Let $f: N \to N$ be a function satisfying the following: $\bullet$ $f(ab) = f(a)f(b)$, whenever the greatest common divisor of $a$ and $b$ is $1$. $\bullet$ $f(p + q) = f(p)+ f(q)$ whenever $p$ and $q$ are primes. Determine all possible values of $f(2002)$. Justify your answers.

Let $P(x) = x^3 - px^2 + qx - r$ be a cubic polynomial with integer roots $a, b, c$. [b](a)[/b] Show that the greatest common divisor of $p, q, r$ is equal to $1$ if the greatest common divisor of $a, b, c$ is equal to $1$. [b](b)[/b] What are the roots of polynomial $Q(x) = x^3-98x^2+98sx-98t$ with $s, t$ positive integers.

Let $m,k,n$ be positive integers. Determine all triples $(m,k,n)$ satisfying the following equation: $3^m5^k=n^3+125$

[asy] size(5cm); pen p=linewidth(3), dark_grey=gray(0.25), ll_grey=gray(0.90), light_grey=gray(0.75); transform dishift(real x) { return shift(x,x); } // Draw the table of latch of table path ell = ((0,0)--(0,-1)--(-0.1,-1)--(-0.1,-0.1)--(-1,-0.1)--(-1,0)--cycle); // the ell shape path corner = dishift(-0.85)*ell; // define the path path table = dishift(-1)*scale(5)*ell; // define the table by scaling the pulley filldraw(corner, ll_grey, light_grey+p); // base of pulley filldraw(table, ll_grey, grey+p); // table real block_size = 1.6; // template for block path block = unitsquare; pair block_center = (0.5,0.5); /* Resting block */ transform rest = shift(-5, -0.9) * scale(block_size); // transformation for resting block filldraw(rest * block, ll_grey, light_grey+p); // draw block draw(rest*(1,0.5)--dir(110), light_grey+p); // rope fr0m midpoint of right block to wheel label("$m$", rest * block_center, fontsize(16)); // label block /* Hanging block */ transform hang = shift(0.2,-4.1) * scale(block_size); // transformation for hanging block draw((1,0)--(1,-2.5), light_grey+p); // string of pulley filldraw(hang * block, ll_grey, light_grey+p); // fill it label("$M$",hang * block_center,fontsize(16)); // label the small m // Draws the actual pulley filldraw(unitcircle, grey, p); // outer boundary of pulley wheel filldraw(scale(0.4)*unitcircle, light_grey, p); // inner boundary of pulley wheel path pulley_body=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; // defines "arm" of pulley filldraw(pulley_body, ll_grey, dark_grey+p); // draws the arm filldraw(scale(0.18)*unitcircle, ll_grey, dark_grey+p); // inner circle of pulley [/asy] A pulley system of two blocks, shown above, is released from rest. The block on the table, which has mass $m=1.0 \, \text{kg}$ slides after the time of release and hits the pulley to come to a dead stop. There was originally a distance of $ 1.0 \, \text{m} $ between the block and the pulley, which the block fully covers during the slide. From the time of release to the time of hitting the pulley, the angle that the rope above the table makes with the horizontal axis is a, nearly constant, $10.0^\circ$. The hanging block has mass $ M = 2.0 \, \text{kg} $. The table has a coefficient of friction of $0.50$ with the block that sits on it. The pulley is frictionless. Also, assume that, during the entire slide, the block never leaves the ground. Let $t$ be the number of seconds in takes for the $1.0\text{-m}$ slide. Find $100t$, rounded to two significant figures. [i](Ahaan Rungta, 4 points)[/i]

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

In rectangular coordinate system, the area which is surrounded by the figure of $f(x)=a\sin ax+\cos ax(a>0)$ on a complete period and the figure of $g(x)=\sqrt{a^2+1}$ is________.

Find the maximal number of crosses with 5 squares that can be placed on 8x8 grid without overlapping.

