In right $ \triangle ABC$ with hypotenuse $ \overline{AB}$, $ AC \equal{} 12$, $ BC \equal{} 35$, and $ \overline{CD}$ is the altitude to $ \overline{AB}$. Let $ \omega$ be the circle having $ \overline{CD}$ as a diameter. Let $ I$ be a point outside $ \triangle ABC$ such that $ \overline{AI}$ and $ \overline{BI}$ are both tangent to circle $ \omega$. The ratio of the perimeter of $ \triangle ABI$ to the length $ AB$ can be expressed in the form $ \displaystyle\frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

Find $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$ where $_iC_j$ is a binominal coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.

The sides of $\triangle ABC$ have lengths $6, 8$ and $10$. A circle with center $P$ and radius $1$ rolls around the inside of $\triangle ABC$, always remaining tangent to at least one side of the triangle. When $P$ first returns to its original position, through what distance has $P$ traveled? [asy] draw((0,0)--(8,0)--(8,6)--(0,0)); draw(Circle((4.5,1),1)); draw((4.5,2.5)..(5.55,2.05)..(6,1), EndArrow); dot((0,0)); dot((8,0)); dot((8,6)); dot((4.5,1)); label("A", (0,0), SW); label("B", (8,0), SE); label("C", (8,6), NE); label("8", (4,0), S); label("6", (8,3), E); label("10", (4,3), NW); label("P", (4.5,1), NW); [/asy] $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 17 $

A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? [asy] draw((0,8)--(0,0)--(4,0)--(4,8)--(0,8)--(3.5,8.5)--(3.5,8)); draw((2,-1)--(2,9),dashed);[/asy] $ \text{(A)}\ \frac{1}{3}\qquad\text{(B)}\ \frac{1}{2}\qquad\text{(C)}\ \frac{3}{4}\qquad\text{(D)}\ \frac{4}{5}\qquad\text{(E)}\ \frac{5}{6} $

A lake contains $250$ trout, along with a variety of other fish. When a marine biologist catches and releases a sample of $180$ fish from the lake, $30$ are identified as trout. Assume that the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake? $\textbf{(A)}~1250\qquad \textbf{(B)}~1500\qquad \textbf{(C)}~1750\qquad \textbf{(D)}~1800\qquad \textbf{(E)}~2000$

In cyclic quadrilateral $ABCD$, $\angle A+ \angle D=\frac{\pi}{2}$. $AC$ intersects $BD$ at ${E}$. A line ${l}$ cuts segment $AB, CD, AE, DE$ at $X, Y, Z, T$ respectively. If $AZ=CE$ and $BE=DT$, prove that the diameter of the circumcircle of $\triangle EZT$ equals $XY$.

For two real numbers $ a$, $ b$, with $ ab\neq 1$, define the $ \ast$ operation by \[ a\ast b=\frac{a+b-2ab}{1-ab}.\] Start with a list of $ n\geq 2$ real numbers whose entries $ x$ all satisfy $ 0<x<1$. Select any two numbers $ a$ and $ b$ in the list; remove them and put the number $ a\ast b$ at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains. $ a.$ Prove that this single number is the same regardless of the choice of pair at each stage. $ b.$ Suppose that the condition on the numbers $ x$ is weakened to $ 0<x\leq 1$. What happens if the list contains exactly one $ 1$?

The length of rectangle $ABCD$ is $5$ inches and its width is $3$ inches. Diagonal $AC$ is dibided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is: $\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac 53 \qquad \text{(C)} \ \frac 52 \qquad \text{(D)} \ \frac13\sqrt{34} \qquad \text{(E)} \ \frac13\sqrt{68}$

For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are in the ratio $ 1: 2: 3: 4: 5: 6$. What is the probability of rolling a total of 7 on the two dice? $ \textbf{(A) } \frac 4{63} \qquad \textbf{(B) } \frac 18 \qquad \textbf{(C) } \frac 8{63} \qquad \textbf{(D) } \frac 16 \qquad \textbf{(E) } \frac 27$

A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units? [asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(1,-1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,-1)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(1,1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,1)*unitcube, white, thick(), nolight);[/asy] $\textbf{(A)} \:1 : 6 \qquad\textbf{ (B)}\: 7 : 36 \qquad\textbf{(C)}\: 1 : 5 \qquad\textbf{(D)}\: 7 : 30\qquad\textbf{ (E)}\: 6 : 25$

In regular triangular pyramid $P-ABC$, $PO$ is its height, $M$ is the midpoint of $PO$. Draw the plane that passes $AM$ and parallel to $BC$. Now the triangular pyramid is divided into two parts. Find the ratio of their volume.

In $ \triangle ABC$ points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$, respectively. If $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT \equal{} 3$ and $ BT/ET \equal{} 4$, what is $ CD/BD$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)); pair A = (0,0); pair C = (2,0); pair B = dir(57.5)*2; pair E = waypoint(C--A,0.25); pair D = waypoint(C--B,0.25); pair T = intersectionpoint(D--A,E--B); label("$B$",B,NW);label("$A$",A,SW);label("$C$",C,SE);label("$D$",D,NE);label("$E$",E,S);label("$T$",T,2*W+N); draw(A--B--C--cycle); draw(A--D); draw(B--E);[/asy]$ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {2}{9}\qquad \textbf{(C)}\ \frac {3}{10}\qquad \textbf{(D)}\ \frac {4}{11}\qquad \textbf{(E)}\ \frac {5}{12}$

Quadrilateral $ABCD$ is both cyclic and circumscribed. Its incircle touches its sides $AB$ and $CD$ at points $X$ and $Y$, respectively. The perpendiculars to $AB$ and $CD$ drawn at $A$ and $D$, respectively, meet at point $U$; those drawn at $X$ and $Y$ meet at point $V$, and finally, those drawn at $B$ and $C$ meet at point $W$. Prove that points $U$, $V$ and $W$ are collinear. [i]Proposed by A. Golovanov[/i]

For positive integers $n$, let $f_2(n)$ denote the number of divisors of $n$ which are perfect squares, and $f_3(n)$ denotes the number of positive divisors which are perfect cubes. Prove that for each positive integer $k$ there exists a positive integer $n$ for which $\frac{f_2(n)}{f_3(n)}=k$.

