A baker uses $6\tfrac{2}{3}$ cups of flour when she prepares $\tfrac{5}{3}$ recipes of rolls. She will use $9\tfrac{3}{4}$ cups of flour when she prepares $\tfrac{m}{n}$ recipes of rolls where m and n are relatively prime positive integers. Find $m + n.$

(a) Is there a natural number $n$ such that the number $2^n$ has last digit $6$ and the sum of the other digits is $2$? b) Are there natural numbers $a$ and $m\ge 3$ such that the number $a^m$ has last digit $6$ and the sum of the other digits is 3?

Let $O$ be the circumcircle of $\triangle ABC$, with $AC=BC$ end let $D=AO\cap BC$. If $BD$ and $CD$ are integer numbers and $AO-CD$ is prime, determine such three numbers.

Circle $ \Gamma_{1},$ with radius $ r,$ is internally tangent to circle $ \Gamma_{2}$ at $ S.$ Chord $ AB$ of $ \Gamma_{2}$ is tangent to $ \Gamma_{1}$ at $ C.$ Let $ M$ be the midpoint of arc $ AB$ (not containing $ S$), and let $ N$ be the foot of the perpendicular from $ M$ to line $ AB.$ Prove that $ AC\cdot CB\equal{}2r\cdot MN.$

Given an integer $n \geqslant 2$. Suppose there is a point $P$ inside a convex cyclic $2n$-gon $A_1 \ldots A_{2n}$ satisfying $$\angle PA_1A_2 = \angle PA_2A_3 = \ldots = \angle PA_{2n}A_1,$$prove that $$ \prod_{i=1}^{n} \left|A_{2i - 1}A_{2i} \right| = \prod_{i=1}^{n} \left|A_{2i}A_{2i+1} \right|,$$where $A_{2n + 1} = A_1$.

A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds $T_{max}$. One end of the cord is attached to a fixed point, the other is attached to an object of mass $3m$. If a second, smaller object of mass m moving at an initial speed $v_0$ strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of $v_f$. All motion occurs on a horizontal, frictionless surface. Find $v_f/v_0$. $ \textbf{(A)}\ 1/\sqrt{12}\qquad\textbf{(B)}\ 1/\sqrt{2}\qquad\textbf{(C)}\ 1/\sqrt{6} \qquad\textbf{(D)}\ 1/\sqrt{3}\qquad\textbf{(E)}\ \text{none of the above} $

Let $S_1$ and $S_2$ be sets of points on the coordinate plane $\mathbb{R}^2$ defined as follows \[S_1={(x,y)\in \mathbb{R}^2:|x+|x||+|y+|y||\le 2}\] \[S_2={(x,y)\in \mathbb{R}^2:|x-|x||+|y-|y||\le 2}\] Find the area of the intersection of $S_1$ and $S_2$

Let $ABC$ be a equilateral triangle and let $P$ be any point on the minor arc $AC$ of the circumcircle of $ABC$.Prove that $PB=PA+PC$

There are 100 apples on the table with total weight of 10 kg. Each apple weighs no less than 25 grams. The apples need to be cut for 100 children so that each of the children gets 100 grams. Prove that you can do it in such a way that each piece weighs no less than 25 grams.

Georg writes the numbers from $1$ to $15$ on different pieces of paper. He attempts to sort these pieces of paper into two stacks so that none of the stacks contains two numbers whose sum is a square number.Prove that this is impossible. (The square numbers are the numbers $0 = 0^2$, $1 = 1^2$, $4 = 2^2$, $9 = 3^2$ etc.)

Pegs are put in a board $ 1$ unit apart both horizontally and vertically. A reubber band is stretched over $ 4$ pegs as shown in the figure, forming a quadrilateral. Its area in square units is [asy] int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<4; j=j+1) { dot((i,j)); }} draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7)); [/asy] $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 4.5 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 5.5 \qquad \textbf{(E)}\ 6$

Calculate the $100$th derivative of the function \[\frac{1}{x^2+3x+2}\] at $x=0$ with $10\%$ relative error.

