[b]p7.[/b] Let $A, B, C$, and $D$ be equally spaced points on a circle $O$. $13$ circles of equal radius lie inside $O$ in the configuration below, where all centers lie on $\overline{AC}$ or $\overline{BD}$, adjacent circles are externally tangent, and the outer circles are internally tangent to $O$. Find the ratio of the area of the region inside $O$ but outside the smaller circles to the total area of the smaller circles. [img]https://cdn.artofproblemsolving.com/attachments/9/7/7ff192baf58f40df0e4cfae4009836eab57094.png[/img] [b]p8.[/b] Find the greatest divisor of $40!$ that has exactly three divisors. [b]p9.[/b] Suppose we have positive integers $a, b, c$ such that $a = 30$, lcm $(a, b) = 210$, lcm $(b, c) = 126$. What is the minimum value of lcm $(a, c)$? PS. You should use hide for answers.

$n$ consecutive positive integers are put down in a row (not necessarily in order) so that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. What is the largest possible value of $n$ if the last number in the row is odd? (A Shapovalov)

$ABC$ is a triangle. The point A' lies on the side opposite to $A$ and $BA'/BC = k$, where $1/2 < k < 1$. Similarly, $B'$ lies on the side opposite to $B$ with $CB'/CA = k$, and $C'$ lies on the side opposite to $C$ with $AC'/AB = k$. Show that the perimeter of $A'B'C'$ is less than $k$ times the perimeter of $ABC$.

A set $S \subseteq \mathbb{N}$ satisfies the following conditions: (a) If $x, y \in S$ (not necessarily distinct), then $x + y \in S$. (b) If $x$ is an integer and $2x \in S$, then $x \in S$. Find the number of pairs of integers $(a, b)$ with $1 \le a, b\le 50$ such that if $a, b \in S$ then $S = \mathbb{N}.$ [i] Proposed by Yang Liu [/i]

Let $\omega = e^{2\pi i /5}$ be a primitive fifth root of unity. Prove that there do not exist integers $a, b, c, d, k$ with $k > 1$ such that \[(a + b \omega + c \omega^2 + d \omega^3)^{k}=1+\omega.\] [i]Carl Lian[/i]

Triangle $ABC$ lies in the first quadrant. Points $A$, $B$, and $C$ are reflected across the line $y=x$ to points $A'$, $B'$, and $C'$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is [u][i]not[/i][/u] always true? $(A)$ Triangle $A'B'C'$ lies in the first quadrant. $(B)$ Triangles $ABC$ and $A'B'C'$ have the same area. $(C)$ The slope of line $AA'$ is $-1$. $(D)$ The slopes of lines $AA'$ and $CC'$ are the same. $(E)$ Lines $AB$ and $A'B'$ are perpendicular to each other.

Let $\{k_1, k_2, \dots , k_m\}$ a set of $m$ integers. Show that there exists a matrix $m \times m$ with integers entries $A$ such that each of the matrices $A + k_jI, 1 \leq j \leq m$ are invertible and their entries have integer entries (here $I$ denotes the identity matrix).

Let $a_1<a_2<a_3$ be positive integers. Prove that there are integers $x_1,x_2,x_3$ such that $\sum_{i=1}^3 |x_i | >0$, $\sum_{i=1}^3 a_ix_i= 0$ and $$\max_{1\le i\le 3} | x_i|<\frac{2}{\sqrt{3}}\sqrt{a_3}+1$$.

Does there exist a set of $100$ triangles in which not one of the triangles can be covered by the other $99$?

Show that for any positive real numbers $a, b, c$ true is inequality: $\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2 \ge 2\sqrt{2}\left(\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} \right)$.

Given $\triangle ABC$ inscribed in $(O)$ an let $I$ and $I_a$ be it's incenter and $A$-excenter ,respectively. Tangent lines to $(O)$ at $C,B$ intersect the angle bisector of $A$ at $M,N$ ,respectively. Second tangent lines through $M,N$ intersect $(O)$ at $X,Y$. Prove that $XYII_a$ is cyclic.

(A.Zaslavsky, 9--11) Given two circles. Their common external tangent is tangent to them at points $ A$ and $ B$. Points $ X$, $ Y$ on these circles are such that some circle is tangent to the given two circles at these points, and in similar way (external or internal). Determine the locus of intersections of lines $ AX$ and $ BY$.

Consider $2011^2$ points arranged in the form of a $2011 \times 2011$ grid. What is the maximum number of points that can be chosen among them so that no four of them form the vertices of either an isosceles trapezium or a rectangle whose parallel sides are parallel to the grid lines?

