Let $C_1$ and $C_2$ be tangent circles internally at point $A$, with $C_2$ inside of $C_1$. Let $BC$ be a chord of $C_1$ that is tangent to $C_2$. Prove that the ratio between the length $BC$ and the perimeter of the triangle $ABC$ is constant, that is, it does not depend of the selection of the chord $BC$ that is chosen to construct the trangle.

Let $ABCD$ be a quadrilateral circumscribed around a circle $\omega$ with center $I$. Assume $P$ and $Q$ are distinct points and are isogonal conjugates such that $P, Q$, and $I$ are collinear. Show that $ABCD$ is a kite, that is, it has two disjoint pairs of consecutive equal sides.

Three cyclists ride along a circular road with radius $1$ km counterclockwise. Their velocities are constant and different. Does there necessarily exist (in a sufficiently long time) a moment when all the three distances between cyclists are greater than $1$ km? by V. Protasov

A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?

Let $\vec{v_1},\vec{v_2},\vec{v_3},\vec{v_4}$ and $\vec{v_5}$ be vectors in three dimensions. Show that for some $i,j$ in $1,2,3,4,5$, $\vec{v_i}\cdot \vec{v_j}\ge 0$.

Prove that the following statement holds for all odd integers $n \ge 3$: If a quadrilateral $ABCD$ can be partitioned by lines into $n$ cyclic quadrilaterals, then $ABCD$ is itself cyclic.

Given the isosceles triangle $ABC$, where $AB = AC$. Let $D$ be a point in the segment $BC$ so that $BD = 2DC$. Suppose also that point $P$ lies on the segment $AD$ such that: $\angle BAC = \angle BP D$. Prove that $\angle BAC = 2\angle DP C$.

In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$

A square is divided by horizontal and vertical lines that form $n^2$ squares each with side $1$. What is the greatest possible value of $n$ such that it is possible to select $n$ squares such that any rectangle with area $n$ formed by the horizontal and vertical lines would contain at least one of the selected $n$ squares.

Danielle draws a point $O$ on the plane and a set of points $\mathcal P = \{P_0, P_1, \ldots , P_{2022}\}$ such that $$\angle{P_0OP_1} = \angle{P_1OP_2} = \cdots = \angle{P_{2021}OP_{2022}} = \alpha, \hspace{5pt} 0 < \alpha < \pi,$$where the angles are measured counterclockwise and for $0 \leq n \leq 2022$, $OP_n = r^n$, where $r > 1$ is a given real number. Then, obtain new sets of points in the plane by iterating the following process: given a set of points $\{A_0, A_1, \ldots , A_n\}$ in the plane, it is built a new set of points $\{B_0, B_1, \ldots , B_{n-1}\}$ such that $A_kA_{k+1}B_k$ is an equilateral triangle oriented clockwise for $0 \leq k \leq n - 1$. After carrying out the process $2022$ times from the set $P$, Danielle obtains a single point $X$. If the distance from $X$ to point $O$ is $d$, show that $$(r-1)^{2022} \leq d \leq (r+1)^{2022}.$$

Let $I$ be the incenter of a nonisosceles triangle $ABC$. Prove that there exists a unique pair of points $M$, $N$ lying on the sides $AC$, $BC$ respectively, such that $\angle AIM = \angle BIN$ and $MN|| AB$.

Let $N$ be the orthocentre of triangle $ABC$ (i .e. the point where the altitudes meet). Prove that the circumscribed circles of triangles $ABN, ACN$ and $BCN$ each have equal radius.

Circle $ \Gamma_{1},$ with radius $ r,$ is internally tangent to circle $ \Gamma_{2}$ at $ S.$ Chord $ AB$ of $ \Gamma_{2}$ is tangent to $ \Gamma_{1}$ at $ C.$ Let $ M$ be the midpoint of arc $ AB$ (not containing $ S$), and let $ N$ be the foot of the perpendicular from $ M$ to line $ AB.$ Prove that $ AC\cdot CB\equal{}2r\cdot MN.$

Let $ABCD$ be a convex quadrilateral with \[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\] The line through \(C\), parallel to \(AD\), intersects the external angle bisector of \(\angle ABC\) at point \(T\). Prove that the angles $\angle ATB$, $\angle TBC$, $\angle BCD$, $\angle CDA$, $\angle DAT$ can be divided into two groups, so that the angles in each group have a sum of $270^{\circ}$. [i] Miroslav Marinov, Bulgaria [/i]

