Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]

The numbers from 1 to $n$ are arranged in some way on a circle. What’s the smallest value of $n$, for which no matter how the numbers are arranged there exist ten consecutively increasing numbers $A_1<A_2<A_3…<A_{10}$ such that $A_1 A_2…A_{10}$ is a convex decagon?

In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$

A line is called $good$ if it bisects perimeter and area of a figure at the same time.Prove that: [i]a)[/i] all of the good lines in a triangle concur. [i]b)[/i] all of the good lines in a regular polygon concur too.

Let $ABC$ be a scalene acute triangle whose orthocenter is $H$. $M$ is the midpoint of segment $BC$. $N$ is the point where the segment $AM$ intersects the circle determined by $B, C$, and $H$. Show that lines $HN$ and $AM$ are perpendicular.

What is the measure of the acute angle formed by the hands of the clock at $4:20$ PM? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

The function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(x^2)f^{\prime\prime}(x)=f^\prime (x)f^\prime (x^2)$ for all real $x$. Given that $f(1)=1$ and $f^{\prime\prime\prime}(1)=8$, determine $f^\prime (1)+f^{\prime\prime}(1)$.

Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$. Proposed by [i]Viktor Simjanoski, Macedonia[/i]

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $X$ and $K$ points over $AC$ and $AB$, respectively, such that $KX = CX$. Bisector of $\angle AKX$ intersects line $BC$ at $Z$. Show that $XZ$ passes through the midpoint of $BK$.

Assume there are $ n\ge3$ points in the plane, Prove that there exist three points $ A,B,C$ satisfying $ 1\le\frac{AB}{AC}\le\frac{n\plus{}1}{n\minus{}1}.$

Let's call this position of the hour and minute hands on the analog clock [i]wonderful[/i], during which the hands change places after some time. Count the total number of wonderful clockwise positions.

Three couples sit for a photograph in $2$ rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible?

Let $ d$ be a line and points $ M,N$ on the $ d$. Circles $ \alpha,\beta,\gamma,\delta$ with centers $ A,B,C,D$ are tangent to $ d$, circles $ \alpha,\beta$ are externally tangent at $ M$, and circles $ \gamma,\delta$ are externally tangent at $ N$. Points $ A,C$ are situated in the same half-plane, determined by $ d$. Prove that if exists an circle, which is tangent to the circles $ \alpha,\beta,\gamma,\delta$ and contains them in its interior, then lines $ AC,BD,MN$ are concurrent or parallel.

Let $a, b, c$ be nonzero integers such that $M = \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $N =\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are both integers. Find $M$ and $N$.

A tourist traveling on a bus from the VIAZUL company arrived at the station in the city of Cienfuegos and immediately set out to take a walking tour of the city. After visiting the Terry theater in this city, decided to return to the station but walking only for blocks traveled an odd number of times in his previous walk. Can he do this on his return regardless. of the initial route?

Find the smallest positive integer $a$ such that for some integers $b$, $c$ the polynomial $ax^2 - bx + c$ has two distinct zeros in the interval $(0,1)$.

Let $ABCD$ be an isoceles trapezoid having parallel bases $\overline{AB}$ and $\overline{CD}$ with $AB>CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\overline{CD}$ and moving clockwise as shown in the diagram below. What is the ratio $\frac{AB}{CD}?$[center][asy]unitsize(100); pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); draw(A--B--C--D--cycle, black); draw(A--P, black); draw(B--P, black); draw(C--P, black); draw(D--P, black); label("$A$",A,(-1,0)); label("$B$",B,(1,0)); label("$C$",C,(1,-0)); label("$D$",D,(-1,0)); label("$2$",E,(0,0)); label("$3$",F,(0,0)); label("$4$",G,(0,0)); label("$5$",H,(0,0)); dot(A^^B^^C^^D^^P); [/asy][/center] $\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}$

Prove that for any positive integer $n$, the arithmetic mean of $\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},\ldots ,\sqrt[n]{n}$ lies in $\left[ 1,1+\frac{2\sqrt{2}}{\sqrt{n}} \right]$ .

A common external tangent to circles $\omega_1$ and $\omega_2$ touches them at points $T_1, T_2$ respectively. Let $A$ be an arbitrary point on the extension of $T_1T_2$ beyond $T_1$, and $B$ be a point on the extension of $T_1T_2$ beyond $T_2$ such that $AT_1 = BT_2$. The tangents from $A$ to $\omega_1$ and from $B$ to $\omega_2$ distinct from $T_1T_2$ meet at point $C$. Prove that all nagelians of triangles $ABC$ from $C$ have a common point.

