Let $ABC$ be a triangle, $O$ its circumcenter and $R=1$ its circumradius. Let $G_1,G_2,G_3$ be the centroids of the triangles $OBC, OAC$ and $OAB.$ Prove that the triangle $ABC$ is equilateral if and only if $$AG_1+BG_2+CG_3=4$$

Let $ABCD$ be a trapezoid with $AD\parallel BC$. A point $M $ is chosen inside the trapezoid, and a point $N$ is chosen inside the triangle $BMC$ such that $AM\parallel CN$, $BM\parallel DN$. Prove that triangles $ABN$ and $CDM$ have equal areas.

[b]Q.[/b]Let $ABC$ be a triangle with incenter $I$ and orthocenter $H$. Let $A_1,A_2$ lie on $\overline{BC}$ such that $\overline{IA_1}\perp \overline{IB},\overline{IA_2}\perp\overline{IC}$. $\overline{AA_1},\overline{AA_2}$ cut $\odot(ABC)$ again at $A_3,A_4$. $\overline{A_3A_4}$ cuts $\overline{BC}$ at $A_0$. Similarly, we have $B_0,C_0$. Prove that $A_0,B_0,C_0$ are collinear on a line which is perpendicular to line $\overline{IH}$.

Show that the planes $ACG$ and $BEH$ defined by the vertices of the cube shown in Figure are parallel. What is their distance if the edge length of the cube is $1$ meter? [img]https://cdn.artofproblemsolving.com/attachments/c/9/21585f6c462e4289161b4a29f8805c3f63ff3e.png[/img]

[b]p1.[/b] The final Big $10$ standings for the $1996$ Women's Softball season were 1. Michigan 2. Minnesota З. Iowa 4. Indiana 5. Michigan State 6. Purdue 7. Northwestern 8. Ohio State 9. Penn State 10. Wisconsin (Illinois does not participate in Women's Softball.) When you compare the $1996$ final standings (above) to the final standings for the $1999$ season, you find that the following pairs of teams changed order relative to each other from $1996$ to $1999$ (there are no ties, and no other pairs changed places): (Iowa, Michigan State) (Indiana, Penn State) (Purdue, Wisconsin) (Iowa, Penn State) (Indiana, Wisconsin) (Northwestern, Penn State) (Indiana, Michigan State) (Michigan State, Penn State) (Northwestern, Wisconsin) (Indiana, Purdue) (Purdue, Northwestern) (Ohio State, Penn State) (Indiana, Northwestern) (Purdue, Penn State) (Ohio State, Penn State) (Indiana, Ohio State) Determine as much as you can about the final Big $10$ standings for the $1999$ Women's Softball season. If you cannot determine the standings, explain why you do not have enough information. You must justify your answer. [b]p2.[/b] a) Take as a given that any expression of the form $A \sin t + B \cos t$ ($A>0$) can be put in the form $C \sin (t + D)$, where $C>0$ and $-\pi /2 <D <\pi /2 $. Determine $C$ and $D$ in terms of $A$ and $B$. b) For the values of $C$ and $D$ found in part a), prove that $A \sin t + B \cos t = C \sin (t + D)$. c) Find the maximum value of $3 \sin t +2 \cos t$. [b]pЗ.[/b] А $6$-bу-$6$ checkerboard is completelу filled with $18$ dominoes (blocks of size $1$-bу-$2$). Prove that some horizontal or vertical line cuts the board in two parts but does not cut anу of the dominoes. [b]p4.[/b] a) The midpoints of the sides of a regular hexagon are the vertices of a new hexagon. What is the ratio of the area of the new hexagon to the area of the original hexagon? Justify your answer and simplify as much as possible. b) The midpoints of the sides of a regular $n$-gon ($n >2$) are the vertices of a new $n$-gon. What is the ratio of the area of the new $n$-gon to that of the old? Justify your answer and simplify as much as possible. [b]p5. [/b] You run a boarding house that has $90$ rooms. You have $100$ guests registered, but on any given night only $90$ of these guests actually stay in the boarding house. Each evening a different random set of $90$ guests will show up. You don't know which $90$ it will be, but they all arrive for dinner before you have to assign rooms for the night. You want to give out keys to your guests so that for any set of $90$ guests, you can assign each to a private room without any switching of keys. a) You could give every guest a key to every room. But this requires $9000$ keys. Find a way to hand out fewer than $9000$ keys so that each guest will have a key to a private room. b) What is the smallest number of keys necessary so that each guest will have a key to a private room? Describe how you would distribute these keys and assign the rooms. Prove that this number of keys is as small as possible. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two circles $\omega$ and $\gamma$ have radii $3$ and $4$ respectively, and their centers are $10$ units apart. Let $x$ be the shortest possible distance between a point on $\omega$ and a point on $\gamma$ , and let$ y$ be the longest possible distance between a point on $\omega$ and a point on $\gamma$ . Find the product $xy$.

