Show that when a prime number is divided by $30$, the remainder is $1$ or a prime number. Shows that if it is divided by $60$ or $90$ the same thing does not happen.

Consider two parallel lines $a$ and $b$. The circles $C,C_1$ are tangent to each other and to the line $a$. The circles $C,C_2$ are tangent to each other and to the line $b$. The circles $C_1,C_2$ are tangent to each other, have radii $R_1=9,R_2=16$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy82L2E1Y2NjMTY0NTNjMzExYmRmOTZjYmZlMWMwNzE4YmNlM2I0YTNkLnBuZw==&rn=Sm9zZXBoIDIwMjAucG5n[/img] What is the radius $R$ of the circle $C$? [i]Proposed by Ovidiu Bagdasar[/i]

A dealer bought $n$ radios for $d$ dollars, $d$ a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $\$8$ on each radio sold. If the overall profit was $\$72$, then the least possible value of $n$ for the given information is: $\textbf{(A)}\ 18\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 11$

All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Marcy could have? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

Let $ n \ge 2009$ be an integer and define the set: \[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce? [asy] //diagram by pog size(8.5cm); usepackage("mathptmx"); defaultpen(mediumgray*0.5+gray*0.5+linewidth(0.63)); add(grid(6,6)); label(scale(0.7)*"$1$", (1,-0.3), black); label(scale(0.7)*"$2$", (2,-0.3), black); label(scale(0.7)*"$3$", (3,-0.3), black); label(scale(0.7)*"$4$", (4,-0.3), black); label(scale(0.7)*"$5$", (5,-0.3), black); label(scale(0.7)*"$1$", (-0.3,1), black); label(scale(0.7)*"$2$", (-0.3,2), black); label(scale(0.7)*"$3$", (-0.3,3), black); label(scale(0.7)*"$4$", (-0.3,4), black); label(scale(0.7)*"$5$", (-0.3,5), black); label(scale(0.75)*rotate(90)*"Price (dollars)", (-1,3.2), black); label(scale(0.75)*"Weight (ounces)", (3.2,-1), black); dot((1,1.2),black); dot((1,1.7),black); dot((1,2),black); dot((1,2.8),black); dot((1.5,2.1),black); dot((1.5,3),black); dot((1.5,3.3),black); dot((1.5,3.75),black); dot((2,2),black); dot((2,2.9),black); dot((2,3),black); dot((2,4),black); dot((2,4.35),black); dot((2,4.8),black); dot((2.5,2.7),black); dot((2.5,3.7),black); dot((2.5,4.2),black); dot((2.5,4.4),black); dot((3,2.5),black); dot((3,3.4),black); dot((3,4.2),black); dot((3.5,3.8),black); dot((3.5,4.5),black); dot((3.5,4.8),black); dot((4,3.9),black); dot((4,5.1),black); dot((4.5,4.75),black); dot((4.5,5),black); dot((5,4.5),black); dot((5,5),black); [/asy] $\textbf{(A)} ~1\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~3\qquad\textbf{(D)} ~4\qquad\textbf{(E)} ~5\qquad$

Let $ABCD$ be a quadrilateral satisfying $\angle BCD=\angle CDA$. Suppose rays $AD$ and $BC$ meet at $E$, and let $\Gamma$ be the circumcircle of $ABE$. Let $\Gamma_1$ be a circle tangent to ray $CD$ past $D$ at $W$, segment $AD$ at $X$, and internally tangent to $\Gamma$. Similarly, let $\Gamma_2$ be a circle tangent to ray $DC$ past $C$ at $Y$, segment $BC$ at $Z$, and internally tangent to $\Gamma$. Let $P$ be the intersection of $WX$ and $YZ$, and suppose $P$ lies on $\Gamma$. If $F$ is the $E$-excenter of triangle $ABE$, and $AB=544$, $AE=2197$, $BE=2299$, then find $m+n$, where $FP=\tfrac{m}{n}$ with $m,n$ relatively prime positive integers. [i]Proposed by Michael Kural[/i]

Evaluate \[\int_0^1 \frac{(e^x+1)\{e^x+1+(1+x+e^x)\ln (1+x+e^x)\}}{1+x+e^x}\ dx\]

All expressions of the form $$\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}$$ (with every possible combination of signs) are multiplied together. Prove that the result is: (a) an integer; (b) the square of an integer. (A Kanel)

