Given a segment of a circle, consisting of a straight edge and an arc. The length of the straight edge is $24$. The length between the midpoint of the straight edge and the midpoint of the arc is $6$. Find the radius of the circle.

Four people were guessing the number, $N$, of jellybeans in a jar. No two guesses were equally close to $N$. The closest guess was 80 jellybeans, the next closest guess was 60 jellybeans, followed by 49 jellybeans, and the furthest guess was 125 jellybeans. Find the sum of all possible values for $N$. [i]Proposed by CJ Quines[/i]

Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$). It is known that the circumcircles of triangles $AEM$ and $CDM$ are tangent. Find the angle $\angle BMC.$

Let $ a_i, b_i$ be positive real numbers, $ i\equal{}1,2,\ldots,n$, $ n\geq 2$, such that $ a_i<b_i$, for all $ i$, and also \[ b_1\plus{}b_2\plus{}\cdots \plus{} b_n < 1 \plus{} a_1\plus{}\cdots \plus{} a_n.\] Prove that there exists a $ c\in\mathbb R$ such that for all $ i\equal{}1,2,\ldots,n$, and $ k\in\mathbb Z$ we have \[ (a_i\plus{}c\plus{}k)(b_i\plus{}c\plus{}k) > 0.\]

Let $A_{1}$, $B_{1}$, $C_{1}$ be the feet of the altitudes of an acute-angled triangle $ABC$ issuing from the vertices $A$, $B$, $C$, respectively. Let $K$ and $M$ be points on the segments $A_{1}C_{1}$ and $B_{1}C_{1}$, respectively, such that $\measuredangle KAM = \measuredangle A_{1}AC$. Prove that the line $AK$ is the angle bisector of the angle $C_{1}KM$.

In racing over a distance $d$ at uniform speed, $A$ can beat $B$ by $20$ yards, $B$ can beat $C$ by $10$ yards, and $A$ can beat $C$ by $28$ yards. Then $d$, in yards, equals: ${{ \textbf{(A)}\ \text{Not determined by the given information} \qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 100 \qquad\textbf{(D)}\ 116}\qquad\textbf{(E)}\ 120} $

Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.

Find all reals $ k$ such that \[ a^3 \plus{} b^3 \plus{} c^3 \plus{} d^3 \plus{} 1\geq k(a \plus{} b \plus{} c \plus{} d) \] holds for all $ a,b,c,d\geq \minus{} 1$. [i]Edited by orl.[/i]

Let $ \sigma \in S_n$ and $ \alpha <2$. Evaluate$ \displaystyle\lim_{n\to\infty} \displaystyle\sum_{k\equal{}1}^{n}\frac{\sigma (k)}{k^{\alpha}}$.

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

For a prime number $p > 3$, define the following irreducible fraction: $$\frac{m}{n} = \frac{p-1}{2} + \frac{p-2}{3} + \ldots + \frac{2}{p-1} - 1$$ Prove that $m$ is divisible by $p$. [i]Proposed by Oleksii Masalitin[/i]

Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.

Let $n\ge 2$ be an integer. They are given $n + 1$ red points in the plane. Prove that there exist $2n$ circles $C_1 , C_2 , \ldots , C_n , D_1 , D_2 , \ldots , D_n$ such that: $\bullet$ $C_1 , C_2 ,\ldots , C_n$ are concentric. $\bullet$ $D_1 , D_2 ,\ldots , D_n$ are concentric. $\bullet$ For $k = 1, 2, 3,\ldots , n$ the circles $C_k$ and $D_k$ are disjoint. $\bullet$ For $k = 1, 2, 3,\ldots , n$ it is true that $C_k$ contains exactly $k$ red dots in its interior and $D_k$ contains exactly $n + 1 - k$ red dots in its interior.

Suppose $ABCD$ is a convex quadrilateral.Points $P,Q,R$ and $S$ are four points on the line segments $AB,BC,CD$ and $DA$ respectively.The line segments $PR$ and $QS$ meet at $T$.Suppose that each of the quadrilaterals $APTS,BQTP,CRTQ$ and $DSTR$ have an incircle.Prove that the quadrilateral $ABCD$ also has an incircle.

Triangle $ABC$ has side lengths $AB=12$, $BC=25$, and $CA=17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ=w$, the area of $PQRS$ can be expressed as the quadratic polynomial \[\text{Area}(PQRS)=\alpha w-\beta\cdot w^2\] Then the coefficient $\beta=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? $ \textbf{(A)}\ \dfrac{2}{5} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{4}{7} \qquad\textbf{(D)}\ \dfrac{3}{5} \qquad\textbf{(E)}\ \dfrac{4}{5} $

In the parallelogram $ABCD$, rays are released from its vertices towards its interior. The rays coming out of the vertices $A{}$ and $D{}$ intersect at $E{}$ and the rays coming out of the vertices $B{}$ and $C{}$ at point $F{}$. It is known that $\angle BAE=\angle BCF$ and $\angle CDE = \angle CBF$. Prove that $AB \parallel EF$. [i]Proposed by V. Eisenstadt[/i]

Let $a$, $b$, $c$, $d$ be real numbers. Suppose that $$\frac{a}{b+c}+\frac{b}{a+d}=\frac{3}{5},\qquad\frac{b}{c+d}+\frac{c}{a+b}=1,\qquad\frac{c}{a+d}+\frac{d}{b+c}=\frac{7}{5}.$$ Find the value of $$\frac{d}{a+b}+\frac{a}{c+d}.$$

Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n?$

Let $x_1, x_2, . . . , x_n$ be positive numbers. Prove that \[ \sum\limits_{i=1}^n \dfrac{x_i ^2}{x_i ^2+x_{i+1}x_{i+2}} \leq n-1 \] Where $x_{n+1}=x_1$ and $x_{n+2}=x_2$.

Find all triples ($a$,$b$,$n$) of positive integers such that $$a^3=b^2+2^n$$

Let $n$ be a given positive integer. Alice and Bob play a game. In the beginning, Alice determines an integer polynomial $P(x)$ with degree no more than $n$. Bob doesn’t know $P(x)$, and his goal is to determine whether there exists an integer $k$ such that no integer roots of $P(x) = k$ exist. In each round, Bob can choose a constant $c$. Alice will tell Bob an integer $k$, representing the number of integer $t$ such that $P(t) = c$. Bob needs to pay one dollar for each round. Find the minimum cost such that Bob can guarantee to reach his goal. [i]Proposed by ltf0501[/i]

Given an integer $k>1$, show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$, all greater than $1$, such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$-th powers of some integers. (Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$.)

Let $ABC$ be a triangle with orthocenter $H$. Let $\ell_b, \ell_c$ be the reflection of lines $AB$ and $AC$ about $AH$ respectively. Suppose $\ell_b$ intersect $CH$ at $P$, and $\ell_c$ intersect $BH$ at $Q$. Prove that $AH, PQ, BC$ are concurrent. [i]Proposed by Ivan Chan Kai Chin[/i]

Does there exist two infinite positive integer sets $S,T$, such that any positive integer $n$ can be uniquely expressed in the form $$n=s_1t_1+s_2t_2+\ldots+s_kt_k$$ ,where $k$ is a positive integer dependent on $n$, $s_1<\ldots<s_k$ are elements of $S$, $t_1,\ldots, t_k$ are elements of $T$?