Found problems: 85335
2008 Moldova Team Selection Test, 4
A non-zero polynomial $ S\in\mathbb{R}[X,Y]$ is called homogeneous of degree $ d$ if there is a positive integer $ d$ so that $ S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y)$ for any $ \lambda\in\mathbb{R}$. Let $ P,Q\in\mathbb{R}[X,Y]$ so that $ Q$ is homogeneous and $ P$ divides $ Q$ (that is, $ P|Q$). Prove that $ P$ is homogeneous too.
2005 IMO Shortlist, 6
Let $ABC$ be a triangle, and $M$ the midpoint of its side $BC$. Let $\gamma$ be the incircle of triangle $ABC$. The median $AM$ of triangle $ABC$ intersects the incircle $\gamma$ at two points $K$ and $L$. Let the lines passing through $K$ and $L$, parallel to $BC$, intersect the incircle $\gamma$ again in two points $X$ and $Y$. Let the lines $AX$ and $AY$ intersect $BC$ again at the points $P$ and $Q$. Prove that $BP = CQ$.
1998 May Olympiad, 4
$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle.
[img]https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif[/img]
2022 Denmark MO - Mohr Contest, 2
A positive integer is a [i]palindrome [/i] if it is written identically forwards and backwards. For example, $285582$ is a palindrome. A six digit number $ABCDEF$, where $A, B, C, D, E, F$ are digits, is called [i]cozy [/i] if $AB$ divides $CD$ and $CD$ divides $EF$. For example, $164896$ is cozy. Determine all cozy palindromes.
2006 Victor Vâlcovici, 2
Let $ ABC $ be a triangle with $ AB=AC $ and chose such that $ \angle BAC <120^{\circ } . $ On the altitude of $ ABC $ from $ A, $ consider the point $ O $ so that $ \angle BOC =120^{\circ } , $ and an arbitrary point $ M\neq O $ in the interior of $ ABC. $ Show that $ MA+MB+MC>OA+OB+OC. $
[i]Gheorghe Bucur[/i]
2014 AMC 8, 25
A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch?
Note: $1$ mile= $5280$ feet
[asy]size(10cm); pathpen=black; pointpen=black;
D(arc((-2,0),1,300,360));
D(arc((0,0),1,0,180));
D(arc((2,0),1,180,360));
D(arc((4,0),1,0,180));
D(arc((6,0),1,180,240));
D((-1.5,1)--(5.5,1));
D((-1.5,0)--(5.5,0),dashed);
D((-1.5,-1)--(5.5,-1));
[/asy]
$\textbf{(A) }\frac{\pi}{11}\qquad\textbf{(B) }\frac{\pi}{10}\qquad\textbf{(C) }\frac{\pi}{5}\qquad\textbf{(D) }\frac{2\pi}{5}\qquad \textbf{(E) }\frac{2\pi}{3}$
2022 HMNT, 9
Alice and Bob play the following "point guessing game." First, Alice marks an equilateral triangle $ABC$ and a point $D$ on segment $BC$ satisfying $BD=3$ and $CD=5$. Then, Alice chooses a point $P$ on line $AD$ and challenges Bob to mark a point $Q\neq P$ on line $AD$ such that $\frac{BQ}{QC}=\frac{BP}{PC}$. Alice wins if and only if Bob is unable to choose such a point. If Alice wins, what are the possible values of $\frac{BP}{PC}$ for the $P$ she chose?
2010 Postal Coaching, 4
$\triangle ABC$ has semiperimeter $s$ and area $F$ . A square $P QRS$ with side length $x$ is inscribed in $ABC$ with $P$ and $Q$ on $BC$, $R$ on $AC$, and $S$ on $AB$. Similarly, $y$ and $z$ are the sides of squares two vertices of which lie on $AC$ and $AB$, respectively. Prove that
\[\frac 1x +\frac 1y + \frac 1z \le \frac{s(2+\sqrt3)}{2F}\]
2005 Junior Balkan Team Selection Tests - Romania, 2
Find the largest positive integer $n>10$ such that the residue of $n$ when divided by each perfect square between $2$ and $\dfrac n2$ is an odd number.
2000 Regional Competition For Advanced Students, 3
We consider two circles $k_1(M_1, r_1)$ and $k_2(M_2, r_2)$ with $z = M_1M_2 > r_1+r_2$ and a common outer tangent with the tangent points $P_1$ and $P2$ (that is, they lie on the same side of the connecting line $M_1M_2$). We now change the radii so that their sum is $r_1+r_2 = c$ remains constant. What set of points does the midpoint of the tangent segment $P_1P_2$ run through, when $r_1$ varies from $0$ to $c$?
