Let us assume that each of the equations $x^7 + x^2 + 1= 0$ and $x^5- x^4 + x^2- x + 1.001 = 0$ has a single root. Which of these roots is larger?

In concurrent computing, two processes may have their steps interwoven in an unknown order, as long as the steps of each process occur in order. Consider the following two processes: \begin{tabular}{c|cc} Process & $A$ & $B$\\ \hline Step 1 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\ Step 2 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$\\ Step 3 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\ Step 4 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$ \end{tabular} One such interweaving is $A1$, $B1$, $A2$, $B2$, $A3$, $B3$, $B4$, $A4$, but $A1$, $A3$, $A2$, $A4$, $B1$, $B2$, $B3$, $B4$ is not since the steps of $A$ do not occur in order. We run $A$ and $B$ concurrently with $x$ initially valued at $6$. Find the minimal possible value of $x$ among all interweavings.

Let $x, y, z$ be nonzero real numbers with $$\frac{x + y}{z}=\frac{y + z}{x}=\frac{z + x}{y}.$$ Determine all possible values of $$\frac{(x + y)(y + z)(z + x)}{xyz}.$$ [i](Walther Janous)[/i]

Find the ordered triple of natural numbers $(x,y,z)$ such that $x \le y \le z$ and $x^x+y^y+z^z = 3382.$ [i]Proposed by Evan Fang

Circles $o_1$ and $o_2$ with centers in $O_1$ and $O_2$, respectively, intersect in two different points $A$ and $B$, wherein angle $O_1AO_2$ is obtuse. Line $O_1B$ intersects circle $o_2$ in point $C \neq B$. Line $O_2B$ intersects circle $o_1$ in point $D \neq B$. Show that point $B$ is incenter of triangle $ACD$.

Pentagon $ABCDE$ consists of a square $ACDE$ and an equilateral triangle $ABC$ that share the side $\overline{AC}$. A circle centered at $C$ has area 24. The intersection of the circle and the pentagon has half the area of the pentagon. Find the area of the pentagon. [asy]/* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(4.26cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -1.52, xmax = 2.74, ymin = -2.18, ymax = 6.72; /* image dimensions */ draw((0,1)--(2,1)--(2,3)--(0,3)--cycle); draw((0,3)--(2,3)--(1,4.73)--cycle); /* draw figures */ draw((0,1)--(2,1)); draw((2,1)--(2,3)); draw((2,3)--(0,3)); draw((0,3)--(0,1)); draw((0,3)--(2,3)); draw((2,3)--(1,4.73)); draw((1,4.73)--(0,3)); draw(circle((0,3), 1.44)); label("$C$",(-0.4,3.14),SE*labelscalefactor); label("$A$",(2.1,3.1),SE*labelscalefactor); label("$B$",(0.86,5.18),SE*labelscalefactor); label("$D$",(-0.28,0.88),SE*labelscalefactor); label("$E$",(2.1,0.8),SE*labelscalefactor); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]

Let $ M \equal{} \{ x_1..., x_{30}\}$ a set which consists of 30 distinct positive numbers, let $ A_n,$ $ 1 \leq n \leq 30,$ the sum of all possible products with $ n$ elements each of the set $ M.$ Prove if $ A_{15} > A_{10},$ then $ A_1 > 1.$

Let $c$ be a constant. The simultaneous equations \begin{align*} x-y = &\ 2 \\ cx+y = &\ 3 \\ \end{align*} have a solution $(x, y)$ inside Quadrant I if and only if $ \textbf{(A)}\ c=-1 \qquad\textbf{(B)}\ c>-1 \qquad\textbf{(C)}\ c<\frac{3}{2} \qquad\textbf{(D)}\ 0<c<\frac{3}{2}\\ \qquad\textbf{(E)}\ -1<c<\frac{3}{2} $

A class has 30 students. To celebrate 'Tu BiShvat' each student chose some dried fruits out of $n$ different kinds. Say two students are friends if they both chose from the same type of fruit. Find the minimal $n$ so that it is possible that each student has exactly \(6\) friends.

Find the largest positive integer $k$ such that we can find a set $A \subseteq \{1,2, \ldots, 100 \}$ with $k$ elements such that, for any $a,b \in A$, $a$ divides $b$ if and only if $s(a)$ divides $s(b)$, where $s(k)$ denotes the sum of the digits of $k$.

Let $f(x)=x^2-2$ and let $f^{(n)}(x)$ denote the $n$-th iteration of $f$. Let $H=\{x:f^{(100)}(x)\leq -1\}$. Find the length of $H$ (the sum of the lengths of the intervals of $H$).

Let $a_1,a_2,\dots$ be a sequence of integers satisfying $a_1=2$ and: $$a_n=\begin{cases}a_{n-1}+1, & \text{ if }n\ne a_k \text{ for some }k=1,2,\dots,n-1; \\ a_{n-1}+2, & \text{ if } n=a_k \text{ for some }k=1,2,\dots,n-1. \end{cases}$$ Find the value of $a_{2022!}$.

Let $ABCD$ be a quadrilateral such that $\angle A=\angle C=90^{\circ}$. If $A, D$ and the midpoints of $BA, BC$ are concyclic, show that the midpoints of $AD, DC$ and $B, C$ are concyclic.

