Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$, and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$. Prove that $DF = AG$.

Fixed points $B$ and $C$ are on a fixed circle $\omega$ and point $A$ varies on this circle. We call the midpoint of arc $BC$ (not containing $A$) $D$ and the orthocenter of the triangle $ABC$, $H$. Line $DH$ intersects circle $\omega$ again in $K$. Tangent in $A$ to circumcircle of triangle $AKH$ intersects line $DH$ and circle $\omega$ again in $L$ and $M$ respectively. Prove that the value of $\frac{AL}{AM}$ is constant. [i]Proposed by Mehdi E'tesami Fard[/i]

Suppose $ABC$ is a triangle with $M$ the midpoint of $BC$. Suppose that $AM$ intersects the incircle at $K,L$. We draw parallel line from $K$ and $L$ to $BC$ and name their second intersection point with incircle $X$ and $Y$. Suppose that $AX$ and $AY$ intersect $BC$ at $P$ and $Q$. Prove that $BP=CQ$.

Let $ABCDEF$ be a convex hexagon with $AB = BC, CD = DE$ and $EF = FA$. Prove that the lines through $C,E,A$ perpendicular to $BD,DF,FB$ are concurrent.

Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio \[\frac{AB \cdot CD}{AC \cdot BD}, \] and prove that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles $ACD$ and $BCD$ at the point $C$ are perpendicular.)

Given is triangle $ABC$ with incenter $I$ and $A$-excenter $J$. Circle $\omega_b$ centered at point $O_b$ passes through point $B$ and is tangent to line $CI$ at point $I$. Circle $\omega_c$ with center $O_c$ passes through point $C$ and touches line $BI$ at point $I$. Let $O_bO_c$ and $IJ$ intersect at point $K$. Find the ratio $IK/KJ$.

Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

Let $ABCD$ be a cyclic quadrilateral of area 8. If there exists a point $O$ in the plane of the quadrilateral such that $OA+OB+OC+OD = 8$, prove that $ABCD$ is an isosceles trapezoid.

Let $I$ be the incenter of $ABC$ and $I_A$ the excenter of the side $BC$, let $M$ be the midpoint of $CB$ and $N$ the midpoint of arc $BC$(with the point $A$). If $T$ is the symmetric of the point $N$ by the point $A$, prove that the quadrilateral $I_AMIT$ is cyclic.

[b]p1.[/b] For positive real numbers $x, y$, the operation $\otimes$ is given by $x \otimes y =\sqrt{x^2 - y}$ and the operation $\oplus$ is given by $x \oplus y =\sqrt{x^2 + y}$. Compute $(((5\otimes 4)\oplus 3)\otimes2)\oplus 1$. [b]p2.[/b] Janabel is cutting up a pizza for a party. She knows there will either be $4$, $5$, or $6$ people at the party including herself, but can’t remember which. What is the least number of slices Janabel can cut her pizza to guarantee that everyone at the party will be able to eat an equal number of slices? [b]p3.[/b] If the numerator of a certain fraction is added to the numerator and the denominator, the result is $\frac{20}{19}$ . What is the fraction? [b]p4.[/b] Let trapezoid $ABCD$ be such that $AB \parallel CD$. Additionally, $AC = AD = 5$, $CD = 6$, and $AB = 3$. Find $BC$. [b]p5.[/b] AtMerrick’s Ice Cream Parlor, customers can order one of three flavors of ice cream and can have their ice cream in either a cup or a cone. Additionally, customers can choose any combination of the following three toppings: sprinkles, fudge, and cherries. How many ways are there to buy ice cream? [b]p6.[/b] Find the minimum possible value of the expression $|x+1|+|x-4|+|x-6|$. [b]p7.[/b] How many $3$ digit numbers have an even number of even digits? [b]p8.[/b] Given that the number $1a99b67$ is divisible by $7$, $9$, and $11$, what are $a$ and $b$? Express your answer as an ordered pair. [b]p9.[/b] Let $O$ be the center of a quarter circle with radius $1$ and arc $AB$ be the quarter of the circle’s circumference. Let $M$,$N$ be the midpoints of $AO$ and $BO$, respectively. Let $X$ be the intersection of $AN$ and $BM$. Find the area of the region enclosed by arc $AB$, $AX$,$BX$. [b]p10.[/b] Each square of a $5$-by-$1$ grid of squares is labeled with a digit between $0$ and $9$, inclusive, such that the sum of the numbers on any two adjacent squares is divisible by $3$. How many such labelings are possible if each digit can be used more than once? [b]p11.[/b] A two-digit number has the property that the difference between the number and the sum of its digits is divisible by the units digit. If the tens digit is $5$, how many different possible values of the units digit are there? [b]p12.[/b] There are $2019$ red balls and $2019$ white balls in a jar. One ball is drawn and replaced with a ball of the other color. The jar is then shaken and one ball is chosen. What is the probability that this ball is red? [b]p13.[/b] Let $ABCD$ be a square with side length $2$. Let $\ell$ denote the line perpendicular to diagonal $AC$ through point $C$, and let $E$ and $F$ be themidpoints of segments $BC$ and $CD$, respectively. Let lines $AE$ and $AF$ meet $\ell$ at points $X$ and $Y$ , respectively. Compute the area of $\vartriangle AXY$ . [b]p14.[/b] Express $\sqrt{21-6\sqrt6}+\sqrt{21+6\sqrt6}$ in simplest radical form. [b]p15.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length two. Let $D$ and $E$ be on $AB$ and $AC$ respectively such that $\angle ABE =\angle ACD = 15^o$. Find the length of $DE$. [b]p16.[/b] $2018$ ants walk on a line that is $1$ inch long. At integer time $t$ seconds, the ant with label $1 \le t \le 2018$ enters on the left side of the line and walks among the line at a speed of $\frac{1}{t}$ inches per second, until it reaches the right end and walks off. Determine the number of ants on the line when $t = 2019$ seconds. [b]p17.[/b] Determine the number of ordered tuples $(a_1,a_2,... ,a_5)$ of positive integers that satisfy $a_1 \le a_2 \le ... \le a_5 \le 5$. [b]p18.[/b] Find the sum of all positive integer values of $k$ for which the equation $$\gcd (n^2 -n -2019,n +1) = k$$ has a positive integer solution for $n$. [b]p19.[/b] Let $a_0 = 2$, $b_0 = 1$, and for $n \ge 0$, let $$a_{n+1} = 2a_n +b_n +1,$$ $$b_{n+1} = a_n +2b_n +1.$$ Find the remainder when $a_{2019}$ is divided by $100$. [b]p20.[/b] In $\vartriangle ABC$, let $AD$ be the angle bisector of $\angle BAC$ such that $D$ is on segment $BC$. Let $T$ be the intersection of ray $\overrightarrow{CB}$ and the line tangent to the circumcircle of $\vartriangle ABC$ at $A$. Given that $BD = 2$ and $TC = 10$, find the length of $AT$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ be a constant. In the $xy$ palne, the curve $C_1:y=\frac{\ln x}{x}$ touches $C_2:y=ax^2$. Find the volume of the solid generated by a rotation of the part enclosed by $C_1,\ C_2$ and the $x$ axis about the $x$ axis. [i]2011 Yokohama National Universty entrance exam/Engineering[/i]

