Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?

Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing. [i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]

Find all polynomials $P$ with real coefficients satisfying that there exist infinitely many pairs $(m, n)$ of coprime positives integer such that $P(\frac{m}{n})=\frac{1}{n}$. [i] Proposed by usjl[/i]

Let $S$ be the subset of rational numbers that can be written in the form $a/b$, where $a$ is any integer and $b$ is an odd integer. Does the sum of two of its elements belong to the $S$ ? And the product? Are there elements in $S$ whose inverse belongs to $S$ ?

Given a fixed rational number $q$. Let's call a number $x$ [i]charismatic [/i] if we can find a natural number $n$ and integers $a_1, a_2,.., a_n$ such that $$x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} \cdot ... \cdot(q + n)^{a_n} .$$ i) Prove that one can find a $q$ such that all positive rational numbers are charismatic. ii) Is it true that for all $q$, if the number $x$ is charismatic, then $x + 1$ is also charismatic?

Let $a_1,a_2, \cdots$ be a sequence of integers that satisfies: $a_1=1$ and $a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1 $. Prove that for all positive $k$, there is $m \geq 1$ such that $k \mid a_m$.

[b]p1.[/b] How many nonzero digits are in the number $(5^{94} + 5^{92})(2^{94} + 2^{92})$? [b]p2.[/b] Suppose $A$ is a set of $2013$ distinct positive integers such that the arithmetic mean of any subset of $A$ is also an integer. Find an example of $A$. [b]p3.[/b] How many minutes until the smaller angle formed by the minute and hour hands on the face of a clock is congruent to the smaller angle between the hands at $5:15$ pm? Round your answer to the nearest minute. [b]p4.[/b] Suppose $a$ and $b$ are positive real numbers, $a + b = 1$, and $$1 +\frac{a^2 + 3b^2}{2ab}=\sqrt{4 +\frac{a}{b}+\frac{3b}{a}}.$$ Find $a$. [b]p5.[/b] Suppose $f(x) = \frac{e^x- 12e^{-x}}{ 2}$ . Find all $x$ such that $f(x) = 2$. [b]p6.[/b] Let $P_1$, $P_2$,$...$,$P_n$ be points equally spaced on a unit circle. For how many integer $n \in \{2, 3, ... , 2013\}$ is the product of all pairwise distances: $\prod_{1\le i<j\le n} P_iP_j$ a rational number? Note that $\prod$ means the product. For example, $\prod_{1\le i\le 3} i = 1\cdot 2 \cdot 3 = 6$. [b]p7.[/b] Determine the value $a$ such that the following sum converges if and only if $r \in (-\infty, a)$ : $$\sum^{\infty}_{n=1}(\sqrt{n^4 + n^r} - n^2).$$ Note that $\sum^{\infty}_{n=1}\frac{1}{n^s}$ converges if and only if $s > 1$. [b]p8.[/b] Find two pairs of positive integers $(a, b)$ with $a > b$ such that $a^2 + b^2 = 40501$. [b]p9.[/b] Consider a simplified memory-knowledge model. Suppose your total knowledge level the night before you went to a college was $100$ units. Each day, when you woke up in the morning you forgot $1\%$ of what you had learned. Then, by going to lectures, working on the homework, preparing for presentations, you had learned more and so your knowledge level went up by $10$ units at the end of the day. According to this model, how long do you need to stay in college until you reach the knowledge level of exactly $1000$? [b]p10.[/b] Suppose $P(x) = 2x^8 + x^6 - x^4 +1$, and that $P$ has roots $a_1$, $a_2$, $...$ , $a_8$ (a complex number $z$ is a root of the polynomial $P(x)$ if $P(z) = 0$). Find the value of $$(a^2_1-2)(a^2_2-2)(a^2_3-2)...(a^2_8-2).$$ [b]p11.[/b] Find all values of $x$ satisfying $(x^2 + 2x-5)^2 = -2x^2 - 3x + 15$. [b]p12.[/b] Suppose $x, y$ and $z$ are positive real numbers such that $$x^2 + y^2 + xy = 9,$$ $$y^2 + z^2 + yz = 16,$$ $$x^2 + z^2 + xz = 25.