Points $B_1,C_1$ are on $AC$ and $AB$ and $B_1C_1 \parallel BC$. Circumcircle of $ABB_1$ intersect $CC_1$ at $L$. Circumcircle $CLB_1$ is tangent to $AL$. Prove $AL \leq \frac{AC+AC_1}{2}$

Let $ABC$ be a triangle where $\angle BAC = 30^\circ$. Construct $D$ in $\triangle ABC$ such that $\angle ABD = \angle ACD = 30^\circ$. Let the circumcircle of $\triangle ABD$ intersect $AC$ at $X$. Let the circumcircle of $\triangle ACD$ intersect $AB$ at $Y$. Given that $DB - DC = 10$ and $BC = 20$, find $AX \cdot AY$.

$ABCD$ is a regular tetrahedron with side length $1$. Find the area of the cross section of $ABCD$ cut by the plane that passes through the midpoints of $AB$, $AC$, and $CD$.

a) The triangle $ABC$ was turned around the centre of the circumscribed circle by the angle less than $180$ degrees and thus was obtained the triangle $A_1B_1C_1$. The corresponding segments $[AB]$ and $[A_1B_1]$ intersect in the point $C_2, [BC]$ and $[B_1C_1]$ -- $A_2, [AC]$ and $[A_1C_1]$ -- $B_2$. Prove that the triangle $A_2B_2C_2$ is similar to the triangle $ABC$. b) The quadrangle $ABCD$ was turned around the centre of the circumscribed circle by the angle less than $180$ degrees and thus was obtained the quadrangle $A_1B_1C_1D_1$. Prove that the points of intersection of the corresponding lines ( $(AB$) and $(A_1B_1), (BC)$ and $(B_1C_1), (CD)$ and $(C_1D_1), (DA)$ and $(D_1A_1)$ ) are the vertices of the parallelogram.

Let $AB$ be a chord of a circle $\Gamma$ and let $C$ be a point on the segment $AB$. Let $r$ be a line through $C$ which intersects $\Gamma$ at the points $D,E$; suppose that $D,E$ lie on different sides with respect to the perpendicular bisector of $AB$. Let $\Gamma_D$ be the circumference which is externally tangent to $\Gamma$ at $D$ and touches the line $AB$ at $F$. Let $\Gamma_E$ be the circumference which is externally tangent to $\Gamma$ at $E$ and touches the line $AB$ at $G$. Prove that $CA=CB$ if and only if $CF=CG$.

Points $X,Y,Z$ lies on a line (in indicated order). Triangles $XAB$, $YBC$, $ZCD$ are regular, the vertices of the first and the third triangle are oriented counterclockwise and the vertices of the second are opposite oriented. Prove that $AC$, $BD$ and $XY$ concur. V.A.Yasinsky

Given is a quadrilateral $ABCD$ in which we can inscribe circle. The segments $AB, BC, CD$ and $DA$ are the diameters of the circles $o1, o2, o3$ and $o4$, respectively. Prove that there exists a circle tangent to all of the circles $o1, o2, o3$ and $o4$.

Let $\triangle ABC$ be an acute-angled triangle. Let $P$ be the midpoint of $BC$, and $K$ the foot of the altitude from $A$ to side $BC$. Let $D$ be a point on segment $AP$ such that $\angle BDC = 90^\circ$. Let $E$ be the second point of intersection of line $BC$ with the circumcircle of $\triangle ADK$. Let $F$ be the second point of intersection of line $AE$ with the circumcircle of $\triangle ABC$. Prove that $\angle AFD = 90^\circ$.

Let $\Gamma_{1}$ be a circle. $AB$ is a diameter, $\ell$ is the tangent at $B$, and $M$ is a point on $\Gamma_{1}$ other than $A$. $\Gamma_{2}$ is a circle tangent to $\ell$, and also to $\Gamma_{1}$ at $M$. a) Determine the point of tangency $P$ of $\ell$ and $\Gamma_{2}$ and find the locus of the center of $\Gamma_{2}$ as $M$ varies. b) Show that there exists a circle that is always orthogonal to $\Gamma_{2}$, regardless of the position of $M$.

Let $ABCD$ be a cyclic quadrilateral. Show that $\vert \overline{AC} - \overline{BD} \vert \le \vert \overline{AB}-\overline{CD} \vert$ and determine when does equality hold.

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

In the triangle $ABC$, the respective mid points of the sides $BC$, $AB$ and $AC$ are $D$, $E$ and $F$. Let $M$ be the point where the internal bisector of the angle $\angle ADB$ intersects the side $AB$, and $N$ the point where the internal bisector of the angle $\angle ADC$ intersects the side $AC$. Also, let $O$ be the intersection point of $AD$ and $MN$, $P$ the intersection point of $AB$ and $FO$, and $R$ the intersection point of $AC$ and $EO$. Prove that $PR=AD$.

