We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even . For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ . Prove that $1986 !! + 1985 !!$ i s divisible by $1987$. (V.V . Proizvolov , Moscow)

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

We want to split the set of natural numbers from $1$ to $3n$, where $n$ is a natural number, into $n$ mutually disjoint sets $\{x,y,z\}$ of three elements such that always holds: $x + y = 3z$. Is this possible for : a) $n = 5$? b) $n=10$? In both cases, provide either such a split or proof that such a split is not possible.

Given an equilateral triangle $ABC$ and a point $P$ so that the distances $P$ to $A$ and to $C$ are not farther than the distances $P$ to $B$. Prove that $PB = PA + PC$ if and only if $P$ lies on the circumcircle of $\vartriangle ABC$.

Given are two tangent circles and a point $P$ on their common tangent perpendicular to the lines joining their centres. Construct with ruler and compass all the circles that are tangent to these two circles and pass through the point $P$.

Find all positive integers $n$ such that $$2^n+15|3^n+200$$

The positive real numbers $a,b,c,$ satisfy: $$\frac{a}{2b+1} + \frac{2b}{3c+1} + \frac{3c}{a+1} = 1$$ $$\frac{1}{a+1} + \frac{1}{2b+1} + \frac{1}{3c+1} = 2$$ What is the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Let a non-isosceles acute triangle ABC with the circumscribed cycle (O) and the orthocenter H. D, E, F are the reflection of O in the lines BC, CA and AB. a) $H_a$ is the reflection of H in BC, A' is the reflection of A at O and $O_a$ is the center of (BOC). Prove that $H_aD$ and OA' intersect on (O). b) Let X is a point satisfy AXDA' is a parallelogram. Prove that (AHX), (ABF), (ACE) have a comom point different than A

In triangle $ABC$, we have $AB=AC=20$ and $BC=14$. Consider points $M$ on $\overline{AB}$ and $N$ on $\overline{AC}$. If the minimum value of the sum $BN + MN + MC$ is $x$, compute $100x$. [i]Proposed by Lewis Chen[/i]

Lucy (mass $\text{33.1 kg}$), Henry (mass $\text{63.7 kg}$), and Mary (mass $\text{24.3 kg}$) sit on a lightweight seesaw at evenly spaced $\text{2.74 m}$ intervals (in the order in which they are listed; Henry is between Lucy and Mary) so that the seesaw balances. Who exerts the most torque (in terms of magnitude) on the seesaw? Ignore the mass of the seesaw. (A) Henry (B) Lucy (C) Mary (D) They all exert the same torque. (E) There is not enough information to answer the question.

We have four wooden triangles with sides $3, 4, 5$ centimeters. How many convex polygons can we make by all of these triangles? (Just draw the polygons without any proof) A convex polygon is a polygon which all of it's angles are less than $180^o$ and there isn't any hole in it. For example: [img]https://1.bp.blogspot.com/-JgvF_B-uRag/W1R4f4AXxTI/AAAAAAAAIzc/Fo3qu3pxXcoElk01RTYJYZNwj0plJaKPQCK4BGAYYCw/s640/igo%2B2015.el1.png[/img]

A sequence of natural numbers $a_1, a_2, a_3,..$ is called [i]periodic modulo[/i] $n$ if there exists a positive integer $k$ such that, for any positive integer $i$, the terms $a_i$ and $a_{i+k}$ are equal modulo $n$. Does there exist a strictly increasing sequence of natural numbers that a) is not periodic modulo finitely many positive integers and is periodic modulo all the other positive integers? b) is not periodic modulo infinitely many positive integers and is periodic modulo infinitely many positive integers?

$x\sqrt8 + \frac{1}{x\sqrt8} = \sqrt8$ has two real solutions $x_1, x_2$. The decimal expansion of $x_1$ has the digit $6$ in place $1994$. What digit does $x_2$ have in place $1994$?

From the vertex $A$ of the parallelogram $ABCD$, the perpendiculars $AM,AN$ on sides $BC,CD$ respectively. $P$ is the intersection point of $BN$ and $DM$. Prove that the lines $AP$ and $MN$ are perpendicular.

The largest factor of $n$ not equal to $n$ is $35$. Compute the largest possible value of $n$.

Let $ABC$ be a triangle with $D$ the midpoint of side $AB, E$ the midpoint of side $BC$, and $F$ the midpoint of side $AC$. Let $k_1$ be the circle passing through points $A, D$, and $F$, let $k_2$ be the circle passing through points $B, E$, and $D$, and let $k_3$ be the circle passing through $C, F$, and $E$. Prove that circles $k_1, k_2$, and $k_3$ intersect in a point.

Consider the inequality \[(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).\] a) Find all $n\ge 3$ such that the inequality is true for positive reals. b) Find all $n\ge 3$ such that the inequality is true for reals.

A car rides along a circular track in the clockwise direction. At noon Peter and Paul took their positions at two different points of the track. Some moment later they simultaneously ended their duties and compared their notes. The car passed each of them at least $30$ times. Peter noticed that each circle was passed by the car $1$ second faster than the preceding one while Paul’s observation was opposite: each circle was passed $1$ second slower than the preceding one. Prove that their duty was at least an hour and a half long.

A cube is divided into $8$ smaller cubes of the same size, as shown in the figure. Then, each of these small cubes is divided again into $8$ smaller cubes of the same size. This process is done $4$ more times to each resulting cube. What is the ratio between the sum of the total areas of all the small cubes resulting from the last division and the total area of the initial cube?