$$ Find $xy + yz + xz$ (the answer is unique). [b]p13.[/b] Suppose that $P(x)$ is a monic polynomial (i.e, the leading coefficient is $1$) with $20$ roots, each distinct and of the form $\frac{1}{3^k}$ for $k = 0,1,2,..., 19$. Find the coefficient of $x^{18}$ in $P(x)$. [b]p14.[/b] Find the sum of the reciprocals of all perfect squares whose prime factorization contains only powers of $3$, $5$, $7$ (i.e. $\frac{1}{1} + \frac{1}{9} + \frac{1}{25} + \frac{1}{419} + \frac{1}{811} + \frac{1}{215} + \frac{1}{441} + \frac{1}{625} + ...$). [b]p15.[/b] Find the number of integer quadruples $(a, b, c, d)$ which also satisfy the following system of equations: $$1+b + c^2 + d^3 =0,$$ $$a + b^2 + c^3 + d^4 =0,$$ $$a^2 + b^3 + c^4 + d^5 =0,$$ $$a^3+b^4+c^5+d^6 =0.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] Find all pairs $(a,b)$ of positive integers with $a > b$ and $a^2 -b^2 =111$. [b]p2.[/b] Alice drives at a constant rate of $2017$ miles per hour. Find all positive values of $x$ such that she can drive a distance of $x^2$ miles in a time of $x$ minutes. [b]p3.[/b] $ABC$ is a right triangle with right angle at $B$ and altitude $BH$ to hypotenuse $AC$. If $AB = 20$ and $BH = 12$, find the area of triangle $\vartriangle ABC$. [u]Round 2[/u] [b]p4.[/b] Regular polygons $P_1$ and $P_2$ have $n_1$ and $n_2$ sides and interior angles $x_1$ and $x_2$, respectively. If $\frac{n_1}{n_2}= \frac75$ and $\frac{x_1}{x_2}=\frac{15}{14}$ , find the ratio of the sum of the interior angles of $P_1$ to the sum of the interior angles of $P_2$. [b]p5.[/b] Joey starts out with a polynomial $f (x) = x^2 +x +1$. Every turn, he either adds or subtracts $1$ from $f$ . What is the probability that after $2017$ turns, $f$ has a real root? [b]p6.[/b] Find the difference between the greatest and least positive integer values $x$ such that $\sqrt[20]{\lfloor \sqrt[17]{x}\rfloor}=1$. [u]Round 3[/u] [b]p7.[/b] Let $ABCD$ be a square and suppose $P$ and $Q$ are points on sides $AB$ and $CD$ respectively such that $\frac{AP}{PB} = \frac{20}{17}$ and $\frac{CQ}{QD}=\frac{17}{20}$ . Suppose that $PQ = 1$. Find the area of square $ABCD$. [b]p8.[/b] If $$\frac{\sum_{n \ge 0} r^n}{\sum_{n \ge 0} r^{2n}}=\frac{1+r +r^2 +r^3 +...}{1+r^2 +r^4 +r^6 +...}=\frac{20}{17},$$ find $r$ . [b]p9.[/b] Let $\overline{abc}$ denote the $3$ digit number with digits $a,b$ and $c$. If $\overline{abc}_{10}$ is divisible by $9$, what is the probability that $\overline{abc}_{40}$ is divisible by $9$? [u]Round 4[/u] [b]p10.[/b] Find the number of factors of $20^{17}$ that are perfect cubes but not perfect squares. [b]p11.[/b] Find the sum of all positive integers $x \le 100$ such that $x^2$ leaves the same remainder as $x$ does upon division by $100$. [b]p12.[/b] Find all $b$ for which the base-$b$ representation of $217$ contains only ones and zeros. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3158514p28715373]here[/url].and 9-12 [url=https://artofproblemsolving.com/community/c3h3162362p28764144]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sequences $ a_0, a_1, \ldots$ and $ b_0, b_1, \ldots$ are defined for $ n \equal{} 0, 1, 2, \ldots$ by the equalities \[ a_0 \equal{} \frac {\sqrt {2}}{2}, \quad a_{n \plus{} 1} \equal{} \frac {\sqrt {2}}{2} \cdot \sqrt {1 \minus{} \sqrt {1 \minus{} a^2_n}} \] and \[ b_0 \equal{} 1, \quad b_{n \plus{} 1} \equal{} \frac {\sqrt {1 \plus{} b^2_n} \minus{} 1}{b_n} \] Prove the inequalities for every $ n \equal{} 0, 1, 2, \ldots$ \[ 2^{n \plus{} 2} a_n < \pi < 2^{n \plus{} 2} b_n. \]

