Let $x, y, z$ be real numbers satisfying $x+y+z=0$ and $x^2+y^2+z^2=6$. Find the maximum value of \[ |(x-y)(y-z)(z-x) | \]

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.

A point $P{}$ is considered on the incircle of the triangle $ABC$. We draw the tangent segments from $P{}$ to the three excircles of $ABC$. Prove that from the obtained three tangent segments it is possible to make a right triangle if and only if the point $P{}$ lies on one of the lines connecting two of the midpoints of the sides of $ABC$.

Let $\Gamma$ be a circumference, and $A,B,C,D$ points of $\Gamma$ (in this order). $r$ is the tangent to $\Gamma$ at point A. $s$ is the tangent to $\Gamma$ at point D. Let $E=r \cap BC,F=s \cap BC$. Let $X=r \cap s,Y=AF \cap DE,Z=AB \cap CD$ Show that the points $X,Y,Z$ are collinear. Note: assume the existence of all above points.

Let $BD$ be the external bisector of a triangle $ABC$ with $AB > BC$; $K$ and $K_1$ be the touching points of side $AC$ with the incircle and the excircle centered at $I$ and $I_1$ respectively. The lines $BK$ and $DI_1$ meet at point $X$, and the lines $BK_1$ and $DI$ meet at point $Y$. Prove that $XY \perp AC$.

All possible $6$-digit numbers, in each of which the digits occur in nonincreasing order (from left to right, e.g. $877550$) are written as a sequence in increasing order. Find the $2005$-th number in this sequence.

At a circular track, $2n$ cyclists started from some point at the same time in the same direction with different constant speeds. If any two cyclists are at some point at the same time again, we say that they meet. No three or more of them have met at the same time. Prove that by the time every two cyclists have met at least once, each cyclist has had at least $n^2$ meetings.

Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? [i]Nanang Susyanto, Jogjakarta[/i]

Juan and Pedro play alternately on the given grid. Each one in turn traces $1$ to $5$ routes different from the ones outlined above, that join $A$ with $B$, moving only to the right and upwards on the grid lines. Juan starts playing. The one who traces a route that passes through $C$ or $D$ loses. Prove that one of them can win regardless of how the other plays. [img]https://cdn.artofproblemsolving.com/attachments/2/7/6a24ca9c4c1c710bd41e44bfcab3d3b61b6d4f.png[/img]

A point $P$ is outside of a circle and the distance to the center is $13$. A secant line from $P$ meets the circle at $Q$ and $R$ so that the exterior segment of the secant, $PQ$, is $9$ and $QR$ is $7$. Find the radius of the circle.

In triangle $ABC$, $AC=7$. $D$ lies on $AB$ such that $AD=BD=CD=5$. Find $BC$.

[hide=F stands for Fermat, E stands for Euler]they had two problem sets under those two names[/hide] [b]F1.[/b] A circle has area $\pi$. Find the circumference of the circle. [b]F2.[/b] In triangle $ABC$, $AB = 5$, $BC = 12$, and $\angle B = 90^o$. Compute $AC$. [b]F3 / E1.[/b] A square has area $2015$. Find the length of the square's diagonal. [b]F4.[/b] I have two cylindrical candles. The first candle has diameter $1$ and height $1$. The second candle has diameter $2$ and height $2$. Both candles are lit at $1:00$ PM and both burn at the same constant rate (volume per time period). The first candle burns out at $1:50$ PM. When does the second candle burn out? Specify AM or PM. [b]F5 / E2.[/b] In triangle $ABC$, $BC$ has length $12$, the altitude from $A$ to $BC$ has length $6$, and the altitude from $C$ to $AB$ has length $8$. Compute the length of $AB$. [b]F6 / E3.[/b] Let $ABC$ be an isosceles triangle with base $AC$. Suppose that there exists a point $D$ on side $AB$ such that $AC = CD = BD$. Find the degree measure of $\angle ABC$. [b]F7 / E6.[/b] In concave quadrilateral $ABCD$, $\angle ABC = 60^o$ and $\angle ADC = 240^o$. If $AD = CD = 4$, compute $BD$. [b]F8 / E7.[/b] A circle of radius $5$ is inscribed in an isosceles trapezoid with legs of length $14$. Compute the area of the trapezoid. [b]E4.[/b] The Egyptian goddess Isil has a staff consisting of a pole with a circle on top. The length of the pole is $32$ inches, and the tangent segment from the bottom of the pole to the circle is $40$ inches. Find the radius of the circle, in inches. [img]https://cdn.artofproblemsolving.com/attachments/5/b/01ea1819aa58c4bde105b9b885f658b3823494.png[/img] [b]E5.[/b] The two concentric circles shown below have radii $1$ and $2$. A chord of the larger circle that is tangent to the smaller circle is drawn. Find the area of the shaded region, bounded by the chord and the larger circle. [img]https://cdn.artofproblemsolving.com/attachments/e/5/425735d2717548552fda8363c201dc8043da13.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For constant $a$, find the differentiable function $f(x)$ satisfying $\int_0^x (e^{-x}-ae^{-t})f(t)dt=0$.

