The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$? [asy] pair A,B,C; A=origin; B=(10,0); C=6.5*dir(15); dot(A); dot(B); dot(C); draw(B--A--C); draw(B--C,dashed); label("$6.5$",3.25*dir(15),NNW); label("$10$",(5,0),S); label("$s$",(8,1),NE); [/asy] $\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$

A program works as follows. If the input is given a natural number $n$ ($n \ge 2$), then the program consecutively performs the following procedure: it determines the greatest proper divisor of the number $ n$ (that is, different from $1$ and $n$) and subtracts it from the number $n$, then applies again the same procedure to the obtained result and so on. If the program cannot find any proper divisor of the given number at a step, then it stops and outputs the total number $m$ of procedures performed (this number can be equal to $0$). The input was given the number $n = 13^{13}$. Determine the respective number $m$ at the output.

A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly $n$ dollars where $n$ is a positive integer. The smallest value of $n$ is $\text{(A)} \ 1 \qquad \text{(B)} \ 13 \qquad \text{(C)} \ 25 \qquad \text{(D)} \ 26 \qquad \text{(E)} \ 100$

Random sequences $a_1, a_2, . . .$ and $b_1, b_2, . . .$ are chosen so that every element in each sequence is chosen independently and uniformly from the set $\{0, 1, 2, 3, . . . , 100\}$. Compute the expected value of the smallest nonnegative integer $s$ such that there exist positive integers $m$ and $n$ with $$s =\sum^m_{i=1} a_i =\sum^n_{j=1}b_j .$$

Define the two sequences $a_0, a_1, a_2, \cdots$ and $b_0, b_1, b_2, \cdots$ by $a_0 = 3$ and $b_0 = 1$ with the recurrence relations $a_{n+1} = 3a_n + b_n$ and $b_{n+1} = 3b_n - a_n$ for all nonnegative integers $n.$ Let $r$ and $s$ be the remainders when $a_{32}$ and $b_{32}$ are divided by $31,$ respectively. Compute $100r + s.$

$(CZS 5)$ A convex quadrilateral $ABCD$ with sides $AB = a, BC = b, CD = c, DA = d$ and angles $\alpha = \angle DAB, \beta = \angle ABC, \gamma = \angle BCD,$ and $\delta = \angle CDA$ is given. Let $s = \frac{a + b + c +d}{2}$ and $P$ be the area of the quadrilateral. Prove that $P^2 = (s - a)(s - b)(s - c)(s - d) - abcd \cos^2\frac{\alpha +\gamma}{2}$

Find all positive integers $k>1$, such that there exist positive integer $n$, such that the number $A=17^{18n}+4.17^{2n}+7.19^{5n}$ is product of $k$ consecutive positive integers.

For breakfast, Mihir always eats a bowl of Lucky Charms cereal, which consists of oat pieces and marshmallow pieces. He denes the luckiness of a bowl of cereal to be the ratio of the number of marshmallow pieces to the total number of pieces. One day, Mihir notices that his breakfast cereal has exactly $90$ oat pieces and $9$ marshmallow pieces, and exclaims, "This is such an unlucky bowl!" How many marshmallow pieces does Mihir need to add to his bowl to double its luckiness?

Segment $\overline{PQ}$ is drawn and squares $ABPQ$ and $CDQP$ are constructed in the plane such that they lie on opposite sides of segment $\overline{PQ}.$ If $PQ=1,$ find $BD.$

Let $ABC$ be the triangle in which $AB> AC$. Circle $\omega_a$ touches the segment of the $BC$ at point $D$, the extension of the segment $AB$ towards point $B$ at the point $F$, and the extension of the segment $AC$ towards point $C$ at the point $E$. The ray $AD$ intersects circle $\omega_a$ for second time at point $M$. Denote the circle circumscribed around the triangle $CDM$ by $\omega$. Circle $\omega$ intersects the segment $DF$ at N. Prove that $FN > ND$.

Find integer $n$ such that both $n-86$ and $n + 86$ are perfect squares.

Let $a,b,c \in [1, 3]$ and satisfy the following conditions: $ max \{a, b, c\}\ge 2$ and $ a + b + c = 5$ What is the smallest possible value of $a^2 + b^2 + c^2$?

