A regular polygon of $n$ sides is inscribed in a circle of radius $R$. The area of the polygon is $3R^2$. Then $n$ equals: ${{ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15}\qquad\textbf{(E)}\ 18} $

The kid cut out of grid paper with the side of the cell $1$ rectangle along the grid lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle. - My rectangle ... - kid sobbed. - There is something strange about this figure! - Nonsense, do not mention it - Carlson said - waving his hand carelessly. - Here you see, in this figure the perimeter is the same as the area of ​​the rectangle was, and the area is the same as was the perimeter! What size square did Carlson cut out?

Find all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that \[xy(f(x)-f(y))|x-f(f(y))\] holds for all positive rationals $x$, $y$ (we define that $a|b$ if and only if exist $n \in \mathbb{Z}$ such that $b=an$) [i]Proposed by supercarry & windleaf1A[/i]

A rectangular piece of paper has the side lengths $12$ and $15$. A corner is bent about as shown in the figure. Determine the area of the gray triangle. [img]https://1.bp.blogspot.com/-HCfqWF0p_eA/XzcIhnHS1rI/AAAAAAAAMYg/KfY14frGPXUvF-H6ZVpV4RymlhD_kMs-ACLcBGAsYHQ/s0/1993%2BMohr%2Bp2.png[/img]

Five real numbers of absolute values not greater than $1$ and having the sum equal to $1$ are written on the circumference of a circle. Prove that one can choose three consecutively disposed numbers $a, b, c$, such that all the sums $a + b,b + c$ and $a + b + c$ are nonnegative.

Acute triangle $ABC$ has circumcircle $\Gamma$. Let $M$ be the midpoint of $BC.$ Points $P$ and $Q$ lie on $\Gamma$ so that $\angle APM = 90^{\circ}$ and $Q \neq A$ lies on line $AM.$ Segments $PQ$ and $BC$ intersect at $S$. Suppose that $BS = 1, CS = 3, PQ = 8\sqrt{\tfrac{7}{37}},$ and the radius of $\Gamma$ is $r$. If the sum of all possible values of $r^2$ can be expressed as $\tfrac ab$ for relatively prime positive integers $a$ and $b,$ compute $100a + b$.

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.

Points $D$ and $E$ are chosen on the sides $AB$ and $AC$ of the triangle $ABC$ in such a way that if $F$ is the intersection point of $BE$ and $CD$, then $AE + EF = AD + DF$. Prove that $AC + CF = AB + BF.$

Let $O$ be a fixed point in the plane. We have $2024$ red points, $2024$ yellow points and $2024$ green points in the plane, where there isn't any three colinear points and all points are distinct from $O$. It is known that for any two colors, the convex hull of the points that are from any of those two colors contains $O$ (it maybe contain it in it's interior or in a side of it). We say that a red point, a yellow point and a green point make a [i]bolivian[/i] triangle if said triangle contains $O$ in its interior or in one of its sides. Determine the greatest positive integer $k$ such that, no matter how such points are located, there is always at least $k$ [i]bolivian[/i] triangles.

Decide whether there exists a set $M$ of positive integers satisfying the following conditions: (i) For any natural number $m>1$ there exist $a, b \in M$ such that $a+b = m.$ (ii) If $a, b, c, d \in M$, $a, b, c, d > 10$ and $a + b = c + d$, then $a = c$ or $a = d.$

Twelve balls are numbered by the numbers $1,2,3,\cdots,12$. Each ball is colored either red or green, so that the following two conditions are satisfied: (i) If two balls marked by different numbers $a$ and $b$ are colored red and $a+b<13$, then the ball marked by the number $a+b$ is colored red, too. (ii) If two balls marked by different numbers $a$ and $b$ are colored green and $a+b<13$, then the ball marked by the number $a+b$ is also colored green. How many ways are there of coloring the balls? Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

Find the derivative of the solution of the equation $\ddot{x} = \dot{x}^2 + x^3$ with initial condition $x(0) = 0$, $\dot{x}(0) = A$ with respect to $A$ for $A = 0$.

How many multiples of $20$ are also divisors of $17!$?

