divide regions

by macves, Jun 1, 2025, 11:47 AM

We are given a geometry problem involving separating 99 red points and 100 blue points placed on the plane in general position (no 3 collinear), using lines that do not pass through any point, such that no region contains both a red and blue point. We are to find the smallest positive integer k such that for every configuration, k lines suffice to ensure all regions created by the lines contain points of only one color.

combinatorics

Old Inequality

by giangtruong13, Jun 1, 2025, 10:05 AM

Let $a,b,c >0$ and $abc=1$ . Prove that: $\sqrt{a^2-a+1}+\sqrt{b^2-b+1} +\sqrt{c^2-c+1} \ge a+b+c$

This post has been edited 1 time. Last edited by giangtruong13, 3 hours ago

inequalities

interesting geometry config (3/3)

by Royal_mhyasd, Jun 1, 2025, 7:06 AM

Let $\triangle ABC$ be an acute triangle, $H$ its orthocenter and $E$ the center of its nine point circle. Let $P$ be a point on the parallel through $C$ to $AB$ such that $\angle CPH = |\angle BAC-\angle ABC|$ and $P$ and $A$ are on different sides of $BC$ and $Q$ a point on the parallel through $B$ to $AC$ such that $\angle BQH = |\angle BAC - \angle ACB|$ and $C$ and $Q$ are on different sides of $AB$ . If $B'$ and $C'$ are the reflections of $H$ over $AC$ and $AB$ respectively, $S$ and $T$ are the intersections of $B'Q$ and $C'P$ respectively with the circumcircle of $\triangle ABC$ , prove that the intersection of lines $CT$ and $BS$ lies on $HE$ .

final problem for this "points on parallels forming strange angles with the orthocenter" config, for now. personally i think its pretty cool

Attachments:

geometry

orthocenter

circumcircle

interesting geo config (2/3)

by Royal_mhyasd, May 31, 2025, 11:36 PM

Let $\triangle ABC$ be an acute triangle and $H$ its orthocenter. Let $P$ be a point on the parallel through $A$ to $BC$ such that $\angle APH = |\angle ABC-\angle ACB|$ . Define $Q$ and $R$ as points on the parallels through $B$ to $AC$ and through $C$ to $AB$ similarly. If $P,Q,R$ are positioned around the sides of $\triangle ABC$ as in the given configuration, prove that $P,Q,R$ are collinear.

Attachments:

Easy P4 combi game with nt flavour

by Maths_VC, May 27, 2025, 8:01 PM

Two players, Alice and Bob, play the following game, taking turns. In the beginning, the number $1$ is written on the board. A move consists of adding either $1$ , $2$ or $3$ to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is $10$ , then in a move a player can't choose the number $2$ , but he can choose either $1$ or $3$ ). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.

combinatorics

Centroid, altitudes and medians, and concyclic points

by BR1F1SZ, May 5, 2025, 9:45 PM

Let $\triangle{ABC}$ be an acute triangle with $BC > AC$ . Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$ . The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$ . Let $M$ be the point where $CS$ intersects $AB$ . The line $SF$ intersects the circle $\gamma$ at a point $Q$ , such that $F$ lies between $S$ and $Q$ . Prove that the points $M,P,Q$ and $F$ lie on a circle.

(Karl Czakler)

geometry

Inversion

homothety

Polyline with increasing links

by NO_SQUARES, May 5, 2025, 5:30 PM

There are $100$ points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?

geometry

combinatorics

combinatorial geometry

A complex FE from Iran

by mojyla222, Aug 29, 2024, 9:23 AM

A surjective function $g: \mathbb{C} \to \mathbb C$ is given. Find all functions $f: \mathbb{C} \to \mathbb C$ such that for all $x,y\in \mathbb C$ we have
$|f(x)+g(y)| = | f(y) + g(x)|.$

Proposed by Mojtaba Zare, Amirabbas Mohammadi

This post has been edited 3 times. Last edited by mojyla222, Dec 27, 2024, 9:44 AM

Find the number of interesting numbers

by WakeUp, May 19, 2011, 2:08 PM

A positive integer $n$ is known as an interesting number if $n$ satisfies
${\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}}$
for all $k=1,2,\ldots 9$ .
Find the number of interesting numbers.

number theory proposed

number theory

combinatorics

Find the perfect squares

by Johann Peter Dirichlet, Mar 18, 2006, 4:47 AM

The sequence $(a_n)_{n \in\mathbb{N}}$ is defined by $a_1 = 8, a_2 = 18, a_{n+2} = a_{n+1}a_{n}$ . Find all terms which are perfect squares.

Submit

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math_exploration

by macves, Jun 1, 2025, 11:47 AM

by giangtruong13, Jun 1, 2025, 10:05 AM

by Royal_mhyasd, Jun 1, 2025, 7:06 AM

by Royal_mhyasd, May 31, 2025, 11:36 PM

by Maths_VC, May 27, 2025, 8:01 PM

by BR1F1SZ, May 5, 2025, 9:45 PM

by NO_SQUARES, May 5, 2025, 5:30 PM

by mojyla222, Aug 29, 2024, 9:23 AM

by WakeUp, May 19, 2011, 2:08 PM

by Johann Peter Dirichlet, Mar 18, 2006, 4:47 AM