Let a triangle $ABC$ , $O$ it's circumcenter , $H$ ortocenter and $M$ the midpoint of $AH$. The perpendicular at $M$ to line $OM$ meets $AB$ and $AC$ at points $P$, respective $Q$. Prove that $MP=MQ$. Babis

Consider the rectangular parallelepiped $ ABCDA'B'C'D' $ and the point $ O $ to be the intersection of $ AB' $ and $ A'B. $ On the edge $ BC, $ pick a point $ N $ such that the plane formed by the triangle $ B'AN $ has to be parallel to the line $ AC', $ and perpendicular to $ DO'. $ Prove, then, that this parallelepiped is a cube.

In the figure below, each side of the rhombus has length 5 centimeters. [asy] import graph; unitsize(2.0cm); real w = sqrt(3); draw((w, 0) -- (0, 1) -- (-w, 0) -- (0, -1) -- cycle); filldraw(Circle((-0.5, 0), 0.8 / sqrt(pi)), gray); label("$60^\circ$", (w - 0.1, 0), W); [/asy] The circle lies entirely within the rhombus. The area of the circle is $n$ square centimeters, where $n$ is a positive integer. Compute the number of possible values of $n$.

Let be $ABC$ an acute-angled triangle. Consider point $P$ lying on the opposite ray to the ray $BC$ such that $|AB|=|BP|$. Similarly, consider point $Q$ on the opposite ray to the ray $CB$ such that $|AC|=|CQ|$. Denote $J$ the excenter of $ABC$ with respect to $A$ and $D,E$ tangent points of this excircle with the lines $AB$ and $AC$, respectively. Suppose that the opposite rays to $DP$ and $EQ$ intersect in $F\neq J$. Prove that $AF\perp FJ$.

Let $P$ be a point and $\omega$ be a circle with center $O$ and radius $r$ . We define the power of the point $P$ with respect to the circle $\omega$ to be $OP^2 - r^2$ , and we denote this by pow $(P, \omega)$. We define the radical axis of two circles $\omega_1$ and $\omega_2$ to be the locus of all points P such that pow $(P,\omega_1) =$ pow $(P,\omega_2)$. It turns out that the pairwise radical axes of three circles are either concurrent or pairwise parallel. The concurrence point is referred to as the radical center of the three circles. In $\vartriangle ABC$, let $I$ be the incenter, $\Gamma$ be the circumcircle, and $O$ be the circumcenter. Let $A_1,B_1,C_1$ be the point of tangency of the incircle of $\vartriangle ABC$ with side $BC,CA, AB$, respectively. Let $X_1,X_2 \in \Gamma$ such that $X_1,B_1,C_1,X_2$ are collinear in this order. Let $M_A$ be the midpoint of $BC$, and define $\omega_A$ as the circumcircle of $\vartriangle X_1X_2M_A$. Define $\omega_B$ ,$\omega_C$ analogously. The goal of this problem is to show that the radical center of $\omega_A$, $\omega_B$, $\omega_C$ lies on line $OI$. (a) Let$ A'_1$ denote the intersection of $B_1C_1$ and $BC$. Show that $\frac{A_1B}{A_1C}=\frac{A'_1B}{A'_1C}$. (b) Prove that $A_1$ lies on $\omega_A$. (c) Prove that $A_1$ lies on the radical axis of $\omega_B$ and $\omega_C$ . (d) Prove that the radical axis of $\omega_B$ and $\omega_C$ is perpendicular to $B_1C_1$. (e) Prove that the radical center of $\omega_A$, $\omega_B$, $\omega_C$ is the orthocenter of $\vartriangle A_1B_1C_1$. (f ) Conclude that the radical center of $\omega_A$, $\omega_B$, $\omega_C$ , $O$, and $I$ are collinear. PS. You had better use hide for answers.

For $p\in (0;\infty)$ find the area of the region bounded by the curves $y^{2}=4px$ and $16py^{2}=5(x-p)^{3}$

Lengths of sides of triangle $ABC$ are positive integers, and smallest side is equal to $2$. Determine the area of triangle $P$ if $v_c = v_a + v_b$, where $v_a$, $v_b$ and $v_c$ are lengths of altitudes in triangle $ABC$ from vertices $A$, $B$ and $C$, respectively.

