On the edges of a convex polyhedra we draw arrows such that from each vertex at least an arrow is pointing in and at least one is pointing out. Prove that there exists a face of the polyhedra such that the arrows on its edges form a circuit. [i]Dan Schwartz[/i]

a) Prove that every function of the form $$f(x)=\frac{a_{0}}{2}+\cos(x)+\sum_{n=2}^{N}a_{n}\cos(nx)$$ with $|a_{0}|<1$ has positive as well as negative values in the period $[0,2\pi)$. b) Prove that the function $$F(x)=\sum_{n=1}^{100}\cos(n^{\frac{3}{2}}x)$$ has at least $40$ zeroes in the interval $(0,1000)$.

Let $X$ and $Y$ be independent random variables with "Saint-Petersburg" distribution, i.e. for any $k=1,2,\ldots$ their value is $2^k$ with probability $\frac{1}{2^k}$. Show that $X$ and $Y$ can be realized on a sufficiently big probability space such that there exists another pair of independent "Saint-Petersburg" random variables $(X', Y')$ on this space with the property that $X+Y=2X'+Y'I(Y'\le X')$ almost surely (here $I(A)$ denotes the indicator function of the event $A$). (translated by L. Erdős)

The measure of a given angle is $\frac{180^{\circ}}{n}$ where $n$ is a positive integer not divisible by $3$. Prove that the angle can be trisected by Euclidean means (straightedge and compasses).

[u]Round 1[/u] [b]p1.[/b] Compute $(1 - 2(3 - 4(5 - 6)))(7 - (8 - 9))$. [b]p2.[/b] How many numbers are in the set $\{20, 21, 22, ..., 88, 89\}$? [b]p3.[/b] Three times the complement of the supplement of an angle is equal to $60$ degrees less than the angle itself. Find the measure of the angle in degrees. [u]Round 2[/u] [b]p4.[/b] A positive number is decreased by $10\%$, then decreased by $20\%$, and finally increased by $30\%$. By what percent has this number changed from the original? Give a positive answer for a percent increase and a negative answer for a percent decrease. [b]p5.[/b] What is the area of the triangle with vertices at $(2, 3)$, $(8, 11)$, and $(13, 3)$? [b]p6.[/b] There are three bins, each containing red, green, and/or blue pens. The first bin has $0$ red, $0$ green, and $3$ blue pens, the second bin has $0$ red, $2$ green, and $4$ blue pens, and the final bin has $1$ red, $5$ green, and $6$ blue pens. What is the probability that if one pen is drawn from each bin at random, one of each color pen will be drawn? [u]Round 3[/u] [b]p7.[/b] If a and b are positive integers and $a^2 - b^2 = 23$, what is the value of $a$? [b]p8.[/b] Find the prime factorization of the greatest common divisor of $2^3\cdot 3^2\cdot 5^5\cdot 7^4$ and $2^4\cdot 3^1\cdot 5^2\cdot 7^6$. [b]p9.[/b] Given that $$a + 2b + 3c = 5$$ $$2a + 3b + c = -2$$ $$3a + b + 2c = 3,$$ find $3a + 3b + 3c$. [u]Round 4[/u] [b]p10.[/b] How many positive integer divisors does $11^{20}$ have? [b]p11.[/b] Let $\alpha$ be the answer to problem $10$. Find the real value of $x$ such that $2^{x-5} = 64^{x/\alpha}$. [b]p12.[/b] Let $\beta$ be the answer to problem $11$. Triangle $LMT$ has a right angle at $M$, $LM = \beta$, and $LT = 4\beta - 3$. If $Z$ is the midpoint of $LT$, what is the length$ MZ$? PS. You should use hide for answers. Rounds 5-8 are [url=https://artofproblemsolving.com/community/c3h3133709p28395558]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134133p28400917]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

A cube side $100$ is divided into a million unit cubes with faces parallel to the large cube. The edges form a lattice. A prong is any three unit edges with a common vertex. Can we decompose the lattice into prongs with no common edges?

