roots of $$\prod^{25}_{n=1} \prod^{2n}_{k=0}(-1)^k \cdot x^k = 0$$ are written in the form $r(\cos \phi + i\sin \phi)$ for $i^2 = -1$, $r > 0$, and $0 \le \phi < 2\pi$. What is the smallest positive value of $\phi$ in radians? [b]p17.[/b] Find the sum of the distinct real roots of the equation $$\sqrt[3]{x^2 - 2x + 1} + \sqrt[3]{x^2 - x - 6} = \sqrt[3]{2x^2 - 3x - 5}.$$ [b]p18.[/b] If $a$ and $b$ satisfy the property that $a2^n + b$ is a square for all positive integers $n$, find all possible value(s) of $a$. [u]Round 7 [/u] [b]p19.[/b] Compute $(1 - \cot 19^o)(1 - \cot 26^o)$. [b]p20.[/b] Consider triangle $ABC$ with $AB = 3$, $BC = 5$, and $\angle ABC = 120^o$. Let point $E$ be any point inside $ABC$. The minimum of the sum of the squares of the distances from $E$ to the three sides of $ABC$ can be written in the form $a/b$ , where a and b are natural numbers such that the greatest common divisor of $a$ and $b$ is $1$. Find $a + b$. [b]p21.[/b] Let $m \ne 1$ be a square-free number (an integer – possibly negative – such that no square divides $m$). We denote $Q(\sqrt{m})$ to be the set of all $a + b\sqrt{m}$ where $a$ and $b$ are rational numbers. Now for a fixed $m$, let $S$ be the set of all numbers $x$ in $Q(\sqrt{m})$ such that x is a solution to a polynomial of the form: $x^n + a_1x^{n-1} + .... + a_n = 0$, where $a_0$, $...$, $a_n$ are integers. For many integers m, $S = Z[\frac{m}] = \{a + b\sqrt{m}\}$ where $a$ and $b$ are integers. Give a classification of the integers for which this is not true. (Hint: It is true for $ m = -1$ and $2$.) PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782002p24434611]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for any prime $p$, there exist integers $x,y,z,$ and $w$ such that $x^2+y^2+z^2-wp=0$ and $0<w<p$

Let $a, b, c$, and $d$ be prime numbers with $a \le b \le c \le d > 0$. Suppose $a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)$. Find $4a + 3b + 2c + d$.

Let $p$ and $q$ be prime numbers of the form $4k+3$. Suppose that there exist integers $x$ and $y$ such that $x^2-pqy^2=1$. Prove that there exist positive integers $a$ and $b$ such that $|pa^2-qb^2|=1$.

Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000 \cdot N$ contains no square of an integer.

Find the sum of all even numbers greater than 100000, that u can make only with the digits 0,2,4,6,8,9 without any digit repeating in any number.

Let $d \geq 2$ be a positive integer. Define the sequence $a_1,a_2,\dots$ by $$a_1=1 \ \text{and} \ a_{n+1}=a_n^d+1 \ \text{for all }n\geq 1.$$ Determine all pairs of positive integers $(p, q)$ such that $a_p$ divides $a_q$.

Triangle $ABC$ has $AB=21$, $AC=22$, and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $x, y$ be positive integers such that $$ x^4=(x-1)\left(y^3-23\right)-1 . $$ Find the maximum possible value of $x+y$.

Let $p$ be a prime and $Q(x)$ be a polynomial with integer coefficients such that $Q(0) = 0, \ Q(1) = 1$ and the remainder of $Q(n)$ is either $0$ or $1$ when divided by $p$, for every $n \in \mathbb{N}$. Prove that $Q(x)$ is of degree at least $p - 1$.

For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

Find all ordered pairs $ (m,n)$ where $ m$ and $ n$ are positive integers such that $ \frac {n^3 \plus{} 1}{mn \minus{} 1}$ is an integer.