In a three-digit positive integer $M$, the number of hundreds is less than the number of tenths and the number of tenths is less than the number of ones. The arithmetic mean of the integer three-digit numbers obtained by arranging the number $M$ and its numbers ends with the number $5$. Find all such three-digit numbers $M$.

$\frac{a}{c}=\frac{b}{d}=\frac{3}{4}$, $\sqrt{a^2+c^2}-\sqrt{b^2+d^2}=15$. Find $ac+bd-ad-bc$.

The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is called to be the power of the triple. What is the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$? $ \textbf{(A)}\ 9000 \qquad\textbf{(B)}\ 8460 \qquad\textbf{(C)}\ 7290 \qquad\textbf{(D)}\ 6150 \qquad\textbf{(E)}\ 6000 $

On the equal $ AC $ and $ BC $ of an isosceles right triangle $ ABC $ , points $ D $ and $ E $ are marked respectively, so that $ CD = CE $. Perpendiculars on the straight line $ AE $, passing through the points $ C $ and $ D $, intersect the side $ AB $ at the points $ P $ and $ Q $.Prove that $ BP = PQ $.

Prove that for any natural $n>1$ there are infinitely many natural numbers $m$ such that for any nonnegative integers $k_1$,$k_2$, $...$,$k_m$, $$m \ne k_1^n+ k_2^n+... k_n^n,$$

Let $ 1\leq a,b,c,d,e,f,g,h,k \leq 9 $ and $ a,b,c,d,e,f,g,h,k $ are different integers, find the minimum value of the expression $ E = a*b*c+d*e*f+g*h*k $ and prove that it is minimum.

[b]p1.[/b] Zion, RJ, Cam, and Tre decide to start learning languages. The four most popular languages that Duke offers are Spanish, French, Latin, and Korean. If each friend wants to learn exactly three of these four languages, how many ways can they pick courses such that they all attend at least one course together? [b]p2. [/b] Suppose we wrote the integers between $0001$ and $2019$ on a blackboard as such: $$000100020003 · · · 20182019.$$ How many $0$’s did we write? [b]p3.[/b] Duke’s basketball team has made $x$ three-pointers, $y$ two-pointers, and $z$ one-point free throws, where $x, y, z$ are whole numbers. Given that $3|x$, $5|y$, and $7|z$, find the greatest number of points that Duke’s basketball team could not have scored. [b]p4.[/b] Find the minimum value of $x^2 + 2xy + 3y^2 + 4x + 8y + 12$, given that $x$ and $y$ are real numbers. Note: calculus is not required to solve this problem. [b]p5.[/b] Circles $C_1, C_2$ have radii $1, 2$ and are centered at $O_1, O_2$, respectively. They intersect at points $ A$ and $ B$, and convex quadrilateral $O_1AO_2B$ is cyclic. Find the length of $AB$. Express your answer as $x/\sqrt{y}$ , where $x, y$ are integers and $y$ is square-free. [b]p6.[/b] An infinite geometric sequence $\{a_n\}$ has sum $\sum_{n=0}^{\infty} a_n = 3$. Compute the maximum possible value of the sum $\sum_{n=0}^{\infty} a_{3n} $. [b]p7.[/b] Let there be a sequence of numbers $x_1, x_2, x_3,...$ such that for all $i$, $$x_i = \frac{49}{7^{\frac{i}{1010}} + 49}.$$ Find the largest value of $n$ such that $$\left\lfloor \sum_{i=1}{n} x_i \right\rfloor \le 2019.$$ [b]p8.[/b] Let $X$ be a $9$-digit integer that includes all the digits $1$ through $9$ exactly once, such that any $2$-digit number formed from adjacent digits of $X$ is divisible by $7$ or $13$. Find all possible values of $X$. [b]p9.[/b] Two $2025$-digit numbers, $428\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}571$ and $571\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}428$ , form the legs of a right triangle. Find the sum of the digits in the hypotenuse. [b]p10.[/b] Suppose that the side lengths of $\vartriangle ABC$ are positive integers and the perimeter of the triangle is $35$. Let $G$ the centroid and $I$ be the incenter of the triangle. Given that $\angle GIC = 90^o$ , what is the length of $AB$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Through a point $P$ exterior to a given circle pass a secant and a tangent to the circle. The secant intersects the circle at $A$ and $B$, and the tangent touches the circle at $C$ on the same side of the diameter through $P$ as the points $A$ and $B$. The projection of the point $C$ on the diameter is called $Q$. Prove that the line $QC$ bisects the angle $\angle AQB$.

For $n \in \mathbb{N}$, consider in $\mathbb{R}^3$ the regular tetrahedron with vertices $O(0, 0, 0)$, $A(n, 9n, 4n)$, $B(9n, 4n, n)$ and $C(4n, n, 9n)$. Show that the number $N$ of points $(x, y, z)$, $[x, y, z \in \mathbb{Z}]$ inside or on the boundary of the tetrahedron $OABC$ is given by$$N=\frac{343n^3}{3}+\frac{35n^2}{2}+\frac{7n}{6}+1$$

What minimal number of numbers from the set $\{1,2,...,1982\}$ should be deleted to provide the property: [i]none of the remained numbers equals to the product of two other remained numbers[/i]?

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

Determine all triples of integers $(x, y, z)$ that satisfy the equation \[x^z + y^z = z.\]