If the length and width of a rectangle are each increased by $ 10 \%$, then the perimeter of the rectangle is increased by \[ \textbf{(A)}\ 1 \% \qquad \textbf{(B)}\ 10 \% \qquad \textbf{(C)}\ 20 \% \qquad \textbf{(D)}\ 21 \% \qquad \textbf{(E)}\ 40 \% \]

Let $ABC$ be a triangle with circumcircle $\omega$ such that $AB = 11$, $AC = 13$, and $\angle A = 30^o$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = 7$ and $AE = 8$. There exists a unique point $F \ne A$ on minor arc $AB$ of $\omega$ such that $\angle F DA = \angle F EA$. Compute $F A^2$.

Let $ABC$ be a triangle with $AB < AC$. Its incircle with centre $I$ touches the sides $BC, CA,$ and $AB$ in the points $D, E,$ and $F$ respectively. The angle bisector $AI$ intersects the lines $DE$ and $DF$ in the points $X$ and $Y$ respectively. Let $Z$ be the foot of the altitude through $A$ with respect to $BC$. Prove that $D$ is the incentre of the triangle $XYZ$.

[b]p1.[/b] In a given tetrahedron the sum of the measures of the three face angles at each of the vertices is $180$ degrees. Prove that all faces of the tetrahedron are congruent triangles. [img]https://cdn.artofproblemsolving.com/attachments/c/c/40f03324fd19f6a5e0a5e541153a2b38faac79.png[/img] [b]p2.[/b] The digital sum $D(n)$ of a positive integer $n$ is defined recursively by: $D(n) = n$ if $1 \le n \le 9$ $D(n) = D(a_0 + a_1 + ... + a_m)$ if $n>9$ where $a_0 , a_1 ,..,a_m$ are all the digits of $n$ expressed in base ten. (For example, $D(959) = D(26) = D(8) = 8$.) Prove that $D(n \times 1234)= D(n)$ fcr all positive integers $n$ . [b]p3.[/b] A right triangle has area $A$ and perimeter $P$ . Find the largest possible value for the positive constant $k$ such that for every such triangle, $P^2 \ge kA$ . [b]p4.[/b] In the accompanying diagram, $\overline{AB}$ is tangent at $A$ to a circle of radius $1$ centered at $O$ . The segment $\overline{AP}$ is equal in length to the arc $AB$ . Let $C$ be the point of intersection of the lines $AO$ and $PB$ . Determine the length of segment $\overline{AC}$ in terms of $a$ , where $a$ is the measure of $\angle AOB$ in radians. [img]https://cdn.artofproblemsolving.com/attachments/e/0/596e269a89a896365b405af7bc6ca47a1f7c57.png[/img] [b]p5.[/b] Let $a_1 = a > 0$ and $a_2 = b >a$. Consider the sequence $\{a_1,a_2,a_3,...\}$ of positive numbers defined by: $a_3=\sqrt{a_1a_2}$, $a_4=\sqrt{a_2a_3}$, $...$ and in general, $a_n=\sqrt{a_{n-2}a_{n-1}}$, for $n\ge 3$ . Develop a formula $a_n$ expressing in terms of $a$, $b$ and $n$ , and determine $\lim_{n \to \infty} a_n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In triangle $ABC$ we have $AC = CB$. On side $AB$ is a point $D$ such that the radius of the incircle of triangle $ACD$ is equal to the radius of the circle tangent to the segment $DB$ and to the extensions of the lines $CD$ and $CB$. Prove that this radius equals a quarter of either of the two equal altitudes of triangle $ABC$.

Let $ \triangle{ABC}$ be a triangle with $ AB\not\equal{}AC$. The incircle with centre $ I$ touches $ BC$, $ CA$, $ AB$ at $ D$, $ E$, $ F$, respectively. Furthermore let $ M$ the midpoint of $ EF$ and $ AD$ intersect the incircle at $ P\not\equal{}D$. Show that $ PMID$ ist cyclic.

A convex hexagon $ABCDEF$ is given whose sides $AB$ and $DE$ are parallel. Each of the diagonals $AD, BE, CF$ divides this hexagon into two quadrilaterals of equal perimeters. Show that these three diagonals intersect at one point.

Let $P$ be a convex polygon in the plane. Show that there exists a circle containing the entire polygon $P$ and having at least three adjacent vertices of $P$ on its boundary.

Let $\triangle ABC$ be connected to the circle $\Gamma$. The angular bisector of $\angle BAC$ intersects $BC$ to $D$. Straight line $BP$ intersects $AC$ to $E$, and straight line $CP$ intersects $AB$ to $F$. Let the tangent of the circle $\Gamma$ at point $A$ intersect the line $EF$ at the point $Q$. Proof: $PQ\parallel BC$.