Let $a_1, a_2, a_3, a_4, a_5$ be positive integers such that $a_1, a_2, a_3$ and $a_3, a_4, a_5$ are both geometric sequences and $a_1, a_3, a_5$ is an arithmetic sequence. If $a_3 = 1575$, find all possible values of $\vert a_4 - a_2 \vert$.

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

[b]p1.[/b] For what integers $x$ is it possible to find an integer $y$ such that $$x(x + 1) (x + 2) (x + 3) + 1 = y^2 ?$$ [b]p2.[/b] Two tangents to a circle are parallel and touch the circle at points $A$ and $B$, respectively. A tangent to the circle at any point $X$, other than $A$ or $B$, meets the first tangent at $Y$ and the second tangent at $Z$. Prove $AY \cdot BZ$ is independent of the position of $X$. [b]p3.[/b] If $a, b, c$ are positive real numbers, prove that $$8abc \le (b + c) (c + a) (a + b)$$ by first verifying the relation in the special case when $c = b$. [b]p4.[/b] Solve the equation $$\frac{x^2}{3}+\frac{48}{x^2}=10 \left( \frac{x}{3}-\frac{4}{x}\right)$$ [b]p5.[/b] Tom and Bill live on the same street. Each boy has a package to deliver to the other boy’s house. The two boys start simultaneously from their own homes and meet $600$ yards from Bill's house. The boys continue on their errand and they meet again $700$ yards from Tom's house. How far apart do the boy's live? [b]p6.[/b] A standard set of dominoes consists of $28$ blocks of size $1$ by $2$. Each block contains two numbers from the set $0,1,2,...,6$. We can denote the block containing $2$ and $3$ by $[2, 3]$, which is the same block as $[3, 2]$. The blocks $[0, 0]$, $[1, 1]$,..., $[6, 6]$ are in the set but there are no duplicate blocks. a) Show that it is possible to arrange the twenty-eight dominoes in a line, end-to-end, with adjacent ends matching, e. g., $... [3, 1]$ $[1, 1]$ $[1, 0]$ $[0, 6] ...$ . b) Consider the set of dominoes which do not contain $0$. Show that it is impossible to arrange this set in such a line. c) Generalize the problem and prove your generalization. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

It is desired to choose $n$ integers from the collection of $2n$ integers, namely, $0,0,1,1,2,2,\ldots,n-1,n-1$ such that the average of these $n$ chosen integers is itself an integer and as minimum as possible. Show that this can be done for each positive integer $n$ and find this minimum value for each $n$.

Do there exist irrational numbers $a,b$ both greater than $1$, such that $\lfloor{a^m}\rfloor\neq \lfloor{b^n}\rfloor$ for all $m,n\in\mathbb{N}$ ?

Consider the sequence in which $a_1 = 1$ and $a_n$ is obtained by juxtaposing the decimal representation of $n$ at the end of the decimal representation of $a_{n-1}$. That is, $a_1 = 1$, $a_2 = 12$, $a_3 = 123$, $\dots$ , $a_9 = 123456789$, $a_{10} = 12345678910$ and so on. Prove that infinitely many numbers of this sequence are multiples of $7$.

Let $A = (0, 0, 0)$ in 3D space. Define the [i]weight[/i] of a point as the sum of the absolute values of the coordinates. Call a point a [i]primitive lattice point[/i] if all of its coordinates are integers whose gcd is 1. Let square $ABCD$ be an [i]unbalanced primitive integer square[/i] if it has integer side length and also, $B$ and $D$ are primitive lattice points with different weights. Prove that there are infinitely many unbalanced primitive integer squares such that the planes containing the squares are not parallel to each other.

