Erasing a and b and replacing them with a - b + 1

by jl_, Apr 23, 2025, 10:38 AM

Ruby writes the numbers $1, 2, 3, . . . , 10$ on the whiteboard. In each move, she selects two distinct numbers, $a$ and $b$ , erases them, and replaces them with $a+b-1$ . She repeats this process until only one number, $x$ , remains. What are all the possible values of $x$ ?

combinatorics

Find maximum area of right triangle

by jl_, Apr 23, 2025, 10:33 AM

Given a right-angled triangle with hypothenuse $2024$ , find the maximal area of the triangle.

algebra

x^3+y^3 is prime

by jl_, Apr 23, 2025, 10:28 AM

Find all pairs of positive integers $(x,y)$ , so that the number $x^3+y^3$ is a prime.

number theory

Sum and product of 5 numbers

by jl_, Apr 23, 2025, 10:14 AM

Ivan bought $50$ cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number $100000$ . How many cats of each breed does he have?

This post has been edited 5 times. Last edited by jl_, 2 hours ago
Reason: Change source

Prove that sum of 1^3+...+n^3 is a square

by jl_, Apr 23, 2025, 10:12 AM

Prove that for all positive integers $n$ , $1^3 + 2^3 + 3^3 +\dots+n^3$ is a perfect square.

This post has been edited 3 times. Last edited by jl_, 2 hours ago
Reason: Change subject and source

interesting function equation (fe) in IR

by skellyrah, Apr 23, 2025, 9:51 AM

find all function F: IR->IR such that $xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy)$

Why is the old one deleted?

by EeEeRUT, Apr 16, 2025, 1:33 AM

For a positive integer $N$ , let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$ . Find all $N \geqslant 3$ such that $\gcd( N, c_i + c_{i+1}) \neq 1$ for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$ . Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$ .

Proposed by Paulius Aleknavičius, Lithuania

This post has been edited 2 times. Last edited by EeEeRUT, Apr 18, 2025, 12:56 AM
Reason: Authorship

number theory

EGMO

EGMO magic square

by Lukaluce, Apr 14, 2025, 11:03 AM

In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$ , and the sum of the numbers in each column is equal to $1$ . Define $r_i$ to be the largest value in row $i$ , and let $R = r_1 + r_2 + ... + r_{2025}$ . Similarly, define $c_i$ to be the largest value in column $i$ , and let $C = c_1 + c_2 + ... + c_{2025}$ .
What is the largest possible value of $\frac{R}{C}$ ?

Proposed by Paulius Aleknavičius, Lithuania

This post has been edited 1 time. Last edited by Lukaluce, Apr 14, 2025, 12:16 PM

board

maximum

EGMO

A colouring game on a rectangular frame

by Tintarn, Mar 17, 2025, 12:25 PM

For integers $m,n \ge 3$ we consider a $m \times n$ rectangular frame, consisting of the $2m+2n-4$ boundary squares of a $m \times n$ rectangle.

Renate and Erhard play the following game on this frame, with Renate to start the game. In a move, a player colours a rectangular area consisting of a single or several white squares. If there are any more white squares, they have to form a connected region. The player who moves last wins the game.

Determine all pairs $(m,n)$ for which Renate has a winning strategy.

combinatorics

combinatorics proposed

game

winning strategy

colouring

Integer a_k such that b - a^n_k is divisible by k

by orl, Jul 13, 2008, 1:41 PM

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$ . Prove that $b = A^n$ for some integer $A$ .

Author: Dan Brown, Canada

This post has been edited 3 times. Last edited by v_Enhance, Jan 18, 2016, 2:13 AM
Reason: Fix obsolete TeX

Telv Cohl's Geometry Blog

by jl_, Apr 23, 2025, 10:38 AM

by jl_, Apr 23, 2025, 10:33 AM

by jl_, Apr 23, 2025, 10:28 AM

by jl_, Apr 23, 2025, 10:14 AM

by jl_, Apr 23, 2025, 10:12 AM

by skellyrah, Apr 23, 2025, 9:51 AM

by EeEeRUT, Apr 16, 2025, 1:33 AM

by Lukaluce, Apr 14, 2025, 11:03 AM

by Tintarn, Mar 17, 2025, 12:25 PM

by orl, Jul 13, 2008, 1:41 PM