Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality: $\frac{1}{\sqrt{(a+2b)(b+2a)}}+\frac{1}{\sqrt{(b+2c)(c+2b)}}+\frac{1}{\sqrt{(c+2a)(a+2c)}} \geq 3$

Find all positive integers $a,b$ such that $b^{619}$ divides $a^{1000}+1$ and $a^{619}$ divides $b^{1000}+1$.

Let $a,b, c$ denote the real numbers such that $1 \le a, b, c\le 2$. Consider $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$. Determine the largest possible value of $T$.

Let $C$ and $D$ be points on the circle with center $O$ and diameter $[AB]$ where $C$ and $D$ are on different semicircles with diameter $[AB]$. Let $H$ be the foot perpendicular from $B$ to $[CD]$. If $|AO|=13$, $|AC|=24$, and $|HD|=12$, what is $\widehat{DCB}$ in degrees? $ \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 45 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 75 \qquad\textbf{(E)}\ 80 $

Rectangle $ABCD$ has side lengths $AB = 6\sqrt3$ and $BC = 8\sqrt3$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Find all positive real numbers $r<1$ such that there exists a set $\mathcal{S}$ with the given properties: i) For any real number $t$, exactly one of $t, t+r$ and $t+1$ belongs to $\mathcal{S}$; ii) For any real number $t$, exactly one of $t, t-r$ and $t-1$ belongs to $\mathcal{S}$.

A uniform circular disc is suspended in a horizontal position on a string attached to its center $ O $. At three different points $ A $, $ B $, $ C $ on the edge of the disc, weights $ p_1 $, $ p_2 $, $ p_3 $ are placed, after which the disc remains in equilibrium. Calculate angles $ AOB $, $ BOC $, and $ COA $.

What is the value of \[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\] $\textbf{(A) } 28 \qquad \textbf{(B) } 68 \qquad \textbf{(C) } 70 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 84$

Let $c_0,\,c_1,\,c_2,\,\ldots$ be a sequence defined so that \[ \frac{1-3x-\sqrt{1-14x+9x^2}}{4}=\sum_{k=0}^\infty c_kx^k \] for sufficiently small $x$. For a positive integer $n$, let $A$ be the $n$-by-$n$ matrix with $i,j$-entry $c_{i+j-1}$ for $i$ and $j$ in $\{1,\,\ldots,\,n\}$. Find the determinant of $A$.

Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations \begin{align*} 13x + by + cz &= 0 \\ ax + 23y + cz &= 0 \\ ax + by + 42z &= 0. \end{align*}Suppose that $ a \ne 13$ and $ x \ne 0$. What is the value of \[ \frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c - 42} \, ?\]

Find all positive integers $x,y$ such that $202x + 4x^2 = y^2$.

We color each number in the set $S = \{1, 2, ..., 61\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$?

Consider the set of all polynomials of degree less than or equal to $4$ with rational coefficients. a) Prove that it has a vector space structure over the field of numbers rational. b) Prove that the polynomials $1, x - 2, (x -2)^2, (x - 2)^3$ and $(x -2)^4$ form a base of this space. c) Express the polynomial $7 + 2x - 45x^2 + 3x^4$ in the previous base.

For any natural number, let $S(n)$ denote sum of digits of $n$. Find the number of $3$ digit numbers for which $S(S(n)) = 2$.

Evaluate \[\lim_{n\to\infty} n\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx\ \ ( n=1,2,\cdots)\]

