Let $k$ be the incircle of the triangle $ABC$ with the center of the incircle $I$. The circle $k$ touches the sides $BC, CA$ and $AB$ in points $D, E$ and $F$. Let $G$ be the intersection of the straight line $AI$ and the circle $k$, which lies between $A$ and $I$. Assume $BE$ and $FG$ are parallel. Show that $BD = EF$.

Triangle $ABC$ has $AC = 3$, $BC = 5$, $AB = 7$. A circle is drawn internally tangent to the circumcircle of $ABC$ at $C$, and tangent to $AB$. Let $D$ be its point of tangency with $AB$. Find $BD - DA$. [asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(6cm); real labelscalefactor = 2.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.5, xmax = 7.01, ymin = -3, ymax = 8.02; /* image dimensions */ /* draw figures */ draw(circle((1.37,2.54), 5.17)); draw((-2.62,-0.76)--(-3.53,4.2)); draw((-3.53,4.2)--(5.6,-0.44)); draw((5.6,-0.44)--(-2.62,-0.76)); draw(circle((-0.9,0.48), 2.12)); /* dots and labels */ dot((-2.62,-0.76),dotstyle); label("$C$", (-2.46,-0.51), SW * labelscalefactor); dot((-3.53,4.2),dotstyle); label("$A$", (-3.36,4.46), NW * labelscalefactor); dot((5.6,-0.44),dotstyle); label("$B$", (5.77,-0.17), SE * labelscalefactor); dot((0.08,2.37),dotstyle); label("$D$", (0.24,2.61), SW * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); label("$7$",(-3.36,4.46)--(5.77,-0.17), NE * labelscalefactor); label("$3$",(-3.36,4.46)--(-2.46,-0.51),SW * labelscalefactor); label("$5$",(-2.46,-0.51)--(5.77,-0.17), SE * labelscalefactor); /* end of picture */ [/asy]

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $AC = 15$. Let$ D$ and $E$ be the feet of the altitudes from $A$ and $B$, respectively. Find the circumference of the circumcircle of $\vartriangle CDE$

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 $DQ$.

For a point $O$ inside a triangle $ABC$, denote by $A_1,B_1, C_1,$ the respective intersection points of $AO, BO, CO$ with the corresponding sides. Let \[n_1 =\frac{AO}{A_1O}, n_2 = \frac{BO}{B_1O}, n_3 = \frac{CO}{C_1O}.\] What possible values of $n_1, n_2, n_3$ can all be positive integers?

Given an acute triangle $ABC$. The inscribed circle of triangle $ABC$ is tangent to $AB$ and $AC$ at $X$ and $Y$ respectively. Let $CH$ be the altitude. The perpendicular bisector of the segment $CH$ intersects the line $XY$ at $Z$. Prove that $\angle BZC = 90^o.$

[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family? [b]p2.[/b] Solve each of the following problems. (1) Find a pair of numbers with a sum of $11$ and a product of $24$. (2) Find a pair of numbers with a sum of $40$ and a product of $400$. (3) Find three consecutive numbers with a sum of $333$. (4) Find two consecutive numbers with a product of $182$. [b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive. [b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$. [b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map. [img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img] On each path $A \to B \to C$ and $B \to C \to D$ there are $10$ hills, on the path $A \to B \to D$ there are $22$ hills, on the path $A \to D \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a unit square. Consider an equilateral triangle $XYZ$ with $X,Y$ as (inner or boundary) points of the square. Determine the locus $M$ of vertices $Z$ of all these triangles $XYZ$ and compute the area of $M.$

In triangle $ABC$ the medians $AM$ and $CN$ to sides $BC$ and $AB$, respectively, intersect in point $O$. $P$ is the midpoint of side $AC$, and $MP$ intersects $CN$ in $Q$. If the area of triangle $OMQ$ is $n$, then the area of triangle $ABC$ is: $\text{(A)}\ 16n\qquad \text{(B)}\ 18n\qquad \text{(C)}\ 21n\qquad \text{(D)}\ 24n\qquad \text{(E)}\ 27n$

Consider the right isosceles $\vartriangle ABC$ at $ A$. Let $L$ be the intersection of the bisector of $\angle ACB$ with $AB$ and $K$ the intersection point of $CL$ with the bisector of $BC$. Let $X$ be the point on line $AK$ such that $\angle KCX = 90^o$ and let $Y$ be the point of intersection of $CX$ with the circumcircle of $\vartriangle ABC$. Let $Y'$ the reflection of point $Y$ wrt $BC$. Prove that $B - K -Y'$. Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.

