Found problems: 85335
1994 Bulgaria National Olympiad, 2
Find all functions $f : R \to R$ such that $x f(x)-y f(y) = (x-y)f(x+y)$ for all $x,y \in R$.
2014 JBMO Shortlist, 5
Let $ABC$ be a triangle with ${AB\ne BC}$; and let ${BD}$ be the internal bisector of $\angle ABC,\ $, $\left( D\in AC \right)$. Denote by ${M}$ the midpoint of the arc ${AC}$ which contains point ${B}$. The circumscribed circle of the triangle ${\vartriangle BDM}$ intersects the segment ${AB}$ at point ${K\neq B}$. Let ${J}$ be the reflection of ${A}$ with respect to ${K}$. If ${DJ\cap AM=\left\{O\right\}}$, prove that the points ${J, B, M, O}$ belong to the same circle.
2007 India IMO Training Camp, 1
Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.
Estonia Open Senior - geometry, 2000.1.3
In the plane, the segments $AB$ and $CD$ are given, while the lines $AB$ and $CD$ intersect. Prove that the set of all points $P$ in the plane such that triangles $ABP$ and $CDP$ have equal areas , form two lines intersecting at the intersection of the lines $AB$ and $CD$.
2015 Romanian Master of Mathematics, 3
A finite list of rational numbers is written on a blackboard. In an [i]operation[/i], we choose any two numbers $a$, $b$, erase them, and write down one of the numbers \[
a + b, \; a - b, \; b - a, \; a \times b, \; a/b \text{ (if $b \neq 0$)}, \; b/a \text{ (if $a \neq 0$)}.
\] Prove that, for every integer $n > 100$, there are only finitely many integers $k \ge 0$, such that, starting from the list \[ k + 1, \; k + 2, \; \dots, \; k + n, \] it is possible to obtain, after $n - 1$ operations, the value $n!$.
2024 239 Open Mathematical Olympiad, 2
There are $2n$ points on the plane, no three of which lie on the same line. Some segments are drawn between them so that they do not intersect at internal points and any segment with ends among the given points intersects some of the drawn segments at an internal point. Is it true that it is always possible to choose $n$ drawn segments having no common ends?
2005 MOP Homework, 1
Given real numbers $x$, $y$, $z$ such that $xyz=-1$, show that
$x^4+y^4+z^4+3(x+y+z) \ge \sum_{sym} \frac{x^2}{y}$.
2019 BMT Spring, 7
(My problem. :D)
Call the number of times that the digits of a number change from increasing to decreasing, or vice versa, from the left to right while ignoring consecutive digits that are equal the [i]flux[/i] of the number. For example, the flux of 123 is 0 (since the digits are always increasing from left to right) and the flux of 12333332 is 1, while the flux of 9182736450 is 8. What is the average value of the flux of the positive integers from 1 to 999, inclusive?
2021 Flanders Math Olympiad, 1
Johnny once saw plums hanging, like eggs so big and numbered according to the first natural numbers. He is the first to pick the plum with number $2$. After that, Jantje picks the plum each time with the smallest number $n$ that satisfies the following two conditions:
$\bullet$ $n$ is greater than all numbers on the already picked plums,
$\bullet$ $n$ is not the product of two equal or different numbers on already picked plums.
We call the numbers on the picked plums plum numbers. Is $100 000$ a plum number?
Justify your answer.
2010 Laurențiu Panaitopol, Tulcea, 2
Find the strictly monotone functions $ f:\{ 0\}\cup\mathbb{N}\longrightarrow\{ 0\}\cup\mathbb{N} $ that satisfy the following two properties:
$ \text{(i)} f(2n)=n+f(n), $ for any nonnegative integers $ n. $
$ \text{(ii)} f(n) $ is a perfect square if and only if $ n $ is a perfect square.
2024 Auckland Mathematical Olympiad, 8
There are $25$ points on the plane, and among any three of them there are two at a distance less than $1$. Prove that there is a circle of radius $1$ containing at least $13$ of these points.
2019 MOAA, 10
Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.
ICMC 6, 3
Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started.