Alice and Bob play the following game. They write some fractions of the form $1/n$, where $n{}$ is positive integer, onto the blackboard. The first move is made by Alice. Alice writes only one fraction in each her turn and Bob writes one fraction in his first turn, two fractions in his second turn, three fractions in his third turn and so on. Bob wants to make the sum of all the fractions on the board to be an integer number after some turn. Can Alice prevent this? [i]Andrey Arzhantsev[/i]

Prove that there exists a positive integer $N$ such that for every polynomial $P(x)$ of degree $2019$, there exist $N$ linear polynomials $p_1,p_2, \ldots p_N$ such that $P(x)=p_1(x)^{2019}+p_2(x)^{2019}+ \ldots + p_N(x)^{2019}$. (Assume all polynomials in this problem have real coefficients, and leading coefficients cannot be zero.)

A convex hexagon is circumscribed about a circle of radius $1$. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number $r$ such that the length of at least one segment is at least $r.$

In kindergarten, nurse took $n>1$ identical cardboard rectangles and distributed them to $n$ children; every child got one rectangle. Every child cut his (her) rectangle into several identical squares (squares of different children could be different). Finally, the total number of squares was prime. Prove that initial rectangles was squares.

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.

All sides of a polygonal billiard table are in one of two perpendicular directions. A tiny billiard ball rolls out of the vertex $A$ of an inner $90^o$ angle and moves inside the billiard table, bouncing off its sides according to the law “angle of reflection equals angle of incidence”. If the ball passes a vertex, it will drop in and srays there. Prove that the ball will never return to $A$.

The negation of the proposition "For all pairs of real numbers $a$, $b$, if $a=0$, then $ab=0$" is: There are real numbers $a,b$ such that $\text{(A)} \ a\ne0,ab\ne 0 ~~\text{(B)} \ a\ne 0, ab=0 ~~ \text{(C)} \ a=0,ab\ne 0$ $\text{(D)} \ ab\ne0,a\ne0 ~~\text{(E)} \ ab=0,a\ne0$

A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$. Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows: A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.

Two positive integers $a$ and $b$ are prime-related if $a = pb$ or $b = pa$ for some prime $p$. Find all positive integers $n$, such that $n$ has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related. Note that $1$ and $n$ are included as divisors.

Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions.

For each prime $p$ of the form $4k+3$ with $k \in \mathbb{Z}^+$, consider the polynomial $$Q(x)=px^{2p} - x^{2p-1} + p^2x^{\frac{3p+1}{2}} - px^{p+1} +2(p^2+1)x^p -px^{p-1}+ p^2 x^{\frac{p-1}{2}} -x + p.$$ Determine all ordered pairs of polynomials $f, g$ with integer coefficients such that $Q(x)=f(x)g(x)$.

Let $S_n = 1 + 2 + ,,, + n$. Define $$T_n =\frac{S_2}{S_2- 1}\cdot \frac{S_3}{S_3 - 1}\cdot ... \cdot \frac{S_n}{S_n - 1}.$$ Find $T_{2015}.$

For how many integers $ n$ is $ \frac{n}{20\minus{}n}$ the square of an integer? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 10$

A [i]regular icosahedron[/i] is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.[asy] size(3cm); pair A=(0.05,0),B=(-.9,-0.6),C=(0,-0.45),D=(.9,-0.6),E=(.55,-0.85),F=(-0.55,-0.85),G=B-(0,1.1),H=F-(0,0.6),I=E-(0,0.6),J=D-(0,1.1),K=C-(0,1.4),L=C+K-A; draw(A--B--F--E--D--A--E--A--F--A^^B--G--F--K--G--L--J--K--E--J--D--J--L--K); draw(B--C--D--C--A--C--H--I--C--H--G--H--L--I--J--I--D^^H--B,dashed); dot(A^^B^^C^^D^^E^^F^^G^^H^^I^^J^^K^^L); [/asy]

Let $f(x) = 3x^2 + 1$. Prove that for any given positive integer $n$, the product $$f(1)\cdot f(2)\cdot\dots\cdot f(n)$$ has at most $n$ distinct prime divisors. [i]Proposed by Géza Kós[/i]

A prime number$ q $is called[b][i] 'Kowai' [/i][/b]number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one [b][i]'Kowai'[/i][/b] number can be found. Find the summation of all [b][i]'Kowai'[/i][/b] numbers.