Found problems: 85335
Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent.
[i]Proposed by Robin Park[/i]
Define $f(x)=\max \{\sin x, \cos x\} .$ Find at how many points in $(-2 \pi, 2 \pi), f(x)$ is not differentiable?
[list=1]
[*] 0
[*] 2
[*] 4
[*] $\infty$
[/list]
Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n))=n+1987$ for all $n$.
Find the area of the quadrilateral with vertices at $(0,0), (2,0), (20,24), (0,2)$ in that order.
The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$.
[i]Proposed by Croatia[/i]
For each positive integer $k$, denote by $P (k)$ the number of all positive integers $4k$-digit numbers which can be composed of the digits $2, 0$ and which are divisible by the number $2 020$. Prove the inequality $$P (k) \ge \binom{2k - 1}{k}^2$$ and determine all $k$ for which equality occurs.
(Note: A positive integer cannot begin with a digit of $0$.)
(Jaromir Simsa)
Find the sum of the coefficients of the polynomial $(63x-61)^4$.
Let $n$ be a positive integer. Let $x_1$, $x_2$, $\ldots\,$, $x_n$ be a sequence of $n$ real numbers. Say that a sequence $a_1$, $a_2$, $\ldots\,$, $a_n$ is [i]unimodular[/i] if each $a_i$ is $\pm 1$. Prove that
\[
\sum a_1 a_2 \ldots a_n (a_1x_1 + a_2x_2 + \cdots + a_nx_n)^n = 2^{n} n!\, x_1 x_2 \ldots x_n ,
\]
where the sum is over all $2^{n}$ unimodular sequences $a_1$, $a_2$, $\ldots\,$, $a_n$.
Bisectors $AA_1$, $BB_1$, and $CC_1$ of triangle $ABC$ meet at point $I$. The perpendicular bisector to $BB_1$ meets $AA_1,CC_1$ at points $A_0,C_0$ respectively. Prove that the circumcircles of triangles $A_0IC_0$ and $ABC$ touch.
Let $ ABC$ be an arbitrary acute angled triangle. For any point $ P$ lying within the triangle, let
$ D$, $ E$, $ F$ denote the feet of the perpendiculars from $ P$ onto the sides $ AB$, $ BC$, $ CA$ respectively.
Determine the set of all possible positions of the point $ P$ for which the triangle $ DEF$ is isosceles.
For which position of $ P$ will the triangle $ DEF$ become equilateral?
Let $P$ be a point on the graph of the function $y=x+\frac{2}{x}(x>0)$. $PA,PB$ are perpendicular to line $y=x$ and $x=0$, respectively, the feet of perpendicular being $A$ and $B$. Find the value of $\overrightarrow{PA}\cdot \overrightarrow{PB}$.
Prove that for every natural number \( n>2 \) there exists an integer \( k \) that can be written as the sum of \( i \) positive perfect squares, for every \( i \) between \( 2 \) and \( n \).
Maxim has guessed a polynomial $f(x)$ of degree $n$. Sasha wants to guess it (knowing $n$). During a turn, Sasha can name a certain segment $[a;b]$ and Maxim will give in response the maximum value of $f(x)$ on the segment $[a;b]$. Will Sasha be able to guess $f(x)$ in a finite number of steps?
[i]M. Didin[/i]
In a triangle $\triangle ABC$ with incenter $I$, the incircle meets lines $BC, CA, AB$ at $D, E, F$ respectively.
Define the circumcenter of $\triangle IAB$ and $\triangle IAC$ $O_1$ and $O_2$ respectively.
Let the two intersections of the circumcircle of $\triangle ABC$ and line $EF$ be $P, Q$.
Prove that the circumcenter of $\triangle DPQ$ lies on the line $O_1O_2$.
The numbers -2, 4, 6, 9 and 12 are rearranged according to these rules:
1. The largest isn't first, but it is in one of the first three places.
2. The smallest isn't last, but it is in one of the last three places.
3. The median isn't first or last.
What is the average of the first and last numbers?
$\textbf{(A)}\: 3.5 \qquad\textbf{(B)} \:5 \qquad\textbf{(C)} \:6.5 \qquad\textbf{(D)} \:7.5 \qquad\textbf{(E)} \:8$
Let $ABCDE$ be a convex pentagon that has $AB < BC$, $AE <ED$ and $AB + CD + EA = BC + DE$. Variable points $F,G$ and $H$ are taken that move on the segments $BC$, $CD$ and $OF$ respectively . $B'$ is defined as the projection of $B$ on $AF$, $C'$ as the projection of $C$ on $FG$, $D'$ as the projection of $D$ on $GH$ and $E'$ as the projection of $E$ onto $HA$. Prove that there is at least one quadrilateral $B'C'D'E'$ when $F,G$ and $H$ move on their sides, which is a parallelogram.
Let $p$ be a prime number, $p> 5$. Determine the non-zero natural numbers $x$ with the property that $5p + x$ divides $5p ^ n + x ^ n$, whatever $n \in N ^ {*} $.
Diameter $AB$ divides a circle into two semicircles. Points $P_1$ , $P_2$, $...$, $P_n$ are given on one of the semicircles in this order. How should a point C be chosen on the other semicircle in order to maximize the sum of the areas of triangles $CP_1P_2$, $CP_2P_3$, $...$,$CP_{n-1}P_n$?
The numbers $1$ to $12$ are arranged in a sequence. The number of ways this can be done equals $12 \times11 \times 10\times ...\times 1$. We impose the condition that in the sequence there should be exactly one number that is smaller than the number directly preceding it.
How many of the $12 \times11 \times 10\times ...\times 1$ sequences satisfy this condition?
Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$
Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$
In the triangle $ ABC$, $ \angle B$ is greater than $ \angle C$. $ T$ is the midpoint of the arc $ BAC$ from the circumcircle of $ ABC$ and $ I$ is the incenter of $ ABC$. $ E$ is a point such that $ \angle AEI\equal{}90^\circ$ and $ AE\parallel BC$. $ TE$ intersects the circumcircle of $ ABC$ for the second time in $ P$. If $ \angle B\equal{}\angle IPB$, find the angle $ \angle A$.
Given integers $a> 0$, $n> 0$, suppose that $a^1 + a^2 +...+ a^n \equiv 1 \mod 10$.
Prove that $a \equiv n \equiv 1 \mod 10$ .
[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.
If $(7+4\sqrt{3})x^2+(2+\sqrt{3})x-2=0$, the larger root minus the smaller root is:
$ \textbf{(A)}\ -2+3\sqrt{3}\qquad\textbf{(B)}\ 2-\sqrt{3}\qquad\textbf{(C)}\ 6+3\sqrt{3}\qquad\textbf{(D)}\ 6-3\sqrt{3}\qquad\textbf{(E)}\ 3\sqrt{3}+2 $