Let be two nonzero real numbers $ a,b, $ and a function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ satisfying the functional equation $$ f(x+a+b)+f(x)=f(x+a)+f(x+b) . $$ [b]1)[/b] Prove that $ f $ is periodic if $ a/b $ is rational. [b]2)[/b] If $ a/b $ is not rational, could $ f $ be nonperiodic?

Let $x_1,x_2,\dots,x_n$ be a sequence of integers, such that $-1\leq x_i\leq 2$, for $i=1,2,\dots,n$, $x_1+x_2+\dots+x_n = 7$ and $x_1^8+x_2^8+\dots x_n^8 =2009$. Let $m$ and $M$ be the minimal and maximal possible value of $x_1^9+x_2^9+\dots x_n^9$, respectively. Find the $\frac{M}{m}$. Round your answer to nearest integer, if necessary.

$f(x) = x^3+3x^2 - 1$. Find the number of real solutions of $f(f(x)) = 0$.

Consider positive integers $m$ and $n$ for which $m> n$ and the number $22 220 038^m-22 220 038^n$ has are eight zeros at the end. Show that $n> 7$.

Is it possible to draw circles on the plane so that every line intersects at least one of them but no more than $100$ of them?

[u]Round 1[/u] [b]p1.[/b] Let $n$ be a two-digit positive integer. What is the maximum possible sum of the prime factors of $n^2 - 25$ ? [b]p2.[/b] Angela has ten numbers $a_1, a_2, a_3, ... , a_{10}$. She wants them to be a permutation of the numbers $\{1, 2, 3, ... , 10\}$ such that for each $1 \le i \le 5$, $a_i \le 2i$, and for each $6 \le i \le 10$, $a_i \le - 10$. How many ways can Angela choose $a_1$ through $a_{10}$? [b]p3.[/b] Find the number of three-by-three grids such that $\bullet$ the sum of the entries in each row, column, and diagonal passing through the center square is the same, and $\bullet$ the entries in the nine squares are the integers between $1$ and $9$ inclusive, each integer appearing in exactly one square. [u]Round 2 [/u] [b]p4.[/b] Suppose that $P(x)$ is a quadratic polynomial such that the sum and product of its two roots are equal to each other. There is a real number $a$ that $P(1)$ can never be equal to. Find $a$. [b]p5.[/b] Find the number of ordered pairs $(x, y)$ of positive integers such that $\frac{1}{x} +\frac{1}{y} =\frac{1}{k}$ and k is a factor of $60$. [b]p6.[/b] Let $ABC$ be a triangle with $AB = 5$, $AC = 4$, and $BC = 3$. With $B = B_0$ and $C = C_0$, define the infinite sequences of points $\{B_i\}$ and $\{C_i\}$ as follows: for all $i \ge 1$, let $B_i$ be the foot of the perpendicular from $C_{i-1}$ to $AB$, and let $C_i$ be the foot of the perpendicular from $B_i$ to $AC$. Find $C_0C_1(AC_0 + AC_1 + AC_2 + AC_3 + ...)$. [u]Round 3 [/u] [b]p7.[/b] If $\ell_1, \ell_2, ... ,\ell_{10}$ are distinct lines in the plane and $p_1, ... , p_{100}$ are distinct points in the plane, then what is the maximum possible number of ordered pairs $(\ell_i, p_j )$ such that $p_j$ lies on $\ell_i$? [b]p8.[/b] Before Andres goes to school each day, he has to put on a shirt, a jacket, pants, socks, and shoes. He can put these clothes on in any order obeying the following restrictions: socks come before shoes, and the shirt comes before the jacket. How many distinct orders are there for Andres to put his clothes on? [b]p9. [/b]There are ten towns, numbered $1$ through $10$, and each pair of towns is connected by a road. Define a backwards move to be taking a road from some town $a$ to another town $b$ such that $a > b$, and define a forwards move to be taking a road from some town $a$ to another town $b$ such that $a < b$. How many distinct paths can Ali take from town $1$ to town $10$ under the conditions that $\bullet$ she takes exactly one backwards move and the rest of her moves are forward moves, and $\bullet$ the only time she visits town $10$ is at the very end? One possible path is $1 \to 3 \to 8 \to 6 \to 7 \to 8 \to 10$. [u]Round 4[/u] [b]p10.[/b] How many prime numbers $p$ less than $100$ have the properties that $p^5 - 1$ is divisible by $6$ and $p^6 - 1$ is divisible by $5$? [b]p11.[/b] Call a four-digit integer $\overline{d_1d_2d_3d_4}$ [i]primed [/i] if 1) $d_1$, $d_2$, $d_3$, and $d_4$ are all prime numbers, and 2) the two-digit numbers $\overline{d_1d_2}$ and $\overline{d_3d_4}$ are both prime numbers. Find the sum of all primed integers. [b]p12.[/b] Suppose that $ABC$ is an isosceles triangle with $AB = AC$, and suppose that $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, with $\overline{DE} \parallel \overline{BC}$. Let $r$ be the length of the inradius of triangle $ADE$. Suppose that it is possible to construct two circles of radius $r$, each tangent to one another and internally tangent to three sides of the trapezoid $BDEC$. If $\frac{BC}{r} = a + \sqrt{b}$ forpositive integers $a$ and $b$ with $b$ squarefree, then find $a + b$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2800986p24675177]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If we write $ |x^2 \minus{} 4| < N$ for all $ x$ such that $ |x \minus{} 2| < 0.01$, the smallest value we can use for $ N$ is: $ \textbf{(A)}\ .0301 \qquad \textbf{(B)}\ .0349 \qquad \textbf{(C)}\ .0399 \qquad \textbf{(D)}\ .0401 \qquad \textbf{(E)}\ .0499 \qquad$

Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.

