Found problems: 85335
Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$.
[i]Alex Zhu.[/i]
Let $ABCDEF$ be a convex hexagon in which diagonals $AD, BE, CF$ are concurrent at $O$. Suppose $[OAF]$ is geometric mean of $[OAB]$ and $[OEF]$ and $[OBC]$ is geometric mean of $[OAB]$ and $[OCD]$. Prove that $[OED]$ is the geometric mean of $[OCD]$ and $[OEF]$.
(Here $[XYZ]$ denotes are of $\triangle XYZ$)
Given a triangle $ABC$ and a point $K$ . The lines $AK$,$BK$,$CK$ hit the opposite side of the triangle at $D,E,F$ respectively. On the exterior of $ABC$, we construct three pairs of similar triangles: $BDM$,$DCN$ on $BD$,$DC$, $CEP$,$EAQ$ on $CE$,$EA$, and $AFR$,$FBS$ on $AF$, $FB$. The lines $MN$,$PQ$,$RS$ intersect each other form a triangle $XYZ$. Prove that $AX$,$BY$,$CZ$ are concurrent.
Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .
The organizers of a mathematical congress found that if they accomodate any participant in a room the rest can be accomodated in double rooms so that 2 persons living in each room know each other. Prove that every participant can organize a round table on graph theory for himself and an even number of other people so that each participant of the round table knows both his neigbours.
[i]Proposed by S. Berlov, S. Ivanov[/i]
Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let
$$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$
(1) If $n = 6063$, find the maximum value of $S$.
(2) If $n= 2021$, find the maximum value of $S$.
Find the maximal value of $c>0$ such that for any $n\ge 1$, and for any $n$ real numbers $x_1, \cdots, x_n$ there exists real numbers $a ,b$ such that $$\{x_i-a\}+\{x_{i+1}-b\}\le \frac{1}{2024}$$ for at least $cn$ indices $i$. Here, $x_{n+1}=x_1$ and $\{x\}$ denotes the fractional part of $x$.
[i]Proposed by Wong Jer Ren[/i]
An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
$\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$
Let $ABC$ be a triangle inscribed in circle $\omega$ and $P$ a point in its interior. The lines $AP,BP$ and $CP$ intersect circle $\omega$ for the second time at $D,E$ and $F,$ respectively. If $A',B',C'$ are the reflections of $A,B,C$ with respect to the lines $EF,FD,DE,$ respectively, prove that the triangles $ABC$ and $A'B'C'$ are similar.
For how many positive integers $n \le 500$ is $n!$ divisible by $2^{n-2}$?
[i]Proposed by Eugene Chen[/i]
An infinite network is partitioned with dominos. Prove there exist three other tilings with dominos, have neither common domino with the existing tiling nor with each other.
Clarifications for network: It means an infinite board consisting of square cells.
Given is the non-equilateral triangle $A_1A_2A_3$. $B_{ij}$ is the symmetric of $A_i$ wrt the inner bisector of $\angle A_j$. Prove that lines $B_{12}B_{21}$, $B_{13}B_{31}$ and $B_{23}B_{32}$ are parallel.
The maximum value of $k$ such that the enequality $\sqrt{x-3}+\sqrt{6-x}\geq k$ has a real solution is
$\text{(A)}\sqrt6-\sqrt3\qquad\text{(B)}\sqrt3\qquad\text{(C)}\sqrt3+\sqrt6\qquad\text{(D)}\sqrt6$
Suppose that $a, b, c > 0$ such that $abc = 1$. Prove that \[ \frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1. \]
[b]p1.[/b] A shape made by joining four identical regular hexagons side-to-side is called a hexo. Two hexos are considered the same if one can be rotated / reflected to match the other. Find the number of different hexos.
[b]p2.[/b] The sequence $1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6,... $ consists of numbers written in increasing order, where every even number $2n$ is written once, and every odd number $2n + 1$ is written $2n + 1$ times. What is the $2019^{th}$ term of this sequence?
[b]p3.[/b] On planet EMCCarth, months can only have lengths of $35$, $36$, or $42$ days, and there is at least one month of each length. Victor knows that an EMCCarth year has $n$ days, but realizes that he cannot figure out how many months there are in an EMCCarth year. What is the least possible value of $n$?
[b]p4.[/b] In triangle $ABC$, $AB = 5$ and $AC = 9$. If a circle centered at $A$ passing through $B$ intersects $BC$ again at $D$ and $CD = 7$, what is $BC$?
[b]p5.[/b] How many nonempty subsets $S$ of the set $\{1, 2, 3,..., 11, 12\}$ are there such that the greatest common factor of all elements in $S$ is greater than $1$?