[u]Round 1[/u] [b]p1.[/b] A positive integer is said to be transcendent if it leaves a remainder of $1$ when divided by $2$. Find the $1010$th smallest positive integer that is transcendent. [b]p2.[/b] The two diagonals of a square are drawn, forming four triangles. Determine, in degrees, the sum of the interior angle measures in all four triangles. [b]p3.[/b] Janabel multiplied $2$ two-digit numbers together and the result was a four digit number. If the thousands digit was nine and hundreds digit was seven, what was the tens digit? [u]Round 2[/u] [b]p4.[/b] Two friends, Arthur and Brandon, are comparing their ages. Arthur notes that $10$ years ago, his age was a third of Brandon’s current age. Brandon points out that in $12$ years, his age will be double of Arthur’s current age. How old is Arthur now? [b]p5.[/b] A farmer makes the observation that gathering his chickens into groups of $2$ leaves $1$ chicken left over, groups of $3$ leaves $2$ chickens left over, and groups of $5$ leaves $4$ chickens left over. Find the smallest possible number of chickens that the farmer could have. [b]p6.[/b] Charles has a bookshelf with $3$ layers and $10$ indistinguishable books to arrange. If each layer must hold less books than the layer below it and a layer cannot be empty, how many ways are there for Charles to arrange his $10$ books? [u]Round 3[/u] [b]p7.[/b] Determine the number of factors of $2^{2019}$. [b]p8.[/b] The points $A$, $B$, $C$, and $D$ lie along a line in that order. It is given that $\overline{AB} : \overline{CD} = 1 : 7$ and $\overline{AC} : \overline{BD} = 2 : 5$. If $BC = 3$, find $AD$. [b]p9.[/b] A positive integer $n$ is equal to one-third the sum of the first $n$ positive integers. Find $n$. [u]Round 4[/u] [b]p10.[/b] Let the numbers $a,b,c$, and $d$ be in arithmetic progression. If $a +2b +3c +4d = 5$ and $a =\frac12$ , find $a +b +c +d$. [b]p11.[/b] Ten people playing brawl stars are split into five duos of $2$. Determine the probability that Jeff and Ephramare paired up. [b]p12.[/b] Define a sequence recursively by $F_0 = 0$, $F_1 = 1$, and for all $n\ge 2$, $$F_n = \left \lceil \frac{F_{n-1}+F_{n-2}}{2} \right \rceil +1,$$ where $\lceil r \rceil$ denotes the least integer greater than or equal to $r$ . Find $F_{2019}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166019p28809679]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166115p28810631]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that in a non-equilateral triangle, the following statements are equivalent: $(a)$ The angles of the triangle are in arithmetic progression. $(b)$ The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.

Given $\triangle ABC$ with $\angle A = 15^{\circ}$, let $M$ be midpoint of $BC$ and let $E$ and $F$ be points on ray $BA$ and $CA$ respectively such that $BE = BM = CF$. Let $R_1$ be the radius of $(MEF)$ and $R_{2}$ be radius of $(AEF)$. If $\frac{R_1^2}{R_2^2}=a-\sqrt{b+\sqrt{c}}$ where $a,b,c$ are integers. Find $a^{b^{c}}$

Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product$$n = f_1\cdot f_2\cdots f_k,$$where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$? $\textbf{(A) } 112 \qquad\textbf{(B) } 128 \qquad\textbf{(C) } 144 \qquad\textbf{(D) } 172 \qquad\textbf{(E) } 184$

Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three [i]aligned[/i] numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?

[b](i)[/b] Solve the equation: \[ \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.\] [b](ii)[/b] Supposing the solutions are in the form of arcs $AB$ with one end at the point $A$, the beginning of the arcs of the trigonometric circle, and $P$ a regular polygon inscribed in the circle with one vertex in $A$, find: 1) The subsets of arcs having the other end in $B$ in one of the vertices of the regular dodecagon. 2) Prove that no solution can have the end $B$ in one of the vertices of polygon $P$ whose number of sides is prime or having factors other than 2 or 3.

Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers: $a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.

Real numbers $x,y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x,y$, and $z$ are within $1$ unit of each other is greater than $\tfrac{1}{2}$. What is the smallest possible value of $n$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $

Let $a_1,a_2,\ldots,a_n$ be a finite sequence of non negative integers, its subsequences are the sequences of the form $a_i,a_{i+1},\ldots,a_j$ with $1\le i\le j \le n$. Two subsequences are said to be equal if they have the same length and have the same terms, that is, two subsequences $a_i,a_{i+1},\ldots,a_j$ and $a_u,a_{u+1},\ldots a_v$ are equal iff $j-i=u-v$ and $a_{i+k}=a_{u+k}$ forall integers $k$ such that $0\le k\le j-1$. Finally, we say that a subsequence $a_i,a_{i+1},\ldots,a_j$ is palindromic if $a_{i+k}=a_{j-k}$ forall integers $k$ such that $0\le k \le j-i$ What is the greatest number of different palindromic subsequences that can a palindromic sequence of length $n$ contain?

Ler $a$ be the number $123456789$. Compare the numbers $$2014^{9^{9^a}}, 2015^{a^{a^9}}$$