Let $a_1, a_2, \ldots, a_n$ ($n\ge 3$) be positive integers such that $gcd(a_1, a_2, \ldots, a_n)=1$ and for each $i\in \lbrace 1,2,\ldots, n \rbrace$ we have $a_i|a_1+a_2+\ldots+a_n$. Prove that $a_1a_2\ldots a_n | (a_1+a_2+\ldots+a_n)^{n-2}$.

Let $a_1,a_2,\ldots,a_{2n}$ be positive real numbers such that $a_ja_{n+j}=1$ for the values $j=1,2,\ldots,n$. [list] a. Prove that either the average of the numbers $a_1,a_2,\ldots,a_n$ is at least 1 or the average of the numbers $a_{n+1},a_{n+2},\ldots,a_{2n}$ is at least 1. b. Assuming that $n\ge2$, prove that there exist two distinct numbers $j,k$ in the set $\{1,2,\ldots,2n\}$ such that \[|a_j-a_k|<\frac{1}{n-1}.\] [/list]

Let $F$ be the midpoint of circle arc $AB$, and let $M$ be a point on the arc such that $AM <MB$. The perpendicular drawn from point $F$ on $AM$ intersects $AM$ at point $T$. Show that $T$ bisects the broken line $AMB$, that is $AT =TM+MB$. KöMaL Gy. 2404. (March 1987), Archimedes of Syracuse

Given a segment $ AB $ and a line $ m $ parallel to this segment. Find the midpoint of the segment $ AB $ using only a ruler, i.e. drawing only straight lines.

Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is positive and $1$ otherwise. Prove that $A = \frac{1}{\pi}\cos^{-1}\left(\frac{x_1}{2}\right)$. [i]Evan O' Dorney.[/i]

Each of the numbers $1, 2, ... , 10$ is colored red or blue. $5$ is red and at least one number is blue. If $m, n$ are different colors and $m+n \le 10$, then $m+n$ is blue. If $m, n$ are different colors and $mn \le 10$, then $mn$ is red. Find all the colors.

$\overline{5654}_b$ is a power of a prime number. Find $b$ if $b > 6$.

Original post by shalomrav, but for some reason the mods locked the problem without any solves :noo: Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let $F_A$ be the (associated with $\Omega$) Feuerbach point of the triangle formed by the tangents to $\Omega$ at $B,C,D$, that is, the point of tangency of $\Omega$ and the nine-point circle of that triangle. Define $F_B, F_C, F_D$ similarly. Let $A'$ be the intersection of the tangents to $\Omega$ at $A$ and $F_A$. Define $B', C', D'$ similarly. Prove that quadrilaterals $ABCD$ and $A'B'C'D'$ are similar

Prove that for any convex quadrilateral it is always possible to cut out three smaller quadrilaterals similar to the original one with the scale factor equal to 1/2. (The angles of a smaller quadrilateral are equal to the corresponding original angles and the sides are twice smaller then the corresponding sides of the original quadrilateral.)

Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many miles from City $A$ will they be when they meet? $\textbf{(A) }20\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

Circles with centers $ A$ and $ B$ have radii 3 and 8, respectively. A common internal tangent intersects the circles at $ C$ and $ D$, respectively. Lines $ AB$ and $ CD$ intersect at $ E$, and $ AE \equal{} 5$. What is $ CD$? [asy]unitsize(2.5mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair A=(0,0), Ep=(5,0), B=(5+40/3,0); pair M=midpoint(A--Ep); pair C=intersectionpoints(Circle(M,2.5),Circle(A,3))[1]; pair D=B+8*dir(180+degrees(C)); dot(A); dot(C); dot(B); dot(D); draw(C--D); draw(A--B); draw(Circle(A,3)); draw(Circle(B,8)); label("$A$",A,W); label("$B$",B,E); label("$C$",C,SE); label("$E$",Ep,SSE); label("$D$",D,NW);[/asy]$ \textbf{(A) } 13\qquad \textbf{(B) } \frac {44}{3}\qquad \textbf{(C) } \sqrt {221}\qquad \textbf{(D) } \sqrt {255}\qquad \textbf{(E) } \frac {55}{3}$

Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$, one has $$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$ Call a positive integer $a$ to be $k$-good if there exists a good sequence such that $a_k = a$. Does there exists a $k$ such that there are exactly $2019$ $k$-good positive integers?