In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear. [i]proposed by Amirhossein Zabeti[/i]

A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of $8^\text{th}$-graders to $6^\text{th}$-graders is $5:3$, and the the ratio of $8^\text{th}$-graders to $7^\text{th}$-graders is $8:5$. What is the smallest number of students that could be participating in the project? $\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 79 \qquad \textbf{(E)}\ 89$

A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $ 2: 1$. The ratio of the rectangle's length to its width is $ 2: 1$. What percent of the rectangle's area is inside the square? $ \textbf{(A)}\ 12.5 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 75 \qquad \textbf{(E)}\ 87.5$

If $ 1 \minus{} y$ is used as an approximation to the value of $ \frac {1}{1 \plus{} y}$, $ |y| < 1$, the ratio of the error made to the correct value is: $ \textbf{(A)}\ y \qquad \textbf{(B)}\ y^2 \qquad \textbf{(C)}\ \frac {1}{1 \plus{} y} \qquad \textbf{(D)}\ \frac {y}{1 \plus{} y} \qquad \textbf{(E)}\ \frac {y^2}{1 \plus{} y}\qquad$

In a convex quadrilateral $ABCD$ we have $\angle ADB=\angle ACD$ and $AC=CD=DB$. If the diagonals $AC$ and $BD$ intersect at $X$, prove that $\frac{CX}{BX}-\frac{AX}{DX}=1$.

In triangle $ ABC$, angle $ C$ is a right angle and $ CB > CA$. Point $ D$ is located on $ \overline{BC}$ so that angle $ CAD$ is twice angle $ DAB$. If $ AC/AD \equal{} 2/3$, then $ CD/BD \equal{} m/n$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$. $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 22\qquad \textbf{(E)}\ 26$

Trapezoid $ ABCD$ has $ AD\parallel{}BC$, $ BD \equal{} 1$, $ \angle DBA \equal{} 23^{\circ}$, and $ \angle BDC \equal{} 46^{\circ}$. The ratio $ BC: AD$ is $ 9: 5$. What is $ CD$? $ \textbf{(A)}\ \frac {7}{9}\qquad \textbf{(B)}\ \frac {4}{5}\qquad \textbf{(C)}\ \frac {13}{15} \qquad \textbf{(D)}\ \frac {8}{9}\qquad \textbf{(E)}\ \frac {14}{15}$

Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$ [i]2010 Hirosaki University School of Medicine entrance exam[/i]

Prove that in triangle $ABC$, radical center of its excircles lies on line $GI$, which $G$ is Centroid of triangle $ABC$, and $I$ is the incenter.

Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.

Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$ [i]Proposed by Ray Li[/i]

Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ ad $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares? [asy] pair A,B,C,D,E,F,G,H,I,J; A = (0.5,2); B = (1.5,2); C = (1.5,1); D = (0.5,1); E = (0,1); F = (0,0); G = (1,0); H = (1,1); I = (2,1); J = (2,0); draw(A--B); draw(C--B); draw(D--A); draw(F--E); draw(I--J); draw(J--F); draw(G--H); draw(A--J); filldraw(A--B--C--I--J--cycle,grey); draw(E--I); dot("$A$", A, NW); dot("$B$", B, NE); dot("$C$", C, NE); dot("$D$", D, NW); dot("$E$", E, NW); dot("$F$", F, SW); dot("$G$", G, S); dot("$H$", H, N); dot("$I$", I, NE); dot("$J$", J, SE);[/asy] $\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac7{24} \qquad \textbf{(C)}\ \frac13 \qquad \textbf{(D)}\ \frac38 \qquad \textbf{(E)}\ \frac5{12}$

Starting with two points A and B, some circles and points are constructed as shown in the figure:[list][*]the circle with centre A through B [*]the circle with centre B through A [*]the circle with centre C through A [*]the circle with centre D through B [*]the circle with centre E through A [*]the circle with centre F through A [*]the circle with centre G through A[/list] [i][size=75](I think the wording is not very rigorous, you should assume intersections from the drawing)[/size][/i] Show that $M$ is the midpoint of $AB$. [img]https://cdn.artofproblemsolving.com/attachments/d/4/2352ab21cc19549f0381e88ddde9dce4299c2e.png[/img]

The bisector of angle $\angle BAC$ of triangle $ABC$ intersects arc $BC$ (not containing point $A$) of the circle circumscribed around this triangle at point $P$. Segment $AP$ is divided by side $BC$ in ratio $k$ (counting from vertex $A$). Find the perimeter of triangle $ABC$ if $BC = a$.

Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives \begin{eqnarray*} &\ & \binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1+\binom{1000}{2}(0.2)^2+\cdots+\binom{1000}{1000}(0.2)^{1000}\\ &\ & = A_0 + A_1 + A_2 + \cdots + A_{1000}, \end{eqnarray*} where $A_k = \binom{1000}{k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For which $k$ is $A_k$ the largest?

Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.

Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$

$ABC$ and $A'B'C'$ are equilateral triangles with parallel sides and the same center, as in the figure. The distance between side $BC$ and side $B'C'$ is $\frac{1}{6}$ the altitude of $\triangle ABC$. The ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$ is [asy] size(170); defaultpen(linewidth(0.7)+fontsize(10)); pair H=origin, B=(1,-(1/sqrt(3))), C=(-1,-(1/sqrt(3))), A=(0,(2/sqrt(3))), E=(2,-(2/sqrt(3))), F=(-2,-(2/sqrt(3))), D=(0,(4/sqrt(3))); draw(A--B--C--A^^D--E--F--D); label("$A'$", A, dir(90)); label("$B'$", B, SE); label("$C'$", C, SW); label("$A$", D, dir(90)); label("$B$", E, dir(0)); label("$C$", F, W); [/asy] $ \textbf{(A)}\ \frac{1}{36}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{\sqrt{3}}{4}\qquad\textbf{(E)}\ \frac{9+8\sqrt{3}}{36} $

A triangle $ABC$ is given. For every positive numbers $p,q,r$, let $A',B',C'$ be the points such that $\overrightarrow{BA'} = p\overrightarrow{AB}, \overrightarrow{CB'}=q\overrightarrow{BC} $, and $\overrightarrow{AC'}=r\overrightarrow{CA}$. Define $f(p,q,r)$ as the ratio of the area of $\vartriangle A'B'C'$ to that of $\vartriangle ABC$. Prove that for all positive numbers $x,y,z$ and every positive integer $n$, $\sum_{k=0}^{n-1}f(x+k,y+k,z+k) = n^3f\left(\frac{x}{n},\frac{y}{n},\frac{z}{n}\right)$.

Let $P(x,y)=(x^2y^3,x^3y^5)$, $P^1=P$ and $P^{n+1}=P\circ P^n$. Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$, and $f(n)$ be the degree of $p_n(x)$. Find \[\lim_{n\to\infty}f(n)^{1/n}\]

Ted flips seven fair coins. there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that Ted flips at least two heads given that he flips at least three tails. Find $m+n$.

On side $AB$ of triangle $ABC$, points $M$ and $K$ are taken ($M$ on segment $AK$). It is known that $AM: MK: MB = a: b: c$. Straight lines $CM$ and $CK$ intersect for the second time the circumscribed circle of the triangle $ABC$ at points $E$ and $F$, respectively. In what ratio does the circumscribed circle of the triangle $BMF$ divide the segment $BE$?

The sides of a triangle are in the ratio $ 6: 8: 9$. Then: $ \textbf{(A)}\ \text{the triangle is obtuse} \qquad\textbf{(B)}\ \text{the angles are in the ratio } 6: 8: 9$ $ \textbf{(C)}\ \text{the triangle is acute}$ $ \textbf{(D)}\ \text{the angle opposite the largest side is double the angle opposite the smallest side}$ $ \textbf{(E)}\ \text{none of these}$

Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders? $\textbf{(A) }\text{The second height is 10\% less than the first.}$ $\textbf{(B) }\text{The first height is 10\% more than the second.}$ $\textbf{(C) }\text{The second height is 21\% less than the first.}$ $\textbf{(D) }\text{The first height is 21\% more than the second.}$ $\textbf{(E) }\text{The second height is 80\% of the first.}$

Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$