On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.

Define $f (n) = n + 1$ if $n = p^k > 1$ is a power of a prime number, and $f (n) =p_1^{k_1}+... + p_r^{k_r}$ for natural numbers $n = p_1^{k_1}... p_r^{k_r}$ ($r > 1, k_i > 0$). Given $m > 1$, we construct the sequence $a_0 = m, a_{j+1} = f (a_j)$ for $j \ge 0$ and denote by $g(m)$ the smallest term in this sequence. For each $m > 1$, determine $g(m)$.

a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$. b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.

Let $f : R^+ \to R^+$ be a function such that for all $x, y \in R^+$, $f(x)f(y) = f(xy) + f\left( \frac{x}{y}\right)$, where $R^+$ represents the positive real numbers. Given that $f(2) = 3$, compute the last two digits of $f(2^{2^{2020}})$. .

Given a prime number $p$ and let $\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}$ be $n$ distinct vectors of length $p$ with integer coordinates in an $\mathbb{R}^3$ Cartesian coordinate system. Suppose that for any $1\leqslant j<k\leqslant n$, there exists an integer $0<\ell <p$ such that all three coordinates of $\overline{v_j} -\ell \cdot \overline{v_k} $ is divisible by $p$. Prove that $n\leqslant 6$.

Find all $m,n\in\mathbb N$ and primes $p\geq 5$ satisfying \[m(4m^2+m+12)=3(p^n-1).\]

Find all prime numbers $p$ and $q$ such that $\frac{(7^{p}-2^{p})(7^{q}-2^{q})}{pq}$ is an integer.

[u]Set 6[/u] [b]p16.[/b] Let $n! = n \times (n - 1) \times ... \times 2 \times 1$. Find the maximum positive integer value of $x$ such that the quotient $\frac{160!}{160^x}$ is an integer. [b]p17.[/b] Let $\vartriangle OAB$ be a triangle with $\angle OAB = 90^o$ . Draw points $C, D, E, F, G$ in its plane so that $$\vartriangle OAB \sim \vartriangle OBC \sim \vartriangle OCD \sim \vartriangle ODE \sim \vartriangle OEF \sim \vartriangle OFG,$$ and none of these triangles overlap. If points $O, A, G$ lie on the same line, then let $x$ be the sum of all possible values of $\frac{OG}{OA }$. Then, $x$ can be expressed in the form $m/n$ for relatively prime positive integers $m, n$. Compute $m + n$. [b]p18.[/b] Let $f(x)$ denote the least integer greater than or equal to $x^{\sqrt{x}}$. Compute $f(1)+f(2)+f(3)+f(4)$. [u]Set 7[/u] The Fibonacci sequence $\{F_n\}$ is defined as $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \ge 0$. [b]p19.[/b] Find the least odd prime factor of $(F_3)^{20} + (F_4)^{20} + (F_5)^{20}$. [b]p20.[/b] Let $$S = \frac{1}{F_3F_5}+\frac{1}{F_4F_6}+\frac{1}{F_5F_7}+\frac{1}{F_6F_8}+...$$ Compute $420S$. [b]p21.[/b] Consider the number $$Q = 0.000101020305080130210340550890144... ,$$ the decimal created by concatenating every Fibonacci number and placing a 0 right after the decimal point and between each Fibonacci number. Find the greatest integer less than or equal to $\frac{1}{Q}$. [u]Set 8[/u] [b]p22.[/b] In five dimensional hyperspace, consider a hypercube $C_0$ of side length $2$. Around it, circumscribe a hypersphere $S_0$, so all $32$ vertices of $C_0$ are on the surface of $S_0$. Around $S_0$, circumscribe a hypercube $C_1$, so that $S_0$ is tangent to all hyperfaces of $C_1$. Continue in this same fashion for $S_1$, $C_2$, $S_2$, and so on. Find the side length of $C_4$. [b]p23.[/b] Suppose $\vartriangle ABC$ satisfies $AC = 10\sqrt2$, $BC = 15$, $\angle C = 45^o$. Let $D, E, F$ be the feet of the altitudes in $\vartriangle ABC$, and let $U, V , W$ be the points where the incircle of $\vartriangle DEF$ is tangent to the sides of $\vartriangle DEF$. Find the area of $\vartriangle UVW$. [b]p24.[/b] A polynomial $P(x)$ is called spicy if all of its coefficients are nonnegative integers less than $9$. How many spicy polynomials satisfy $P(3) = 2019$? [i]The next set will consist of three estimation problems.[/i] [u]Set 9[/u] Points will be awarded based on the formulae below. Answers are nonnegative integers that may exceed $1,000,000$. [b]p25.[/b] Suppose a circle of radius $20192019$ has area $A$. Let s be the side length of a square with area $A$. Compute the greatest integer less than or equal to $s$. If $n$ is the correct answer, an estimate of $e$ gives $\max \{ 0, \left\lfloor 1030 ( min \{ \frac{n}{e},\frac{e}{n}\}^{18}\right\rfloor -1000 \}$ points. [b]p26.[/b] Given a $50 \times 50$ grid of squares, initially all white, define an operation as picking a square and coloring it and the four squares horizontally or vertically adjacent to it blue, if they exist. If a square is already colored blue, it will remain blue if colored again. What is the minimum number of operations necessary to color the entire grid blue? If $n$ is the correct answer, an estimate of $e$ gives $\left\lfloor \frac{180}{5|n-e|+6}\right\rfloor$ points. [b]p27.[/b] The sphere packing problem asks what percent of space can be filled with equally sized spheres without overlap. In three dimensions, the answer is $\frac{\pi}{3\sqrt2} \approx 74.05\%$ of space (confirmed as recently as $2017!$), so we say that the packing density of spheres in three dimensions is about $0.74$. In fact, mathematicians have found optimal packing densities for certain other dimensions as well, one being eight-dimensional space. Let d be the packing density of eight-dimensional hyperspheres in eightdimensional hyperspace. Compute the greatest integer less than $10^8 \times d$. If $n$ is the correct answer, an estimate of e gives $\max \left\{ \lfloor 30-10^{-5}|n - e|\rfloor, 0 \right\}$ points. PS. You had better use hide for answers. First sets have be posted [url=https://artofproblemsolving.com/community/c4h2777330p24370124]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $p,p^2+2$ are both primes, how many divisors does $p^5+2p^2$ have? [i](Zhuge Liang)[/i]

