Heather and Kyle need to mow a lawn and paint a room. If Heather does both jobs by herself, it will take her a total of nine hours. If Heather mows the lawn and, after she finishes, Kyle paints the room, it will take them a total of eight hours. If Kyle mows the lawn and, after he finishes, Heather paints the room, it will take them a total of seven hours. If Kyle does both jobs by himself, it will take him a total of six hours. It takes Kyle twice as long to paint the room as it does for him to mow the lawn. The number of hours it would take the two of them to complete the two tasks if they worked together to mow the lawn and then worked together to paint the room is a fraction $\tfrac{m}{n}$where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Kelvin the Frog has a pair of standard fair $8$-sided dice (each labelled from $1$ to $8$). Alex the sketchy Kat also has a pair of fair $8$-sided dice, but whose faces are labelled differently (the integers on each Alex's dice need not be distinct). To Alex's dismay, when both Kelvin and Alex roll their dice, the probability that they get any given sum is equal! Suppose that Alex's two dice have $a$ and $b$ total dots on them, respectively. Assuming that $a \neq b$, find all possible values of $\min \{a,b\}$.

Let $\triangle ABC$ be a right-angled triangle and $BC > AC$. $M$ is a point on $BC$ such that $BM = AC$ and $N$ is a point on $AC$ such that $AN = CM$. Find the angle between $BN$ and $AM$.

On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.

Let $m$ be a fixed positive integer. The infinite sequence $\{a_n\}_{n\geq 1}$ is defined in the following way: $a_1$ is a positive integer, and for every integer $n\geq 1$ we have $$a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}$$ For each $m$, determine all possible values of $a_1$ such that every term in the sequence is an integer.

A prefrosh is participating in Caltech’s “Rotation.” They must rank Caltech’s $8$ houses, which are Avery, Page, Lloyd, Venerable, Ricketts, Blacker, Dabney, and Fleming, each a distinct integer rating from $1$ to $8$ inclusive. The conditions are that the rating $x$ they give to Fleming is at most the average rating $y$ given to Ricketts, Blacker, and Dabney, which is in turn at most the average rating $z$ given to Avery, Page, Lloyd, and Venerable. Moreover $x, y, z$ are all integers. How many such rankings can the prefrosh provide?

In a plane we choose a cartesian system of coordinates. A point $A(x,y)$ in the plane is called an integer point if and only if both $x$ and $y$ are integers. An integer point $A$ is called invisible if on the segment $(OA)$ there is at least one integer point. Prove that for each positive integer $n$ there exists a square of side $n$ in which all the interior integer points are invisible.

Let $a_1,a_2,\cdots ,a_n$ be $n$ positive numbers such that $\sum \limits_{i=1}^n\sqrt{a_i}=\sqrt{n}$. Then$$\prod \limits_{i=1}^{n-1}\left(1+\frac{1}{a_i}\right)^{a_{i+1}}\left(1+\frac{1}{a_n}\right)^{a_1}\geq 1+\frac{n}{\sum \limits_{i=1}^na_i}$$

The symmetric difference of two homothetic triangles $T_1$ and $T_2$ consists of six triangles $t_1, \ldots, t_6$ with circumcircles $\omega_1, \omega_2, \ldots, \omega_6$ (counterclockwise, no two intersect). Circle $\Omega_1$ with center $O_1$ is externally tangent to $\omega_1, \omega_3,$ and $\omega_5$; circle $\Omega_2$ with center $O_2$ is externally tangent to $\omega_2, \omega_4,$ and $\omega_6$; circle $\Omega_3$ with center $O_3$ is internally tangent to $\omega_1, \omega_3,$ and $\omega_5$; circle $\Omega_4$ with center $O_4$ is internally tangent to $\omega_2, \omega_4,$ and $\omega_6$. Prove that $O_1O_3 = O_2O_4$. [i]Proposed by Ilya Zamotorin[/i]

Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? $ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$

Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.

Show that there exists a sequence of positive integers $x_1, x_2,…x_n,…$ that satisfies the following two conditions: (i) Every positive integer appears exactly once, (ii) For every $n=1,2,…$ the partial sum $x_1+x_2+…+x_n$ is divisible by $n^n$.

A unit cube $C$ is rotated around one of its diagonals for the angle $\pi /3$ to form a cube $C'$. Find the volume of the intersection of $C$ and $C'$.

Two circles $(O_1,R_1)$,$(O_2,R_2)$ lie each external to the other. Find : a) the minimum length of the segment connecting points of the circles b) the max length of the segment connecting points of the circles

Find the least possible number of elements which can be deleted from the set $\{1,2,...,20\}$ so that the sum of no two different remaining numbers is not a perfect square. N. Sedrakian , I.Voronovich

Three circle of unit radius passing through the point $P$ and one of the points of $A, B$ and $C$ each. What can be the radius of the circumcircle of the triangle $ABC$?

A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5, $ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$? $\textbf{(A)}\ {-}18.5 \qquad\textbf{(B)}\ {-}13.5 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 13.5 \qquad\textbf{(E)}\ 18.5$

A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\frac{1}{3}$ and on region $B$ is $\frac{1}{2}$. The probability of the arrow stopping on region $C$ is: $\text{(A)}\ \frac{1}{12} \qquad \text{(B)}\ \frac{1}{6} \qquad \text{(C)}\ \frac{1}{5} \qquad \text{(D)}\ \frac{1}{3} \qquad \text{(E)}\ \frac{2}{5}$

In the triangle $ABC$, the angle - bisector $AD$ ($D \in BC$) and the median $BE$ ($E \in AC$) intersect at point $P$. Lines $AB$ and $CP$ intesect at point $F$. The parallel through $B$ to $CF$ intersects $DF$ at point $M$. Prove that $DM = BF$

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

Consider the figure below, not drawn to scale. In this figure, assume that$AB \perp BE$ and $AD \perp DE$. Also, let $AB = \sqrt6$ and $\angle BED =\frac{\pi}{6}$ . Find $AC$. [img]https://cdn.artofproblemsolving.com/attachments/2/d/f87ac9f111f02e261a0b5376c766a615e8d1d8.png[/img]

Consider the equation $$FORTY + TEN + TEN = SIXTY$$ , where each of the ten letters represents a distinct digit from $0$ to $9$. Find all possible values of $SIXTY$ .

[asy] real t=pi/8;real u=7*pi/12;real v=13*pi/12; real ct=cos(t);real st=sin(t);real cu=cos(u);real su=sin(u); draw(unitcircle); draw((ct,st)--(-ct,st)--(cos(v),sin(v))); draw((cu,su)--(cu,st)); label("A",(-ct,st),W);label("B",(ct,st),E); label("M",(cu,su),N);label("P",(cu,st),S); label("C",(cos(v),sin(v)),W); //Credit to Zimbalono for the diagram[/asy] In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to $\textbf{(A) }3x+2\qquad\textbf{(B) }3x+1\qquad\textbf{(C) }2x+3\qquad\textbf{(D) }2x+2\qquad \textbf{(E) }2x+1$

Let triangle $ABC$ with incenter $I$ satisfy $AB = 3$, $AC = 4$, and $BC = 5$. Suppose that $D$ and $E$ lie on $AB$ and $AC$, respectively, such that $D$, $I$, and $E$ are collinear and $DE \perp AI$. Points $P$ and $Q$ lie on side $BC$ such that $IP = BP$ and $IQ = CQ$, and lines $DP$ and $EQ$ meet at $S$. Compute $SI^2$.