The plane has a semicircle with center $O$ and diameter $AB$. Chord $CD$ is parallel to the diameter $AB$ and $\angle AOC = \angle DOB = \frac{7}{16}$ (radians). Which of the two parts it divides into a semicircle is larger area?

How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.

How many square inches of paint are needed to fully paint a regular $6$-sided die with side length $2$ inches, except for the $\frac13$-inch diameter circular dots marking $1$ through $6$ (a different number per side)? The paint has negligible thickness, and the circular dots are non-overlapping.

Given a triangle $ABC$, points $K,L,M$ are the midpoints of the sides $BC,CA,AB$, and points $X,Y,Z$ are the midpoints of the arcs $BC,CA,AB$ of the circumcircle not containing $A,B,C$ respectively. If $R$ denotes the circumradius and $r$ the inradius of the triangle, show that $r+KX+LY+MZ=2R$.

Let $n$ be an integer with $n\geq 3$ and assume that $2n$ vertices of a regular $(4n + 1)-$gon are coloured. Show that there must exist three of the coloured vertices forming an isosceles triangle.

Three circles $k_1,k_2$ and $k_3$ go through the points $A$ and $B$. A secant through $A$ intersects the circles $k_1,k_2$ and $k_3$ again in the points $C,D$ resp. $E$. Prove that the ratio $|CD|:|DE|$ does not depend on the choice of the secant.

Today was realized the National Olimpiad in Cuba, this is the 3rd problem of the second day: Prof that we can color the lattice points in the plane with two color so that every rectangle with vertices in the lattice points and edges parallels to the co-ordinate axis that have area 2^n is not monocromatic [/img]

Given a triangle $ABC$ with $AB=AC$ and circumcenter $O$. Let $D$ and $E$ be midpoints of $AC$ and $AB$ respectively, and let $DE$ intersect $AO$ at $F$. Denote $\omega$ to be the circle $(BOE)$. Let $BD$ intersect $\omega$ again at $X$ and let $AX$ intersect $\omega$ again at $Y$. Suppose the line parallel to $AB$ passing through $O$ meets $CY$ at $Z$. Prove that the lines $FX$ and $BZ$ meet at $\omega$. [i]Proposed by Ivan Chan Kai Chin[/i]

The polyhedron $PABCDQ$ has the form shown in the figure. It is known that $ABCD$ is parallelogram, the planes of the triangles of the $PAC$ and $PBD$ mutually perpendicular, and also mutually perpendicular are the planes of triangles $QAC$ and $QBC$. Each face of this polyhedron is painted black or white so that the faces that have a common edge are painted in different colors. Prove that the sum of the squares of the areas of the black faces is equal to the sum of the squares of the areas of the white faces. [img]https://1.bp.blogspot.com/-UM5PKEGGWqc/X1V2cXAFmwI/AAAAAAAAMdw/V-Qr94tZmqkj3_q-5mkSICGF1tMu-b_VwCLcBGAsYHQ/s0/2007.5%2Bchampions%2Btourn.png[/img]

[asy]size(100); draw((0,0)--(1,0)--(1,1)--(1,2)--(2,2)--(2,3)--(2,4)--(1,4)--(1,3)--(0,3)--(-1,3)--(-1,2)--(0,2)--(0,1)--cycle); draw((0,1)--(1,1)); draw((0,2)--(1,2)); draw((0,2)--(0,3)); draw((1,2)--(1,3)); draw((1,3)--(2,3)); label("Z",(0.5,0.2),N); label("X",(0.5,1.2),N); label("V",(0.5,2.2),N); label("U",(-0.5,2.2),N); label("W",(1.5,2.2),N); label("Y",(1.5,3.2),N);[/asy] A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $ \text{X}$ is: \[ \textbf{(A)}\ \text{Z} \qquad \textbf{(B)}\ \text{U} \qquad \textbf{(C)}\ \text{V} \qquad \textbf{(D)}\ \text{W} \qquad \textbf{(E)}\ \text{Y} \]

For any two points $A (x_1 , y_1)$ and $B (x_2, y_2)$, the distance $r (A, B)$ between them is determined by the equality $r(A, B) = max\{| x_1- x_2 | , | y_1 - y_2 |\}$. Prove that the triangle inequality $r(A, C) + r(C, B) \ge r(A, B)$. holds for the distance introduced in this way . Let $A$ and $B$ be two points of the plane . Find the locus of points $C$ for which a) $r(A, C) + r(C, B) = r(A, B)$ b) $r(A, C) = r(C, B).$

Let $ABC$ be an obtuse triangle with $\angle A>90^{\circ}$, and let $r$ and $R$ denote its inradius and circumradius. Prove that \[\frac{r}{R} \le \frac{a\sin A}{a+b+c}.\]