Found problems: 85335
2015 BMT Spring, 10
Quadratics $g(x) = ax^2 + bx + c$ and $h(x) = dx^2 + ex + f$ are such that the six roots of $g,h$, and $g - h$ are distinct real numbers (in particular, they are not double roots) forming an arithmetic progression in some order. Determine all possible values of $a/d$.
2018 Iran MO (3rd Round), 2
Find all functions $f: \mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that:
$f(x^3+xf(xy))=f(xy)+x^2f(x+y) \forall x,y \in \mathbb{R}^{\ge 0}$
2022 China Team Selection Test, 2
Given a non-right triangle $ABC$ with $BC>AC>AB$. Two points $P_1 \neq P_2$ on the plane satisfy that, for $i=1,2$, if $AP_i, BP_i$ and $CP_i$ intersect the circumcircle of the triangle $ABC$ at $D_i, E_i$, and $F_i$, respectively, then $D_iE_i \perp D_iF_i$ and $D_iE_i = D_iF_i \neq 0$. Let the line $P_1P_2$ intersects the circumcircle of $ABC$ at $Q_1$ and $Q_2$. The Simson lines of $Q_1$, $Q_2$ with respect to $ABC$ intersect at $W$.
Prove that $W$ lies on the nine-point circle of $ABC$.
2014 ELMO Shortlist, 8
In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$.
[i]Proposed by Sammy Luo[/i]
2011 Romania National Olympiad, 2
The numbers $x, y, z, t, a$ and $b$ are positive integers, so that $xt-yz = 1$ and $$\frac{x}{y} \ge \frac{a}{b} \ge \frac{z}{t} .$$Prove that $$ab \le (x + z) (y +t)$$
2015 Azerbaijan IMO TST, 2
Alex and Bob play a game 2015 x 2015 checkered board by the following rules.Initially the board is empty: the players move in turn, Alex moves first. By a move, a player puts either red or blue token into any unoccopied square. If after a player's move there appears a row of three consecutive tokens of the same color( this row may be vertical,horizontal, or dioganal), then this player wins. If all the cells are occupied by tokens, but no such row appears, then a draw is declared.Determine whether Alex, Bob, or none of them has winning strategy.
2012 Estonia Team Selection Test, 1
Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.
2004 Iran MO (3rd Round), 3
Suppose $V= \mathbb{Z}_2^n$ and for a vector $x=(x_1,..x_n)$ in $V$ and permutation $\sigma$.We have $x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})$
Suppose $ n=4k+2,4k+3$ and $f:V \to V$ is injective and if $x$ and $y$ differ in more than $n/2$ places then $f(x)$ and $f(y)$ differ in more than $n/2$ places.
Prove there exist permutaion $\sigma$ and vector $v$ that $f(x)=x_{\sigma}+v$
2011 Romanian Master of Mathematics, 3
A triangle $ABC$ is inscribed in a circle $\omega$.
A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$).
Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$.
Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$.
[i](Russia) Vasily Mokin and Fedor Ivlev[/i]
2000 Estonia National Olympiad, 1
There are three candidates in the Hundilaane forest governor elections: $A, B$ and $C$. For each of the $20$ forest dwellers, the names of all three candidates were written on the ballot paper in the order of their preference. Examination of the ballots revealed that $11$ forest dwellers prefer $A$, $12$ $B$ and $14$ $C$. Which of the candidates will be marked first on the largest number of ballot papers when it is known that each possible the order of the candidates appears on at least one ballot?
2014 Balkan MO Shortlist, N1
$\boxed{N1}$Let $n$ be a positive integer,$g(n)$ be the number of positive divisors of $n$ of the form $6k+1$ and $h(n)$ be the number of positive divisors of $n$ of the form $6k-1,$where $k$ is a nonnegative integer.Find all positive integers $n$ such that $g(n)$ and $h(n)$ have different parity.
2012 Albania Team Selection Test, 4
Find all couples of natural numbers $(a,b)$ not relatively prime ($\gcd(a,b)\neq\ 1$) such that
\[\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.\]
2003 JHMMC 8, 21
The surface area and the volume of a cube are numerically equal. Find the cube’s volume.
2007 Rioplatense Mathematical Olympiad, Level 3, 5
Divide each side of a triangle into $50$ equal parts, and each point of the division is joined to the opposite vertex by a segment. Calculate the number of intersection points determined by these segments.
Clarification : the vertices of the original triangle are not considered points of intersection or division.