2024 Taiwan TST Round 2, 3
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f\colon \mathbb{N}\to \mathbb{N}$ such that $mf(m)+(f(f(m))+n)^2$ divides $4m^4+n^2f(f(n))^2$ for all positive integers $m$ and $n$.
1999 Putnam, 6
The sequence $(a_n)_{n\geq 1}$ is defined by $a_1=1,a_2=2,a_3=24,$ and, for $n\geq 4,$ \[a_n=\dfrac{6a_{n-1}^2a_{n-3}-8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.\] Show that, for all $n$, $a_n$ is an integer multiple of $n$.
2020 Durer Math Competition Finals, 2
What number should we put in place of the question mark such that the following statement becomes true?
$$11001_? = 54001_{10}$$
A number written in the subscript means which base the number is in.
2021 Harvard-MIT Mathematics Tournament., 9
Let scalene triangle $ABC$ have circumcenter $O$ and incenter $I$. Its incircle $\omega$ is tangent to sides $BC,CA,$ and $AB$ at $D,E,$ and $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $EF$, and let line $DP$ intersect $\omega$ again at $Q \ne D$. The line $OI$ intersects the altitude from $A$ to$ BC$ at $T$. Given that $OI \|BC,$ show that $PQ=PT$.
2021 China Team Selection Test, 2
Given distinct positive integer $ a_1,a_2,…,a_{2020} $. For $ n \ge 2021 $, $a_n$ is the smallest number different from $a_1,a_2,…,a_{n-1}$ which doesn't divide $a_{n-2020}...a_{n-2}a_{n-1}$. Proof that every number large enough appears in the sequence.
2016 Nigerian Senior MO Round 2, Problem 8
If $a, b, c, d$ are the solutions of the equation $x^4-kx-15=0$, find the equation whose solutions are $\frac{a+b+c}{d^2}, \frac{a+b+d}{c^2}, \frac{a+c+d}{b^2}, \frac{b+c+d}{a^2}$.
2014 Harvard-MIT Mathematics Tournament, 8
The numbers $2^0, 2^1, \dots , 2{}^1{}^5, 2{}^1{}^6 = 65536$ are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one form the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left?
1975 Putnam, A6
Given three points in space forming an acute-angled triangle, show that we can find two further points such that no three of the five points are collinear and the line through any two is normal to the plane through the other three.
2003 Olympic Revenge, 1
Let $ABC$ be a triangle with circumcircle $\Gamma$. $D$ is the midpoint of arc $BC$ (this arc does not contain $A$). $E$ is the common point of $BC$ and the perpendicular bisector of $BD$. $F$ is the common point of $AC$ and the parallel to $AB$ containing $D$. $G$ is the common point of $EF$ and $AB$. $H$ is the common point of $GD$ and $AC$. Show that $GAH$ is isosceles.
2012 Belarus Team Selection Test, 2
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that
\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]
[i]Proposed by Alexey Gladkich, Israel[/i]
1991 IMTS, 5
The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?
2017 F = ma, 15
An object starting from rest can roll without slipping down an incline.
Which of the following four objects, each a uniform solid sphere released from rest, would have the largest speed after the center of mass has moved through a vertical distance h?
$\textbf{(A)}\text{a sphere of mass M and radius R}$
$\textbf{(B)}\text{a sphere of mass 2M and radius} \frac{R}{2}$
$\textbf{(C)}\text{a sphere of mass }\frac{M}{2} \text{ and radius 2R}$
$\textbf{(D)}\text{a sphere of mass 3M and radius 3R}$
$\textbf{(E)}\text{All objects would have the same speed}$
2018 Math Prize for Girls Problems, 9
How many 3-term geometric sequences $a$, $b$, $c$ are there where $a$, $b$, and $c$ are positive integers with $a < b < c$ and $c = 8000$?
2022 JBMO Shortlist, G6
Let $ABC$ be a right triangle with hypotenuse $BC$. The tangent to the circumcircle of triangle $ABC$ at $A$ intersects the line $BC$ at $T$. The points $D$ and $E$ are chosen so that $AD = BD, AE = CE,$ and $\angle CBD = \angle BCE < 90^{\circ}$. Prove that $D, E,$ and $T$ are collinear.
Proposed by [i]Nikola Velov, Macedonia[/i]
2017 Finnish National High School Mathematics Comp, 1
By dividing the integer $m$ by the integer $n, 22$ is the quotient and $5$ the remainder.
As the division of the remainder with $n$ continues, the new quotient is $0.4$ and the new remainder is $0.2$.
Find $m$ and $n$.