Given triangle $ABC$, let $D$ be an inner point of segment $BC$. Let $P$ and $Q$ be distinct inner points of the segment $AD$. Let $K=BP\cap AC, L=CP\cap AB, E=BQ\cap AC, F=CQ\cap AB$. Given that $KL\parallel EF$, find all possible values of the ratio $BD:DC$.

Let $ABC$ be a triangle $D\in[BC]$ (different than $A$ and $B$).$E$ is the midpoint of $[CD]$. $F\in[AC]$ such that $\widehat{FEC}=90$ and $|AF|.|BC|=|AC|.|EC|.$ Circumcircle of $ADC$ intersect $[AB]$ at $G$ different than $A$.Prove that tangent to circumcircle of $AGF$ at $F$ is touch circumcircle of $BGE$ too.

Prove that there are infinitely many positive integers $n$ such that $n \times n \times n$ can not be filled completely with 2 x 2 x 2 and 3 x 3 x 3 solid cubes.

Inside a circle $c$ there are circles $c_1, c_2$ and $c_3$ which are tangent to $c$ at points $A, B$ and $C$ correspondingly, which are all different. Circles $c_2$ and $c_3$ have a common point $K$ in the segment $BC$, circles $c_3$ and $c_1$ have a common point $L$ in the segment $CA$, and circles $c_1$ and $c_2$ have a common point $M$ in the segment $AB$. Prove that the circles $c_1, c_2$ and $c_3$ intersect in the center of the circle $c$.

Let $ABC$ be a triangle with $\angle ABC=90^{\circ}$. The square $BDEF$ is inscribed in $\triangle ABC$, such that $D,E,F$ are in the sides $AB,CA,BC$ respectively. The inradius of $\triangle EFC$ and $\triangle EDA$ are $c$ and $b$, respectively. Four circles $\omega_1,\omega_2,\omega_3,\omega_4$ are drawn inside the square $BDEF$, such that the radius of $\omega_1$ and $\omega_3$ are both equal to $b$ and the radius of $\omega_2$ and $\omega_4$ are both equal to $a$. The circle $\omega_1$ is tangent to $ED$, the circle $\omega_3$ is tangent to $BF$, $\omega_2$ is tangent to $EF$ and $\omega_4$ is tangent to $BD$, each one of these circles are tangent to the two closest circles and the circles $\omega_1$ and $\omega_3$ are tangents. Determine the ratio $\frac{c}{a}$.

In a circle of diameter $1$ consider $65$ points, no three of them collinear. Prove that there exist three among these points which are the vertices of a triangle with area less than or equal to $\frac{1}{72}$.

[b]p1.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. His drink contains $45\%$ of orange juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $60\%$ of orange juice? [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] For one particular number $a > 0$ the function f satisfies the equality $f(x + a) =\frac{1 + f(x)}{1 - f(x)}$ for all $x$. Show that $f$ is a periodic function. (A function $f$ is periodic with the period $T$ if $f(x + T) = f(x)$ for any $x$.) [b]p4.[/b] If $a, b, c, x, y, z$ are numbers so that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}= 1$ and $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}= 0$. Show that $\frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} = 1$ [b]p5.[/b] Is it possible that a four-digit number $AABB$ is a perfect square? (Same letters denote the same digits). [b]p6.[/b] A finite number of arcs of a circle are painted black (see figure). The total length of these arcs is less than $\frac15$ of the circumference. Show that it is possible to inscribe a square in the circle so that all vertices of the square are in the unpainted portion of the circle. [img]https://cdn.artofproblemsolving.com/attachments/2/c/bdfa61917a47f3de5dd3684627792a9ebf05d5.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If the length of a diagonal of a square is $ a \plus{} b$, then the area of the square is: $ \textbf{(A)}\ (a \plus{} b)^2 \qquad\textbf{(B)}\ \frac {1}{2}(a \plus{} b)^2 \qquad\textbf{(C)}\ a^2 \plus{} b^2$ $ \textbf{(D)}\ \frac {1}{2}(a^2 \plus{} b^2) \qquad\textbf{(E)}\ \text{none of these}$

Let $ A,B$ be different points on a parabola. Prove that we can find $ P_1,P_2,\dots,P_{n}$ between $ A,B$ on the parabola such that area of the convex polygon $ AP_1P_2\dots P_nB$ is maximum. In this case prove that the ratio of $ S(AP_1P_2\dots P_nB)$ to the sector between $ A$ and $ B$ doesn't depend on $ A$ and $ B$, and only depends on $ n$.