For positive constant $a$, let $C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})$. Denote by $l(t)$ the length of the part $a\leq y\leq t$ for $C$ and denote by $S(t)$ the area of the part bounded by the line $y=t\ (a<t)$ and $C$. Find $\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.$

A value $x \in [0, 1]$ is selected uniformly at random. A point $(a, b)$ is called [i]friendly [/i] to $x$ if there exists a circle between the lines $y = 0$ and $y = 1$ that contains both $(a, b)$ and $(0, x)$. Find the area of the region of the plane determined by possible locations of friendly points.

A $9\times 9$ board has a number written on each square: all squares in the first row have $1$, all squares in the second row have $2$, $\ldots$, all squares in the ninth row have $9$. We will call [i]special [/i] rectangle any rectangle of $2\times 3$ or $3\times 2$ or $4\times 5$ or $5\times 4$ on the board. The permitted operations are: $\bullet$ Simultaneously add $1$ to all the numbers located in a special rectangle. $\bullet$ Simultaneously subtract $1$ from all numbers located in a special rectangle. Demonstrate that it is possible to achieve, through a succession of permitted operations, that $80$ squares to have $0$ (zero). What number is left in the remaining box?

Let $A_1,A_2, \dots A_k$ be points on the unit circle.Prove that: $\sum\limits_{1\le i<j \le k} d(A_i,A_j)^2 \le k^2 $ Where $d(A_i,A_j)$ denotes the distance between $A_i,A_j$.

A circle $K$ with radius $r$, a point $D$ on $K$, and a convex angle with vertex $S$ and rays $a$ and $b$ are given in the plane. Construct a parallelogram $ABCD$ such that $A$ and $B$ lie on $a$ and $b$ respectively, $SA+SB=r$, and $C$ lies on $K$.

Let $n$ be an integer greater than $2$. Consider a set of $n$ different points, with no three collinear, in the plane. Prove that we can label the points $P_1,~ P_2, \dots , P_n$ such that $P_1P_2 \dots P_n$ is not a self-intersecting polygon. ([i]A polygon is self-intersecting if one of its side intersects the interior of another side. The polygon is not necessarily convex[/i] )

Let $ABCD$ be a parallelogram with diagonal $AC = 10$ such that the distance from $A$ to line $CD$ is $6$ and the distance from $A$ to line $BC$ is $7$. There are two non-congruent configurations of $ABCD$ that satisfy these conditions. The sum of the areas of these two parallelograms is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

A square $ABCD$ is inscribed in a circle. If $M$ is a point on the shorter arc $AB$, prove that \[MC \cdot MD > 3\sqrt{3} \cdot MA \cdot MB.\]

Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies. [i]Proposed by T. Vitanov, E. Kolev[/i]

Let $n > 2$ be a fixed positive integer. For a set $S$ of $n$ points in the plane, let $P(S)$ be the set of perpendicular bisectors of pairs of distinct points in $S$. Call set $S$ [i]complete[/i] if no two (distinct) pairs of points share the same perpendicular bisector, and every pair of lines in $P(S)$ intersects. Let $f(S)$ be the number of distinct intersection points of pairs of lines in $P(S)$. (a) Find all complete sets $S$ such that $f(S) = 1$. (b) Let $S$ be a complete set with $n$ points. Show that if $f(S)>1$, then $f(S) \geq n$.

Paint the maximum number of vertices of the cube red so that you cannot select three of the red vertices that form an equilateral triangle.

Triangles $ABC$ and $DEF$, having a common incircle of radius $R$, intersect at points $X_1, X_2, \ldots , X_6$ and form six triangles (see the figure below). Let $r_1, r_2,\ldots, r_6$ be the radii of the inscribed circles of these triangles, and let $R_1, R_2, \ldots , R_6$ be the radii of the inscribed circles of the triangles $AX_1F, FX_2B, BX_3D, DX_4C, CX_5E$ and $EX_6A$ respectively. [img]https://i.ibb.co/BspgdHB/Image.jpg[/img] Prove that \[ \sum_{i=1}^{6} \frac{1}{r_i} < \frac{6}{R}+\sum_{i=1}^{6} \frac{1}{R_i} \] [i]U. Maksimenkau[/i]

Consider the sequence $ a_n\equal{}\sqrt[3]{n^3\plus{}3n^2\plus{}2n\plus{}1}\plus{}a\sqrt[5]{n^5\plus{}5n^4\plus{}1}\plus{}\frac{ln(e^{n^2}\plus{}n\plus{}2)}{n\plus{}2}\plus{}b$. Find $ a,b \in \mathbb{R}$ such that $ \displaystyle\lim_{n\to\infty}a_n\equal{}5$.