[i]Proposed by Dylan Toh[/i]
2018 JBMO Shortlist, G4
Let $ABC$ be a triangle with side-lengths $a, b, c$, inscribed in a circle with radius $R$ and let $I$ be ir's incenter. Let $P_1, P_2$ and $P_3$ be the areas of the triangles $ABI, BCI$ and $CAI$, respectively. Prove that $$\frac{R^4}{P_1^2}+\frac{R^4}{P_2^2}+\frac{R^4}{P_3^2}\ge 16$$
2001 Junior Balkan Team Selection Tests - Romania, 3
Let $ABCD$ be a quadrilateral inscribed in the circle $O$. For a point $E\in O$, its projections $K,L,M,N$ on the lines $DA,AB,BC,CD$, respectively, are considered. Prove that if $N$ is the orthocentre of the triangle $KLM$ for some point $E$, different from $A,B,C,D$, then this holds for every point $E$ of the circle.
2007 Croatia Team Selection Test, 5
Let there be two circles. Find all points $M$ such that there exist two points, one on each circle such that $M$ is their midpoint.
Novosibirsk Oral Geo Oly VII, 2023.5
One convex quadrilateral is inside another. Can it turn out that the sum of the lengths of the diagonals of the outer quadrilateral is less than the sum of the lengths of the diagonals of the inner?
2020 BMT Fall, 8
Let $ABCD$ be a unit square and let $E$ and $F$ be points inside $ABCD$ such that the line containing $\overline{EF}$ is parallel to $\overline{AB}$. Point $E$ is closer to $\overline{AD}$ than point $F$ is to $\overline{AD}$. The line containing $\overline{EF}$ also bisects the square into two rectangles of equal area. Suppose $[AEF B] = [DEFC] = 2[AED] = 2[BFC]$. The length of segment $\overline{EF}$ can be expressed as $m/n$ , where m and $n$ are relatively prime positive integers. Compute $m + n$.
2011 Kosovo National Mathematical Olympiad, 2
Is it possible that by using the following transformations:
\[ f(x) \mapsto x^2 \cdot f \left(\frac{1}{x}+1 \right) \ \ \ \text{or} \ \ \ f(x) \mapsto (x-1)^2 \cdot f\left(\frac{1}{x-1} \right)\]
the function $f(x)=x^2+5x+4$ is sent to the function $g(x)=x^2+10x+8$ ?
1997 Romania Team Selection Test, 1
Let $ABCDEF$ be a convex hexagon, and let $P= AB \cap CD$, $Q = CD \cap EF$, $R = EF \cap AB$, $S = BC \cap DE$, $T = DE \cap FA$, $U = FA \cap BC$. Prove that
$\frac{PQ}{CD} = \frac{QR}{EF} = \frac{RP}{AB}$ if and only if $\frac{ST}{DE} = \frac{TU}{FA} = \frac{US}{BC}$
2009 Romania Team Selection Test, 1
Given two (identical) polygonal domains in the Euclidean plane, it is not possible in general to superpose the two using only translations and rotations. Prove that this can however be achieved by splitting one of the domains into a finite number of polygonal subdomains which then fit together, via translations and rotations in the plane, to recover the other domain.
1974 IMO Longlists, 4
Let $K_a,K_b,K_c$ with centres $O_a,O_b,O_c$ be the excircles of a triangle $ABC$, touching the interiors of the sides $BC,CA,AB$ at points $T_a,T_b,T_c$ respectively.
Prove that the lines $O_aT_a,O_bT_b,O_cT_c$ are concurrent in a point $P$ for which $PO_a=PO_b=PO_c=2R$ holds, where $R$ denotes the circumradius of $ABC$. Also prove that the circumcentre $O$ of $ABC$ is the midpoint of the segment $PI$, where $I$ is the incentre of $ABC$.
1999 Bundeswettbewerb Mathematik, 4
A natural number is called [i]bright [/i] if it is the sum of a perfect square and a perfect cube.
Prove that if $r$ and $s$ are any two positive integers, then
(a) there exist infinitely many positive integers $n$ such that both $r+n$ and $s+n$ are [i]bright[/i],
(b) there exist infinitely many positive integers $m$ such that both rm and sm are [i]bright[/i].
1987 Traian Lălescu, 1.3
Let $ A'\neq A $ be the intersection of the bisector of $ \angle BAC $ with the circumcircle of the triangle $ ABC. $
Prove that $ AA'>\frac{AB+AC}{2}. $
2005 Italy TST, 1
Suppose that $f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}$ satisfies $f(1)=1$ and
\[f^{2005}(x)=x\quad\text{for}\ x=1,2,\ldots ,1600. \]
$(a)$ Prove that $f$ has a fixed point different from $1$.
$(b)$ Find all $n>1600$ such that any $f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}$ satisfying the above condition has at least two fixed points.