$ABC$ is a triangle, $G$ its centroid and $A',B',C'$ the midpoints of its sides $BC,CA,AB$, respectively. Prove that if the quadrilateral $AC'GB'$ is cyclic then $AB \cdot CC' = AC \cdot BB'$:

Construct outside the acute-angled triangle $ABC$ the isosceles triangles $ABA_B, ABB_A , ACA_C,ACC_A ,BCB_C$ and $BCC_B$, so that $$AB = AB_A = BA_B, AC = AC_A=CA_C, BC = BC_B = CB_C$$ and $$\angle BAB_A = \angle ABA_B =\angle CAC_A=\angle ACA_C= \angle BCB_C =\angle CBC_B = a < 90^o$$. Prove that the perpendiculars from $A$ to $B_AC_A$, from $B$ to $A_BC_B$ and from $C$ to $A_CB_C$ are concurrent

Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$. (A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.) [i]Note. Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha > 1/3$.[/i] [i]Proposed by Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan[/i]

Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.

An even number of circles are nested, starting with a radius of $1$ and increasing by $1$ each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius $2$ but outside the circle of radius $1.$ An example showing $8$ circles is displayed below. What is the least number of circles needed to make the total shaded area at least $2023\pi$?

Two circles have radius $2$ and $3$, and the distance between their centers is $10$. Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$. (A [i]common external tangent[/i] is a tangent line to two circles such that the circles are on the same side of the line, while a [i]common internal tangent[/i] is a tangent line to two circles such that the circles are on opposite sides of the line). [i]Proposed by Connor Gordon)[/i]

Denote by $B_1$ and $C_1$, the midpoints of the sides $AB$ and $AC$ of the triangle $ABC$. Let the circles circumscribed around the triangles $ABC_1$ and $AB_1C$ intersect at points $A$ and $P$, and let the line $AP$ intersect the circle circumscribed around the triangle $ABC$ at points $A$ and $Q$. Find the ratio $\frac{AQ}{AP}$.

On semicircle, with diameter $|AB|=d$, are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$

A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.

A circle has radius $52$ and center $O$. Points $A$ is on the circle, and point $P$ on $\overline{OA}$ satisfies $OP = 28$. Point $Q$ is constructed such that $QA = QP = 15$, and point $B$ is constructed on the circle so that $Q$ is on $\overline{OB}$. Find $QB$. [i]Proposed by Justin Hsieh[/i]

A rectangle with side lengths $1{ }$ and $3,$ a square with side length $1,$ and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n?$ [asy] size(8cm); draw((0,0)--(10,0)); draw((0,0)--(0,10)); draw((10,0)--(10,10)); draw((0,10)--(10,10)); draw((1,6)--(0,9)); draw((0,9)--(3,10)); draw((3,10)--(4,7)); draw((4,7)--(1,6)); draw((0,3)--(1,6)); draw((1,6)--(10,3)); draw((10,3)--(9,0)); draw((9,0)--(0,3)); draw((6,13/3)--(10,22/3)); draw((10,22/3)--(8,10)); draw((8,10)--(4,7)); draw((4,7)--(6,13/3)); label("$3$",(9/2,3/2),N); label("$3$",(11/2,9/2),S); label("$1$",(1/2,9/2),E); label("$1$",(19/2,3/2),W); label("$1$",(1/2,15/2),E); label("$1$",(3/2,19/2),S); label("$1$",(5/2,13/2),N); label("$1$",(7/2,17/2),W); label("$R$",(7,43/6),W); [/asy] $(\textbf{A})\: 14\qquad(\textbf{B}) \: 23\qquad(\textbf{C}) \: 46\qquad(\textbf{D}) \: 59\qquad(\textbf{E}) \: 67$

Two sequences of integers, $ a_1, a_2, a_3, \ldots$ and $ b_1, b_2, b_3, \ldots$, satisfy the equation \[ (a_n \minus{} a_{n \minus{} 1})(a_n \minus{} a_{n \minus{} 2}) \plus{} (b_n \minus{} b_{n \minus{} 1})(b_n \minus{} b_{n \minus{} 2}) \equal{} 0 \] for each integer $ n$ greater than $ 2$. Prove that there is a positive integer $ k$ such that $ a_k \equal{} a_{k \plus{} 2008}$.

In the convex pentagon $ABCDE$, $AE = AD$, $AB = AC$, and angle $CAD$ equals the sum of angles $AEB$ and $ABE$. Prove that segment $CD$ is double the length of median $AM$ of triangle $ABE$.

An ellipse $\Gamma_1$ with foci at the midpoints of sides $AB$ and $AC$ of a triangle $ABC$ passes through $A$, and an ellipse $\Gamma_2$ with foci at the midpoints of $AC$ and $BC$ passes through $C$. Prove that the common points of these ellipses and the orthocenter of triangle $ABC$ are collinear.

Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)); draw(scale(4)*unitsquare); draw((0,3)--(4,3)); draw((1,3)--(1,4)); draw((2,3)--(2,4)); draw((3,3)--(3,4));[/asy]$ \textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {4}{3} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

Let \( ABC \) be a triangle with \( AB < AC \). Let \( M \) be the midpoint of \( AC \), and let \( D \) be a point on segment \( AC \) such that \( DB = DC \). Let \( E \) be the point of intersection, different from \( B \), of the circumcircle of triangle \( ABM \) and line \( BD \). Define \( P \) and \( Q \) as the points of intersection of line \( BC \) with \( EM \) and \( AE \), respectively. Prove that \( P \) is the midpoint of \( BQ \).

The figure shows a square and a circle with a common center $O$, with equal areas of striped shapes. Find the value of $\cos a$. [img]https://2.bp.blogspot.com/-7uwa0H42ELg/XnmsSoPMgcI/AAAAAAAALgk/pHNBqtbsdKgMhcvIRYLm_8JRpOeIYcUeACK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs2.4.png[/img]

Let $ABCD$ be a rectangle of sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $A$ cuts $BD$ at point $H$. We call $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the measure of the segment $MN$.

In the triangle $ABC$, the angle $A$ is equal to $60^o$, and the median $BD$ is equal to the altitude $CH$. Prove that this triangle is equilateral.

Let $ABC$ be a triangle with $\angle{BAC} > 90$. Let $D$ be a point on the segment $BC$ and $E$ be a point on line $AD$ such that $AB$ is tangent to the circumcircle of triangle $ACD$ at $A$ and $BE$ is perpendicular to $AD$. Given that $CA=CD$ and $AE=CE$. Determine $\angle{BCA}$ in degrees.

Miguel has a square piece of paper $ABCD$ that he folded along a line $EF$, $E$ on $AB$, and $F$ on $CD$. This fold sent $A$ to point $A'$ on $BC$, distinct from $B$ and $C$. Also, it brought $D$ to point $D'$. $G$ is the intersection of $A'D'$ and $DC$. Prove that the inradius of $GCA'$ is equal to the sum of the inradius of $D'GF$ and $A'BE$.

Let $ABC$ be an acute triangle with $AB<AC<BC, c$ it's circumscribed circle and $D,E$ be the midpoints of $AB,AC$ respectively. With diameters the sides $AB,AC$, we draw semicircles, outer of the triangle, which are intersected by line $D$ at points $M$ and $N$ respectively. Lines $MB$ and $NC$ intersect the circumscribed circle at points $T,S$ respectively. Lines $MB$ and $NC$ intersect at point $H$. Prove that: a) point $H$ lies on the circumcircle of triangle $AMN$ b) lines $AH$ and $TS$ are perpedicular and their intersection, let it be $Z$, is the circimcenter of triangle $AMN$

Tiles I, II, III and IV are translated so one tile coincides with each of the rectangles $A, B, C$ and $D$. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C$? [asy] size(400); defaultpen(linewidth(0.8)); path p=origin--(8,0)--(8,6)--(0,6)--cycle; draw(p^^shift(8.5,0)*p^^shift(8.5,10)*p^^shift(0,10)*p); draw(shift(20,2)*p^^shift(28,2)*p^^shift(20,8)*p^^shift(28,8)*p); label("8", (4,6+10), S); label("6", (4+8.5,6+10), S); label("7", (4,6), S); label("2", (4+8.5,6), S); label("I", (4,6+10), N); label("II", (4+8.5,6+10), N); label("III", (4,6), N); label("IV", (4+8.5,6), N); label("3", (0,3+10), E); label("4", (0+8.5,3+10), E); label("1", (0,3), E); label("9", (0+8.5,3), E); label("7", (4,10), N); label("2", (4+8.5,10), N); label("0", (4,0), N); label("6", (4+8.5,0), N); label("9", (8,3+10), W); label("3", (8+8.5,3+10), W); label("5", (8,3), W); label("1", (8+8.5,3), W); label("A", (24,10), N); label("B", (32,10), N); label("C", (24,4), N); label("D", (32,4), N); [/asy] $\mathrm{(A)}\ I \qquad \mathrm{(B)}\ II \qquad \mathrm{(C)}\ III \qquad \mathrm{(D)}\ IV \qquad \mathrm{(E)}\text{ cannot be determined}$

Let $H{}$ be the orthocenter of the acute-angled triangle $ABC$. In the triangle $BHC$, the median $HM$ and the symedian $HL$ are drawn. The point $K{}$ is marked on the line $LH$ so that $\angle AKL=90^\circ$. Prove that the circumcircles of the triangles $ABC$ and $KLM$ are tangent.

Symedians of an acute-angled non-isosceles triangle $ABC$ intersect at a point at point $L$, and $AA_1$, $BB_1$ and $CC_1$ are its altitudes. Prove that you can construct equilateral triangles $A_1B_1C'$, $B_1C_1A'$ and $C_1A_1B'$ not lying in the plane $ABC$, so that lines $AA' , BB'$ and $CC'$ and also perpendicular to the plane $ABC$ at point $L$ intersected at one point.