Prove that for every non-negative integer $n$ there exist integers $x, y, z$ with $gcd(x, y, z) = 1$, such that $x^2 + y^2 + z^2 = 3^{2^n}$.(Poland)

A cylindrical tank with radius $ 4$ feet and height $ 9$ feet is lying on its side. The tank is filled with water to a depth of $ 2$ feet. What is the volume of the water, in cubic feet? $ \textbf{(A)}\ 24\pi \minus{} 36 \sqrt {2} \qquad \textbf{(B)}\ 24\pi \minus{} 24 \sqrt {3} \qquad \textbf{(C)}\ 36\pi \minus{} 36 \sqrt {3} \qquad \textbf{(D)}\ 36\pi \minus{} 24 \sqrt {2} \\ \textbf{(E)}\ 48\pi \minus{} 36 \sqrt {3}$

In $ \bigtriangleup ABC $, $ AB = 86 $, and $ AC = 97 $. A circle with center $ A $ and radius $ AB $ intersects $ \overline{BC} $ at points $ B $ and $ X $. Moreover $ \overline{BX} $ and $ \overline{CX} $ have integer lengths. What is $ BC $? $ \textbf{(A)} \ 11 \qquad \textbf{(B)} \ 28 \qquad \textbf{(C)} \ 33 \qquad \textbf{(D)} \ 61 \qquad \textbf{(E)} \ 72 $

Given that $x$ and $y$ are positive integers such that $xy^{10}$ is perfect 33rd power of a positive integer, prove that $x^{10}y$ is also a perfect 33rd power!

Let $ f : (0,\infty) \to \mathbb R$ be a continous function such that the sequences $ \{f(nx)\}_{n\geq 1}$ are nondecreasing for any real number $ x$. Prove that $ f$ is nondecreasing.

Three circular arcs $\gamma_1, \gamma_2,$ and $\gamma_3$ connect the points $A$ and $C.$ These arcs lie in the same half-plane defined by line $AC$ in such a way that arc $\gamma_2$ lies between the arcs $\gamma_1$ and $\gamma_3.$ Point $B$ lies on the segment $AC.$ Let $h_1, h_2$, and $h_3$ be three rays starting at $B,$ lying in the same half-plane, $h_2$ being between $h_1$ and $h_3.$ For $i, j = 1, 2, 3,$ denote by $V_{ij}$ the point of intersection of $h_i$ and $\gamma_j$ (see the Figure below). Denote by $\widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}}$ the curved quadrilateral, whose sides are the segments $V_{ij}V_{il},$ $V_{kj}V_{kl}$ and arcs $V_{ij}V_{kj}$ and $V_{il}V_{kl}.$ We say that this quadrilateral is $circumscribed$ if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals $\widehat{V_{11}V_{21}}\widehat{V_{22}V_{12}}, \widehat{V_{12}V_{22}}\widehat{V_{23}V_{13}},\widehat{V_{21}V_{31}}\widehat{V_{32}V_{22}}$ are circumscribed, then the curved quadrilateral $\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}$ is circumscribed, too. [i]Proposed by Géza Kós, Hungary[/i] [asy] pathpen=black; size(400); pair A=(0,0), B=(4,0), C=(10,0); draw(L(A,C,0.3)); MP("A",A); MP("B",B); MP("C",C); pair X=(5,-7); path G1=D(arc(X,C,A)); pair Y=(5,7), Z=(9,6); draw(Z--B--Y); struct T {pair C;real r;}; T f(pair X, pair B, pair Y, pair Z) { pair S=unit(Y-B)+unit(Z-B); real s=abs(sin(angle((Y-B)/(Z-B))/2)); real t=10, r=abs(X-A); pair Q; for(int k=0;k<30;++k) { Q=B+t*S; t-=(abs(X-Q)-r)/abs(S)-s*t; } T T=new T; T.C=Q; T.r=s*t*abs(S); return T; } void g(pair Q, real r) { real t=0; for(int k=0;k<30;++k) { X=(5,t); t+=(abs(X-Q)+r-abs(X-A)); } } pair Z1=(1.07,6); draw(B--Z1); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G2=D(arc(X,C,A)); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G3=D(arc(X,C,A)); pen p=black+fontsize(8); MC("\gamma_1",G1,0.85,p); MC("\gamma_2",G2,0.85,NNW,p); MC("\gamma_3",G3,0.85,WNW,p); MC("h_1",B--Z1,0.95,E,p); MC("h_2",B--Y,0.95,E,p); MC("h_3",B--Z,0.95,E,p); path[] G={G1,G2,G3}; path[] H={B--Z1,B--Y,B--Z}; pair[][] al={{S+SSW,S+SSW,3*S},{SE,NE,NW},{2*SSE,2*SSE,2*E}}; for(int i=0;i<3;++i) for(int j=0;j<3;++j) MP("V_{"+string(i+1)+string(j+1)+"}",IP(H[i],G[j]),al[i][j],fontsize(8));[/asy]