For positive integer $k,$ we say that it is a [i]Taurus integer[/i] if we can delete one element from the set $M_k=\{1,2,\cdots,k\},$ such that the sum of remaining $k-1$ elements is a positive perfect square. For example, $7$ is a Taurus integer, because if we delete $3$ from $M_7=\{1,2,3,4,5,6,7\},$ the sum of remaining $6$ elements is $25,$ which is a positive perfect square. $(1)$ Determine whether $2021$ is a Taurus integer. $(2)$ For positive integer $n,$ determine the number of Taurus integers in $\{1,2,\cdots,n\}.$

Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that \[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\] for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that \[a_{k+2011}=a_{k+2011^{2011}}.\]

Let $a, b$ and $k$ be positive integers with $k> 1$ such that $lcm (a, b) + gcd (a, b) = k (a + b)$. Prove that $a + b \geq 4k$

Call an ordered pair $(a, b)$ of positive integers [i]fantastic [/i] if and only if $a, b \le 10^4$ and $$gcd(a \cdot n! - 1, a \cdot (n + 1)! + b) > 1$$ for infinitely many positive integers $n$. Find the sum of $a + b$ across all fantastic pairs $(a, b)$.

Determine all composite positive integers $n$ with the following property: If $1 = d_1 < d_2 < \cdots < d_k = n$ are all the positive divisors of $n$, then $$(d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1:2: \cdots :(k-1)$$ (Walther Janous)

Define the sequence $\{a_n\}$ in the following manner: $a_1=1$ $a_2=3$ $a_{n+2}=2a_{n+1}a_{n}+1$ ; for all $n\geq1$ Prove that the largest power of $2$ that divides $a_{4006}-a_{4005}$ is $2^{2003}.$