[b]p1.[/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 square $ABCD$ is $14$ cm. [img]https://cdn.artofproblemsolving.com/attachments/1/1/0f80fc5f0505fa9752b5c9e1c646c49091b4ca.png[/img] [b]p2.[/b] If $a, b$, and $c$ are numbers so that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 1$. Compute $a^4 + b^4 + c^4$. [b]p3.[/b] A given fraction $\frac{a}{b}$ ($a, b$ are positive integers, $a \ne b$) is transformed by the following rule: first, $1$ is added to both the numerator and the denominator, and then the numerator and the denominator of the new fraction are each divided by their greatest common divisor (in other words, the new fraction is put in simplest form). Then the same transformation is applied again and again. Show that after some number of steps the denominator and the numerator differ exactly by $1$. [b]p4.[/b] A goat uses horns to make the holes in a new $30\times 60$ cm large towel. Each time it makes two new holes. Show that after the goat repeats this $61$ times the towel will have at least two holes whose distance apart is less than $6$ cm. [b]p5.[/b] You are given $555$ weights weighing $1$ g, $2$ g, $3$ g, $...$ , $555$ g. Divide these weights into three groups whose total weights are equal. [b]p6.[/b] Draw on the regular $8\times 8$ chessboard a circle of the maximal possible radius that intersects only black squares (and does not cross white squares). Explain why no larger circle can satisfy the condition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the acute-angled triangle $ABC$, the altitudes $BP$ and $CQ$ were drawn, and the point $T$ is the intersection point of the altitudes of $\Delta PAQ$. It turned out that $\angle CTB = 90 {} ^ \circ$. Find the measure of $\angle BAC$. (Mikhail Plotnikov)

Assume that $ x$ is a positive real number. Which is equivalent to $ \sqrt[3]{x\sqrt{x}}$? $ \textbf{(A)}\ x^{1/6} \qquad \textbf{(B)}\ x^{1/4} \qquad \textbf{(C)}\ x^{3/8} \qquad \textbf{(D)}\ x^{1/2} \qquad \textbf{(E)}\ x$

Let $ ABC$ be an isosceles triangle with $ AC \equal{} BC.$ Its incircle touches $ AB$ in $ D$ and $ BC$ in $ E.$ A line distinct of $ AE$ goes through $ A$ and intersects the incircle in $ F$ and $ G.$ Line $ AB$ intersects line $ EF$ and $ EG$ in $ K$ and $ L,$ respectively. Prove that $ DK \equal{} DL.$

In a scalene triangle $ABC$ $I$ is the incentr and $CN$ is the bisector of angle $C$. The line $CN$ meets the circumcircle of $ABC$ again at $M$. The line $l$ is parallel to $AB$ and touches the incircle of $ABC$. The point $R$ on $l$ is such. That $CI \bot IR$. The circumcircle of $MNR$ meets the line $IR$ again at S. Prpve that $AS=BS$.

$A,B,C$ are points that three complex numbers $z_0=a\text{i},z_1=\frac{1}{2}+b\text{i},z_2=1+c\text{i}(a,b,c\in\mathbb{R})$ refer to on complex plane (not collinear). Prove that curve $Z=Z_0\cos^4t+2Z_1\cos^2t\sin^2t+Z_2\sin^4t(t\in\mathbb{R})$ has only one common point with the perpendicular bisector of $AC$, and find the point.

