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Calculate the distance of chess king!!
egxa   3
N an hour ago by egxa
Source: All Russian 2025 9.4
A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
3 replies
egxa
Apr 18, 2025
egxa
an hour ago
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real+ FE
pomodor_ap   4
N an hour ago by jasperE3
Source: Own, PDC001-P7
Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that
$$f(x)f(x^2 + y f(y)) = f(x)f(y^2) + x^3$$for all $x, y \in \mathbb{R}^+$. Determine all such functions $f$.
4 replies
pomodor_ap
Yesterday at 11:24 AM
jasperE3
an hour ago
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FE solution too simple?
Yiyj1   8
N an hour ago by lksb
Source: 101 Algebra Problems from the AMSP
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality $$f(f(x)+y) = f(x^2-y)+4f(x)y$$holds for all pairs of real numbers $(x,y)$.

My solution

I feel like my solution is too simple. Is there something I did wrong or something I missed?
8 replies
Yiyj1
Apr 9, 2025
lksb
an hour ago
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Polynomials in Z[x]
BartSimpsons   16
N 2 hours ago by bin_sherlo
Source: European Mathematical Cup 2017 Problem 4
Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $$P^n(m)\cdot P^m(n)$$is a square of an integer for all nonnegative integers $n, m$.

Remark: For a nonnegative integer $k$ and an integer $n$, $P^k(n)$ is defined as follows: $P^k(n) = n$ if $k = 0$ and $P^k(n)=P(P(^{k-1}(n))$ if $k >0$.

Proposed by Adrian Beker.
16 replies
BartSimpsons
Dec 27, 2017
bin_sherlo
2 hours ago
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Why is the old one deleted?
EeEeRUT   13
N 2 hours ago by EVKV
Source: EGMO 2025 P1
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
13 replies
EeEeRUT
Apr 16, 2025
EVKV
2 hours ago
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Factor sums of integers
Aopamy   2
N 2 hours ago by cadaeibf
Let $n$ be a positive integer. A positive integer $k$ is called a benefactor of $n$ if the positive divisors of $k$ can be partitioned into two sets $A$ and $B$ such that $n$ is equal to the sum of elements in $A$ minus the sum of the elements in $B$. Note that $A$ or $B$ could be empty, and that the sum of the elements of the empty set is $0$.

For example, $15$ is a benefactor of $18$ because $1+5+15-3=18$.

Show that every positive integer $n$ has at least $2023$ benefactors.
2 replies
Aopamy
Feb 23, 2023
cadaeibf
2 hours ago
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Least integer T_m such that m divides gauss sum
Al3jandro0000   33
N 2 hours ago by NerdyNashville
Source: 2020 Iberoamerican P2
Let $T_n$ denotes the least natural such that
$$n\mid 1+2+3+\cdots +T_n=\sum_{i=1}^{T_n} i$$Find all naturals $m$ such that $m\ge T_m$.

Proposed by Nicolás De la Hoz
33 replies
Al3jandro0000
Nov 17, 2020
NerdyNashville
2 hours ago
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Estonian Math Competitions 2005/2006
STARS   2
N 2 hours ago by jasperE3
Source: Juniors Problem 4
A $ 9 \times 9$ square is divided into unit squares. Is it possible to fill each unit square with a number $ 1, 2,..., 9$ in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
2 replies
STARS
Jul 30, 2008
jasperE3
2 hours ago
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Sum of whose elements is divisible by p
nntrkien   43
N 3 hours ago by lpieleanu
Source: IMO 1995, Problem 6, Day 2, IMO Shortlist 1995, N6
Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$?
43 replies
nntrkien
Aug 8, 2004
lpieleanu
3 hours ago
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Arrangement of integers in a row with gcd
egxa   2
N 3 hours ago by Qing-Cloud
Source: All Russian 2025 10.5 and 11.5
Let \( n \) be a natural number. The numbers \( 1, 2, \ldots, n \) are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the \( n - 1 \) GCDs obtained?
2 replies
egxa
Apr 18, 2025
Qing-Cloud
3 hours ago
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Two sets
steven_zhang123   6
N Apr 16, 2025 by lgx57
Given \(0 < b < a\), let
\[ A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\}, \]and
\[ B = \left[2\sqrt{ab}, a + b\right]. \]
Prove that \(A = B\).
6 replies
steven_zhang123
Apr 16, 2025
lgx57
Apr 16, 2025
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Two sets
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steven_zhang123
411 posts
steven_zhang123
#1 h  Apr 16, 2025, 7:44 AM • 1 Y
Y by cubres
Given \(0 < b < a\), let
\[ A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\}, \]and
\[ B = \left[2\sqrt{ab}, a + b\right]. \]
Prove that \(A = B\).
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lgx57
26 posts
lgx57
#2 h  Apr 16, 2025, 7:48 AM
Y by
steven_zhang123 wrote:
Given \(0 < b < a\), let
\[ A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\}, \]and
\[ B = \left[2\sqrt{ab}, a + b\right]. \]
Prove that \(A = B\).

Just prove that the min of $\frac{a}{3}(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})+b(xyz)^{\frac{1}{3}}$ is $2\sqrt{ab}$ and the max is $a+b$
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lgx57
26 posts
lgx57
#3 h  Apr 16, 2025, 7:52 AM
Y by
steven_zhang123 wrote:
Given \(0 < b < a\), let
\[ A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\}, \]and
\[ B = \left[2\sqrt{ab}, a + b\right]. \]
Prove that \(A = B\).

$r \ge \frac{a}{3}(\frac{3}{\sqrt[3]{xyz}})+b\sqrt[3]{xyz} \ge 2\sqrt{ab}$
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lgx57
26 posts
lgx57
#4 h  Apr 16, 2025, 8:01 AM
Y by
steven_zhang123 wrote:
Given \(0 < b < a\), let
\[ A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\}, \]and
\[ B = \left[2\sqrt{ab}, a + b\right]. \]
Prove that \(A = B\).

For the max:
$r \le \frac{a}{3}(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})+\frac{b}{3}(x+y+z)$
Then use local adjustment method to prove only $x=y=z$ can set $r$ to max
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GeoMorocco
34 posts
GeoMorocco
#5 h  Apr 16, 2025, 8:15 AM
Y by
You have only proved that A is included in B. Now you need to prove that B is included in A. It is not very hard, but if you don't do it, you will lose half the points of this question.
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Filipjack
872 posts
Filipjack
#6 h  Apr 16, 2025, 11:56 AM
Y by
This is Romania National Olympiad 2012 ($x,y,z$ were replaced here by $a/x, a/y, a/z$).
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lgx57
26 posts
lgx57
#7 h  Apr 16, 2025, 2:55 PM
Y by
GeoMorocco wrote:
You have only proved that A is included in B. Now you need to prove that B is included in A. It is not very hard, but if you don't do it, you will lose half the points of this question.

You are right, but this is obvious.
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