$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle. [img]https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif[/img]

[b]p1.[/b] Find a real number $x$ for which $x\lfloor x \rfloor = 1234.$ Note: $\lfloor x\rfloor$ is the largest integer less than or equal to $x$. [b]p2.[/b] Let $C_1$ be a circle of radius $1$, and $C_2$ be a circle that lies completely inside or on the boundary of $C_1$. Suppose$ P$ is a point that lies inside or on $C_2$. Suppose $O_1$, and $O_2$ are the centers of $C_1$, and $C_2$, respectively. What is the maximum possible area of $\vartriangle O_1O_2P$? Prove your answer. [b]p3.[/b] The numbers $1, 2, . . . , 99$ are written on a blackboard. We are allowed to erase any two distinct (but perhaps equal) numbers and replace them by their nonnegative difference. This operation is performed until a single number $k$ remains on the blackboard. What are all the possible values of $k$? Prove your answer. Note: As an example if we start from $1, 2, 3, 4$ on the board, we can proceed by erasing $1$ and $2$ and replacing them by $1$. At that point we are left with $1, 3, 4$. We may then erase $3$ and $4$ and replacethem by $1$. The last step would be to erase $1$, $1$ and end up with a single $0$ on the board. [b]p4.[/b] Let $a, b$ be two real numbers so that $a^3 - 6a^2 + 13a = 1$ and $b^3 - 6b^2 + 13b = 19$. Find $a + b$. Prove your answer. [b]p5.[/b] Let $m, n, k$ be three positive integers with $n \ge k$. Suppose $A =\prod_{1\le i\le j\le m} gcd(n + i, k + j) $ is the product of $gcd(n + i, k + j)$, where $i, j$ range over all integers satisfying $1\le i\le j\le m$. Prove that the following fraction is an integer $$\frac{A}{(k + 1) \dots(k + m)}{n \choose k}.$$ Note: $gcd(a, b)$ is the greatest common divisor of $a$ and $b$, and ${n \choose k}= \frac{n!}{k!(n - k)!}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the sides $AB$ and $AC$ of a triangle $ABC$ select points $D$ and $E$ respectively, such that $AB = 6$, $AC = 9$, $AD = 4$ and $AE = 6$. It is known that the circumscribed circle of $\vartriangle ADE$ interects the side $BC$ at points $F, G$ , where $BF < BG$. Knowing that the point of intersection of lines $DF$ and $EG$ lies on the circumscribed circle of $\vartriangle ABC$ , find the ratio $BC:FG$.

In the right-angled triangle $ABC$ with a right angle at $C$, the side $BC$ is longer than the side $AC$. The perpendicular bisector of $AB$ intersects the line $BC$ at point $D$ and the line $AC$ at point $E$. The segments $DE$ has the same length as the side $AB$. Find the measures of the angles of the triangle $ABC$. (R. Henner, Vienna)

Find the minimum possible sum of lengths of edges of a prism all of whose edges are tangent of a unit sphere. [Muller-Pfeiffer].

Let $ABC$ be a triangle, $O$ its circumcenter and $\Gamma$ its circumcircle. Let $E$ and $F$ be points on $AB$ and $AC$, respectively, such that $O$ is the midpoint of $EF$. Let $A'=AO\cap \Gamma$, with $A'\ne A$. Finally, let $P$ be the point on line $EF$ such that $A'P\perp EF$. Prove that the lines $EF,BC$ and the tangent to $\Gamma$ at $A'$ are concurrent and that $\angle BPA' = \angle CPA'$.

In the exterior of the triangle $ABC$ with $m(\angle B) > 45^o$, $m(\angle C) >45°^o$ one constructs the right isosceles triangles $ACM$ and $ABN$ such that $m(\angle CAM) = m(\angle BAN) = 90^o$ and, in the interior of $ABC$, the right isosceles triangle $BCP$, with $m(\angle P) = 90^o$. Show that the triangle $MNP$ is a right isosceles triangle.

Two circles $\Gamma_1$ and $\Gamma_2$ intersect at $A,B$. Points $C,D$ lie on $\Gamma_1$, points $E,F$ lie on $\Gamma_2$ such that $A,B$ lies on segments $CE,DF$ respectively and segments $CE,DF$ do not intersect. Let $CF$ meet $\Gamma_1,\Gamma_2$ again at $K,L$ respectively, and $DE$ meet $\Gamma_1,\Gamma_2$ at $M,N$ respectively. If the circumcircles of $\triangle ALM$ and $\triangle BKN$ are tangent, prove that the radii of these two circles are equal.

Lisa and Bart are playing a game. A round table has $n$ lights evenly spaced around its circumference. Some of the lights are on and some of them off; the initial configuration is random. Lisa wins if she can get all of the lights turned on; Bart wins if he can prevent this from happening. On each turn, Lisa chooses the positions at which to flip the lights, but before the lights are flipped, Bart, knowing Lisa’s choices, can rotate the table to any position that he chooses (or he can leave the table as is). Then the lights in the positions that Lisa chose are flipped: those that are off are turned on and those that are on are turned off. Here is an example turn for $n = 5$ (a white circle indicates a light that is on, and a black circle indicates a light that is off): [asy] size(250); defaultpen(linewidth(1)); picture p = new picture; real r = 0.2; pair s1=(0,-4), s2=(0,-8); int[][] filled = {{1,2,3},{1,2,5},{2,3,4,5}}; draw(p,circle((0,0),1)); for(int i = 0; i < 5; ++i) { pair P = dir(90-72*i); filldraw(p,circle(P,r),white); label(p,string(i+1),P,2*P,fontsize(10)); } add(p); add(shift(s1)*p); add(shift(s2)*p); for(int j = 0; j < 3; ++j) for(int i = 0; i < filled[j].length; ++i) filldraw(circle(dir(90-72*(filled[j][i]-1))+j*s1,r)); label("$\parbox{15em}{Initial Position.}$", (-4.5,0)); label("$\parbox{15em}{Lisa says ``1,3,4.'' \\ Bart rotates the table one \\ position counterclockwise. }$", (-4.5,0)+s1); label("$\parbox{15em}{Lights in positions 1,3,4 are \\ flipped.}$", (-4.5,0)+s2);[/asy] Lisa can take as many turns as she needs to win, or she can give up if it becomes clear to her that Bart can prevent her from winning. (a) Show that if $n = 7$ and initially at least one light is on and at least one light is off, then Bart can always prevent Lisa from winning. (b) Show that if $n = 8$, then Lisa can always win in at most 8 turns.

