Combi Algorithm/PHP/..

by CatalanThinker, May 28, 2025, 5:47 AM

5. [Czech and Slovak Republics 1997]
Each side and diagonal of a regular n-gon (n ≥ 3) is colored blue or green. A move consists of choosing a vertex and
switching the color of each segment incident to that vertex (from blue to green or vice versa). Prove that regardless of the initial coloring, it is possible to make the number of blue segments incident to each vertex even by following a sequence of moves. Also show that the final configuration obtained is uniquely determined by the initial coloring.

algorithm

pigeonhole principle

Proof

combinatorics

Combi Proof Math Algorithm

by CatalanThinker, May 28, 2025, 5:38 AM

3. [Russia 1961]
Real numbers are written in an $m \times n$ table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.

This post has been edited 2 times. Last edited by CatalanThinker, 2 hours ago

combinatorics

Proof

Unexpecredly Quick-Solve Inequality

by Primeniyazidayi, May 28, 2025, 5:18 AM

If $a, b, c>0$ , prove that $\frac{a^5}{b^2}+\frac{b}{c}+\frac{c^3}{a^2}>2a$

inequalities

exponential diophantine in integers

by skellyrah, May 27, 2025, 7:04 PM

find all integers x,y,z such that $45^x = 5^y + 2000^z$

Turkish JMO 2025?

by bitrak, May 27, 2025, 2:04 PM

Let p and q be prime numbers. Prove that if pq(p+ 1)(q + 1)+ 1 is a perfect square, then pq + 1 is also a perfect square.

number theory

Strange circles in an orthocenter config

by VideoCake, May 26, 2025, 5:10 PM

Let $\overline{AD}$ and $\overline{BE}$ be altitudes in an acute triangle $ABC$ which meet at $H$ . Suppose that $DE$ meets the circumcircle of $ABC$ at $P$ and $Q$ such that $P$ lies on the shorter arc of $BC$ and $Q$ lies on the shorter arc of $CA$ . Let $AQ$ and $BE$ meet at $S$ . Show that the circumcircles of $BPE$ and $QHS$ and the line $PH$ concur.

geometry

Problem 7

by SlovEcience, May 14, 2025, 11:03 AM

Consider the sequence $(u_n)$ defined by $u_0 = 5$ and
$u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.$ a) Prove that there exist infinitely many positive integers $n$ such that $u_n > 2020n$ .

b) Compute
$\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.$

arithmetic sequence

algebra

x^2+y^2+z^2+xy+yz+zx=6xyz diophantine

by parmenides51, Mar 2, 2024, 7:46 PM

Prove that there are infinite triples of positive integers $(x,y,z)$ such that
$x^2+y^2+z^2+xy+yz+zx=6xyz.$

This post has been edited 1 time. Last edited by parmenides51, Mar 2, 2024, 7:46 PM

number theory

Diophantine equation

diophantine

Easy Taiwanese Geometry

by USJL, Jan 31, 2024, 6:27 AM

Suppose $O$ is the circumcenter of $\Delta ABC$ , and $E, F$ are points on segments $CA$ and $AB$ respectively with $E, F \neq A$ . Let $P$ be a point such that $PB = PF$ and $PC = PE$ .
Let $OP$ intersect $CA$ and $AB$ at points $Q$ and $R$ respectively. Let the line passing through $P$ and perpendicular to $EF$ intersect $CA$ and $AB$ at points $S$ and $T$ respectively. Prove that points $Q, R, S$ , and $T$ are concyclic.

Proposed by Li4 and usjl

This post has been edited 1 time. Last edited by USJL, Jan 31, 2024, 6:28 AM

Taiwan

geometry

IMO 2017 Problem 4

by Amir Hossein, Jul 19, 2017, 4:30 PM

Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$ . Point $T$ is such that $S$ is the midpoint of the line segment $RT$ . Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$ . Line $AJ$ meets $\Omega$ again at $K$ . Prove that the line $KT$ is tangent to $\Gamma$ .

Proposed by Charles Leytem, Luxembourg

This post has been edited 1 time. Last edited by djmathman, Jun 16, 2020, 4:13 AM