In acute angled triangle $ABC$ we denote by $a,b,c$ the side lengths, by $m_a,m_b,m_c$ the median lengths and by $r_{b}c,r_{ca},r_{ab}$ the radii of the circles tangents to two sides and to circumscribed circle of the triangle, respectively. Prove that $$\frac{m_a^2}{r_{bc}}+\frac{m_b^2}{r_{ab}}+\frac{m_c^2}{r_{ab}} \ge \frac{27\sqrt3}{8}\sqrt[3]{abc}$$

Find the number of ordered pairs of positive integers $(a,b)$ with $a+b$ prime, $1\leq a, b \leq 100$, and $\frac{ab+1}{a+b}$ is an integer. [i]Author: Alex Zhu[/i]

The lengths in inches of the three sides of each of four triangles I, II, III, and IV are as follows: \[ \begin{array}{rlrl} \hbox {I}& 3,\ 4,\ \hbox{and}\ 5 \qquad & \hbox{III}& 7,\ 24,\ \hbox{and}\ 25 \\ \hbox{II}& 4,\ 7\frac{1}{2},\ \hbox{and}\ 8\frac{1}{2} \qquad & \hbox{IV}& 3\frac{1}{2},\ 4\frac{1}{2},\ \hbox{and}\ 5\frac{1}{2}. \end{array} \] Of these four given triangles, the only right triangles are \[ \begin{tabular}{rlrlrl} (A) & I and II \qquad & (B) & I and III \qquad & (C) & I and IV \\ (D) & I, II, and III \qquad & (E) & I, II, and IV & \end{tabular} \]

[u]Round 5 [/u] [b]p13.[/b] A palindrome is a number that reads the same forward as backwards; for example, $121$ and $36463$ are palindromes. Suppose that $N$ is the maximal possible difference between two consecutive three-digit palindromes. Find the number of pairs of consecutive palindromes $(A, B)$ satisfying $A < B$ and $B - A = N$. [b]p14.[/b] Suppose that $x, y$, and $z$ are complex numbers satisfying $x +\frac{1}{yz} = 5$, $y +\frac{1}{zx} = 8$, and $z +\frac{1}{xy} = 6$. Find the sum of all possible values of $xyz$. [b]p15.[/b] Let $\Omega$ be a circle with radius $25\sqrt2$ centered at $O$, and let $C$ and $J$ be points on $\Omega$ such that the circle with diameter $\overline{CJ}$ passes through $O$. Let $Q$ be a point on the circle with diameter $\overline{CJ}$ satisfying $OQ = 5\sqrt2$. If the area of the region bounded by $\overline{CQ}$, $\overline{QJ}$, and minor arc $JC$ on $\Omega$ can be expressed as $\frac{a\pi-b}{c}$ for integers $a, b$, and $c$ with $gcd \,\,(a, c) = 1$, then find $a + b + c$. [u]Round 6[/u] [b]p16.[/b] Veronica writes $N$ integers between $2$ and $2020$ (inclusive) on a blackboard, and she notices that no number on the board is an integer power of another number on the board. What is the largest possible value of $N$? [b]p17.[/b] Let $ABC$ be a triangle with $AB = 12$, $AC = 16$, and $BC = 20$. Let $D$ be a point on $AC$, and suppose that $I$ and $J$ are the incenters of triangles $ABD$ and $CBD$, respectively. Suppose that $DI = DJ$. Find $IJ^2$. [b]p18.[/b] For each positive integer $a$, let $P_a = \{2a, 3a, 5a, 7a, . . .\}$ be the set of all prime multiples of $a$. Let $f(m, n) = 1$ if $P_m$ and $P_n$ have elements in common, and let $f(m, n) = 0$ if $P_m$ and $P_n$ have no elements in common. Compute $$\sum_{1\le i<j\le 50} f(i, j)$$ (i.e. compute $f(1, 2) + f(1, 3) + ,,, + f(1, 50) + f(2, 3) + f(2, 4) + ,,, + f(49, 50)$.) [u]Round 7[/u] [b]p19.[/b] How many ways are there to put the six letters in “$MMATHS$” in a two-by-three grid such that the two “$M$”s do not occupy adjacent squares and such that the letter “$A$” is not directly above the letter “$T$” in the grid? (Squares are said to be adjacent if they share a side.) [b]p20.[/b] Luke is shooting basketballs into a hoop. He makes any given shot with fixed probability $p$ with $p < 1$, and he shoots n shots in total with $n \ge 2$. Miraculously, in $n$ shots, the probability that Luke makes exactly two shots in is twice the probability that Luke makes exactly one shot in! If $p$ can be expressed as $\frac{k}{100}$ for some integer $k$ (not necessarily in lowest terms), find the sum of all possible values for $k$. [b]p21.[/b] Let $ABCD$ be a rectangle with $AB = 24$ and $BC = 72$. Call a point $P$ [i]goofy [/i] if it satisfies the following conditions: $\bullet$ $P$ lies within $ABCD$, $\bullet$ for some points $F$ and $G$ lying on sides $BC$ and $DA$ such that the circles with diameter $BF$ and $DG$ are tangent to one another, $P$ lies on their common internal tangent. Find the smallest possible area of a polygon that contains every single goofy point inside it. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2800971p24674988]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x$ be a variable that can take any positive real value. For certain positive real constants $s$ and $t$, the value of $x^2 + \frac{s}{x}$ is minimized at $x = t$, and the value of $t^2\ln(2 + tx) + \frac{1}{x^2}$ is minimized at $x = s$. Compute the ordered pair $(s, t)$.