[b]p6.[/b] Jasmine rolls a fair $6$-sided die, with faces labeled from $1$ to $6$, and a fair $20$-sided die, with faces labeled from $1$ to $20$. What is the probability that the product of these two rolls, added to the sum of these two rolls, is a multiple of $3$?
[b]p7.[/b] Let $\{a_n\}$ be a sequence such that $a_n$ is either $2a_{n-1}$ or $a_{n-1} - 1$. Given that $a_1 = 1$ and $a_{12} = 120$, how many possible sequences $a_1$, $a_2$, $...$, $a_{12}$ are there?
[b]p8.[/b] A tetrahedron has two opposite edges of length $2$ and the remaining edges have length $10$. What is the volume of this tetrahedron?
[b]p9.[/b] In the garden of EMCCden, there is a tree planted at every lattice point $-10 \le x, y \le 10$ except the origin. We say that a tree is visible to an observer if the line between the tree and the observer does not intersect any other tree (assume that all trees have negligible thickness). What fraction of all the trees in the garden of EMCCden are visible to an observer standing at the origin?
[b]p10.[/b] Point $P$ lies inside regular pentagon $\zeta$, which lies entirely within regular hexagon $\eta$. A point $Q$ on the boundary of pentagon $\zeta$ is called projective if there exists a point $R$ on the boundary of hexagon $\eta$ such that $P$, $Q$, $R$ are collinear and $2019 \cdot \overline{PQ} = \overline{QR}$. Given that no two sides of $\zeta$ and $\eta$ are parallel, what is the maximum possible number of projective points on $\zeta$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
(Eisentein's Criterion) Let $f(x)=a_{n}x^{n} +\cdots +a_{1}x+a_{0}$ be a nonconstant polynomial with integer coefficients. If there is a prime $p$ such that $p$ divides each of $a_{0}$, $a_{1}$, $\cdots$,$a_{n-1}$ but $p$ does not divide $a_{n}$ and $p^2$ does not divide $a_{0}$, then $f(x)$ is irreducible in $\mathbb{Q}[x]$.
In a tetrahedral $ABCD$, given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$, $AD=BD=3$, and $CD=2$. Find the radius of the circumsphere of $ABCD$.
For any real numbers $x,y$ that satisfies the equation $$x+y-xy=155$$ and $$x^2+y^2=325$$, Find $|x^3-y^3|$
In the parallelogram $ABCD$, all sides are equal, and $\angle A = 60^o$. Let $F$ be a point on line $AD, H$ a point on line $DC$, and $G$ a point on diagonal $AC$ such that $DFGH$ is a parallelogram. Show that then $\vartriangle BHF$ is equilateral.
Equations $x^2+ax+b=0$ and $x^2+px+q=0$ have a common root.Find quadratic equation roots of which are two other roots.
In an $8\times 8$ chessboard, the rows are numbers from $1$ to $8$ and the columns are labelled from $a$ to $h$. In a two-player game on this chessboard, the fi rst player has a White Rook which starts on the square $b2$, and the second player has a Black Rook which starts on the square $c4$. The two players take turns moving their rooks. In each move, a rook lands on another square in the same row or the same column as its starting square. However, that square cannot be under attack by the other rook, and cannot have been landed on before by either rook. The player without a move loses the game. Which player has a winning strategy?
For every pair $(m, n)$ of positive integers, a positive real number $a_{m, n}$ is given. Assume that
\[a_{m+1, n+1} = \frac{a_{m, n+1} a_{m+1, n} + 1}{a_{m, n}}\]
for all positive integers $m$ and $n$. Suppose further that $a_{m, n}$ is an integer whenever $\min(m, n) \le 2$. Prove that $a_{m, n}$ is an integer for all positive integers $m$ and $n$.
Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$
Let $ \triangle{ABC}$ be a triangle with $ AB\not\equal{}AC$. The incircle with centre $ I$ touches $ BC$, $ CA$, $ AB$ at $ D$, $ E$, $ F$, respectively. Furthermore let $ M$ the midpoint of $ EF$ and $ AD$ intersect the incircle at $ P\not\equal{}D$.
Show that $ PMID$ ist cyclic.
Show that for each prime $p \geq 7$, there exist a positive integer $n$ and integers $x_{i}$, $y_{i}$ $(i = 1, . . . , n)$, not divisible by $p$, such that $x_{i}^{2}+ y_{i}^{2}\equiv x_{i+1}^{2}\pmod{p}$ where $x_{n+1} = x_{1}$