Consider a (right) square pyramid $ABCDV$ with the apex $V$ and the base (square) $ABCD$. Denote $d=AB/2$ and $\varphi$ the dihedral angle between planes $VAD$ and $ABC$. (1) Consider a line $XY$ connecting the skew lines $VA$ and $BC$, where $X$ lies on line $VA$ and $Y$ lies on line $BC$. Describe a construction of line $XY$ such that the segment $XY$ is of the smallest possible length. Compute the length of segment $XY$ in terms of $d,\varphi$. (2) Compute the distance $v$ between points $V$ and $X$ in terms of $d,\varphi.$

Consider two solid spherical balls, one centered at $(0, 0, \frac{21}{2} )$ with radius $6$, and the other centered at $(0, 0, 1)$ with radius $\frac 92$ . How many points $(x, y, z)$ with only integer coordinates (lattice points) are there in the intersection of the balls? $\text{(A)}\ 7 \qquad \text{(B)}\ 9 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15$

Let $ABC$ be a scalene triangle, $M$ be the midpoint of $BC,P$ be the common point of $AM$ and the incircle of $ABC$ closest to $A$, and $Q$ be the common point of the ray $AM$ and the excircle farthest from $A$. The tangent to the incircle at $P$ meets $BC$ at point $X$, and the tangent to the excircle at $Q$ meets $BC$ at $Y$. Prove that $MX=MY$.

Mari and Juri ordered a round pizza. Juri cut the pizza into four pieces by two straight cuts, none of which passed through the centre point of the pizza. Mari can choose two pieces not aside of these four, and Juri gets the rest two pieces. Prove that if Mari chooses the piece that covers the centre point of the pizza, she will get more pizza than Juri.

The sides of a convex $200$-gon $A_1 A_2 \dots A_{200}$ are colored red and blue in an alternating fashion. Suppose the extensions of the red sides determine a regular $100$-gon, as do the extensions of the blue sides. Prove that the $50$ diagonals $\overline{A_1A_{101}},\ \overline{A_3A_{103}},\ \dots, \ \overline{A_{99}A_{199}}$ are concurrent. [i]Proposed by: [b]Ankan Bhattacharya[/b][/i]

The area of the triangle bounded by the lines $y = x, y = -x$ and $y = 6$ is $ \mathbf{(A)}\; 12\qquad \mathbf{(B)}\; 12\sqrt2\qquad \mathbf{(C)}\; 24\qquad \mathbf{(D)}\; 24\sqrt2\qquad \mathbf{(E)}\; 36$

Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$

We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic.

Let $ABC$ be an acute triangle with circumcentre $O$. Let $\mathit{\Gamma}_B$ be the circle through $A$ and $B$ that is tangent to $AC$, and let $\mathit{\Gamma}_C$ be the circle through $A$ and $C$ that is tangent to $AB$. An arbitrary line through $A$ intersects $\mathit{\Gamma}_B$ again in $X$ and $\mathit{\Gamma}_C$ again in $Y$. Prove that $|OX|=|OY|$.

Quadrilateral $ABCD$ inscribed in circle with center $O$. Let lines $AD$ and $BC$ intersects at $M$, lines $AB$ and $CD$- at $N$, lines $AC$ and $BD$ -at $P$, lines $OP$ and $MN$ at $K$. Proved that $ \angle AKP = \angle PKC$. As I know, this problem was very short solution by polars, but in olympiad for this solution gives maximum 4 balls (in marking schemes written, that needs to prove all theorems about properties of polars)

You are given a convex hexagon with parallel opposite sides. For each pair of opposite sides, a line is drawn parallel to these sides and equidistant from them. Prove that the three lines thus obtained intersect at one point if and only if the lengths of the opposite sides are equal. [i]Proposed by Nazar Serdyuk[/i]

The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region? [asy] pen white = gray(1); pen gray = gray(0.5); draw((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle); fill((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle, gray); draw((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle); fill((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle, white); draw((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle); fill((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle, white); draw((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle); fill((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle, white); draw((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle); fill((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle, white); [/asy] $\textbf{(A) } 4\qquad\textbf{(B) }12 - 4\sqrt{3} \qquad\textbf{(C) } 3\sqrt{3} \qquad \textbf{(D) }4\sqrt{3}\qquad \textbf{(E) }16 - \sqrt{3}$

Let $ ABCD $ be a rhombus with center $ O. $ $ P $ is a point lying on the side $ AB. $ Let $ I, $ $ J, $ and $ L $ be the incenters of triangles $ PCD, $ $ PAD, $ and $PBC, $ respectively. Let $ H $ and $ K $ be orthocenters of triangles $ PLB $ and $ PJA, $ respectively. Prove that $ OI \perp HK. $ [i]Proposed by buratinogigle[/i]

Consider an isosceles triangle $ABC$ with $AB=AC$, and a circle $\omega$ which is tangent to the sides $AB$ and $AC$ of this triangle and intersects the side $BC$ at the points $K$ and $L$. The segment $AK$ intersects the circle $\omega$ at a point $M$ (apart from $K$). Let $P$ and $Q$ be the reflections of the point $K$ in the points $B$ and $C$, respectively. Show that the circumcircle of triangle $PMQ$ is tangent to the circle $\omega$.

Suppose $ABCD$ and $A_1B_1C_1D_1$ be quadrilaterals with corresponding angles equal. Also $AB=A_1B_1$, $AC=A_1C_1$, $BD=B_1D_1$. Are the quadrilaterals necessarily congruent?

Let $ABC$ be a triangle, $AD$ be the altitude from $A$ on to $BC$. Draw perpendiculars $DD_1$ and $DD_2$ from $D$ on to $AB$ and $AC$ respectively and let $p(A)$ be the length of the segment $D_1D_2$. Similarly define $p(B)$ and $p(C)$. Prove that $\frac{p(A)p(B)p(C)}{s^3}\le \frac18$ , where s is the semi-perimeter of the triangle $ABC$.