$ABCD$ is a trapezium such that $AB\parallel DC$ and $\frac{AB}{DC}=\alpha >1$. Suppose $P$ and $Q$ are points on $AC$ and $BD$ respectively, such that \[\frac{AP}{AC}=\frac{BQ}{BD}=\frac{\alpha -1}{\alpha+1}\] Prove that $PQCD$ is a parallelogram.

Given an acute angled triangle $ ABC$ , $ O$ is the circumcenter and $ H$ is the orthocenter.Let $ A_1$,$ B_1$,$ C_1$ be the midpoints of the sides $ BC$,$ AC$ and $ AB$ respectively. Rays $ [HA_1$,$ [HB_1$,$ [HC_1$ cut the circumcircle of $ ABC$ at $ A_0$,$ B_0$ and $ C_0$ respectively.Prove that $ O$,$ H$ and $ H_0$ are collinear if $ H_0$ is the orthocenter of $ A_0B_0C_0$

Let $ ABC$ be triangle, $ I$ its in-center; $ A_1,B_1,C_1$ be the reflections of $ I$ in $ BC, CA, AB$ respectively. Suppose the circum-circle of triangle $ A_1B_1C_1$ passes through $ A$. Prove that $ B_1,C_1,I,I_1$ are concylic, where $ I_1$ is the in-center of triangle $ A_1,B_1,C_1$.

The ratio of the measures of two acute angles is $5:4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles? $\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270$

p1. Let $A = \{(x, y)|3x + 5y\ge 15, x + y^2\le 25, x\ge 0, x, y$ integer numbers $\}$. Find all pairs of $(x, zx)\in A$ provided that $z$ is non-zero integer. p2. A shop owner wants to be able to weigh various kinds of weight objects (in natural numbers) with only $4$ different weights. (For example, if he has weights $ 1$, $2$, $5$ and $10$. He can weighing $ 1$ kg, $2$ kg, $3$ kg $(1 + 2)$, $44$ kg $(5 - 1)$, $5$ kg, $6$ kg, $7$ kg, $ 8$ kg, $9$ kg $(10 - 1)$, $10$ kg, $11$ kg, $12$ kg, $13$ kg $(10 + 1 + 2)$, $14$ kg $(10 + 5 -1)$, $15$ kg, $16$ kg, $17$ kg and $18$ kg). If he wants to be able to weigh all the weight from $ 1$ kg to $40$ kg, determine the four weights that he must have. Explain that your answer is correct. p3. Given the following table. [img]https://cdn.artofproblemsolving.com/attachments/d/8/4622407a72656efe77ccaf02cf353ef1bcfa28.png[/img] Table $4\times 4$ ​​is a combination of four smaller table sections of size $2\times 2$. This table will be filled with four consecutive integers such that: $\bullet$ The horizontal sum of the numbers in each row is $10$ . $\bullet$ The vertical sum of the numbers in each column is $10$ $\bullet$ The sum of the four numbers in each part of $2\times 2$ which is delimited by the line thickness is also equal to $10$. Determine how many arrangements are possible. p4. A sequence of real numbers is defined as following: $U_n=ar^{n-1}$, if $n = 4m -3$ or $n = 4m - 2$ $U_n=- ar^{n-1}$, if $n = 4m - 1$ or $n = 4m$, where $a > 0$, $r > 0$, and $m$ is a positive integer. Prove that the sum of all the $ 1$st to $2009$th terms is $\frac{a(1+r-r^{2009}+r^{2010})}{1+r^2}$ 5. Cube $ABCD.EFGH$ is cut into four parts by two planes. The first plane is parallel to side $ABCD$ and passes through the midpoint of edge $BF$. The sceond plane passes through the midpoints $AB$, $AD$, $GH$, and $FG$. Determine the ratio of the volumes of the smallest part to the largest part.

A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<1000$. Find $r+s+t$.

A pyramid with the base $ABCD$ and the top $V$ is inscribed in a sphere. Let $AD = 2BC$ and let the rays $AB$ and $DC$ intersect in point $E$. Compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$.

The vertices of a regular $2012$-gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$, then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled $A_1$ and $A_4$. How is the tenth vertex labeled? [i]Proposed by A. Golovanov[/i]

The points with integer coordinates are painted by red if the product of $x$ and $y$ coordinates is divisible by $6$. Otherwise the points with integer coordinates are painted by white. Consider a very big square whose sides are parallel to the axis of the $xy-$plane. The ratio of white points over red points inside this square will be closer to $\textbf{(A)}\ \frac75 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac43 \qquad\textbf{(E)}\ \frac54$

Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that: \[PX || AC \ , \ PY ||AB \] Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$

Let $ ABCD$ be a parellogram with $ AB>AD$. Suposse the ratio between diagonals $ AC$ and $ BD$ is $ \frac {AC} {BD}\equal{}3$. Let $ r$ be the line symmetric to $ AD$ with respect to $ AC$ and $ s$ the line symmetric to $ BC$ with respect to $ BD$. If $ r$ and $ s$ intersect at $ P$ , find the ratio $ \frac {PA} {PB}$ Daniel