You are given a right triangle $ABC$ with $\angle ACB = 90^\circ$. Let $W_A , W_B$ respectively be the midpoints of the smaller arcs $BC$ and $AC$ of the circumcircle of $\triangle ABC$, and $N_A , N_B$ respectively be the midpoints of the larger arcs $BC$ and $AC$ of this circle. Denote by $P$ and $Q$ the points of intersection of segment $AB$ with the lines $N_AW_B$ and $N_BW_A$, respectively. Prove that $AP = BQ$. [i]Proposed by Oleksiy Masalitin[/i]

Find $\sum_{n=1}^{\infty} \frac{1}{2^n}\int_{-1}^1 (1-x)^2(1+x)^n dx\ (n\geq 1).$

Delete all perfect squares in $1,2,3,\cdots$, then the 2003rd number is $\text{(A)}2046\qquad\text{(B)}2047\qquad\text{(C)}2048\qquad\text{(D)}2049$

Let $n$ a positive integer and let $x_1, \ldots, x_n, y_1, \ldots, y_n$ real positive numbers such that $x_1+\ldots+x_n=y_1+\ldots+y_n=1$. Prove that: $$|x_1-y_1|+\ldots+|x_n-y_n|\leq 2-\underset{1\leq i\leq n}{min} \;\dfrac{x_i}{y_i}-\underset{1\leq i\leq n}{min} \;\dfrac{y_i}{x_i}$$

Let $ABC $ be an acute triangle such that $AB\neq AC$ ,with circumcircle $ \Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $ \Gamma$ such that $AD \perp BC$. let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $ \Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $ \Gamma$ at point $X\neq E$. Prove that the point $A,M$ and $X$ are collinear.

[b]p1.[/b] If $A = 0$, $B = 1$, $C = 2$, $...$, $Z = 25$, then what is the sum of $A + B + M+ C$? [b]p2.[/b] Eric is playing Tetris against Bryan. If Eric wins one-fifth of the games he plays and he plays $15$ games, find the expected number of games Eric will win. [b]p3.[/b] What is the sum of the measures of the exterior angles of a regular $2023$-gon in degrees? [b]p4.[/b] If $N$ is a base $10$ digit of $90N3$, what value of $N$ makes this number divisible by $477$? [b]p5.[/b] What is the rightmost non-zero digit of the decimal expansion of $\frac{1}{2^{2023}}$ ? [b]p6.[/b] if graphs of $y = \frac54 x + m$ and $y = \frac32 x + n$ intersect at $(16, 27)$, what is the value of $m + n$? [b]p7.[/b] Bryan is hitting the alphabet keys on his keyboard at random. If the probability he spells out ABMC at least once after hitting $6$ keys is $\frac{a}{b^c}$ , for positive integers $a$, $b$, $c$ where $b$, $c$ are both as small as possible, find $a+b+c$. Note that the letters ABMC must be adjacent for it to count: AEBMCC should not be considered as correctly spelling out ABMC. [b]p8.[/b] It takes a Daniel twenty minutes to change a light bulb. It takes a Raymond thirty minutes to change a light bulb. It takes a Bryan forty-five minutes to change a light bulb. In the time that it takes two Daniels, three Raymonds, and one and a half Bryans to change $42$ light bulbs, how many light bulbs could half a Raymond change? Assume half a person can work half as productively as a whole person. [b]p9.[/b] Find the value of $5a + 4b + 3c + 2d + e$ given $a, b, c, d, e$ are real numbers satisfying the following equations: $$a^2 = 2e + 23$$ $$b^2 = 10a - 34$$ $$c^2 = 8b - 23$$ $$d^2 = 6c - 14$$ $$e^2 = 4d - 7.$$ [b]p10.[/b] How many integers between $1$ and $1000$ contain exactly two $1$’s when written in base $2$? [b]p11.[/b] Joe has lost his $2$ sets of keys. However, he knows that he placed his keys in one of his $12$ mailboxes, each labeled with a different positive integer from $1$ to $12$. Joe plans on opening the $2$ mailbox labeled $1$ to see if any of his keys are there. However, a strong gust of wind blows by, opening mailboxes $11$ and $12$, revealing that they are empty. If Joe decides to open one of the mailboxes labeled $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ , or $10$, the probability that he finds at least one of his sets of keys can be expressed as $\frac{a}{b}$, where a and b are relatively prime positive integers. Find the sum $a + b$. Note that a single mailbox can contain $0$, $1$, or $2$ sets of keys, and the mailboxes his sets of keys were placed in are determined independently at random. [b]p12.[/b] As we all know, the top scientists have recently proved that the Earth is a flat disc. Bob is standing on Earth. If he takes the shortest path to the edge, he will fall off after walking $1$ meter. If he instead turns $90$ degrees away from the shortest path and walks towards the edge, he will fall off after $3$ meters. Compute the radius of the Earth. [b]p13.[/b] There are $999$ numbers that are repeating decimals of the form $0.abcabcabc...$ . The sum of all of the numbers of this form that do not have a $1$ or $2$ in their decimal representation can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Find $a + b$. [b]p14.[/b] An ant is crawling along the edges of a sugar cube. Every second, it travels along an edge to another adjacent vertex randomly, interested in the sugar it notices. Unfortunately, the cube is about to be added to some scalding coffee! In $10$ seconds, it must return to its initial vertex, so it can get off and escape. If the probability the ant will avoid a tragic doom can be expressed as $\frac{a}{3^{10}}$ , where $a$ is a positive integer, find $a$. Clarification: The ant needs to be on its initial vertex in exactly $10$ seconds, no more or less. [b]p15.[/b] Raymond’s new My Little Pony: Friendship is Magic Collector’s book arrived in the mail! The book’s pages measure $4\sqrt3$ inches by $12$ inches, and are bound on the longer side. If Raymond keeps one corner in the same plane as the book, what is the total area one of the corners can travel without ripping the page? If the desired area in square inches is $a\pi+b\sqrt{c}$ where $a$, $b$, and $c$ are integers and $c$ is squarefree, find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].