Quadratic + cubic residue => 6th power residue?

by Miquel-point, May 16, 2025, 6:02 PM

Decide whether the following statement is true: if an infinite arithmetic sequence of positive integers includes both a perfect square and a perfect cube, then it also includes a perfect $6$ th power.

Proposed by Sándor Róka, Nyíregyháza

number theory

II_a - r_a = R - r implies A = 60

by Miquel-point, May 16, 2025, 5:55 PM

The incenter and the inradius of the acute triangle $ABC$ are $I$ and $r$ , respectively. The excenter and exradius relative to vertex $A$ is $I_a$ and $r_a$ , respectively. Let $R$ denote the circumradius. Prove that if $II_a=r_a+R-r$ , then $\angle BAC=60^\circ$ .

Proposed by Class 2024C of Fazekas M. Gyak. Ált. Isk. és Gimn., Budapest

geometry

Cheating effectively in game of luck

by Miquel-point, May 16, 2025, 5:53 PM

Ádám, the famous conman signed up for the following game of luck. There is a rotating table with a shape of a regular $13$ -gon, and at each vertex there is a black or a white cap. (Caps of the same colour are indistinguishable from each other.) Under one of the caps $1000$ dollars are hidden, and there is nothing under the other caps. The host rotates the table, and then Ádám chooses a cap, and take what is underneath. Ádám's accomplice, Béla is working at the company behind this game. Béla is responsible for the placement of the $1000$ dollars under the caps, however, the colors of the caps are chosen by a different collegaue. After placing the money under a cap, Béla

has to change the color of the cap,
is allowed to change the color of the cap, but he is not allowed to touch any other cap.

Can Ádám and Béla find a strategy in part a. and in part b., respectively, so that Ádám can surely find the money? (After entering the casino, Béla cannot communicate with Ádám, and he also cannot influence his colleague choosing the colors of the caps on the table.)

Proposed by Gábor Damásdi, Budapest

combinatorics

"Eulerian" closed walk with of length less than v+e

by Miquel-point, May 16, 2025, 4:56 PM

Show that a connected graph $G=(V, E)$ has a closed walk of length at most $|V|+|E|-1$ passing through each edge of $G$ at least once.

Proposed by Radu Bumbăcea

combinatorics

graph theory

Trigonometric Product

by Henryfamz, May 13, 2025, 4:52 PM

Compute $\prod_{n=1}^{45}\sin(2n-1)$

algebra

Gcd(m,n) and Lcm(m,n)&F.E.

by Jackson0423, May 13, 2025, 4:12 PM

Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for all positive integers $m, n$ ,
$f(mn) = \mathrm{lcm}(m, n) \cdot \gcd(f(m), f(n)),$ where $\mathrm{lcm}(m, n)$ and $\gcd(m, n)$ denote the least common multiple and the greatest common divisor of $m$ and $n$ , respectively.

algebra

algebra unsolved

number theory

Anything real in this system must be integer

by Assassino9931, May 9, 2025, 9:26 AM

Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
$\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}$ and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.

Marek Maruin, Slovakia

This post has been edited 1 time. Last edited by Assassino9931, May 9, 2025, 9:26 AM

algebra

Al-Khwarizmi IJMO

IMO Genre Predictions

by ohiorizzler1434, May 3, 2025, 6:51 AM

Everybody, with IMO upcoming, what are you predictions for the problem genres?

Personally I predict: predict

Good Permutations in Modulo n

by swynca, Apr 27, 2025, 2:03 PM

An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$ , such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$ ;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$ .
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.

This post has been edited 2 times. Last edited by swynca, Apr 27, 2025, 4:15 PM

BMO

number theory

Concurrency from isogonal Mittenpunkt configuration

by MarkBcc168, Apr 28, 2020, 7:07 AM

Let $\triangle ABC$ be a scalene triangle with circumcenter $O$ , incenter $I$ , and incircle $\omega$ . Let $\omega$ touch the sides $\overline{BC}$ , $\overline{CA}$ , and $\overline{AB}$ at points $D$ , $E$ , and $F$ respectively. Let $T$ be the projection of $D$ to $\overline{EF}$ . The line $AT$ intersects the circumcircle of $\triangle ABC$ again at point $X\ne A$ . The circumcircles of $\triangle AEX$ and $\triangle AFX$ intersect $\omega$ again at points $P\ne E$ and $Q\ne F$ respectively. Prove that the lines $\text{[math]}$ , $FP$ , and $OI$ are concurrent.

Proposed by MarkBcc168.

This post has been edited 1 time. Last edited by MarkBcc168, Apr 28, 2020, 7:08 AM

geometry