Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and \[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\] for all natural $n \geq 2$. [i]Peter Boyvalenkov[/i]

Let $k$ and $m$ be positive integers. Show that \[S(m, k)=\sum_{n=1}^{\infty}\frac{1}{n(mn+k)}\] is rational if and only if $m$ divides $k$.

Find all positive integers $n$ for which $1 - 5^n + 5^{2n+1}$ is a perfect square.

We took a non-equilateral acute-angled triangle and marked $4$ wonderful points in it: the centers of the inscribed and circumscribed circles, the center of gravity (the point of intersection of the medians) and the intersection point of altitudes. Then the triangle itself was erased. It turned out that it was impossible to establish which of the centers corresponds to each of the marked points. Find the angles of the triangle

Consider an arbitrary (optional convex) polygon. It's [i]chord [/i] is a segment whose ends lie on the boundary of the polygon, and itself belongs entirely to the polygon. Will there always be a chord of a polygon that divides it into two equal parts? Is it true that any polygon can be divided by some chord into parts, the area of each of which is not less than $\frac13$ the area of the polygon?

The infinite sequence $a_1,a_2,a_3,\ldots$ of positive integers is defined as follows: $a_1=1$, and for each $n \ge 2$, $a_n$ is the smallest positive integer, distinct from $a_1,a_2, \ldots , a_{n-1}$ such that: $$\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}}$$ is an integer. Prove that all positive integers appear on the sequence $a_1,a_2,a_3,\ldots$

$p(m)$ is the number of distinct prime divisors of a positive integer $m>1$ and $f(m)$ is the $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$ th smallest prime divisor of $m$. Find all positive integers $n$ satisfying the equation: $$f(n^2+2) + f(n^2+5) = 2n-4$$

[b]p9.[/b] The frozen yogurt machine outputs yogurt at a rate of $5$ froyo$^3$/second. If the bowl is described by $z = x^2+y^2$ and has height $5$ froyos, how long does it take to fill the bowl with frozen yogurt? [b]p10.[/b] Prankster Pete and Good Neighbor George visit a street of $2021$ houses (each with individual mailboxes) on alternate nights, such that Prankster Pete visits on night $1$ and Good Neighbor George visits on night $2$, and so on. On each night $n$ that Prankster Pete visits, he drops a packet of glitter in the mailbox of every $n^{th}$ house. On each night $m$ that Good Neighbor George visits, he checks the mailbox of every $m^{th}$ house, and if there is a packet of glitter there, he takes it home and uses it to complete his art project. After the $2021^{th}$ night, Prankster Pete becomes enraged that none of the houses have yet checked their mail. He then picks three mailboxes at random and takes out a single packet of glitter to dump on George’s head, but notices that all of the mailboxes he visited had an odd number of glitter packets before he took one. In how many ways could he have picked these three glitter packets? Assume that each of these three was from a different house, and that he can only visit houses in increasing numerical order. [b]p11. [/b]The taxi-cab length of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $|x_1 - x_2| + |y_1- y_2|$. Given a series of straight line segments connected head-to-tail, the taxi-cab length of this path is the sum of the taxi-cab lengths of its line segments. A goat is on a rope of taxi-cab length $\frac72$ tied to the origin, and it can’t enter the house, which is the three unit squares enclosed by $(-2, 0)$,$(0, 0)$,$(0, -2)$,$(-1, -2)$,$(-1, -1)$,$(-2, -1)$. What is the area of the region the goat can reach? (Note: the rope can’t ”curve smoothly”-it must bend into several straight line segments.) [b]p12.[/b] Parabola $P$, $y = ax^2 + c$ has $a > 0$ and $c < 0$. Circle $C$, which is centered at the origin and lies tangent to $P$ at $P$’s vertex, intersects $P$ at only the vertex. What is the maximum value of a, possibly in terms of $c$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an equilateral triangle, and point $P$ on its circumcircle. Let $PA$ and $BC$ intersect at $D$, $PB$ and $AC$ intersect at $E$, and $PC$ and $AB$ intersect at $F$. Prove that the area of $\triangle DEF$ is twice the area of $\triangle ABC$. [i]Proposed by Titu Andreescu, Luis Gonzales, Cosmin Pohoata[/i]

Let [i]Revolution[/i]$(x) = x^3 +Ux^2 +Sx + A$, where $U$, $S$, and $A$ are all integers and $U +S + A +1 = 1773$. Given that [i]Revolution[/i] has exactly two distinct nonzero integer roots $G$ and $B$, find the minimum value of $|GB|$. [i]Proposed by Jacob Xu[/i] [hide=Solution] [i]Solution.[/i] $\boxed{392}$ Notice that $U + S + A + 1$ is just [i]Revolution[/i]$(1)$ so [i]Revolution[/i]$(1) = 1773$. Since $G$ and $B$ are integer roots we write [i]Revolution[/i]$(X) = (X-G)^2(X-B)$ without loss of generality. So Revolution$(1) = (1-G)^2(1-B) = 1773$. $1773$ can be factored as $32 \cdot 197$, so to minimize $|GB|$ we set $1-G = 3$ and $1-B = 197$. We get that $G = -2$ and $B = -196$ so $|GB| = \boxed{392}$. [/hide]

Define a sequence $\{u_n\}_{n=0}^{\infty}$ by $u_0=u_1=u_2=1,$ and thereafter by the condition that $\det\begin{vmatrix} u_n & u_{n+1} \\ u_{n+2} & u_{n+3} \end{vmatrix}=n!$ for all $n\ge 0.$ Show that $u_n$ is an integer for all $n.$ (By convention, $0!=1$.)