Wild Bill goes to Las Vejas and takes part in a special lottery called [i]Reverse Yrettol[/i]. In this lottery, a player may buy a ticket on which he or she may select $5$ distinct numbers from $1-20$ (inclusive). Then, $5$ distinct numbers from $1-20$ are drawn at random. A player wins if his or her ticket contains [i]none[/i] of the numbers which were drawn. If Wild Bill buys a ticket, what is the probability that he will win? [i]2017 CCA Math Bonanza Lightning Round #1.4[/i]

Given a circle, construct a chord that is trisected by two given noncollinear radii.

Given an integer $ n>6$. Consider those integers $ k\in (n(n\minus{}1),n^{2})$ which are coprime with $ n$. Prove that the greatest common divisor of the considered numbers is $ 1$.

Let $ABC$ be a triangle, $D$ the midpoint of side $BC$ and $E$ the intersection point of the bisector of angle $\angle BAC$ with side $BC$. The perpendicular bisector of $AE$ intersects the bisectors of angles $\angle CBA$ and $\angle CDA$ at $M$ and $N$, respectively. The bisectors of angles $\angle CBA$ and $\angle CDA$ intersect at $P$ . Prove that points $A, M, N, P$ are concyclic.

Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign "$+$" or a multiplication sign "$\times$" between each pair of adjacent digits. (For example, her expression could be $4 + 3\times 3$, with value 13.) Each positive digit is equally likely, each arithmetic sign ("$+$" or "$\times$") is equally likely, and all choices are independent. What is the expected value (average value) of her expression?

Misha solved the equation $x^2 + ax + b = 0$ and told Dima the set of four numbers - two roots and two coefficients of this equation (but not said which of them are roots and which are coefficients). Will he be able to Dima, find out what equation Misha solved if all the numbers in the set turned out to be different?

Let $S$ be a finite set of real numbers such that given any three distinct elements $x,y,z\in\mathbb{S}$, at least one of $x+y$, $x+z$, or $y+z$ is also contained in $S$. Find the largest possible number of elements that $S$ could have.

An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given. a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$; b) Calculate the distance of $p$ from either base; c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.

Several crocodiles, dragons and snakes were left on an island. Animals were eating each other according to the following rules. Every day at the breakfast, each snake ate one dragon; at the lunch, each dragon ate one crocodile; and at the dinner, each crocodile ate one snake. On the Saturday after the dinner, only one crocodile and no snakes and dragons remained on the island. How many crocodiles, dragons and snakes were there on the Monday in the same week before the breakfast?

Let $a, b, c, d$ be odd positive integers and pairwise coprime. For a positive integer $n$, let $$f(n) = \left[\frac{n}{a} \right]+\left[\frac{n}{b}\right]+\left[\frac{n}{c}\right]+\left[\frac{n}{d}\right]$$ Prove that $$\sum_{n=1}^{abcd}(-1)^{f(n)}=1$$

Let $ \alpha ,\beta $ be real numbers. Find the greatest value of the expression $$ |\alpha x +\beta y| +|\alpha x-\beta y| $$ in each of the following cases: [b]a)[/b] $ x,y\in \mathbb{R} $ and $ |x|,|y|\le 1 $ [b]b)[/b] $ x,y\in \mathbb{C} $ and $ |x|,|y|\le 1 $

Given the vectors $ v_ {1}, \dots, v_ {n} $ and $ w_ {1}, \dots, w_ {n} $ in the plane with the following properties: for every $ 1 \leq i \leq n $ ,$ \left | v_{i} -w_{i} \right | \leq 1, $ and for every $ 1 \leq i <j \leq n $ ,$ \left | v_{i} -v_{j} \right | \ge 3 $ and $ v_{i} -w_ {i} \ne v_ {j} -w_ {j} $. Prove that for sets $ V = \left \{v_ {1}, \dots, v_{n } \right \} $ and $ W = \left \{w_ {1}, \dots, w_ {n} \right \}$, the set of $ V + (V \cup W) $ must have at least $ cn^{3/2} $ elements ,for some universal constant $ c>0 $ .