Let $a$, $b$, $c$, $d$ be four integers. Prove that $$\left(b-a\right)\left(c-a\right)\left(d-a\right)\left(d-c\right)\left(d-b\right)\left(c-b\right)$$ is divisible by $12$.

What is the greatest number of parts that $5$ spheres can divide the space into?

A cube $A_1A_2A_3A_4A_5A_6A_7A_8$ is given in space. We will mark its center with the letter $S$ (intersection of solid diagonals). Find all natural numbers $k$ for which there exists a plane not containing the point $S$ and intersecting just $k$ of the rays $SA_1, SA_2, .. SA_8$

An acute angled triangle \(ABC\) has circumcenter \(O\). The lines \(AO\) and \(BC\) intersect at \(D\), while \(BO\) and \(AC\) intersect at \(E\) and \(CO\) and \(AB\) intersect at \(F\). Show that if the triangles \(ABC\) and \(DEF\) are similar(with vertices in that order), than \(ABC\) is equilateral.

In triangle $ABC$ the median $BM$ is equal to half of the side $BC$. Show that $\angle ABM = \angle BCA + \angle BAC$. [i](Proposed by Anton Trygub)[/i]

Find all positive integers $ n$ such that acute-angled $ \triangle ABC$ with $ \angle BAC<\frac{\pi}{4}$ could be divided into $ n$ quadrilateral. Every quadrilateral is inscribed in circle and radiuses of circles are in geometric progression. [hide] be carefull ! :lol: [/hide]

Let $g(x)$ be a function that has a continuous first derivative $g'(x)$. Suppose that $g(0)=0$ and $|g'(x)| \leq |g(x)|$ for all values of $x.$ Prove that $g(x)$ vanishes identically.

Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le3$, let $D_i $, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$. Suppose that the areas of the triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, $D_1D_2D_3$, $E_1E_2E_3$, and $F_1F_2F_3$ are $2$, $3$, $4$, $20$, $21$, and $2020$, respectively. Compute the largest possible area of $G_1G_2G_3$.

For all $n \in {\mathbb Z^+}$ we define $$I_n=\{\frac{0}{n},\frac{1}{n},\frac{2}{n},\dotsm,\frac{n-1}{n},\frac{n}{n},\frac{n+1}{n},\dotsm\}$$ infinite cluster. For whichever $x$ and $y$ real number, we say $\mid{x-y}\mid$ is between distance of the $x$ and $y$. [b]a[/b]) For all $n$'s we find a number in $I_n$ such that, the between distance of the this number and $\sqrt 2$ is less than $\frac{1}{2n}$ [b]b[/b]) We find a $n$ such that, between distance of the a number in $I_n$ and $\sqrt 2$ is less than $\frac{1}{2011n}$

Given a finite sequence of complex numbers $c_1, c_2, \ldots , c_n$, show that there exists an integer $k$ ($1 \leq k \leq n$) such that for every finite sequence $a_1, a_2, \ldots, a_n$ of real numbers with $1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$, the following inequality holds: \[\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.\]

The points with integer coordinates in a plane are painted in two colors – blue and red. Prove that there exist an infinite monochromatic subset that is symmetrical on some point.

Let $a_1$, $a_2$,...,$a_n$ be constant real numbers and $x$ be variable real number $x$. Let $f(x)=cos(a_1+x)+\frac{cos(a_2+x)}{2}+\frac{cos(a_3+x)}{2^2}+...+\frac{cos(a_n+x)}{2^{n-1}}$. If $f(x_1)=f(x_2)=0$, prove that $x_1-x_2=m\pi$, where $m$ is integer.

A positive integer n is called [i]primary divisor [/i] if for every positive divisor $d$ of $n$ at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is divisor primary, because its positive divisors $1$, $2$, $4$, and $8$ each differ by $1$ from a prime number ($2$, $3$, $5$, and $7$, respectively), while $9$ is not divisor primary, because the divisor $9$ does not differ by $1$ from a prime number (both $8$ and $10$ are composite). Determine the largest primary divisor number.

Let $ a,b,c$ be positive real numbers which satisfy $ 5(a^2\plus{}b^2\plus{}c^2)<6(ab\plus{}bc\plus{}ca)$. Prove that these three inequalities hold: $ a\plus{}b>c$, $ b\plus{}c>a$, $ c\plus{}a>b$.