Given a circle with four circles that intersect in pairs as shown in the figure. The "internal" the points of intersection are $A, B, C$ and $D$, while the ‘outer’ points of intersection are $E, F, G$ and $H$. Prove that the quadrilateral $ABCD$ is cyclic if also the quadrilateral $EFGH$ is also cyclic. [img]https://cdn.artofproblemsolving.com/attachments/0/0/6a369c93e37eefd57775fd8586bdff393e1914.png[/img]

Two real sequence $ \{x_{n}\}$ and $ \{y_{n}\}$ satisfies following recurrence formula; $ x_{0}\equal{} 1$, $ y_{0}\equal{} 2007$ $ x_{n\plus{}1}\equal{} x_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(y_{n}\plus{}y_{n\plus{}1})$, $ y_{n\plus{}1}\equal{} y_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(x_{n}\plus{}x_{n\plus{}1})$ Then show that for all nonnegative integer $ n$, $ {x_{n}}^{2}\leq 2007$.

On side \( AB \) of an isosceles trapezoid \( ABCD \) (\( AD \parallel BC \)), points \( E \) and \( F \) are chosen such that a circle can be inscribed in quadrilateral \( CDEF \). Prove that the circumcircles of triangles \( ADE \) and \( BCF \) are tangent to each other. [i]Proposed by Matthew Kurskyi[/i]

In right triangle $ ABC$ with right angle at $ C$, $ \angle BAC < 45$ degrees and $ AB \equal{} 4$. Point $ P$ on $ AB$ is chosen such that $ \angle APC \equal{} 2\angle ACP$ and $ CP \equal{} 1$. The ratio $ \frac{AP}{BP}$ can be represented in the form $ p \plus{} q\sqrt{r}$, where $ p,q,r$ are positive integers and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$.

In parallelogram $ABCD$, point $K$ is marked on diagonal $AC$. Circle $s_1$ passes through point $K$ and touches lines $AB$ and $AD$ ($s_1$ intersects the diagonal $AC$ for the second time on the segment $AK$). Circle $s_2$ passes through point $K$ and touches lines $CB$ and $CD$ ($s_2$ intersects for the second time diagonal $AC$ on segment $KC$). Prove that for all positions of the point $K$ on the diagonal $AC$, the straight lines connecting the centers of circles $ s_1$ and $s_2$, will be parallel to each other.

$ABCD$ is a cyclic quadrilateral with circumcenter $O$. The lines $AC, BD$ intersect at $E$ and $AD, BC$ intersect at $F$. $O_1$ and $O_2$ are the circumcenters of $\triangle ABE$ and $\triangle CDE$, respectively. Assume that $(ABCD)$ and $(OO_1O_2)$ intersect at two points $P, Q$. Prove that $P, Q, F$ are collinear.

Julia is placing identical $1$-by-$1$ tiles on the $2$-by-$2$ grid pictured, one piece at a time, so that every piece she places after the first is adjacent to, but not on top of, some piece she’s already placed. Determine the number of ways that Julia can complete the grid. [img]https://cdn.artofproblemsolving.com/attachments/4/6/4a585593b9301ddb0e4ac3ceced212c378c9f8.png[/img]

Let $H$ be a regular hexagon with area 360. Three distinct vertices $X$, $Y$, and $Z$ are picked randomly, with all possible triples of distinct vertices equally likely. Let $A$, $B$, and $C$ be the unpicked vertices. What is the expected value (average value) of the area of the intersection of $\triangle ABC$ and $\triangle XYZ$?

Consider a triangle $ABC$. The midpoints of the sides $BC, CA$, and $AB$ are denoted by $D, E$, and $F$, respectively. Assume that the median $AD$ is perpendicular to the median $BE$ and that their lengths are given by $AD = 18$ and $BE = 13.5$. Compute the length of the third median $CF$. (K. Czakler, Vienna)

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs. Juniors) How many vertical sides can there be in this set? Seniors) How many ways are there to do that? [asy] size(120); defaultpen(linewidth(0.8)); path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle; for(int i=0;i<=3;i=i+1) { for(int j=0;j<=2;j=j+1) { real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2; draw(shift(shiftx,shifty)*hex); } } [/asy] [i](T. Doslic)[/i]

We say that a set $S$ of $3$ unit squares is \textit{commutable} if $S = \{s_1,s_2,s_3\}$ for some $s_1,s_2,s_3$ where $s_2$ shares a side with each of $s_1,s_3$. How many ways are there to partition a $3\times 3$ grid of unit squares into $3$ pairwise disjoint commutable sets? [i]Proposed by Srinivasan Sathiamurthy[/i]

There are six small circles in the figure with a radius of $1$ and tangent to a large circle and the sides of the $ABC$ of an equilateral triangle, where touch points are $K, L$ and $M$ respectively with the midpoints of sides $AB, BC$ and $AC$. Find the radius of the large circle and the side of the triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/3/0/f858dcc5840759993ea2722fd9b9b15c18f491.png[/img]

Let $ABCD$ be a tetrahedron and let $E,F,G,H,K,L$ be points lying on the edges $AB,BC,CD$ $,DA,DB,DC$ respectively, in such a way that \[AE \cdot BE = BF \cdot CF = CG \cdot AG= DH \cdot AH=DK \cdot BK=DL \cdot CL.\] Prove that the points $E,F,G,H,K,L$ all lie on a sphere.