Weird Geo

by Anto0110, Apr 20, 2025, 9:24 PM

In a trapezium $ABCD$ , the sides $AB$ and $CD$ are parallel and the angles $\angle ABC$ and $\angle BAD$ are acute. Show that it is possible to divide the triangle $ABC$ into 4 disjoint triangle $X_1. . . , X_4$ and the triangle $ABD$ into 4 disjoint triangles $Y_1,. . . , Y_4$ such that the triangles $X_i$ and $Y_i$ are congruent for all $i$ .

geometry

Is the geometric function injective?

by Project_Donkey_into_M4, Apr 20, 2025, 6:23 PM

A non-degenerate triangle $\Delta ABC$ is given in the plane, let $S$ be the set of points which lie strictly inside it. Also let $\mathfrak{C}$ be the set of circles in the plane. For a point $P \in S$ , let $A_P, B_P, C_P$ be the reflection of $P$ in sides $\overline{BC}, \overline{CA}, \overline{AB}$ respectively. Define a function $\omega: S \rightarrow \mathfrak{C}$ such that $\omega(P)$ is the circumcircle of $A_PB_PC_P$ . Is $\omega$ injective?

Note: The function $\omega$ is called injective if for any $P, Q \in S$ , $\omega(P) = \omega(Q) \Leftrightarrow P = Q$

This post has been edited 1 time. Last edited by Project_Donkey_into_M4, 4 hours ago

algebra

geometry

Apple sharing in Iran

by mojyla222, Apr 20, 2025, 4:17 AM

Ali is hosting a large party. Together with his $n-1$ friends, $n$ people are seated around a circular table in a fixed order. Ali places $n$ apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).

Find all values of $n$ such that after some number of steps, the situation reaches a point where each person has exactly one apple.

combinatorics

confusing inequality

by giangtruong13, Apr 18, 2025, 2:07 PM

Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$ . Prove that: $\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$

This post has been edited 3 times. Last edited by giangtruong13, Today at 3:02 PM

inequalities

3 knightlike moves is enough

by sarjinius, Mar 9, 2025, 3:38 PM

An ant is on the Cartesian plane. In a single move, the ant selects a positive integer $k$ , then either travels

$k$ units vertically (up or down) and $2k$ units horizontally (left or right); or
$k$ units horizontally (left or right) and $2k$ units vertically (up or down).

Thus, for any $k$ , the ant can choose to go to one of eight possible points.
Prove that, for any integers $a$ and $b$ , the ant can travel from $(0, 0)$ to $(a, b)$ using at most $3$ moves.

Dear Sqing: So Many Inequalities...

by hashtagmath, Oct 30, 2024, 5:52 AM

I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you

inequalities

Sqing

another functional inequality?

by Scilyse, Jul 17, 2024, 12:07 PM

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that $x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)$ for every $x, y \in \mathbb R_{>0}$ .

This post has been edited 4 times. Last edited by Scilyse, Feb 11, 2025, 9:51 AM

algebra

Functional inequality

IMO Shortlist

AZE IMO TST

winning strategy, vertices of regular n-gon

by parmenides51, Sep 4, 2022, 5:01 PM

The vertices of a regular polygon with $N$ sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following:
$\bullet$ join two vertices with a segment, without cutting another already marked segment; or
$\bullet$ delete a vertex that does not belong to any marked segment.
The player who cannot take any action on his turn loses the game. Determine which of the two players can guarantee victory:
a) if $N=28$
b) if $N=29$

combinatorics

winning strategy

game strategy

numbers at vertices of triangle / tetrahedron, consecutive and gcd related

by parmenides51, Sep 4, 2022, 4:59 PM

a) A positive integer is written at each vertex of a triangle. Then on each side of the triangle the greatest common divisor of its ends is written. It is possible that the numbers written on the sides be three consecutive integers, in some order?
b) A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron is written the greatest common divisor of its ends . It is possible that the numbers written in the edges are six consecutive integers, in some order?

combinatorics

number theory

GCD and LCM

red squares in a 7x7 board

by parmenides51, Sep 4, 2022, 4:44 PM

In a $7\times7$ board, some squares are painted red. Let $a$ be the number of rows that have an odd number of red squares and let $b$ be the number of columns that have an odd number of red squares. Find all possible values of $a+b$ . For each value found, give a example of how the board can be painted.

This post has been edited 3 times. Last edited by parmenides51, Dec 10, 2022, 3:39 AM

Coloring

combinatorics

board

Art of Problem Solving Blog

by Anto0110, Apr 20, 2025, 9:24 PM

by Project_Donkey_into_M4, Apr 20, 2025, 6:23 PM

by mojyla222, Apr 20, 2025, 4:17 AM

by giangtruong13, Apr 18, 2025, 2:07 PM

by sarjinius, Mar 9, 2025, 3:38 PM

by hashtagmath, Oct 30, 2024, 5:52 AM

by Scilyse, Jul 17, 2024, 12:07 PM

by parmenides51, Sep 4, 2022, 5:01 PM

by parmenides51, Sep 4, 2022, 4:59 PM

by parmenides51, Sep 4, 2022, 4:44 PM