Steve stands in the middle of a field of an infinitely large chessboard, all of which are fields square and one square meter. Every second it randomly wanders into the middle one of the four neighboring fields, each of which has the same probability. How high is the probability that after $2015$ steps, he will have taken exactly five meters way from his starting square?

Given are finitely many points in the plane, no three on a line. They are painted in four colours, with at least one point of each colour. Prove that there exist three triangles, distinct but not necessarily disjoint, such that the three vertices of each triangle have different colours, and none of them contains a coloured point in its interior.

An equilateral triangle with side length $6$ has a square of side length $6$ attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked $A$ and $B$ in the figure) can be written as $m + \sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$. [asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle); draw((1,0)--(1+sqrt(3)/2,1/2)--(1/2+sqrt(3)/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2)); draw((0,0)--(-sqrt(3)/2,1/2)--(-sqrt(3)/2+1/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2)); dot((-sqrt(3)/2+1/2,1/2+sqrt(3)/2)); label("A", (-sqrt(3)/2+1/2,1/2+sqrt(3)/2), N); draw((1,0)--(1,-1)--(0,-1)--(0,0)); dot((1,-1)); label("B", (1,-1), SE); [/asy]

How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c,a\neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}$.) $\textbf{(A) } 3 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } \text{infinitely many}$

Prove that for positive real numbers $a,b,c$ satisfying $abc=1$ the following inequality holds: $$ \frac{a}{b(1+c)} +\frac{b}{c(1+a)}+\frac{c}{a(1+b)} \ge \frac{3}{2} $$

For a given integer $d$, let us define $S = \{m^{2}+dn^{2}| m, n \in\mathbb{Z}\}$. Suppose that $p, q$ are two elements of $S$ , where $p$ is prime and $p | q$. Prove that $r = q/p$ also belongs to $S$ .

Suppose there are $n$ ordered pairs of positive integers $(a_i,b_i)$ such that $a_i+b_i=2020$ and $a_ib_i$ is a multiple of $2020$, where $1\le i \le n$. Compute the sum \[\sum_{i=1}^{n} a_i+b_i.\]

Let $ABC$ be a triangle such that $\angle A=90^{\circ }$ and $\angle B<\angle C$. The tangent at $A$ to the circumcircle $\omega$ of triangle $ABC$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ in the line $BC$, let $X$ be the foot of the perpendicular from $A$ to $BE$, and let $Y$ be the midpoint of the segment $AX$. Let the line $BY$ intersect the circle $\omega$ again at $Z$. Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$. [hide="comment"] [i]Edited by Orl.[/i] [/hide]

$P(x)$ and $Q(x)$ are two polynomials with integer coefficients such that $P(x)|Q(x)^2+1$. [b]a)[/b] Prove that there exists polynomials $A(x)$ and $B(x)$ with rational coefficients and a rational number $c$ such that $P(x)=c(A(x)^2+B(x)^2)$. [b]b)[/b] If $P(x)$ is a monic polynomial with integer coefficients, Prove that there exists two polynomials $A(x)$ and $B(x)$ with integer coefficients such that $P(x)$ can be written in the form of $A(x)^2+B(x)^2$. [i]Proposed by Mohammad Gharakhani[/i]

On the $1$ km long ridge of Mount SPAM, there are $2006$ lemmings. In the beginning, each of them walks along the ridge in one of the two possible directions with speed $1$ m/s . When two lemmings meet, they both reverse the directions they walk but keep their walking speed. When some lemming reaches the end of the ridge, he falls down and dies. Find the least upper bound for the time it can take until all the lemmings are dead.

Consider ten concentric circles and ten rays as in the following figure. At the points where the inner circle is intersected by the rays write successively, in direction clockwise, the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. In the next circle we write the numbers $11, 12, 13, 14, 15, 16, 17, 18, 19,20$ successively, and so on successively until the last round were we write the numbers $91, 92, 93, 94, 95, 96, 97, 98, 99, 100$ successively. In this orde, the numbers $1, 11, 21, 31, 41, 51, 61, 71, 81, 91$ are in the same ray, and similarly for the other rays. In front of $50$ of those $100$ numbers, we use the sign ''$-$'' such as: a) in each of the ten rays, exist exactly $5$ signs ''$-$'' , and also b) in each of the ten concentric circles, to be exactly $5$ signs ''$-$''. Prove that the sum of the $100$ signed numbers that occur, equals zero. [img]https://cdn.artofproblemsolving.com/attachments/9/d/ffee6518fcd1b996c31cf06d0ce484a821b4ae.gif[/img]

Find all positive integers $n=p_1p_2 \cdots p_k$ which divide $(p_1+1)(p_2+1)\cdots (p_k+1)$ where $p_1 p_2 \cdots p_k$ is the factorization of $n$ into prime factors (not necessarily all distinct).