2005 JHMT, 2
Regular hexagon $ABCDEF$ is inscribed in rectangle $PQRS$ with $AB = 1$, A and $B$ on side $PQ$, $C$ on side $QR$, $D$ and $E$ on side $RS$, and $F$ on side $SP$. What is the area of $PQRS$?
2012 National Olympiad First Round, 11
The number of real quadruples $(x,y,z,w)$ satisfying $x^3+2=3y, y^3+2=3z, z^3+2=3w, w^3+2=3x$ is
$ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \text{None}$
2016 Romanian Master of Mathematics Shortlist, N1
Determine all integers $n \ge 3$ whose decimal expansion has less than $20$ digits, such that every quadratic non-residue modulo $n$ is a primitive root modulo $n$.
[i]An integer $a$ is a quadratic non-residue modulo $n$, if there is no integer $b$ such that $a - b^2$ is divisible by $n$. An integer $a$ is a primitive root modulo $n$, if for every integer $b$ relatively prime to n there is a positive integer $k$ such that $a^k - b$ is divisible by $n$.[/i]
2022 China Northern MO, 3
Let $\{a_n\}$ be a sequence of positive terms such that $a_{n+1}=a_n+ \frac{n^2}{a_n}$ . Let $b_n =a_n-n$ .
(1) Are there infinitely many $n$ such that $b_n \ge 0$ ?
(2) Prove that there is a positive number $M$ such that $\sum^{\infty}_{n=3} \frac{b_n}{n+1}<M$.
2020-2021 OMMC, 3
Define $f(x)$ as $\frac{x^2-x-2}{x^2+x-6}$. $f(f(f(f(1))))$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p,q$. Find $10p+q$.
2004 Iran MO (3rd Round), 21
$ a_1, a_2, \ldots, a_n$ are integers, not all equal. Prove that there exist infinitely many prime numbers $ p$ such that for some $ k$
\[ p\mid a_1^k \plus{} \dots \plus{} a_n^k.\]
2014 Bosnia And Herzegovina - Regional Olympiad, 1
Solve logarithmical equation $x^{\log _{3} {x-1}} + 2(x-1)^{\log _{3} {x}}=3x^2$
2010 National Chemistry Olympiad, 13
A $2.00 \text{L}$ balloon at $20.0^{\circ} \text{C}$ and $745 \text{mmHg}$ floats to an altitude where the temperature is $10.0^{\circ} \text{C}$ and the air pressure is $700 \text{mmHg}$. What is the new volume of the balloon?
$ \textbf{(A)}\hspace{.05in}0.94 \text{L}\qquad\textbf{(B)}\hspace{.05in}1.06 \text{L}\qquad\textbf{(C)}\hspace{.05in}2.06 \text{L}\qquad\textbf{(D)}\hspace{.05in}2.20 \text{L}\qquad $
2006 Romania National Olympiad, 2
Let $\displaystyle ABC$ and $\displaystyle DBC$ be isosceles triangle with the base $\displaystyle BC$. We know that $\displaystyle \measuredangle ABD = \frac{\pi}{2}$. Let $\displaystyle M$ be the midpoint of $\displaystyle BC$. The points $\displaystyle E,F,P$ are chosen such that $\displaystyle E \in (AB)$, $\displaystyle P \in (MC)$, $\displaystyle C \in (AF)$, and $\displaystyle \measuredangle BDE = \measuredangle ADP = \measuredangle CDF$. Prove that $\displaystyle P$ is the midpoint of $\displaystyle EF$ and $\displaystyle DP \perp EF$.
2016 Czech And Slovak Olympiad III A, 2
Let us denote successively $r$ and $r_a$ the radii of the inscribed circle and the exscribed circle wrt to side BC of triangle $ABC$. Prove that if it is true that $r+r_a=|BC|$ , then the triangle $ABC$ is a right one
2019 IMAR Test, 2
Let $ f_1,f_2,f_3,f_4 $ be four polynomials with real coefficients, having the property that
$$ f_1 (1) =f_2 (0), \quad f_2 (1) =f_3 (0),\quad f_3 (1) =f_4 (0),\quad f_4 (1) =f_1 (0) . $$
Prove that there exists a polynomial $ f\in\mathbb{R}[X,Y] $ such that
$$ f(X,0)=f_1(X),\quad f(1,Y) =f_2(Y) ,\quad f(1-X,1) =f_3(X),\quad f(0,1-Y)=f_4(Y) . $$