Found problems: 85335
2011 HMNT, 4
Toward the end of a game of Fish, the $2$ through $7$ of spades, inclusive, remain in the hands of three distinguishable players: DBR, RB, and DB, such that each player has at least one card. If it is known that DBR either has more than one card or has an even-numbered spade, or both, in how many ways can the players’ hands be distributed?
2014 District Olympiad, 2
Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer.
[list=a]
[*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number?
[*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]
V Soros Olympiad 1998 - 99 (Russia), 11.4
Given a triangular pyramid in which all the plane angles at one of the vertices are right. It is known that there is a point in space located at a distance of $3$ from the indicated vertex and at distances $\sqrt5, \sqrt6, \sqrt7$ from three other vertices. Find the radius of the sphere circumscribed around this pyramid. (The circumscribed sphere for a pyramid is the sphere containing all its vertices.)
1986 Austrian-Polish Competition, 2
The monic polynomial $P(x) = x^n + a_{n-1}x^{n-1} +...+ a_0$ of degree $n > 1$ has $n$ distinct negative roots. Prove that $a_1P(1) > 2n^2a_o$
2022 JHMT HS, 8
In equilateral $\triangle ABC$, point $D$ lies on $\overline{BC}$ such that the radius of the circumcircle $\Gamma_1$ of $\triangle ACD$ is $7$ and the radius of the incircle $\Gamma_2$ of $\triangle{ABD}$ is $2$. Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at points $X$ and $Y$. Find $XY$.
2012 China Girls Math Olympiad, 5
As shown in the figure below, the in-circle of $ABC$ is tangent to sides $AB$ and $AC$ at $D$ and $E$ respectively, and $O$ is the circumcenter of $BCI$. Prove that $\angle ODB = \angle OEC$.
[asy]import graph; size(5.55cm); pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-5.76,xmax=4.8,ymin=-3.69,ymax=3.71;
pen zzttqq=rgb(0.6,0.2,0), wwwwqq=rgb(0.4,0.4,0), qqwuqq=rgb(0,0.39,0);
pair A=(-2,2.5), B=(-3,-1.5), C=(2,-1.5), I=(-1.27,-0.15), D=(-2.58,0.18), O=(-0.5,-2.92);
D(A--B--C--cycle,zzttqq); D(arc(D,0.25,-104.04,-56.12)--(-2.58,0.18)--cycle,qqwuqq); D(arc((-0.31,0.81),0.25,-92.92,-45)--(-0.31,0.81)--cycle,qqwuqq);
D(A--B,zzttqq); D(B--C,zzttqq); D(C--A,zzttqq); D(CR(I,1.35),linewidth(1.2)+dotted+wwwwqq); D(CR(O,2.87),linetype("2 2")+blue); D(D--O); D((-0.31,0.81)--O);
D(A); D(B); D(C); D(I); D(D); D((-0.31,0.81)); D(O);
MP( "A", A, dir(110)); MP("B", B, dir(140)); D("C", C, dir(20)); D("D", D, dir(150)); D("E", (-0.31, 0.81), dir(60)); D("O", O, dir(290)); D("I", I, dir(100));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]
1999 Balkan MO, 3
Let $ABC$ be an acute-angled triangle of area 1. Show that the triangle whose vertices are the feet of the perpendiculars from the centroid $G$ to
$AB$, $BC$, $CA$ has area between $\frac 4{27}$ and $\frac 14$.
1997 Canadian Open Math Challenge, 6
The triangle ABC has sides AB = 137, AC = 241, and BC =200. There is a point D, on BC, such that both incircles of triangles ABD and ACD touch AD at the same point E. Determine the length of CD.
[asy]
pair A = (2,6);
pair B = (0,0);
pair C = (10,0);
pair D = (3.5,0) ;
pair E = (3.1,2);
draw(A--B);
draw(B--C);
draw(C--A);
draw (A--D);
dot ((3.1,1.7));
label ("E", E, dir(45));
label ("A", A, dir(45));
label ("B", B, dir(45));
label ("C", C, dir(45));
label ("D", D, dir(45));
draw(circle((1.8,1.3),1.3));
draw(circle((4.9,1.7),1.75));
[/asy]
2009 JBMO Shortlist, 5
$\boxed{\text{A5}}$ Let $x,y,z$ be positive reals. Prove that $(x^2+y+1)(x^2+z+1)(y^2+x+1)(y^2+z+1)(z^2+x+1)(z^2+y+1)\geq (x+y+z)^6$
2012 AIME Problems, 14
Complex numbers $a$, $b$ and $c$ are the zeros of a polynomial $P(z) = z^3+qz+r$, and $|a|^2+|b|^2+|c|^2=250$. The points corresponding to $a$, $b$, and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h$. Find $h^2$.
2005 Junior Balkan MO, 1
Find all positive integers $x,y$ satisfying the equation \[ 9(x^2+y^2+1) + 2(3xy+2) = 2005 . \]
2022 Oral Moscow Geometry Olympiad, 6
In a tetrahedron, segments connecting the midpoints of heights with the orthocenters of the faces to which these heights are drawn intersect at one point. Prove that in such a tetrahedron all faces are equal or there are perpendicular edges.