Circles $\omega_1$, $\omega_2$, and $\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\omega_1$, $\omega_2$, and $\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_iP_{i+1}$ is tangent to $\omega_i$ for each $i=1,2,3$, where $P_4 = P_1$. See the figure below. The area of $\triangle P_1P_2P_3$ can be written in the form $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b$. What is $a+b$? [asy] unitsize(12); pair A = (0, 8/sqrt(3)), B = rotate(-120)*A, C = rotate(120)*A; real theta = 41.5; pair P1 = rotate(theta)*(2+2*sqrt(7/3), 0), P2 = rotate(-120)*P1, P3 = rotate(120)*P1; filldraw(P1--P2--P3--cycle, gray(0.9)); draw(Circle(A, 4)); draw(Circle(B, 4)); draw(Circle(C, 4)); dot(P1); dot(P2); dot(P3); defaultpen(fontsize(10pt)); label("$P_1$", P1, E*1.5); label("$P_2$", P2, SW*1.5); label("$P_3$", P3, N); label("$\omega_1$", A, W*17); label("$\omega_2$", B, E*17); label("$\omega_3$", C, W*17); [/asy] $\textbf{(A) }546\qquad\textbf{(B) }548\qquad\textbf{(C) }550\qquad\textbf{(D) }552\qquad\textbf{(E) }554$

In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$

Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.

For a point $ P $ situated in the plane determined by a triangle $ ABC, $ prove the following inequality: $$ BC\cdot PB\cdot PC+AC\cdot PC\cdot PA +AB\cdot PA\cdot PB\ge AB\cdot BC\cdot CA $$

Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.

Let $a, b, c$ be sides of a triangle whose perimeter does not exceed $2 \cdot \pi.$, Prove that $\sin a, \sin b, \sin c$ are sides of a triangle.

Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge. (a) Describe one polyhedron with the above property. (b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.

Let $D$ be a point inside an acute triangle $ABC$, such that $\angle BAD = \angle DBC$ and $\angle DAC = \angle BCD$. Let $P$ be a point on the circumcircle of the triangle $ADB$. Suppose $P$ are itself outside the triangle $ABC$. A line through $P$ intersects the ray $BA$ in $X$ and ray $CA$ in $Y$, so that $\angle XPB = \angle PDB$. Show that $BY$ and $CX$ intersect on $AD$.

A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$? [i]Proposed by Dylan Toh[/i]

A cyclic quadrilateral $ABCD$ has circumcenter $O$, its diagonals $AC$ and $BD$ intersect at $G$. Let $P, Q, R, S$ be the circumcenters of $\vartriangle AGB, \vartriangle BGC, \vartriangle CGD, \vartriangle DGA$ respectively. Lines $P R$ and $QS$ intersect at $M$. Show that $M$ is the midpoint of $OG$.

[b]p1.[/b] There are $5$ weights of masses $1,2,3,5$, and $10$ grams. One of the weights is counterfeit (its weight is different from what is written, it is unknown if the weight is heavier or lighter). How to find the counterfeit weight using simple balance scales only twice? [b]p2.[/b] There are $998$ candies and chocolate bars and $499$ bags. Each bag may contain two items (either two candies, or two chocolate bars, or one candy and one chocolate bar). Ann distributed candies and chocolate bars in such a way that half of the candies share a bag with a chocolate bar. Helen wants to redistribute items in the same bags in such a way that half of the chocolate bars would share a bag with a candy. Is it possible to achieve that? [b]p3.[/b] Insert in sequence $2222222222$ arithmetic operations and brackets to get the number $999$ (For instance, from the sequence $22222$ one can get the number $45$: $22*2+2/2 = 45$). [b]p4.[/b] Put numbers from $15$ to $23$ in a $ 3\times 3$ table in such a way to make all sums of numbers in two neighboring cells distinct (neighboring cells share one common side). [b]p5.[/b] All integers from $1$ to $200$ are colored in white and black colors. Integers $1$ and $200$ are black, $11$ and $20$ are white. Prove that there are two black and two white numbers whose sums are equal. [b]p6.[/b] Show that $38$ is the sum of few positive integers (not necessarily, distinct), the sum of whose reciprocals is equal to $1$. (For instance, $11=6+3+2$, $1/16+1/13+1/12=1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].