[u]Set 6[/u] [b]B16.[/b] Let $\ell_r$ denote the line $x + ry + r^2 = 420$. Jeffrey draws the lines $\ell_a$ and $\ell_b$ and calculates their single intersection point. [b]B17.[/b] Let set $L$ consist of lines of the form $3x + 2ay = 60a + 48$ across all real constants a. For every line $\ell$ in $L$, the point on $\ell$ closest to the origin is in set $T$ . The area enclosed by the locus of all the points in $T$ can be expressed in the form nπ for some positive integer $n$. Compute $n$. [b]B18.[/b] What is remainder when the $2020$-digit number $202020 ... 20$ is divided by $275$? [u]Set 7[/u] [b]B19.[/b] Consider right triangle $\vartriangle ABC$ where $\angle ABC = 90^o$, $\angle ACB = 30^o$, and $AC = 10$. Suppose a beam of light is shot out from point $A$. It bounces off side $BC$ and then bounces off side $AC$, and then hits point $B$ and stops moving. If the beam of light travelled a distance of $d$, then compute $d^2$. [b]B20.[/b] Let $S$ be the set of all three digit numbers whose digits sum to $12$. What is the sum of all the elements in $S$? [b]B21.[/b] Consider all ordered pairs $(m, n)$ where $m$ is a positive integer and $n$ is an integer that satisfy $$m! = 3n^2 + 6n + 15,$$ where $m! = m \times (m - 1) \times ... \times 1$. Determine the product of all possible values of $n$. [u]Set 8[/u] [b]B22.[/b] Compute the number of ordered pairs of integers $(m, n)$ satisfying $1000 > m > n > 0$ and $6 \cdot lcm(m - n, m + n) = 5 \cdot lcm(m, n)$. [b]B23.[/b] Andrew is flipping a coin ten times. After every flip, he records the result (heads or tails). He notices that after every flip, the number of heads he had flipped was always at least the number of tails he had flipped. In how many ways could Andrew have flipped the coin? [b]B24.[/b] Consider a triangle $ABC$ with $AB = 7$, $BC = 8$, and $CA = 9$. Let $D$ lie on $\overline{AB}$ and $E$ lie on $\overline{AC}$ such that $BCED$ is a cyclic quadrilateral and $D, O, E$ are collinear, where $O$ is the circumcenter of $ABC$. The area of $\vartriangle ADE$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. What is $m + n + p$? [u]Set 9[/u] [i]This set consists of three estimation problems, with scoring schemes described.[/i] [b]B25.[/b] Submit one of the following ten numbers: $$3 \,\,\,\, 6\,\,\,\, 9\,\,\,\, 12\,\,\,\, 15\,\,\,\, 18\,\,\,\, 21\,\,\,\, 24\,\,\,\, 27\,\,\,\, 30.$$ The number of points you will receive for this question is equal to the number you selected divided by the total number of teams that selected that number, then rounded up to the nearest integer. For example, if you and four other teams select the number $27$, you would receive $\left\lceil \frac{27}{5}\right\rceil = 6$ points. [b]B26.[/b] Submit any integer from $1$ to $1,000,000$, inclusive. The standard deviation $\sigma$ of all responses $x_i$ to this question is computed by first taking the arithmetic mean $\mu$ of all responses, then taking the square root of average of $(x_i -\mu)^2$ over all $i$. More, precisely, if there are $N$ responses, then $$\sigma =\sqrt{\frac{1}{N} \sum^N_{i=1} (x_i -\mu)^2}.$$ For this problem, your goal is to estimate the standard deviation of all responses. An estimate of $e$ gives $\max \{ \left\lfloor 130 ( min \{ \frac{\sigma }{e},\frac{e}{\sigma }\}^{3}\right\rfloor -100,0 \}$ points. [b]B27.[/b] For a positive integer $n$, let $f(n)$ denote the number of distinct nonzero exponents in the prime factorization of $n$. For example, $f(36) = f(2^2 \times 3^2) = 1$ and $f(72) = f(2^3 \times 3^2) = 2$. Estimate $N = f(2) + f(3) +.. + f(10000)$. An estimate of $e$ gives $\max \{30 - \lfloor 7 log_{10}(|N - e|)\rfloor , 0\}$ points. PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777391p24371239]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sum $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Two robots are programmed to communicate numbers using different bases. The first robot states: "I communicate in base 10, which interestingly is a perfect square. You communicate in base 16, which is not a perfect square." The second robot states: "I find it more interesting that the sum of our bases is the factorial of an integer." The second robot is referring to the factorial of which integer?