If $ABC$ is acute triangle, prove distance from each vertex to corresponding excentre is less than sum of two greatest side of triangle

Four squares of side length $4, 7, 9,$ and $10$ are arranged in increasing size order so that their left edges and bottom edges align. The squares alternate in color white-gray-white-gray, respectively, as shown in the figure. What is the area of the visible gray region in square units? [asy] size(150); filldraw((0,0)--(10,0)--(10,10)--(0,10)--cycle,gray(0.7),linewidth(1)); filldraw((0,0)--(9,0)--(9,9)--(0,9)--cycle,white,linewidth(1)); filldraw((0,0)--(7,0)--(7,7)--(0,7)--cycle,gray(0.7),linewidth(1)); filldraw((0,0)--(4,0)--(4,4)--(0,4)--cycle,white,linewidth(1)); draw((11,0)--(11,4),linewidth(1)); draw((11,6)--(11,10),linewidth(1)); label("$10$",(11,5),fontsize(14pt)); draw((10.75,0)--(11.25,0),linewidth(1)); draw((10.75,10)--(11.25,10),linewidth(1)); draw((0,11)--(3,11),linewidth(1)); draw((5,11)--(9,11),linewidth(1)); draw((0,11.25)--(0,10.75),linewidth(1)); draw((9,11.25)--(9,10.75),linewidth(1)); label("$9$",(4,11),fontsize(14pt)); draw((-1,0)--(-1,1),linewidth(1)); draw((-1,3)--(-1,7),linewidth(1)); draw((-1.25,0)--(-0.75,0),linewidth(1)); draw((-1.25,7)--(-0.75,7),linewidth(1)); label("$7$",(-1,2),fontsize(14pt)); draw((0,-1)--(1,-1),linewidth(1)); draw((3,-1)--(4,-1),linewidth(1)); draw((0,-1.25)--(0,-.75),linewidth(1)); draw((4,-1.25)--(4,-.75),linewidth(1)); label("$4$",(2,-1),fontsize(14pt)); [/asy] $\textbf{(A)}\ 42 \qquad \textbf{(B)}\ 45\qquad \textbf{(C)}\ 49\qquad \textbf{(D)}\ 50\qquad \textbf{(E)}\ 52$

Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$. Tangent lines to $(O)$ at $B,C$ meet at $T$. A line passes through $T$ cuts segments $AB$ at $D$ and cuts ray $CA$ at $E$. Take $M$ as midpoint of $DE$ and suppose that $MA$ cuts $(O)$ again at $K$. Prove that $(MKT)$ is tangent to $(O)$.

[b]p1.[/b] You are on a flat planet. There are $100$ cities at points $x = 1, ..., 100$ along the line $y = -1$, and another $100$ cities at points $x = 1, ... , 100$ along the line $y = 1$. The planet’s terrain is scalding hot, and you cannot walk over it directly. Instead, you must cross archways from city to city. There are archways between all pairs of cities with different $y$ coordinates, but no other pairs: for instance, there is an archway from $(1, -1)$ to $(50, 1)$, but not from $(1, -1)$ to $(50, -1)$. The amount of “effort” necessary to cross an archway equals the square of the distance between the cities it connects. You are at $(1, -1)$, and you want to get to $(100, -1)$. What is the least amount of effort this journey can take? [b]p2.[/b] Let $f(x) = x^4 + ax^3 + bx^2 + cx + 25$. Suppose $a, b, c$ are integers and $f(x)$ has $4$ distinct integer roots. Find $f(3)$. [b]p3.[/b] Frankenstein starts at the point $(0, 0, 0)$ and walks to the point $(3, 3, 3)$. At each step he walks either one unit in the positive $x$-direction, one unit in the positive $y$-direction, or one unit in the positive $z$-direction. How many distinct paths can Frankenstein take to reach his destination? [b]p4.[/b] Let $ABCD$ be a rectangle with $AB = 20$, $BC = 15$. Let $X$ and $Y$ be on the diagonal $\overline{BD}$ of $ABCD$ such that $BX > BY$ . Suppose $A$ and $X$ are two vertices of a square which has two sides on lines $\overline{AB}$ and $\overline{AD}$, and suppose that $C$ and $Y$ are vertices of a square which has sides on $\overline{CB}$ and $\overline{CD}$. Find the length $XY$ . [img]https://cdn.artofproblemsolving.com/attachments/2/8/a3f7706171ff3c93389ff80a45886e306476d1.png[/img] [b]p5.[/b] $n \ge 2$ kids are trick-or-treating. They enter a haunted house in a single-file line such that each kid is friends with precisely the kids (or kid) adjacent to him. Inside the haunted house, they get mixed up and out of order. They meet up again at the exit, and leave in single file. After leaving, they realize that each kid (except the first to leave) is friends with at least one kid who left before him. In how many possible orders could they have left the haunted house? [b]p6.[/b] Call a set $S$ sparse if every pair of distinct elements of S differ by more than $1$. Find the number of sparse subsets (possibly empty) of $\{1, 2,... , 10\}$. [b]p7.[/b] How many ordered triples of integers $(a, b, c)$ are there such that $1 \le a, b, c \le 70$ and $a^2 + b^2 + c^2$ is divisible by $28$? [b]p8.[/b] Let $C_1$, $C_2$ be circles with centers $O_1$, $O_2$, respectively. Line $\ell$ is an external tangent to $C_1$ and $C_2$, it touches $C_1$ at $A$ and $C_2$ at $B$. Line segment $\overline{O_1O_2}$ meets $C_1$ at $X$. Let $C$ be the circle through $A, X, B$ with center $O$. Let $\overline{OO_1}$ and $\overline{OO_2}$ intersect circle $C$ at $D$ and $E$, respectively. Suppose the radii of $C_1$ and $C_2$ are $16$ and $9$, respectively, and suppose the area of the quadrilateral $O_1O_2BA$ is $300$. Find the length of segment $DE$. [b]p9.[/b] What is the remainder when $5^{5^{5^5}}$ is divided by $13$? [b]p10.[/b] Let $\alpha$ and $\beta$ be the smallest and largest real numbers satisfying $$x^2 = 13 + \lfloor x \rfloor + \left\lfloor \frac{x}{2} \right\rfloor +\left\lfloor \frac{x}{3} \right\rfloor + \left\lfloor \frac{x}{4} \right\rfloor .$$ Find $\beta - \alpha$ . ($\lfloor a \rfloor$ is defined as the largest integer that is not larger than $a$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A$ and $B$ be two points inside a given circle $k$. Prove that there exist (infinitely many) circles through $A$ and $B$ which lie entirely in $k$.