The planet Caramelo is a cube with a $1$ km edge. This planet is going to be wrapped with foam anti-gluttons in order to prevent the presence of greedy ships less than $500$ meters from the planet. What the minimum volume of foam that must surround the planet?

Let $\triangle ABC$ be a triangle with $AB =AC$ and $\angle BAC = 20^{\circ}$. Let $D$ be point on the side $AB$ such that $\angle BCD = 70^{\circ}$. Let $E$ be point on the side $AC$ such that $\angle CBE = 60^{\circ}$. Determine the value of angle $\angle CDE$.

The incircle of triangle $ABC$ touches its sides at points $A'$, $B'$, $C'$. $I$ is its center. Straight line $B'I$ intersects segment $A'C'$ at point $P$. Prove that straight line $BP$ passes through the midpoint of $AC$.

A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold: [list=1] [*] each triangle from $T$ is inscribed in $\omega$; [*] no two triangles from $T$ have a common interior point. [/list] Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

Let $ABC$ be an acute scalene triangle. $D$ and $E$ are variable points in the half-lines $AB$ and $AC$ (with origin at $A$) such that the symmetric of $A$ over $DE$ lies on $BC$. Let $P$ be the intersection of the circles with diameter $AD$ and $AE$. Find the locus of $P$ when varying the line segment $DE$.

The orthocenter $ H $ of a triangle $ ABC $ is distinct from its vertices and its circumcenter $ O. $ $ M,N,P $ are the circumcenters of the triangles $ HBC,HCA, $ respectively, $ HAB. $ Prove that $ AM,BN,CP $ and $ OH $ are concurrent.

Above the segments $AB$ and $BC$ we drew a semicircle at each. $F_1$ bisects $AB$ and $F_2$ bisects $BC$. Above the segments $AF_2$ and $F_1C$ we also drew a semicircle at each. Segments $P Q$ and $RS$ touch the corresponding semicircles as shown in the figure. Prove that $P Q \parallel RS$ and $|P Q| = 2 \cdot |RS|$. [img]https://cdn.artofproblemsolving.com/attachments/8/2/570e923b91e9e630e3880a014cc6df4dc33aa2.png[/img]

In the following, the abbreviation $g \cap h$ will mean the point of intersection of two lines $g$ and $h$. Let $ABCDE$ be a convex pentagon. Let $A^{\prime}=BD\cap CE$, $B^{\prime}=CE\cap DA$, $C^{\prime}=DA\cap EB$, $D^{\prime}=EB\cap AC$ and $E^{\prime}=AC\cap BD$. Furthermore, let $A^{\prime\prime}=AA^{\prime}\cap EB$, $B^{\prime\prime}=BB^{\prime}\cap AC$, $C^{\prime\prime}=CC^{\prime}\cap BD$, $D^{\prime\prime}=DD^{\prime}\cap CE$ and $E^{\prime\prime}=EE^{\prime}\cap DA$. Prove that: \[ \frac{EA^{\prime\prime}}{A^{\prime\prime}B}\cdot\frac{AB^{\prime\prime}}{B^{\prime\prime}C}\cdot\frac{BC^{\prime\prime}}{C^{\prime\prime}D}\cdot\frac{CD^{\prime\prime}}{D^{\prime\prime}E}\cdot\frac{DE^{\prime\prime}}{E^{\prime\prime}A}=1. \] Darij

A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$, where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK -SH) = 1$, compute the area of the concave quadrilateral $ABHC$.

Does there exist a convex polygon such that all its sidelengths are equal and all triangle formed by its vertices are obtuse-angled?

Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

Let $F$ be the boundary and $M,N$ be any interior points of a triangle $ABC$. Consider the function $f_{M,N}: F \to R$ defined by $f_{M,N}(P) = MP^2 +NP^2$ and let $\eta_{M,N}$ be the number of points $P$ for which $f{M,N}$ attains its minimum. (a) Prove that $1 \le \eta_{M,N} \le 3$. (b) If $M$ is fixed, find the locus of $N$ for which $\eta_{M,N} > 1$. (c) Prove that the locus of $M$ for which there are points $N$ such that $\eta_{M,N} = 3$ is the interior of a tangent hexagon.

[b]U[/b]ndecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years. (The sizes of the space stations and the base are negligible.) Determine the volume, in cubic light years, of the set of all possible locations for the Admiral’s base.

I have $8$ unit cubes of different colors, which I want to glue together into a $2\times 2\times 2$ cube. How many distinct $2\times 2\times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.

A convex hexagon is given. Let $ s$ be the sum of the lengths of the three segments connecting the midpoints of its opposite sides. Prove that there is a point in the hexagon such that the sum of its distances to the lines containing the sides of the hexagon does not exceed $ s.$ [i]Author: N. Sedrakyan[/i]

There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles: a) points of intersection of heights, b) the intersection points of the medians, c) the centers of the circumscribed circles.