[b]p1.[/b] Let $f(x) = \frac{3x^3+7x^2-12x+2}{x^2+2x-3}$ . Find all integers $n$ such that $f(n)$ is an integer. [b]p2.[/b] How many ways are there to arrange $10$ trees in a line where every tree is either a yew or an oak and no two oak trees are adjacent? [b]p3.[/b] $20$ students sit in a circle in a math class. The teacher randomly selects three students to give a presentation. What is the probability that none of these three students sit next to each other? [b]p4.[/b] Let $f_0(x) = x + |x - 10| - |x + 10|$, and for $n \ge 1$, let $f_n(x) = |f_{n-1}(x)| - 1$. For how many values of $x$ is $f_{10}(x) = 0$? [b]p5.[/b] $2$ red balls, $2$ blue balls, and $6$ yellow balls are in a jar. Zion picks $4$ balls from the jar at random. What is the probability that Zion picks at least $1$ red ball and$ 1$ blue ball? [b]p6.[/b] Let $\vartriangle ABC$ be a right-angled triangle with $\angle ABC = 90^o$ and $AB = 4$. Let $D$ on $AB$ such that $AD = 3DB$ and $\sin \angle ACD = \frac35$ . What is the length of $BC$? [b]p7.[/b] Find the value of of $$\dfrac{1}{1 +\dfrac{1}{2+ \dfrac{1}{1+ \dfrac{1}{2+ \dfrac{1}{1+ ...}}}}}$$ [b]p8.[/b] Consider all possible quadrilaterals $ABCD$ that have the following properties; $ABCD$ has integer side lengths with $AB\parallel CD$, the distance between $\overline{AB}$ and $\overline{CD}$ is $20$, and $AB = 18$. What is the maximum area among all these quadrilaterals, minus the minimum area? [b]p9.[/b] How many perfect cubes exist in the set $\{1^{2018},2^{2017}, 3^{2016},.., 2017^2, 2018^1\}$? [b]p10.[/b] Let $n$ be the number of ways you can fill a $2018\times 2018$ array with the digits $1$ through $9$ such that for every $11\times 3$ rectangle (not necessarily for every $3 \times 11$ rectangle), the sum of the $33$ integers in the rectangle is divisible by $9$. Compute $\log_3 n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine whether there is a natural number $n$ for which $8^n + 47$ is prime.

Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares.

A Pythagorean triple is a solution of the equation $x^2 + y^2 = z^2$ in positive integers such that $x < y$. Given any non-negative integer $n$ , show that some positive integer appears in precisely $n$ distinct Pythagorean triples.

Find all pairs of positive integers $(a, b)$, such that $S(a^{b+1})=a^b$, where $S(m)$ denotes the digit sum of $m$.

[u]Round 1[/u] [b]p1.[/b] Alice has a pizza with eight slices. On each slice, she either adds only salt, only pepper, or leaves it plain. Determine how many ways there are for Alice to season her entire pizza. [b]p2.[/b] Call a number almost prime if it has exactly three divisors. Find the number of almost prime numbers less than $100$. [b]p3.[/b] Determine the maximum number of points of intersection between a circle and a regular pentagon. [u]Round 2[/u] [b]p4.[/b] Let $d(n)$ denote the number of positive integer divisors of $n$. Find $d(d(20^{18}))$. [b]p5.[/b] $20$ chubbles are equal to $19$ flubbles. $20$ flubbles are equal to $18$ bubbles. How many bubbles are $1000$ chubbles worth? [b]p6.[/b] Square $ABCD$ and equilateral triangle $EFG$ have equal area. Compute $\frac{AB}{EF}$ . [u]Round 3[/u] [b]p7.[/b] Find the minimumvalue of $y$ such that $y = x^2 -6x -9$ where x is a real number. [b]p8.[/b] I have $2$ pairs of red socks, $5$ pairs of white socks, and $7$ pairs of blue socks. If I randomly pull out one sock at a time without replacement, how many socks do I need to draw to guarantee that I have drawn $3$ pairs of socks of the same color? [b]p9. [/b]There are $23$ paths from my house to the school, $29$ paths from the school to the library, and $3$ paths fromthe library to town center. Additionally, there are $6$ paths directly from my house to the library. If I have to pass through the library to get to town center, howmany ways are there to travel from my house all the way to the town center? [u]Round 4[/u] [b]p10.[/b] A circle of radius $25$ and a circle of radius $4$ are externally tangent. A line is tangent to the circle of radius $25$ at $A$ and the circle of radius $4$ at $B$, where $A \ne B$. Compute the length of $AB$. [b]p11.[/b] A gambler spins two wheels, one numbered $1$ to $4$ and another numbered $1$ to $5$, and the amount of money he wins is the sum of the two numbers he spins in dollars. Determine the expected amount of money he will win. [b]p12.[/b] Find the remainder when $20^{19}$ is divided by $18$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166012p28809547]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166099p28810427]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the difference between the greatest and least values of $lcm (a,b,c)$, where $a$, $b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive.