Let $x,y,z\in \mathbb{R}^*$ such that $xy,yz,zx\in \mathbb{Q}$. a) Prove that $x^2+y^2+z^2$ is rational; b) If $x^3+y^3+z^3$ is rational, prove that $x,y,z$ are rational. [i]Marius Ghergu[/i]

In a right triangle, the altitude from a vertex to the hypotenuse splits the hypotenuse into two segments of lengths $a$ and $b$. If the right triangle has area $T$ and is inscribed in a circle of area $C$, find $ab$ in terms of $T$ and $C$.

[b]p1.[/b] The number $2020$ is very special: the sum of its digits is equal to the product of its nonzero digits. How many such four digit numbers are there? (Numbers with only one nonzero digit, like $3000$, also count) [b]p2.[/b] A locker has a combination which is a sequence of three integers between $ 0$ and $49$, inclusive. It is known that all of the numbers in the combination are even. Let the total of a lock combination be the sum of the three numbers. Given that the product of the numbers in the combination is $12160$, what is the sum of all possible totals of the locker combination? [b]p3.[/b] Given points $A = (0, 0)$ and $B = (0, 1)$ in the plane, the set of all points P in the plane such that triangle $ABP$ is isosceles partitions the plane into $k$ regions. The sum of the areas of those regions that are bounded is $s$. Find $ks$. [b]p4.[/b] Three families sit down around a circular table, each person choosing their seat at random. One family has two members, while the other two families have three members. What is the probability that every person sits next to at least one person from a different family? [b]p5.[/b] Jacob and Alexander are walking up an escalator in the airport. Jacob walks twice as fast as Alexander, who takes $18$ steps to arrive at the top. Jacob, however, takes $27$ steps to arrive at the top. How many of the upward moving escalator steps are visible at any point in time? [b]p6.[/b] Points $A, B, C, D, E$ lie in that order on a circle such that $AB = BC = 5$, $CD = DE = 8$, and $\angle BCD = 150^o$ . Let $AD$ and $BE$ intersect at $P$. Find the area of quadrilateral $PBCD$. [b]p7.[/b] Ivan has a triangle of integers with one number in the first row, two numbers in the second row, and continues up to eight numbers in the eighth row. He starts with the first $8$ primes, $2$ through $19$, in the bottom row. Each subsequent row is filled in by writing the least common multiple of two adjacent numbers in the row directly below. For example, the second last row starts with$ 6, 15, 35$, etc. Let P be the product of all the numbers in this triangle. Suppose that P is a multiple of $a/b$, where $a$ and $b$ are positive integers and $a > 1$. Given that $b$ is maximized, and for this value of $b, a$ is also maximized, find $a + b$. [b]p8.[/b] Let $ABCD$ be a cyclic quadrilateral. Given that triangle $ABD$ is equilateral, $\angle CBD = 15^o$, and $AC = 1$, what is the area of $ABCD$? [b]p9.[/b] Let $S$ be the set of all integers greater than $ 1$. The function f is defined on $S$ and each value of $f$ is in $S$. Given that $f$ is nondecreasing and $f(f(x)) = 2x$ for all $x$ in $S$, find $f(100)$. [b]p10.[/b] An [i]origin-symmetric[/i] parallelogram $P$ (that is, if $(x, y)$ is in $P$, then so is $(-x, -y)$) lies in the coordinate plane. It is given that P has two horizontal sides, with a distance of $2020$ between them, and that there is no point with integer coordinates except the origin inside $P$. Also, $P$ has the maximum possible area satisfying the above conditions. The coordinates of the four vertices of P are $(a, 1010)$, $(b, 1010)$, $(-a, -1010)$, $(-b, -1010)$, where a, b are positive real numbers with $a < b$. What is $b$? [b]p11.[/b] What is the remainder when $5^{200} + 5^{50} + 2$ is divided by $(5 + 1)(5^2 + 1)(5^4 + 1)$? [b]p12.[/b] Let $f(n) = n^2 - 4096n - 2045$. What is the remainder when $f(f(f(... f(2046)...)))$ is divided by $2047$, where the function $f$ is applied $47$ times? [b]p13.[/b] What is the largest possible area of a triangle that lies completely within a $97$-dimensional hypercube of side length $1$, where its vertices are three of the vertices of the hypercube? [b]p14.[/b] Let $N = \left \lfloor \frac{1}{61} \right \rfloor + \left \lfloor\frac{3}{61} \right \rfloor+\left \lfloor \frac{3^2}{61} \right \rfloor+... +\left \lfloor\frac{3^{2019}}{61} \right \rfloor$. Given that $122N$ can be expressed as $3^a - b$, where $a, b$ are positive integers and $a$ is as large as possible, find $a + b$. Note: $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$. [b]p15.[/b] Among all ordered triples of integers $(x, y, z)$ that satisfy $x + y + z = 8$ and $x^3 + y^3 + z^3 = 134$, what is the maximum possible value of $|x| + |y| + |z|$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that a positive integer $m$ is a sum of two triangular numbers if and only if $4m+1$ is a sum of two squares.

An ellipse with foci $B_0,B_1$ intersects $AB_i$ at $C_i$ $(i=0,1)$. Let $P_0$ be a point on ray $AB_0$. $Q_0$ is a point on ray $C_1B_0$ such that $B_0P_0=B_0Q_0$; $P_1$ is on ray $B_1A$ such that $C_1Q_0=C_1P_1$; $Q_1$ is on ray $B_1C_0$ such that $B_1P_1=B_1Q_1$; $P_2$ is on ray $AB_0$ such that $C_0Q_1=C_0Q_2$. Prove that $P_0=P_2$ and that the four points $P_0,Q_0,Q_1,P_1$ are concyclic.

$BB_1$ and $CC_1$ are altitudes in $\Delta ABC$. Let $B_1 C_1$ intersect the circumscribed circle of $\Delta ABC$ in points $E$ and $F$. Let $k$ be a circle passing through $E$ and $F$ in such way that the center of $k$ lies on the arc $\widehat{BAC}$. We denote with $M$ the middle point of $BC$. $X$ and $Y$ are the points on $k$ for which $MX$ and $MY$ are tangent to $k$. Let $EX\cap FY=S_1,EY\cap FX=S_2,BX\cap CY=U,$ and $BY\cap CX=V$. Prove that $S_1 S_2$ and $UV$ intersect in the orthocenter of $\Delta ABC$.

Let $a, b, c, d, e \in \mathbb{R}^+$ and $f:\{(x, y) \in (\mathbb{R}^+)^2|c-dx-ey > 0\}\to \mathbb{R}^+$ be given by $f(x, y) = (ax)(by)(c- dx- ey)$. Find the maximum value of $f$.

Rio likes fruit, and one day she decides to pick persimmons. She picks a total of $12$ persimmons from the first $5$ trees she sees. Rio has $5$ more trees to pick persimmons from. If she wants to pick an average of $4$ persimmons per tree overall, what is the average number of persimmons that she must pick from each of the last $5$ trees for her goal?

When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$? $\textbf{(A)}~6\qquad\textbf{(B)}~8\qquad\textbf{(C)}~9\qquad\textbf{(D)}~10\qquad\textbf{(E)}~11$

An equilateral triangle has side length $ 6$. What is the area of the region containing all points that are outside the triangle and not more than $ 3$ units from a point of the triangle? $ \textbf{(A)}\ 36\plus{}24\sqrt{3} \qquad \textbf{(B)}\ 54\plus{}9\pi \qquad \textbf{(C)}\ 54\plus{}18\sqrt{3}\plus{}6\pi \qquad \textbf{(D)}\ \left(2\sqrt{3}\plus{}3\right)^2\pi \\ \textbf{(E)}\ 9\left(\sqrt{3}\plus{}1\right)^2\pi$

Let $a, b, m, n$ be positive integers such that $gcd(a, b) = 1$ and $a > 1$. Prove that if $$a^m+b^m \mid a^n+b^n$$then $m \mid n$.

Consider the figure below, not drawn to scale. In this figure, assume that$AB \perp BE$ and $AD \perp DE$. Also, let $AB = \sqrt6$ and $\angle BED =\frac{\pi}{6}$ . Find $AC$. [img]https://cdn.artofproblemsolving.com/attachments/2/d/f87ac9f111f02e261a0b5376c766a615e8d1d8.png[/img]

A six dimensional "cube" (a $6$-cube) has $64$ vertices at the points $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3).$ This $6$-cube has $192\text{ 1-D}$ edges and $240\text{ 2-D}$ edges. This $6$-cube gets cut into $6^6=46656$ smaller congruent "unit" $6$-cubes that are kept together in the tightly packaged form of the original $6$-cube so that the $46656$ smaller $6$-cubes share 2-D square faces with neighbors ($\textit{one}$ 2-D square face shared by $\textit{several}$ unit $6$-cube neighbors). How many 2-D squares are faces of one or more of the unit $6$-cubes?

As usual, let $\sigma (N)$ denote the sum of all the (positive integral) divisors of $N.$ (Included among these divisors are $1$ and $N$ itself.) For example, if $p$ is a prime, then $\sigma (p)=p+1.$ Motivated by the notion of a "perfect" number, a positive integer $N$ is called "quasiperfect" if $\sigma (N) =2N+1.$ Prove that every quasiperfect number is the square of an odd integer.