Suppose $O$ is the circumcenter of $\Delta ABC$, and $E, F$ are points on segments $CA$ and $AB$ respectively with $E, F \neq A$. Let $P$ be a point such that $PB = PF$ and $PC = PE$. Let $OP$ intersect $CA$ and $AB$ at points $Q$ and $R$ respectively. Let the line passing through $P$ and perpendicular to $EF$ intersect $CA$ and $AB$ at points $S$ and $T$ respectively. Prove that points $Q, R, S$, and $T$ are concyclic. [i]Proposed by Li4 and usjl[/i]

What is the product of all odd positive integers less than 10000? $ \textbf{(A)} \ \frac {10000!}{(5000!)^2} \qquad \textbf{(B)} \ \frac {10000!}{2^{5000}} \ \qquad \textbf{(C)} \ \frac {9999!}{2^{5000}} \qquad \textbf{(D)} \ \frac {10000!}{2^{5000} \cdot 5000!} \qquad \textbf{(E)} \ \frac {5000!}{2^{5000}}$

[b]p1.[/b] Find the smallest positive integer $N$ such that $N!$ is a multiple of $10^{2010}$. [b]p2.[/b] An equilateral triangle $T$ is externally tangent to three mutually tangent unit circles, as shown in the diagram. Find the area of $T$. [b]p3. [/b]The polynomial $p(x) = x^3 + ax^2 + bx + c$ has the property that the average of its roots, the product of its roots, and the sum of its coefficients are all equal. If $p(0) = 2$, find $b$. [b]p4.[/b] A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs AiBj , for $1 \le i \le 5$ and $1 \le j \le 4$. Find the maximum of $f(P, S)$ over all pairs of shapes. [b]p5.[/b] Let $ a, b, c$ be three three-digit perfect squares that together contain each nonzero digit exactly once. Find the value of $a + b + c$. [b]p6. [/b]There is a big circle $P$ of radius $2$. Two smaller circles $Q$ and $R$ are drawn tangent to the big circle $P$ and tangent to each other at the center of the big circle $P$. A fourth circle $S$ is drawn externally tangent to the smaller circles $Q$ and $R$ and internally tangent to the big circle $P$. Finally, a tiny fifth circle $T$ is drawn externally tangent to the $3$ smaller circles $Q, R, S$. What is the radius of the tiny circle $T$? [b]p7.[/b] Let $P(x) = (1 +x)(1 +x^2)(1 +x^4)(1 +x^8)(...)$. This infinite product converges when $|x| < 1$. Find $P\left( \frac{1}{2010}\right)$. [b]p8.[/b] $P(x)$ is a polynomial of degree four with integer coefficients that satisfies $P(0) = 1$ and $P(\sqrt2 + \sqrt3) = 0$. Find $P(5)$. [b]p9.[/b] Find all positive integers $n \ge 3$ such that both roots of the equation $$(n - 2)x^2 + (2n^2 - 13n + 38)x + 12n - 12 = 0$$ are integers. [b]p10.[/b] Let $a, b, c, d, e, f$ be positive integers (not necessarily distinct) such that $$a^4 + b^4 + c^4 + d^4 + e^4 = f^4.$$ Find the largest positive integer $n$ such that $n$ is guaranteed to divide at least one of $a, b, c, d, e, f$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle. Let $D, E$ be a points on the segment $BC$ such that $BD =DE = EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine $BP:PQ$.

Find all functions $f: (0, \infty) \to (0, \infty)$ such that \begin{align*} f(y(f(x))^3 + x) = x^3f(y) + f(x) \end{align*} for all $x, y>0$. [i]Proposed by Jason Prodromidis, Greece[/i]

We color all vertexs of a convex polygon with $10$ vertexs by $2$ colors: red and blue $($each vertex is colored by $1$ color$).$ How many ways to color all the vertexs such that there are no $2$ adjacent vertex that are both colored red?

Let $a_1 = 20$, $a_2 = 16$, and for $k \ge 3$, let $a_k = \sqrt[3]{k-a_{k-1}^3-a_{k-2}^3}$. Compute $a_1^3+a_2^3+\cdots + a_{10}^3$.

Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.

A wedding is going to be held in a city with $25$ types of meals, to which some of the $2017$ citizens will be invited. All of the citizens like some meals and each meal is liked by at least one person. A "$suitable$ $list$" is a set of citizens, such that each meal is liked by at least one person in the set. A "$kamber$ $group$" is a set that contains at least one person from each "$suitable$ $list$". Given a "$kamber$ $group$", which has no subset (other than itself) that is also a "$kamber$ $group$", prove that there exists a meal, which is liked by everyone in the group.

Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad $

Let n is a natural number,for which $\sqrt{1+12n^2}$ is a whole number.Prove that $2+2\sqrt{1+12n^2}$ is perfect square.

Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? $ \textbf{(A)}\ \frac {5\sqrt {2} \minus{} 7}{3}\qquad \textbf{(B)}\ \frac {10 \minus{} 7\sqrt {2}}{3}\qquad \textbf{(C)}\ \frac {3 \minus{} 2\sqrt {2}}{3}\qquad \textbf{(D)}\ \frac {8\sqrt {2} \minus{} 11}{3}\qquad \textbf{(E)}\ \frac {6 \minus{} 4\sqrt {2}}{3}$