(Yu. Blinkov)
2018 Hong Kong TST, 1
Does there exist a polynomial $P(x)$ with integer coefficients such that $P(1+\sqrt[3]{2})=1+\sqrt[3]{2}$ and $P(1+\sqrt5)=2+3\sqrt5$?
2017 South East Mathematical Olympiad, 5
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $O$, where $AC\perp BD$. $M,N$ are the midpoint of arc $ADC,ABC$. $DO$ and $AN$ intersect each other at $G$, the line passes through $G$ and parellel to $NC$ intersect $CD$ at $K$. Prove that $AK\perp BM$.
2016 NIMO Summer Contest, 6
A positive integer $n$ is lucky if $2n+1$, $3n+1$, and $4n+1$ are all composite numbers. Compute the smallest lucky number.
[i]Proposed by Michael Tang[/i]
2009 China Western Mathematical Olympiad, 4
Prove that for every given positive integer $k$, there exist infinitely many $n$, such that $2^{n}+3^{n}-1, 2^{n}+3^{n}-2,\ldots, 2^{n}+3^{n}-k$ are all composite numbers.
2018 Moscow Mathematical Olympiad, 10
$ABC$ is acute-angled triangle, $AA_1,CC_1$ are altitudes. $M$ is centroid. $M$ lies on circumcircle of $A_1BC_1$. Find all values of $\angle B$
1949 Putnam, B6
Let $C$ be a closed convex curve with a continuously turning tangent and let $O$ be a point inside $C.$ For each point $P$ on $C$ we define $T(P)$ as follows:
Draw the tangent to $C$ at $P$ and from $O$ drop the perpendicular to that tangent. Then $T(P)$ is the point at which $C$ intersects this perpendicular.
Starting now with a point $P_{0}$ on $C$, define points $P_n$ by $P_n =T(P_{n-1}).$ Prove that the points $P_{n}$ approach a limit and characterize all possible limit points. (You may assume that $T$ is continuous.)
2019 Saudi Arabia Pre-TST + Training Tests, 2.1
Let pairwise different positive integers $a,b, c$ with gcd$(a,b,c) = 1$ are such that $a | (b - c)^2, b | (c- a)^2, c | (a - b)^2$. Prove, that there is no non-degenerate triangle with side lengths $a, b$ and $c$.
2014 ASDAN Math Tournament, 3
Let $ABC$ be a triangle and $I$ its incenter. Suppose $AI=\sqrt{2}$, $BI=\sqrt{5}$, $CI=\sqrt{10}$ and the inradius is $1$. Let $A'$ be the reflection of $I$ across $BC$, $B'$ the reflection across $AC$, and $C'$ the reflection across $AB$. Compute the area of triangle $A'B'C'$.
2025 India National Olympiad, P2
Let $n\ge 2$ be a positive integer. The integers $1,2,\cdots,n$ are written on a board. In a move, Alice can pick two integers written on the board $a\neq b$ such that $a+b$ is an even number, erase both $a$ and $b$ from the board and write the number $\frac{a+b}{2}$ on the board instead. Find all $n$ for which Alice can make a sequence of moves so that she ends up with only one number remaining on the board.
[b]Note.[/b] When $n=3$, Alice changes $(1,2,3)$ to $(2,2)$ and can't make any further moves.
[i]Proposed by Rohan Goyal[/i]
2022 Iranian Geometry Olympiad, 4
We call two simple polygons $P, Q$ $\textit{compatible}$ if there exists a positive integer $k$ such that each of $P, Q$ can be partitioned into $k$ congruent polygons similar to the other one. Prove that for every two even integers $m, n \geq 4$, there are two compatible polygons with $m$ and $n$ sides. (A simple polygon is a polygon that does not intersect itself.)
[i]Proposed by Hesam Rajabzadeh[/i]
2015 India Regional MathematicaI Olympiad, 5
Two circles \(\Gamma\) and \(\Sigma\) intersect at two distinct points \(A\) and \(B\). A line through \(B\) intersects \(\Gamma\) and \(\Sigma\) again at \(C\) and \(D\), respectively. Suppose that \(CA=CD\). Show that the centre of \(\Sigma\) lies on \(\Gamma\).
2014 Contests, 3
Let $a\# b$ be defined as $ab-a-3$. For example, $4\#5=20-4-3=13$ Compute $(2\#0)\#(1\#4)$.
1953 AMC 12/AHSME, 3
The factors of the expression $ x^2\plus{}y^2$ are:
$ \textbf{(A)}\ (x\plus{}y)(x\minus{}y) \qquad\textbf{(B)}\ (x\plus{}y)^2 \qquad\textbf{(C)}\ (x^{\frac{2}{3}}\plus{}y^{\frac{2}{3}})(x^{\frac{4}{3}}\plus{}y^{\frac{4}{3}}) \\
\textbf{(D)}\ (x\plus{}iy)(x\minus{}iy) \qquad\textbf{(E)}\ \text{none of these}$