Find all sets of six points in the plane, no three collinear, such that if we partition the set into two sets, then the obtained triangles are congruent.

Let $\leftarrow$ denote the left arrow key on a standard keyboard. If one opens a text editor and types the keys "ab$\leftarrow$ cd $\leftarrow \leftarrow$ e $\leftarrow \leftarrow$ f", the result is "faecdb". We say that a string $B$ is [i]reachable[/i] from a string $A$ if it is possible to insert some amount of $\leftarrow$'s in $A$, such that typing the resulting characters produces $B$. So, our example shows that "faecdb" is reachable from "abcdef". Prove that for any two strings $A$ and $B$, $A$ is reachable from $B$ if and only if $B$ is reachable from $A$.

Let be a finite set $ S. $ Determine the number of functions $ f:S\rightarrow S $ that satisfy $ f\circ f=f. $

Julius has a set of five positive integers whose mean is 100. If Julius removes the median of the set of five numbers, the mean of the set increases by 5, and the median of the set decreases by 5. Find the maximum possible value of the largest of the five numbers Julius has.

Let $ n$ be a $ 5$-digit number, and let $ q$ and $ r$ be the quotient and remainder, respectively, when $ n$ is divided by $ 100$. For how many values of $ n$ is $ q \plus{} r$ divisible by $ 11$? $ \textbf{(A)}\ 8180 \qquad \textbf{(B)}\ 8181 \qquad \textbf{(C)}\ 8182 \qquad \textbf{(D)}\ 9000 \qquad \textbf{(E)}\ 9090$

Let $ABC$ be a triangle and let $I$ and $O$ denote its incentre and circumcentre respectively. Let $\omega_A$ be the circle through $B$ and $C$ which is tangent to the incircle of the triangle $ABC$; the circles $\omega_B$ and $\omega_C$ are defined similarly. The circles $\omega_B$ and $\omega_C$ meet at a point $A'$ distinct from $A$; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA',BB'$ and $CC'$ are concurrent at a point on the line $IO$. [i](Russia) Fedor Ivlev[/i]

For a given triangle ABC, let X be a variable point on the line BC such that the point C lies between the points B and X. Prove that the radical axis of the incircles of the triangles ABX and ACX passes through a point independent of X. This is a slight extension of the [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=41033]IMO Shortlist 2004 geometry problem 7[/url] and can be found, together with the proposed solution, among the files uploaded at http://www.mathlinks.ro/Forum/viewtopic.php?t=15622 . Note that the problem was proposed by Russia. I could not find the names of the authors, but I have two particular persons under suspicion. Maybe somebody could shade some light on this... Darij

Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$, while $BD$ and $CE$ meet at $Q$. Find the area of $APQD$.

A circle which is tangent to the sides $ [AB]$ and $ [BC]$ of $ \triangle ABC$ is also tangent to its circumcircle at the point $ T$. If $ I$ is the incenter of $ \triangle ABC$ , show that $ \widehat{ATI}\equal{}\widehat{CTI}$

Suppose a real number $a$ is a root of a polynomial with integer coefficients $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$. Let $G=|a_n|+|a_{n-1}|+...+|a_1|+|a_0|$. We say that $G$ is a [i]gingado [/i] of $a$. For example, as $2$ is root of $P(x)=x^2-x-2$, $G=|1|+|-1|+|-2|=4$, we say that $4$ is a [i]gingado[/i] of $2$. What is the fourth largest real number $a$ such that $3$ is a [i]gingado [/i] of $a$?

Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60^\circ$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$, respectively. (a) Prove that line $OH$ intersects both segments $AB$ and $AC$. (b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$.

For a real number $ x$, define $ \heartsuit (x)$ to be the average of $ x$ and $ x^2$. What is $ \heartsuit(1) \plus{} \heartsuit(2) \plus{}\heartsuit(3)$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 20$

Let $BCDK$ be a convex quadrilateral such that $BC=BK$ and $DC=DK$. $A$ and $E$ are points such that $ABCDE$ is a convex pentagon such that $AB=BC$ and $DE=DC$ and $K$ lies in the interior of the pentagon $ABCDE$. If $\angle ABC=120^{\circ}$ and $\angle CDE=60^{\circ}$ and $BD=2$ then determine area of the pentagon $ABCDE$.