Given triangle $ABC$ such that angles $A$, $B$, $C$ satisfy \[ \frac{\cos A}{20}+\frac{\cos B}{21}+\frac{\cos C}{29}=\frac{29}{420} \] Prove that $ABC$ is right angled triangle

In the acute triangle $ABC$, let $D$ be the foot of the perpendicular from $A$ to $BC$, let $E$ be the foot of the perpendicular from $D$ to $AC$, and let $F$ be a point on the line segment $DE$. Prove that $AF$ is perpendicular to $BE$ if and only if $FE/FD = BD/DC$

The angle bisector of vertex $A$ of $\triangle ABC$ cuts $[BC]$ at $D$. The circle passing through $A$ and touching to $BC$ at $D$ meets $[AB]$ and $[AC]$ at $P$ and $Q$, respectively. $AD$ and $PQ$ meet at $T$. If $|AB|=5, |BC|=6, |CA|=7$, then $\frac{|AT|}{|TD|}=?$ $ \textbf{(A)}\ \frac75 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac72 \qquad \textbf{(E)}\ 4$

A tangent line to circumcircle of acute triangle $ABC$ ($AC>AB$) at $A$ intersects with the extension of $BC$ at $P$. $O$ is the circumcenter of triangle $ABC$.Point $X$ lying on $OP$ such that $\measuredangle AXP=90^\circ$.Points $E$ and $F$ lying on $AB$ and $AC$,respectively,and they are in one side of line $OP$ such that $ \measuredangle EXP=\measuredangle ACX $ and $\measuredangle FXO=\measuredangle ABX $. $K$,$L$ are points of intersection $EF$ with circumcircle of triangle $ABC$.prove that $OP$ is tangent to circumcircle of triangle $KLX$. Author:Mehdi E'tesami Fard , Iran

For $A\subset\mathbb Z$ and $a,b\in\mathbb Z$. We define $aA+b: =\{ax+b|x\in A\}$. If $a\neq0$ then we calll $aA+b$ and $A$ to similar sets. In this question the Cantor set $C$ is the number of non-negative integers that in their base-3 representation there is no $1$ digit. You see \[C=(3C)\dot\cup(3C+2)\ \ \ \ \ \ (1)\] (i.e. $C$ is partitioned to sets $3C$ and $3C+2$). We give another example $C=(3C)\dot\cup(9C+6)\dot\cup(3C+2)$. A representation of $C$ is a partition of $C$ to some similiar sets. i.e. \[C=\bigcup_{i=1}^{n}C_{i}\ \ \ \ \ \ (2)\] and $C_{i}=a_{i}C+b_{i}$ are similar to $C$. We call a representation of $C$ a primitive representation iff union of some of $C_{i}$ is not a set similar and not equal to $C$. Consider a primitive representation of Cantor set. Prove that a) $a_{i}>1$. b) $a_{i}$ are powers of 3. c) $a_{i}>b_{i}$ d) (1) is the only primitive representation of $C$.

Is it possible to write a natural number in every cell of an infinite chessboard in such a manner that for all integers $m, n > 100$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n \ ?$

Let $ABC$ be an acute-angled triangle. The tangents to its circumcircle at $A, B, C$ form a triangle $PQR$ with $C \in PQ$ and $B \in PR$. Let $C_{1}$ be the foot of the altitude from $C$ in $\Delta ABC$ . Prove that $CC_{1}$ bisects $\widehat{QC_{1}P}$ .

In an inscribed quadrilateral $ABCD$, we have $BC=CD$ but $AB\neq AD$. Points $I$ and $J$ are the incenters of triangles $ABC$ and $ACD$ respectively. Point $K$ was chosen on segment $AC$ so that $IK=JK$. Points $M$ and $N$ are the incenters of triangles $AIK$ and $AJK$. Prove that the perpendicular to $CD$ at $D$ and the perpendicular to $KI$ at $I$ intersect on the circumcircle of $MAN$.

Point $D$ is on side $CB$ of triangle $ABC$. If \[ \angle{CAD} = \angle{DAB} = 60^\circ,\quad AC = 3\quad\mbox{ and }\quad AB = 6, \] then the length of $AD$ is $\text{(A)} \ 2 \qquad \text{(B)} \ 2.5 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 3.5 \qquad \text{(E)} \ 4$