Jae and Yoon are playing SunCraft. The probability that Jae wins the $n$-th game is $\frac{1}{n+2}.$ What is the probability that Yoon wins the first six games, assuming there are no ties?

Let $p, q$ be integers and $p, q > 1$ , $gcd(p, \,6q)=1$. Prove that:$$\sum_{k=1}^{q-1}\left \lfloor \frac{pk}{q}\right\rfloor^2 \equiv 2p \sum_{k=1}^{q-1}k\left\lfloor \frac{pk}{q} \right\rfloor (mod \, q-1)$$

From the vertex $C$ in triangle $ABC$, draw a straight line that bisects the median from $A$. In what ratio does this line divide the segment $AB$? [img]https://1.bp.blogspot.com/-SxWIQ12DIvs/XzcJv5xoV0I/AAAAAAAAMY4/Ezfe8bd7W-Mfp2Qi4qE_gppbh9Fzvb4XwCLcBGAsYHQ/s0/1995%2BMohr%2Bp3.png[/img]

Find all integers $n>1$ such that any prime divisor of $n^6-1$ is a divisor of $(n^3-1)(n^2-1)$.

Julia and Hansol are having a math-off. Currently, Julia has one more than twice as many points as Hansol. If Hansol scores $6$ more points in a row, he will tie Julia’s score. How many points does Julia have?

Sheila is making a regular-hexagon-shaped sign with side length $ 1$. Let $ABCDEF$ be the regular hexagon, and let $R, S,T$ and U be the midpoints of $FA$, $BC$, $CD$ and $EF$, respectively. Sheila splits the hexagon into four regions of equal width: trapezoids $ABSR$, $RSCF$ , $FCTU$, and $UTDE$. She then paints the middle two regions gold. The fraction of the total hexagon that is gold can be written in the form $m/n$ , where m and n are relatively prime positive integers. Compute $m + n$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9lLzIwOTVmZmViZjU3OTMzZmRlMzFmMjM1ZWRmM2RkODMyMTA0ZjNlLnBuZw==&rn=MjAyMCBCTVQgSW5kaXZpZHVhbCAxMy5wbmc=[/img]

Let $ABC$ be a triangle, where $AB = AC$ and $BC = 12$. Let $D$ be the midpoint of $BC$. Let $E$ be a point in $AC$ such that $DE \perp AC$. Let $F$ be a point in $AB$ such that $EF \parallel BC$. If $EC = 4$, determine the length of $EF$.

24) A ball of mass m moving at speed $v$ collides with a massless spring of spring constant $k$ mounted on a stationary box of mass $M$ in free space. No mechanical energy is lost in the collision. If the system does not rotate, what is the maximum compression $x$ of the spring? A) $x = v\sqrt{\frac{mM}{(m + M)k}}$ B) $x = v\sqrt{\frac{m}{k}}$ C) $x = v\sqrt{\frac{M}{k}}$ D) $x = v\sqrt{\frac{m + M}{k}}$ E) $x = v\sqrt{\frac{(m + M)^3}{mMk}}$

[u]Version for Nordic Countries[/u] Six pine trees grow on the shore of a circular lake. It is known that a treasure is submerged at the mid-point $T$ between the intersection points of the altitudes of two triangles, the vertices of one being at three of the $6$ pines, and the vertices of the second one at the other three pines. At how many points $T$ must one dive to find the treasure? [u]Version for Tropical Countries[/u] A captain finds his way to Treasure Island, which is circular in shape. He knows that there is treasure buried at the midpoint of the segment joining the orthocentres of triangles $ABC$ and $DEF$, where $A$, $B$, $C$, $D$, $E$ and $F$ are six palm trees on the shore of the island, not necessarily in cyclic order. He finds the trees all right, but does not know which tree is denoted by which letter. What is the maximum number of points at which the captain has to dig in order to recover the treasure? (S Markelov)

Find all linear homogeneous differential equations with continuous coefficients (on the whole real line) such that for any solution $ f(t)$ and any real number $ c,f(t\plus{}c)$ is also a solution.

There is number $1$ on the blackboard initially. The first step is to erase $1$ and write two nonnegative reals whose sum is $1$. Call the smaller number of the two $L_2$. For integer $k \ge 2$, the ${k}$ the step is to erase a number on the blackboard arbitrarily and write two nonnegative reals whose sum is the number erased just now. Call the smallest number of the $k+1$ on the blackboard $L_{k+1}$. Find the maximum of $L_2+L_3+\cdots+L_{2024}$.