Find all integers $ n \ge 2 $ for which it is possible to divide any triangle $ T $ in triangles $ T_1, T_2, \cdots, T_n $ and choose medians $ m_1, m_2, \cdots, m_n $, one in each of these triangles, so that these $ n $ medians have equal length.

Let $a, b, c \in \mathbb{R}$ be such that $$a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1.$$ We say that a function $f$ is a [i]Palić function[/i] if $f: \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous and satisfies $$f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz)$$ for all $x, y, z \in \mathbb{R}.$ Prove that any Palić function is infinitely many times differentiable and find all Palić functions.

Determine all polynomial $P(x)\in \mathbb{R}[x]$ satisfying the following two conditions: (a) $P(2017)=2016$ and (b) $(P(x)+1)^2=P(x^2+1)$ for all real number $x$.

What is the largest positive integer $n$ for which there are no [i]positive[/i] integers $a,b$ such that $8a+11b=n$? [i]2019 CCA Math Bonanza Lightning Round #2.2[/i]

Let $P (x)$ be a polynomial with integer coefficients. Given that for some integer $a$ and some positive integer $n$, where \[\underbrace{P(P(\ldots P}_{\text{n times}}(a)\ldots)) = a,\] is it true that $P (P (a)) = a$?

You’re given the complex number $\omega = e^{2i\pi/13} + e^{10i\pi/13} + e^{16i\pi/13} + e^{24i\pi/13}$, and told it’s a root of a unique monic cubic $x^3 +ax^2 +bx+c$, where $a, b, c$ are integers. Determine the value of $a^2 + b^2 + c^2$.

There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$

Let $ABC$ be a triangle. A point $P$ varies inside $BC$. Let $Q, R$ be the points on $AC, AB$ in that order, such that $PQ\parallel AB, PR\parallel AC$. 1. Prove that, when $P$ varies, the circumcircle of triangle $AQR$ always passes through a fixed point $X$ other than $A$. 2. Extend $AX$ so that it cuts the circumcircle of $ABC$ a second time at point $K$. Prove that $AX=XK$.

Given two natural numbers $a < b$, Xavier and Ze play the following game. First, Xavier writes $a$ consecutive numbers of his choice; then, repeat some of them, also of his choice, until he has $b$ numbers, with the condition that the sum of the $b$ numbers written is an even number. Ze wins the game if he manages to separate the numbers into two groups with the same amount. Otherwise, Xavier wins. For example, for $a = 4$ and $b = 7$, if Xavier wrote the numbers $3,4,5,6,3,3,4$, Ze could win, separating these numbers into groups $3,3 ,4,4$ and $3,5,6$. For what values of $a$ and $b$ can Xavier guarantee victory?

Let $a,p,q\in \mathbb{Z}_{\ge 1}$ be such that $a$ is a perfect square, $a=pq$ and $$2021~|~p^3+q^3+p^2q+pq^2.$$ Prove that $2021$ divides $\sqrt a$.\\ \\ [i](Cosmin Gavrilă)[/i]

In the Cartesian plane, all points with both integer coordinates are painted blue. Blue colon they are said to be [b]mutually visible[/b] if the line segment connecting them has no other blue dots. Prove that There is a set of $ 2019$ blue dots that are mutually visible two by two.

It takes Clea $60$ seconds to walk down an escalator when it is not operating and only $24$ seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it? $ \textbf{(A)}\ 36\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52 $

Let be a sequence of positive real numbers $ \left( a_n\right)_{n\ge 1} $ defined by the recurrence relation $ a_{n+1}=\ln \left(1+a_n\right) . $ Show that: [b]1)[/b] $ \lim_{n\to\infty } a_n=0 $ [b]2)[/b] $ \lim_{n\to\infty } na_n=2 $ [b]3[/b] $ \lim_{n\to\infty } \frac{n(na_n-2)}{\ln n}=2/3 $ [i]Dorel Duca[/i] and [i]Dorian Popa[/i]

Prove that for every $ n = 2, 3, \ldots $ and any real numbers $ t_1, t_2, \ldots, t_n $, $ s_1, s_2, \ldots, s_n $, if $$ \sum_{i=1}^n t_i = 0, \text{ to } \sum_{i=1}^n\sum_{j=1}^n t_it_j |s_i-s_j| \leq 0.$$

Let $p$ and $q$ two positive integers. Determine the greatest value of $n$ for which there exists sets $A_1,\ A_2,\ldots,\ A_n$ and $B_1,\ B_2,\ldots,\ B_n$ such that: [LIST] [*] The sets $A_1,\ A_2,\ldots,\ A_n$ have $p$ elements each one. [/*] [*] The sets $B_1,\ B_2,\ldots,\ B_n$ have $q$ elements each one. [/*] [*] For all $1\leq i,\ j \leq n$, sets $A_i$ and $B_j$ are disjoint if and only if $i=j$. [/LIST]

A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13,19,20,25$ and $31$, although this is not necessarily their order around the pentagon. The area of the pentagon is $\textbf{(A)}\ 459 \qquad \textbf{(B)}\ 600 \qquad \textbf{(C)}\ 680 \qquad \textbf{(D)}\ 720\qquad \textbf{(E)}\ 745$

Given a positive integer $k\geq2$, set $a_1=1$ and, for every integer $n\geq 2$, let $a_n$ be the smallest solution of equation \[x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor\] that exceeds $a_{n-1}$. Prove that all primes are among the terms of the sequence $a_1,a_2,\ldots$