Two circles are tangent to each other internally at a point $\ T $. Let the chord $\ AB $ of the larger circle be tangent to the smaller circle at a point $\ P $. Prove that the line $\ TP $ bisects $\ \angle ATB $.

In a tetrahedron $ABCD$, all angles at vertex $D$ are equal to $\alpha$ and all dihedral angles between faces having $D$ as a vertex are equal to $\phi$. Prove that there exists a unique $\alpha$ for which $\phi=2\alpha$.

Given is a triangle $ABC$ and let $AA_1$, $BB_1$ be angle bisectors. It turned out that $\angle AA_1B=24^{\circ}$ and $\angle BB_1A=18^{\circ}$. Find the ratio $\angle BAC:\angle ACB:\angle ABC$.

For which arrangements of two infinite circular cylinders does their intersection lie in a plane?

Suppose that $ABCD$ is a convex quadrilateral with no parallel sides. Make a parallelogram on each two consecutive sides. Show that among these $4$ new points, there is only one point inside the quadrilateral $ABCD$. by Morteza Saghafian

Three points $K, L$ and $M$ are given in the plane. It is known that they are the midpoints of three successive sides of an erased quadrilateral and that these three sides have the same length. Reconstruct the quadrilateral.

Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac{1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let \(ABC\) be an isosceles triangle with \(AB = BC\). Let \(D\) be a point on segment \(AB\), \(E\) be a point on segment \(BC\), and \(P\) be a point on segment \(DE\) such that \(AD = DP\) and \(CE = PE\). Let \(M\) be the midpoint of \(DE\). The line parallel to \(AB\) through \(M\) intersects \(AC\) at \(X\) and the line parallel to \(BC\) through \(M\) intersects \(AC\) at \(Y\). The lines \(DX\) and \(EY\) intersect at \(F\). Prove that \(FP\) is perpendicular to \(DE\).

Let $\omega_1$ and $\omega_2$ be two different circles that intersect at two different points, $X$ and $Y$. Let lines $l_1$ and $l_2$ be common tangent lines of these circles such that $l_1$ is tangent $\omega_1$ at $A$ and $\omega_2$ at $C$ and $l_2$ is tangent $\omega_1$ at $B$ and $\omega_2$ at $D$. Let $Z$ be the reflection of $Y$ respect to $l_1$ and let $BC$ and $\omega_1$ meet at $K$ for the second time. Let $AD$ and $\omega_2$ meet at $L$ for the second time. Prove that the line tangent to $\omega_1$ and passes through $K$ and the line tangent to $\omega_2$ and passes through $L$ meet on the line $XZ$.

Prove that every integer can be written as sum of $5$ third powers of integers.

Let $ABC$ be a triangle in which $\measuredangle{A}=135^{\circ}$. The perpendicular to the line $AB$ erected at $A$ intersects the side $BC$ at $D$, and the angle bisector of $\angle B$ intersects the side $AC$ at $E$. Find the measure of $\measuredangle{BED}$.