Let $S_n$ be number of ordered sets of natural numbers $(a_1;a_2;....;a_n)$ for which $\frac{1}{a_1}+\frac{1}{a_2}+....+\frac{1}{a_n}=1$. Determine 1)$S_{10} mod(2)$. 2)$S_7 mod(2)$. (1) is first problem in 10 grade, (2)- third in 9 grade.

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

An equilateral triangle $T$ and a circle $C$ are on the same plane. Suppose each side length of $T$ is $6\sqrt3$ and the radius of $C$ is $2.$ The distance between the centers of $T$ and $C$ is $15.$ For every two points $X$ on $T$ and $Y$ on $C,$ let $M(X, Y)$ be the midpoint of segment $\overline{XY}.$ The points $M(X, Y)$ as $X$ varies on $T$ and $Y$ varies on $C$ create a region whose area is $A.$ Find $A.$ \[\mathrm a. ~\pi + 14\sqrt3 \qquad \mathrm b. ~3\pi + 10\sqrt3 \qquad \mathrm c. ~4\pi+9\sqrt3 \qquad\mathrm d. ~\pi + 15\sqrt3 \qquad\mathrm e. ~4\pi+6\sqrt3\]

Let $\{a_n\}$ be the sequence such that $a_0=2019$ and $$a_n=-\frac{2020}{n}\sum_{k=0}^{n-1}a_k.$$ Compute the last three digits of $\sum_{n=1}^{2020}2020^na_nn$.

The square was cut into $n^2$ rectangles with sides $a_i \times b_j, i , j= 1,..., n$. For what is the smallest $n$ in the set $\{a_1, b_1, ..., a_n, b_n\}$ all the numbers can be different?

The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle. [asy] defaultpen(linewidth(0.7)); draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); draw((34,36)--(34,45)--(25,45)); draw((36,36)--(36,38)--(34,38)); draw((36,38)--(41,38)); draw((34,45)--(41,45));[/asy]

([b]4[/b]) Let $ f(x) \equal{} \sin^6\left(\frac {x}{4}\right) \plus{} \cos^6\left(\frac {x}{4}\right)$ for all real numbers $ x$. Determine $ f^{(2008)}(0)$ (i.e., $ f$ differentiated $ 2008$ times and then evaluated at $ x \equal{} 0$).

Disjoint circles $ o_1, o_2$, with centers $ I_1, I_2$ respectively, are tangent to the line $ k$ at $ A_1, A_2$ respectively and they lie on the same side of this line. Point $ C$ lies on segment $ I_1I_2$ and $ \angle A_1CA_2 \equal{} 90^{\circ}$. Let $ B_1$ be the second intersection of $ A_1C$ with $ o_1$, and let $ B_2$ be the second intersection of $ A_2C$ with $ o_2$. Prove that $ B_1B_2$ is tangent to the circles $ o_1, o_2$.

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be the foot of the altitude from $A$ to $BC$. The inscribed circles of triangles $ABD$ and $ACD$ are tangent to $AD$ at $P$ and $Q$, respectively, and are tangent to $BC$ at $X$ and $Y$ , respectively. Let $PX$ and $QY$ meet at $Z$. Determine the area of triangle $XY Z$.

Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.

A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? $\textbf{(A)}\,\frac{1}{36} \qquad\textbf{(B)}\,\frac{1}{12} \qquad\textbf{(C)}\,\frac{1}{6} \qquad\textbf{(D)}\,\frac{1}{4} \qquad\textbf{(E)}\,\frac{5}{18}$

Let $\triangle{ABC}$ has circumcircle $\Gamma$, drop the perpendicular line from $A$ to $BC$ and meet $\Gamma$ at point $D$, similarly, altitude from $B$ to $AC$ meets $\Gamma$ at $E$. Prove that if $AB=DE, \angle{ACB}=60^{\circ}$ (sorry it is from my memory I can't remember the exact problem, but it means the same)

Given a positive integer $a,$ prove that $n!$ is divisible by $n^2 + n + a$ for infinitely many positive integers $n.{}$ [i]Proposed by Andrei Bâra[/i]

In a square board of size 1001 x 1001, we color some $m$ cells in such a way that: i. Of any two cells that share an edge, at least one is colored. ii. Of any 6 consecutive cells in a column or a row, at least 2 consecutive ones are colored. Determine the smallest possible value of $m$.