[u]Round 5[/u] [b]p13.[/b] Simplify $\frac11+\frac13+\frac16+\frac{1}{10}+\frac{1}{15}+\frac{1}{21}$. [b]p14.[/b] Given that $x + y = 7$ and $x^2 + y^2 = 29$, what is the sum of the reciprocals of $x$ and $y$? [b]p15.[/b] Consider a rectangle $ABCD$ with side lengths $AB = 3$ and $BC = 4$. If circles are inscribeδ in triangles $ABC$ and $BCD$, how far are the centers of the circles from each other? [u]Round 6[/u] [b]p16.[/b] Evaluate $\frac{2!}{1!} +\frac{3!}{2!} +\frac{4!}{3!} + ... +\frac{99!}{98!}+\frac{100!}{99!}$ . [b]p17.[/b] Let $ABCD$ be a square of side length $2$. A semicircle is drawn with diameter $\overline{AC}$ that passes through point $B$. Find the area of the region inside the semicircle but outside the square. [b]p18.[/b] For how many positive integer values of $k$ is $\frac{37k - 30}{k}$ a positive integer? [u]Round 7[/u] [b]p19.[/b] Two parallel planar slices across a sphere of radius $25$ create cross sections of area $576\pi$ and $225\pi$. What is the maximum possible distance between the two slices? [b]p20.[/b] How many positive integers cannot be expressed in the form $3\ell + 4m + 5t$, where $\ell$, $m$, and $t$ are nonnegative integers? [b]p21.[/b] In April, a fool is someone who is fooled by a classmate. In a class of $30$ students, $14$ people were fooled by someone else and $29$ people fooled someone else. What is the largest positive integer $n$ for which we can guarantee that at least one person was fooled by at least $n$ other people? [u]Round 8[/u] [b]p22.[/b] Let $$S = 4 + \dfrac{12}{4 +\dfrac{ 12}{4 +\dfrac{ 12}{4+ ...}}}.$$ Evaluate $4 +\frac{ 12}{S}.$ [b]p23.[/b] Jonathan is buying bananagram sets for $\$11$ each and flip-flops for $\$17$ each. If he spends $\$227$ on purchases for bananagram sets and flip-flops, what is the total number of bananagram sets and flip-flops he bought? [b]p24.[/b] Alan has a $3 \times 3$ array of squares. He starts removing the squares one at a time such that each time he removes one square, all remaining squares share a side with at least two other remaining squares. What is the maximum number of squares Alan can remove? PS. You should use hide for answers. Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h2952214p26434209]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134133p28400917]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $(a_i)_{i\in \mathbb{N}}$ a sequence of positive integers, such that for any finite, non-empty subset $S$ of $\mathbb{N}$, the integer$$\Pi_{k\in S} a_k -1$$is prime. Prove that the number of $a_i$'s with $i\in \mathbb{N}$ such that $a_i$ has less than $m$ distincts prime factors is finite.

Let $a_1, a_2, a_3,\ldots$ be an infinite sequence of positive integers such that $a_2 \ne 2a_1$, and for all positive integers $m$ and $n$, the sum $m + n$ is a divisor of $a_m + a_n$. Prove that there exists an integer $M$ such that for all $n > M$, we have $a_n \ge n^3$.

A positive natural number $n$ is said to be [i]comico[/i] if its prime factorization is $n = p_1p_2...p_k$, with $k\ge 3$, and also the primes $p_1,..., p_k$ they fulfill that $p_1 + p_2 = c^2_1$ $p_1 + p_2 + p_3 = c^2_2$ $...$ $p_1 + p_2 + ...+ p_n = c^2_{n-1}$ where $c_1, c_2, ..., c_{n-1}$ are positive integers where $c_1$ is not divisible by $7$. Find all comico numbers less than $10,000$.

For some positive integer $n$, consider the usual prime factorization \[n = \displaystyle \prod_{i=1}^{k} p_{i}^{e_{i}}=p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k},\] where $k$ is the number of primes factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by \[ Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \text{ and } R(n) = \prod_{i=1}^{k} e_{i}^{e_{i}}. \] For how many $1 \leq n \leq 70$ does $R(n)$ divide $Q(n)$?