Calculate the following indefinite integrals. [1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$ [2] $\int x\log_{10} x dx$ [3] $\int \frac{x}{\sqrt{2x-1}}dx$ [4] $\int (x^2+1)\ln x dx$ [5] $\int e^x\cos x dx$

We say that a function $f: \mathbb{Z}_{\ge 0} \times \mathbb{Z}_{\ge 0} \to \mathbb{Z}$ is [i]great[/i] if for any nonnegative integers $m$ and $n$, \[f(m + 1, n + 1) f(m, n) - f(m + 1, n) f(m, n + 1) = 1.\] If $A = (a_0, a_1, \dots)$ and $B = (b_0, b_1, \dots)$ are two sequences of integers, we write $A \sim B$ if there exists a great function $f$ satisfying $f(n, 0) = a_n$ and $f(0, n) = b_n$ for every nonnegative integer $n$ (in particular, $a_0 = b_0$). Prove that if $A$, $B$, $C$, and $D$ are four sequences of integers satisfying $A \sim B$, $B \sim C$, and $C \sim D$, then $D \sim A$. [i]Ankan Bhattacharya[/i]

Call a number [i]feared [/i] if it contains the digits $13$ as a contiguous substring and [i]fearless [/i] otherwise. (For example, $132$ is feared, while $123$ is fearless.) Compute the smallest positive integer $n$ such that there exists a positive integer $a < 100$ such that $n$ and $n + 10a$ are fearless while $n +a$, $n + 2a$, $. . . $, $n + 9a$ are all feared.

The product $ (1.8)(40.3\plus{}.07)$ is closest to \[ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 42 \qquad \textbf{(C)}\ 74 \qquad \textbf{(D)}\ 84 \qquad \textbf{(E)}\ 737 \]

Consider the family of nonisosceles triangles $ABC$ satisfying the property $AC^2 + BC^2 = 2 AB^2$. Points $M$ and $D$ lie on side $AB$ such that $AM = BM$ and $\angle ACD = \angle BCD$. Point $E$ is in the plane such that $D$ is the incenter of triangle $CEM$. Prove that exactly one of the ratios \[ \frac{CE}{EM}, \quad \frac{EM}{MC}, \quad \frac{MC}{CE} \] is constant.

[asy] //rectangles above problem statement size(15cm); for(int i=0;i<5;++i){ draw((6*i-14,-1.2)--(6*i-14,1.2)--(6*i-10,1.2)--(6*i-10,-1.2)--cycle); } label("$A$", (-12,2.25)); label("$B$", (-6,2.25)); label("$C$", (0,2.25)); label("$D$", (6,2.25)); label("$E$", (12,2.25)); //top numbers label("$1$", (-12,1.25),dir(-90)); label("$0$", (-6,1.25),dir(-90)); label("$8$", (0,1.25),dir(-90)); label("$5$", (6,1.25),dir(-90)); label("$2$", (12,1.25),dir(-90)); //bottom numbers label("$9$", (-12,-1.25),dir(90)); label("$6$", (-6,-1.25),dir(90)); label("$2$", (0,-1.25),dir(90)); label("$8$", (6,-1.25),dir(90)); label("$0$", (12,-1.25),dir(90)); //left numbers label("$4$", (-14,0),dir(0)); label("$1$", (-8,0),dir(0)); label("$3$", (-2,0),dir(0)); label("$7$", (4,0),dir(0)); label("$9$", (10,0),dir(0)); //right numbers label("$6$", (-10,0),dir(180)); label("$3$", (-4,0),dir(180)); label("$5$", (2,0),dir(180)); label("$4$", (8,0),dir(180)); label("$7$", (14,0),dir(180)); [/asy] Each of the sides of the five congruent rectangles is labeled with an integer, as shown above. These five rectangles are placed, without rotating or reflecting, in positions $I$ through $V$ so that the labels on coincident sides are equal. [asy] //diagram below problem statement size(7cm); for(int i=-3;i<=1;i+=2){ for(int j=-1;j<=0;++j){ if(i==1 && j==-1) continue; draw((i,j)--(i+2,j)--(i+2,j-1)--(i,j-1)--cycle); }} label("$I$",(-2,-0.5)); label("$II$",(0,-0.5)); label("$III$",(2,-0.5)); label("$IV$",(-2,-1.5)); label("$V$",(0,-1.5)); [/asy] Which of the rectangles is in position $I$? $\textbf{(A)} \ A \qquad \textbf{(B)} \ B \qquad \textbf{(C)} \ C \qquad \textbf{(D)} \ D \qquad \textbf{(E)} \ E$

A convex quadrilateral $ABC$ is given. $A',B',C',D'$ are the orthocenters of triangles $BCD, CDA, DAB, ABC$ respectively. Prove that in the quadrilaterals $ABCP$ and $A'B'C'D'$, the corresponding diagonals share the intersection points in the same ratio.

Given three natural numbers greater than $100$, that are pairwise coprime and such that the square of the difference of any two of them is divisible by the third and any of them is less than the product of the other two. Prove that these numbers are squares of some natural numbers.