Found problems: 85335
2016 Postal Coaching, 5
A real polynomial of odd degree has all positive coefficients. Prove that there is a (possibly trivial) permutation of the coefficients such that the resulting polynomial has exactly one real zero.
2003 Estonia National Olympiad, 5
Is it possible to cover an $n \times n$ chessboard which has its center square cut out with tiles shown in the picture (each tile covers exactly $4$ squares, tiles can be rotated and turned around) if
a) $n = 5$,
b) $n = 2003$?
[img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]
2008 Sharygin Geometry Olympiad, 18
(A.Abdullayev, 9--11) Prove that the triangle having sides $ a$, $ b$, $ c$ and area $ S$ satisfies the inequality
\[ a^2\plus{}b^2\plus{}c^2\minus{}\frac12(|a\minus{}b|\plus{}|b\minus{}c|\plus{}|c\minus{}a|)^2\geq 4\sqrt3 S.\]
2018 Sharygin Geometry Olympiad, 6
Let $ABCD$ be a circumscribed quadrilateral. Prove that the common point of the diagonals, the incenter of triangle $ABC$ and the centre of excircle of triangle $CDA$ touching the side $AC$ are collinear.
1993 Poland - First Round, 4
Given is a circle with center $O$, point $A$ inside the circle and a chord $PQ$ which is not a diameter and passing through $A$. The lines $p$ and $q$ are tangent to the given circle at $P$ and $Q$ respectively. The line $l$ passing through $A$ and perpendicular to $OA$ intersects the lines $p$ and $q$ at $K$ and $L$ respectively. Prove that $|AK| = |AL|$.
1998 Bosnia and Herzegovina Team Selection Test, 1
Let $P_1$, $P_2$, $P_3$, $P_4$ and $P_5$ be five different points which are inside $D$ or on the border of figure $D$. Let $M=min\left\{P_iP_j \mid i \neq j\right\}$ be minimal distance between different points $P_i$. For which configuration of points $P_i$, value $M$ is at maximum, if :
$a)$ $D$ is unit square
$b)$ $D$ is equilateral triangle with side equal $1$
$c)$ $D$ is unit circle, circle with radius $1$
2019 Denmark MO - Mohr Contest, 2
Two distinct numbers a and b satisfy that the two equations $x^{2019} + ax + 2b = 0$ and $x^{2019}+ bx + 2a = 0$ have a common solution. Determine all possible values of $a + b$.
2008 IMO Shortlist, 5
Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that
\[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\]
[i]Proposed by Pavel Novotný, Slovakia[/i]
2017 Danube Mathematical Olympiad, 4
Determine all triples of positive integers $(x,y,z)$ such that $x^4+y^4 =2z^2$ and $x$ and $y$ are relatively prime.
2014 Contests, 3
Let $ABC$ be a triangle with $AB < AC$ and incentre $I$. Let $E$ be the point on the side $AC$ such that $AE = AB$. Let $G$ be the point on the line $EI$ such that $\angle IBG = \angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$.
Prove that the line $AI$, the line perpendicular to $AE$ at $E$, and the bisector of the angle $\angle BGI$ are concurrent.
2018 Serbia JBMO TST, 2
Show that for $a,b,c > 0$ the following inequality holds:
$\frac{\sqrt{ab}}{a+b+2c}+\frac{\sqrt{bc}}{b+c+2a}+\frac{\sqrt{ca}}{c+a+2b} \le \frac {3}{4}$.
2004 Purple Comet Problems, 2
If $h(a, b, c) = \frac{abc}{a+b+c}$, find $h(3\sqrt{5}, 6 \sqrt{5}, 9 \sqrt{5})$.
2012 National Olympiad First Round, 36
$k$ stones are put into $2012$ boxes in such a way that each box has at most $20$ stones. We are chosing some of the boxes. We are throwing some of the stones of the chosen boxes. Whatever the first arrangement of the stones inside the boxes is, if we can guarantee that there are equal stones inside the chosen boxes and the sum of them is at least $100$, then $k$ can be at least?
$ \textbf{(A)}\ 500 \qquad \textbf{(B)}\ 450 \qquad \textbf{(C)}\ 420 \qquad \textbf{(D)}\ 349 \qquad \textbf{(E)}\ 296$
2005 Czech And Slovak Olympiad III A, 2
Determine for which $m$ there exist exactly $2^{15}$ subsets $X$ of $\{1,2,...,47\}$ with the following property: $m$ is the smallest element of $X$, and for every $x \in X$, either $x+m \in X$ or $x+m > 47$.
2005 Iran Team Selection Test, 2
Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.
2004 National Olympiad First Round, 2
How many pairs of integers $(x,y)$ are there such that $2x+5y=xy-1$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 12
$
2024 AMC 12/AHSME, 4
What is the least value of $n$ such that $n!$ is a multiple of $2024$?
$
\textbf{(A) }11 \qquad
\textbf{(B) }21 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }23 \qquad
\textbf{(E) }253 \qquad
$
2023 AIME, 5
Let $P$ be a point on the circumcircle of square $ABCD$ such that $PA \cdot PC = 56$ and $PB \cdot PD = 90.$ What is the area of square $ABCD?$
2014 National Olympiad First Round, 32
There are $k$ stones on the table. Alper, Betul and Ceyhun take one or two stones from the table one by one. The player who cannot make a move loses the game and then the game finishes. The game is played once for each $k=5,6,7,8,9$. If Alper is always the first player, for how many of the games can Alper guarantee that he does not lose the game?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
PEN M Problems, 22
Let $\, a$, and $b \,$ be odd positive integers. Define the sequence $\{f_n\}_{n\ge 1}$ by putting $\, f_1 = a,$ $f_2 = b, \,$ and by letting $\, f_n \,$ for $\, n \geq 3 \,$ be the greatest odd divisor of $\, f_{n-1} + f_{n-2}$. Show that $\, f_n \,$ is constant for sufficiently large $\, n \,$ and determine the eventual value as a function of $\, a \,$ and $\, b$.
1990 Czech and Slovak Olympiad III A, 1
Let $(a_n)_{n\ge1}$ be a sequence given by
\begin{align*}
a_1 &= 1, \\
a_{2^k+j} &= -a_j\text{ for any } k\ge0,1\le j\le 2^k.
\end{align*}
Show that the sequence is not periodic.
1987 Traian Lălescu, 2.1
Any polynom, with coefficients in a given division ring, that is irreducible over it, is also irreducible over a given extension skew ring of it that's finite. Prove that the ring and its extension coincide.
1950 AMC 12/AHSME, 8
If the radius of a circle is increased $100\%$, the area is increased:
$\textbf{(A)}\ 100\% \qquad
\textbf{(B)}\ 200\% \qquad
\textbf{(C)}\ 300\% \qquad
\textbf{(D)}\ 400\% \qquad
\textbf{(E)}\ \text{By none of these}$
2010 LMT, 22
Two circles, $\omega_1$ and $\omega_2,$ intersect at $X$ and $Y.$ The segment between their centers intersects $\omega_1$ and $\omega_2$ at $A$ and $B$ respectively, such that $AB=2.$ Given that the radii of $\omega_1$ and $\omega_2$ are $3$ and $4,$ respectively, find $XY.$
2011 Princeton University Math Competition, A8
Calculate the sum of the coordinates of all pairs of positive integers $(n, k)$ such that $k\equiv 0, 3\pmod 4$, $n > k$, and $\displaystyle\sum^n_{i = k + 1} i^3 = (96^2\cdot3 - 1)\displaystyle\left(\sum^k_{i = 1} i\right)^2 + 48^2$