There is a circle $c$ centered about the origin of radius $ 1$. There are circles $c_1$,$ . . .$ ,$c_6$, each of radius $r_1$, such that each circle is completely inside c and is tangent to it, and $c_2$ is tangent to $c_1$, $c_3$ is tangent to $c_2$, . . ., and $c_1$ is tangent to $c_6$. There is a circle $d$ which is tangent to $c$, $c_1$, and $c_2$, but does not intersect any of these circles. What is the radius of circle $d$? Express your answer in the form $\frac{a+b\sqrt{c}}{d}$ , where $a,b,c,d$ are integers, $d$ is positive and as small as possible, and $c$ is squarefree.

Let $ n\ge 2 $ be a natural number. Prove the following propositions: [b]a)[/b] $ a_1,a_2,\ldots ,a_n\in\mathbb{R}\wedge a_1+\cdots +a_n=a_1^2+\cdots +a_n^2\implies a_1+\cdots +a_n\le a_n. $ [b]b)[/b] $ x\in [1,n]\implies\exists b_1,b_2,\ldots ,b_n\in\mathbb{R}_{\ge 0}\quad x=b_1+\cdots +b_n=b_1^2 +\cdots +b_n^2 . $

A circle $\omega$ not passing through any vertex of $\triangle ABC$ intersects each of the segments $AB$, $BC$, $CA$ in 2 distinct points. Prove that the incenter of $\triangle ABC$ lies inside $\omega$. [i]Evan O' Dorney.[/i]

$P(x)$ is polynomial with degree $n\geq 2$ and nonnegative coefficients. $a,b,c$ - sides for some triangle. Prove, that $\sqrt[n]{P(a)},\sqrt[n]{P(b)},\sqrt[n]{P(c)}$ are sides for some triangle too.

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

Let $ABCD$ be a convex quadrilateral with \[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\] The line through \(C\), parallel to \(AD\), intersects the external angle bisector of \(\angle ABC\) at point \(T\). Prove that the angles $\angle ATB$, $\angle TBC$, $\angle BCD$, $\angle CDA$, $\angle DAT$ can be divided into two groups, so that the angles in each group have a sum of $270^{\circ}$. [i] Miroslav Marinov, Bulgaria [/i]

Let $ABC$ be an isosceles triangle with $AB=AC$. Points $D$ and $E$ lie on the sides $AB$ and $AC$, respectively. The line passing through $B$ and parallel to $AC$ meets the line $DE$ at $F$. The line passing through $C$ and parallel to $AB$ meets the line $DE$ at $G$. Prove that \[\frac{[DBCG]}{[FBCE]}=\frac{AD}{DE} \]

On the plane are given four distinct points $A,B,C,D$ on a line $g$ in this order, at the mutual distances $AB = a, BC = b, CD = c$. (a) Construct (if possible) a point $P$ outside line $g$ such that $\angle APB =\angle BPC =\angle CPD$. (b) Prove that such a point $P$ exists if and only if $ (a+b)(b+c) < 4ac$

Let $ a,$ $ b,$ $ c,$ $ d,$ and $ e$ be five consecutive terms in an arithmetic sequence, and suppose that $ a \plus{} b \plus{} c \plus{} d \plus{} e \equal{} 30.$ Which of the following can be found? $ \textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$

If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be $ \textbf{(A)}\ 200\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 202\qquad\textbf{(D)}\ 203\qquad\textbf{(E)}\ 204 $

$\textbf{a) }$ Let $p \geq 2$ be a natural number and $G_p = \bigcup\limits_{n \in \mathbb{N}} \lbrace z \in \mathbb{C} \mid z^{p^n}=1 \rbrace.$ Prove that $(G_p, \cdot)$ is a subgroup of $(\mathbb{C}^*, \cdot).$ $\textbf{b) }$ Let $(H, \cdot)$ be an infinite subgroup of $(\mathbb{C}^*, \cdot).$ Prove that all proper subgroups of $H$ are finite if and only if $H=G_p$ for some prime $p.$

In a triangle, $AB=BC$ and $M$ is the midpoint of $AC$. $H$ is chosen on $BC$ so that $MH$ is perpendicular to $BC$. $P$ is the midpoint of $MH$. Prove that $AH$ is perpendicular to $BP$.

Each side of triangle $ABC$ is $12$ units. $D$ is the foot of the perpendicular dropped from $A$ on $BC$, and $E$ is the midpoint of $AD$. The length of $BE$, in the same unit, is: ${{ \textbf{(A)}\ \sqrt{18} \qquad\textbf{(B)}\ \sqrt{28} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ \sqrt{63} }\qquad\textbf{(E)}\ \sqrt{98} } $