Let $O{}$ be the center of the circumscribed sphere of the tetrahedron $ABCD$. Let $L,M,N$ respectively be the midpoints of the segments $BC,CA,AB$. It is known that $AB+BC=AD+CD$, $BC+CA=BD+AD$, $CA+AB=CD+BD$. Prove that $\angle LOM=\angle MON=\angle NOL$. Find their value.

Determine the smallest real number $a$ such that there is a square of side $a$ such that contains $5$ unit circles inside it without common interior points in pairs.

A $3\times 1$ paper rectangle is folded twice to give a square side $1$. The square is folded along a diagonal to give a right-angled triangle. A needle is driven through an interior point of the triangle, making $6$ holes in the paper. The paper is then unfolded. Where should the point be in order to maximise the smallest distance between any two holes?

The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.

Let $ABCD$ be a parallelogram. Let $X$ and $Y$ be arbitrary points on sides $BC$ and $CD$, respectively. Segments $BY$ and $DX$ intersect at $P$. Prove that the line going through the midpoints of segments $BD$ and $XY$ is either parallel to or coincides with line $AP$.

What should the angle at the vertex of an isosceles triangle be so that it is possible to construct a triangle with sides equal to the height, base, and one of the other sides of the isosceles triangle?

Let triangle $ABC$ be an acute scalene triangle with orthocenter $H$. The foot of perpendicular from $A$ to $BC$ is $O$, and denote $K,L$ by the midpoints of $AB, AC$, respectively. For a point $D(\neq O,B,C)$ on segment $BC$, let $E,F$ be the orthocenters of triangles $ABD, ACD$, respectively, and denote $M,N$ by the midpoints of $DE,DF$. The perpendicular line from $M$ to $KH$ cuts the perpendicular line from $N$ to $LH$ at $P$. If $Q$ is the midpoint of $EF$, and $S$ is the orthocenter of triangle $HPQ$, then prove that as $D$ varies on $BC$, the ratio $\frac{OS}{OH}$, $\frac{OQ}{OP}$ remains constant.

Let $ABC$ be a triangle with circumcenter $O$ and $\angle BAC = 60^{\circ}$. The internal angle bisector of $\angle BAC$ meets line $BC$ and the circumcircle of $\triangle ABC$ in points $M,L$ respectively. Let $K$ denote the reflection of $BL\cap AC$ over the line $BC$. Let $D$ be on the line $CO$ with $DM$ perpendicular to $KL$. Prove that points $K,A,D$ are collinear. [i]Proposed by Sanjana Philo Chacko[/i]

Two parabolas, $y = ax^2 + bx + c$ and $y = -ax^2- bx - c$, intersect at $x = 2$ and $x = -2$. If the $y$-intercepts of the two parabolas are exactly $2$ units apart from each other, compute $|a+b+c|$.

$ 5 $ points are situated in the plane so that any three of them form a triangle of area at most $ 1. $ Prove that there is a trapezoid of area at most $ 3 $ which contains all these points ('including' here means that the points can also be on the sides of the trapezoid).

In a tetrahedron $ABCD$, the medians of the faces $ABD$, $ACD$, $BCD$ from $D$ make equal angles with the corresponding edges $AB$, $AC$, $BC$. Prove that each of these faces has area less than or equal to the sum of the areas of the other two faces. [hide="Comment"][i]Equivalent version of the problem:[/i] $ABCD$ is a tetrahedron. $DE$, $DF$, $DG$ are medians of triangles $DBC$, $DCA$, $DAB$. The angles between $DE$ and $BC$, between $DF$ and $CA$, and between $DG$ and $AB$ are equal. Show that: area $DBC$ $\leq$ area $DCA$ + area $DAB$. [/hide]

Rectangular sheet of paper $ABCD$ is folded as shown in the figure. Find the rato $DK: AB$, given that $C_1$ is the midpoint of $AD$. [img]https://3.bp.blogspot.com/-9EkSdxpGnPU/W6dWD82CxwI/AAAAAAAAJHw/iTkEOejlm9U6Dbu427vUJwKMfEOOVn0WwCK4BGAYYCw/s400/Yasinsky%2B2017%2BVIII-IX%2Bp1.png[/img]

For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows. (1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$. (2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$ (3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.

Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points?