Number Theory

by AnhQuang_67, Apr 16, 2025, 4:42 PM

Find all pairs of positive integers $(m,n)$ satisfying $2^m+21^n$ is a perfect square

number theory

Simply equation but hard

by giangtruong13, Apr 16, 2025, 3:29 PM

Find all integer pairs $(x,y)$ satisfy that: $(x^2+y)(y^2+x)=(x-y)^3$

number theory

Hard Polynomial Problem

by MinhDucDangCHL2000, Apr 16, 2025, 2:44 PM

Let $P(x)$ be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs $(a,b)$ such that $P(a) + P(b) = 0$ . Prove that the graph of $P(x)$ is symmetric about a point (i.e., it has a center of symmetry).

Set summed with itself

by Math-Problem-Solving, Apr 16, 2025, 1:59 AM

Let $A = \{1, 4, \ldots, n^2\}$ be the set of the first $n$ perfect squares of nonzero integers. Suppose that $A \subset B + B$ for some $B \subset \mathbb{Z}$ . Here $B + B$ stands for the set $\{b_1 + b_2 : b_1, b_2 \in B\}$ . Prove that $|B| \geq |A|^{2/3 - \epsilon}$ holds for every $\epsilon > 0$ .

combinatorics

Bounding

For positive integers $ a, b, c $, find all possible positive integer values o

by Jackson0423, Apr 13, 2025, 8:35 AM

For positive integers $a, b, c$ , find all possible positive integer values of
$\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.$

number theory

number theory unsolved

A drunk frog jumping ona grid in a weird way

by Tintarn, Nov 16, 2024, 5:18 PM

A frog is located on a unit square of an infinite grid oriented according to the cardinal directions. The frog makes moves consisting of jumping either one or two squares in the direction it is facing, and then turning according to the following rules:
i) If the frog jumps one square, it then turns $90^\circ$ to the right;
ii) If the frog jumps two squares, it then turns $90^\circ$ to the left.

Is it possible for the frog to reach the square exactly $2024$ squares north of the initial square after some finite number of moves if it is initially facing:
a) North;
b) East?

combinatorics

grid

combinatorics proposed

(x+y) f(2yf(x)+f(y))=x^3 f(yf(x)) for all x,y\in R^+

by parmenides51, Aug 5, 2019, 3:27 PM

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $(x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)), \ \ \ \forall x,y\in \mathbb{R}^{+}.$
(Albania)

functional equation

algebra

function

24 Aug FE problem

by nicky-glass, Aug 24, 2016, 7:34 AM

$f:\mathbb R\setminus \{0\} \to \mathbb R$
(i) $f(1)=1$ ,
(ii) $\forall x,y,x+y \neq 0:f(\frac{1}{x+y})=f(\frac{1}{x})+f(\frac{1}{y}) : P(x,y)$
(iii) $\forall x,y,x+y \neq 0:(x+y)f(x+y)=xyf(x)f(y) :Q(x,y)$
$f=?$

function

algebra

Advanced topics in Inequalities

by va2010, Mar 7, 2015, 4:43 AM

So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!

Attachments:

advanced-topics-inequalities (8).pdf (139kb)

inequalities

inequalities proposed

IMO LongList 1985 CYP2 - System of Simultaneous Equations

by Amir Hossein, Sep 10, 2010, 10:57 PM

Solve the system of simultaneous equations
$\sqrt x - \frac 1y - 2w + 3z = 1,$ $x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,$ $x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,$ $x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.$