Twisted easy Number Theory

by reni_wee, May 23, 2025, 5:29 PM

Let $m$ and $n$ be positive integers posessing the following property: the equation
$\gcd(11k-1, m) = \gcd(11k-1 , n)$ holds for all positive integers $k$ . Prove that $m = 11^rn$ for some integer $r$ .

number theory

modular arithmetic

graph thory

by o.k.oo, May 23, 2025, 5:14 PM

There are 10 people at a party. None of the 3 friends of each person are friends with each other. What is the maximum number of friends at this party?

This post has been edited 1 time. Last edited by o.k.oo, an hour ago

graph theory

Consecutive squares are floors

by ICE_CNME_4, May 22, 2025, 1:50 PM

Determine how many positive integers $n$ have the property that both
$\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor$ are consecutive perfect squares.

algebra

Computing functions

by BBNoDollar, May 18, 2025, 5:25 PM

Let $f : [0, \infty) \to [0, \infty)$ , $f(x) = \dfrac{ax + b}{cx + d}$ , with $a, d \in (0, \infty)$ , $b, c \in [0, \infty)$ . Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
$f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.$ (For $n \in \mathbb{N}^*$ and $x \geq 0$ , the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$ . )

function

algebra

Simson lines on OH circle

by DVDTSB, May 13, 2025, 12:10 PM

Let $ABC$ and $DEF$ be two triangles inscribed in the same circle, centered at $O$ , and sharing the same orthocenter $H \ne O$ . The Simson lines of the points $D, E, F$ with respect to triangle $ABC$ form a non-degenerate triangle $\Delta$ .
Prove that the orthocenter of $\Delta$ lies on the circle with diameter $OH$ .

Note. Assume that the points $A, F, B, D, C, E$ lie in this order on the circle and form a convex, non-degenerate hexagon.

Proposed by Andrei Chiriță

This post has been edited 2 times. Last edited by DVDTSB, May 13, 2025, 12:24 PM

geometry

orhocenter

Simson line

real functional equation

by DottedCaculator, Nov 2, 2023, 11:15 PM

Find all functions $f:\mathbb R \to \mathbb R$ (from the set of real numbers to itself) where $f(x-y)+xf(x-1)+f(y)=x^2$ for all reals $x,y.$

Proposed by cj13609517288

ommc

algebra

functional equation

Fixed line

by TheUltimate123, Jun 29, 2023, 1:39 AM

Let $D$ be a point on segment $PQ$ . Let $\omega$ be a fixed circle passing through $D$ , and let $A$ be a variable point on $\omega$ . Let $X$ be the intersection of the tangent to the circumcircle of $\triangle ADP$ at $P$ and the tangent to the circumcircle of $\triangle ADQ$ at $Q$ . Show that as $A$ varies, $X$ lies on a fixed line.

Proposed by Elliott Liu and Anthony Wang

Elmo

geometry

Finding all possible $n$ on a strange division condition!!

by MathLuis, Nov 12, 2021, 12:23 AM

Find the sum of all positive integers $n$ such that
$\frac{n+11}{\sqrt{n-1}}$ is an integer.

This post has been edited 2 times. Last edited by MathLuis, Nov 12, 2021, 12:24 AM

IMO Shortlist 2012, Geometry 8

by lyukhson, Jul 29, 2013, 12:30 PM

Let $ABC$ be a triangle with circumcircle $\omega$ and $\ell$ a line without common points with $\omega$ . Denote by $P$ the foot of the perpendicular from the center of $\omega$ to $\ell$ . The side-lines $BC,CA,AB$ intersect $\ell$ at the points $X,Y,Z$ different from $P$ . Prove that the circumcircles of the triangles $AXP$ , $BYP$ and $CZP$ have a common point different from $P$ or are mutually tangent at $P$ .

Proposed by Cosmin Pohoata, Romania

This post has been edited 3 times. Last edited by djmathman, Aug 11, 2017, 2:28 PM
Reason: added source

geometry

circumcircle

IMO Shortlist

IMO 2012 P5

by mathmdmb, Jul 11, 2012, 7:03 PM

Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$ , and let $D$ be the foot of the altitude from $C$ . Let $X$ be a point in the interior of the segment $CD$ . Let $K$ be the point on the segment $AX$ such that $BK=BC$ . Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$ . Let $M$ be the point of intersection of $AL$ and $BK$ .

Show that $MK=ML$ .

Proposed by Josef Tkadlec, Czech Republic

This post has been edited 3 times. Last edited by Eternica, Jun 19, 2024, 10:03 AM

Robots in Space

by rrusczyk, Aug 25, 2011, 1:00 AM

I should have shared this sooner; crossposted from the AoPS blog:

AoPS blog wrote:

The folks at TopCoder are working with MIT, NASA, and DARPA to offer a robotics/programming competition for high school students. Winning teams may have their programs run by astronauts on the International Space Station! More details here. The deadline for registering is September 5.

contests

programming

robotics