[u]Round 9[/u] [b]p25.[/b] What is the largest integer that cannot be expressed as the sum of nonnegative multiples of $7$, $11$, and $13$? [b]p26.[/b] Evaluate $12{3 \choose3}+ 11{4\choose 3}+ 10{5\choose 3}+ ...+ 2{13\choose 3}+{14 \choose 3}$. [b]p27.[/b] Worker Bob drives to work at $30$ mph half the time and $60$ mph half the time. He returns home along the same route at $30$ mph half the distance and $60$ mph half the distance. What is his average speed along the entire trip, in mph? [u]Round 10[/u] [b]p28.[/b] In quadrilateral $ABCD$, diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$ with $BP = 4$, $P D = 6$, $AP = 8$, $P C = 3$, and $AB = 6$. What is the length of $AD$? [b]p29.[/b] Find all positive integers $x$ such that$ x^2 + 17x + 17$ is a square number. [b]p30.[/b] Zach has ten weighted coins that turn up heads with probabilities $\frac{2}{11^2}$ ,$\frac{2}{10^2}$ ,$\frac{2}{9^2}$ $, . . $.,$\frac{2}{2^2}$ . If he flips all ten coins simultaneously, then what is the probability that he will get an even number of heads? [u]Round 11[/u] [b]p31.[/b] Given a sequence $a_1, a_2, . . .$ such that $a_1 = 3$ and $a_{n+1} = a^2_n - 2a_n + 2$ for $n \ge 1$, find the remainder when the product a1a2 · · · a2012 is divided by 100. [b]p32.[/b] Let $ABC$ be an equilateral triangle and let $O$ be its circumcircle. Let $D$ be a point on $\overline{BC}$, and extend $\overline{AD}$ to intersect $O$ at $P$. If $BP = 5$ and $CP = 4$, then what is the value of $DP$? [b]p33.[/b] Surya and Hao take turns playing a game on a calendar. They start with the date January $1$ and they can either increase the month to a later month or increase the day to a later day in that month but not both. The first person to adjust the date to December $31$ is the winner. If Hao goes first, then what is the first date that he must choose to ensure that he does not lose? [u]Round 12[/u] [b]p34.[/b] On May $5$, $1868$, exactly $144$ years before today, Memorial Day in the United States was officially proclaimed. The first Memorial Day took place that year on May $30$ at Waterloo, New York. On May $5$, $2012$, at $12:00$ PM, how many results did the search “memorial day” on Google return? The search phrase is in quotes, so Google will only return sites that have the words memorial and day next to each other in that order. Let $N = max-\{0, \rfloor 15.5 \times \frac{ Your\,\,\, Answer}{Actual \,\,\,Answer} \rfloor \}$. You will earn the number of points equal to $min\{N, max\{0, 30 - N\}\}$. [b]p35.[/b] Estimate the side length of a regular pentagon whose area is $2012$. You will earn the number of points equal to $max\{0, 15 - \lfloor 5 \times |Your \,\,\,Answer - Actual \,\,\,Answer| \rfloor \}$. [b]p36.[/b] Write down one integer between $1$ and $15$, inclusive. (If you do not, then you will receive $0$ points.) Let the number that you submit be $x$. Let $\overline{x}$ be the arithmetic mean of all of the valid numbers submitted by all of the teams. If $x > \overline{x}$, then you will receive $0$ points; otherwise, you will receive $x$ points. PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134177p28401527]here [/url] and 6-8 [url=https://artofproblemsolving.com/community/c3h3134466p28406321]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Inside a square of side $16$ are placed $1000$ points. Show that it is possible to put a equilateral triangle of side $2\sqrt3$ in the plane so that it covers at least $16$ of these points.

On the sides $AB$ and $BC$ arbitrarily mark points $M$ and $N$, respectively. Let $P$ be the point of intersection of segments $AN$ and $BM$. In addition, we note the points $Q$ and $R$ such that quadrilaterals $MCNQ$ and $ACBR$ are parallelograms. Prove that the points $P,Q$ and $R$ lie on one line.

Let $I$ be the center of a circle inscribed in triangle $ABC$, in which $\angle BAC = 60 ^o$ and $AB \ne AC$. The points $D$ and $E$ were marked on the rays $BA$ and $CA$ so that $BD = CE = BC$. Prove that the line $DE$ passes through the point $I$.

Find all ordered pairs $(a,b)$ of positive integers that satisfy $a>b$ and the equation $(a-b)^{ab}=a^bb^a$.

Let Pascal triangle be an equilateral triangular array of number, consists of $2019$ rows and except for the numbers in the bottom row, each number is equal to the sum of two numbers immediately below it. How many ways to assign each of numbers $a_0, a_1,...,a_{2018}$ (from left to right) in the bottom row by $0$ or $1$ such that the number $S$ on the top is divisible by $1019$.

For $0 \leq p \leq 1/2,$ let $X_1, X_2, \ldots$ be independent random variables such that $$X_i=\begin{cases} 1 & \text{with probability } p, \\ -1 & \text{with probability } p, \\ 0 & \text{with probability } 1-2p, \end{cases} $$ for all $i \geq 1.$ Given a positive integer $n$ and integers $b,a_1, \ldots, a_n,$ let $P(b, a_1, \ldots, a_n)$ denote the probability that $a_1X_1+ \ldots + a_nX_n=b.$ For which values of $p$ is it the case that $$P(0, a_1, \ldots, a_n) \geq P(b, a_1, \ldots, a_n)$$ for all positive integers $n$ and all integers $b, a_1,\ldots, a_n?$

A positive integer is called [i]lonely [/i] if the sum of the inverses of its positive divisors (including $1$ and itself) is not equal with the some of the inverses of the positive divisors of any other positive integer. a) Show that any prime number is lonely. b) Prove that there are infinitely many numbers that are not lonely

Prove that $\forall n\in\mathbb{Z}^+_0:(\exists b\in\mathbb{Z}^+_0:(\forall m\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(x+m = b))\lor(\exists s\in\mathbb{Z}^+_0:(\exists p\in\mathbb{Z}^+_0:((\neg(\exists x\in\mathbb{Z}^+_0:(p+x = 1)))\land(\neg(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:(p = (x+2) \cdot (y+2)))))\land(\exists x\in\mathbb{Z}^+_0:(p = m+x+1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + m) + y))))))\land(\forall u\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(u = p+x))\lor(u = 0)\lor(u = n+1)\lor(\neg(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + u) + y)))))))\lor(\exists v\in\mathbb{Z}^+_0:(\exists k\in\mathbb{Z}^+_0:((\neg(v = 0))\land((u = v \cdot (k+2))\lor(u = v \cdot (k+2) + 1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + v) + y)))))))))))))))))$. Proposed by [i]Warren Bei[/i]

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

Compare the magnitude of the following three numbers. $$ \sqrt[3]{\frac{25}{3}} ,\, \sqrt[3]{\frac{1148}{135}} ,\, \frac{\sqrt[3]{25}}{3} + \sqrt[3]{\frac{6}{5}} $$