Let $a, b$ and $c$ denote positive real numbers. Prove that $\frac{a}{c}+\frac{c}{b}\ge \frac{4a}{a + b}$ . When does equality hold? (Walther Janous)

Let $f : [0, 1] \to [0, 1]$ satisfy $f(0) = 0, f(1) = 1$ and \[f(x + y) - f(x) = f(x) - f(x - y)\] for all $x, y \geq 0$ with $x - y, x + y \in [0, 1].$ Prove that $f(x) = x$ for all $x \in [0, 1].$

Prove that there doesn't exist a function $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $(m+f(n))^2 \geq 3f(m)^2+n^2$ for all $m, n \in \mathbb{N}$.

A steak temperature $5^\circ$ is put into an oven. After $15$ minutes, it has temperature $45^\circ$. After another $15$ minutes it has temperature $77^\circ$. The oven is at a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature.

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\] where $n\in \mathbb{N}.$

Prove that there exists a real $c<\frac{3}{4}$, such that for each sequence $x_1, x_2, \ldots$ satisfying $0 \leq x_i \leq 1$ for all $i$, there exist infinitely many $(m, n)$ with $m>n$, such that $$|x_m-x_n|\leq \frac{c} {m}.$$

For every positive integer $k$ let $a_{k,1},a_{k,2},\ldots$ be a sequence of positive integers. For every positive integer $k$ let sequence $\{a_{k+1,i}\}$ be the difference sequence of $\{a_{k,i}\}$, i.e. for all positive integers $k$ and $i$ the following holds: $a_{k,i+1}-a_{k,i}=a_{k+1,i}$. Is it possible that every positive integer appears exactly once among numbers $a_{k,i}$? [i]Proposed by Dávid Matolcsi, Berkeley[/i]

Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$. Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set. [i]Dan Schwarz, Romania[/i]

The Den has two deals on chicken wings. The first deal is $4$ chicken wings for $3$ dollars, and the second deal is $11$ chicken wings for $ 8$ dollars. If Jeremy has $18$ dollars, what is the largest number of chicken wings he can buy?

Prove that for the arbitrary positive $a_1, a_2, ... , a_n$ the following inequality is held $$\frac{a_1}{a_2+a_3}+\frac{a_2}{a_3+a_4}+....+\frac{a_{n-1}}{a_n+a_1}+\frac{a_n}{a_1+a_2}>\frac{n}{4}$$

Define an such that $a_1 =\sqrt3$ and for all integers $i$, $a_{i+1} = a^2_i - 2$. What is $a_{2016}$?

Find the largest natural number $n$ such that for all real numbers $a, b, c, d$ the following holds: $$(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)$$

Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$, $\rho_2$, $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$. Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$, $C(\rho_2, 0)$, $D(\rho_3, 0)$. If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$, find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained. [i]Brazitikos Silouanos, Greece[/i]

Find all functions $f : \mathbb{Q}[x] \to \mathbb{Q}[x]$ such that two following conditions holds : $$\forall P , Q \in \mathbb{Q}[x] : f(P+Q)=f(P)+f(Q)$$ $$\forall P \in \mathbb{Q}[x] : gcd(P , f(P))=1 \iff$$ $P$ is square-free. Which a square-free polynomial with rational coefficients is a polynomial such that there doesn't exist square of a non-constant polynomial with rational coefficients that divides it. [i]Proposed by Sina Azizedin[/i]

Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).