Let $\xi_{(k_1, k_2)}, k_1, k_2 \in\mathbb N$ be random variables uniformly bounded. Let $c_l, l\in\mathbb N$ be a positive real strictly increasing infinite sequence such that $c_{l+1}/ c_l$ is bounded. Let $d_l=\log \left(c_{l+1}/c_l\right), l\in\mathbb N$ and suppose that $D_n=\sum_{l=1}^n d_l\uparrow \infty$ when $n\to\infty$ Suppose there exist $C>0$ and $\varepsilon>0$ such that $$\left| \mathbb E \left\{ \xi_{(k_1,k_2)}\xi_{(l_1,l_2)}\right\}\right| \leq C\prod_{i=1}^2 \left\{ \log_+\log_+\left( \frac{c_{\max\{ k_i, l_i\}}}{c_{\min\{ k_i, l_i\}}}\right)\right\}^{-(1+\varepsilon)}$$ for each $(k_1, k_2), (l_1,l_2)\in\mathbb N^2$ ($\log_+$ is the positive part of the natural logarithm). Show that $$\lim_{\substack{n_1\to\infty \\ n_2\to\infty}} \frac{1}{D_{n_1}D_{n_2}}\sum_{k_1=1}^{n_1} \sum_{k_2=1}^{n_2} d_{k_1}d_{k_2}\xi_{(k_1,k_2)}=0$$ almost surely. (translated by j___d)

The numbers $1, 2, \ldots, n$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number remains. Prove that this number is at least $\frac{4}{9}n^3$. [i]Brian Hamrick.[/i]

Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? $ \textbf{(A)}\ \frac{1}{5} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{2}{5} \qquad \textbf{(D)}\ \frac{2}{3} \qquad \textbf{(E)}\ \frac{4}{5}$

The four circumcircles of the four faces of a tetrahedron have equal radii. Prove that the four faces of the tetrahedron are congruent triangles.

Once upon a time, there was a tribe called the Goblin Tribe, and their regular game was ''The ATM Game (Level Giveaway)'' . The game stats with a number of Goblin standing in a circle. Then the Chieftain assigns a Level to each Goblin, which can be the same or different (Level is a number which is a non-negative integer). Start play by selecting a Goblin with Level $k$ ($k \not=). 0$) comes up. Let's assume Goblin $A$. Goblin $A$ explodes itself. Goblin A's Level becomes $0$. After that, Level of Goblin $k$ next to Goblin $A$ clockwise gets Level $1$. Prove that: 1.) If after that Goblin $k$ next to Goblin $A$ explodes itself and keep doing this, $k'$ next to that Goblin clockwise explodes itself. Prove that the level of each Goblin will be the same again. 2) 2.) If after that we can choose any Goblin whose level is not $0$ to explode itself. And keep doing this. Prove that no matter what the initial level is, we can make each level the way we want. But there is a condition that the sum of all Goblin's levels must be equal to the beginning. [i](gools)[/i]

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

Let $ABC$ be a triangle with circumcircle $\omega$. Let $l_B$ and $l_C$ be two lines through the points $B$ and $C$, respectively, such that $l_B \parallel l_C$. The second intersections of $l_B$ and $l_C$ with $\omega$ are $D$ and $E$, respectively. Assume that $D$ and $E$ are on the same side of $BC$ as $A$. Let $DA$ intersect $l_C$ at $F$ and let $EA$ intersect $l_B$ at $G$. If $O$, $O_1$ and $O_2$ are circumcenters of the triangles $ABC$, $ADG$ and $AEF$, respectively, and $P$ is the circumcenter of the triangle $OO_1O_2$, prove that $l_B \parallel OP \parallel l_C$. [i]Proposed by Stefan Lozanovski, Macedonia[/i]

Given is a triangle $ABC$ and points $D$ and $E$, respectively on $] BC [$ and $] AB [$. $F$ it is intersection of lines $AD$ and $CE$. We denote as $| CD | = a, | BD | = b, | DF | = c$ and $| AF | = d$. Determine the ratio $\frac{| BE |}{|AE |}$ in terms of $a, b, c$ and $d$ [img]https://cdn.artofproblemsolving.com/attachments/5/7/856c97045db2d9a26841ad00996a2b809addaa.png[/img]

For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$.

the edges of triangle $ABC$ are $a,b,c$ respectively,$b<c$,$AD$ is the bisector of $\angle A$,point $D$ is on segment $BC$. (1)find the property $\angle A$,$\angle B$,$\angle C$ have,so that there exists point $E,F$ on $AB,AC$ satisfy $BE=CF$,and $\angle NDE=\angle CDF$ (2)when such $E,F$ exist,express $BE$ with $a,b,c$

Find the least value of the expression $(x+y)(y+z)$, under the conditionthat $x,y,z$ are positive numbers satisfying the equation $xyz(x + y + z) = 1$.