The Most Difficult Functional Equation in the World

by EthanWYX2009, Jul 12, 2025, 10:04 AM

Determine all functions $f:\mathbb N_+\to\mathbb N_+$, such that for any positive integers $x$, $y$,
\[f(x)^2+y^2\mid\sum_{i=0}^{2023}(xf(x))^{2023-i}\left(f^{(i)}(y)\right)^{2i}.\]Created by Yuxing Ye
This post has been edited 1 time. Last edited by EthanWYX2009, 32 minutes ago

Sums of 1/i with Resticted Legrendre Symbol

by EthanWYX2009, Jul 12, 2025, 9:28 AM

Let prime number $p\equiv 1\pmod 8$, show that
$$\sum_{\substack{1\le i\le\frac{p-1}2\\\left(\frac ip\right)=1}}\frac 1i\equiv \sum_{\substack{1\le i\le\frac{p-1}2\\\left(\frac ip\right)=-1}}\frac 1i\pmod{p^2}.$$Created by Mucong Sun

Perfect Hexagon!

by zqy648, Jul 12, 2025, 8:59 AM

In a convex hexagon \( ABCDEF \), \( AB = BC \), \( CD = DE \), \( EF = AF \), \( BD = DF \), \( AB \neq AF \), and \( \angle ABC = \angle EFA = 2\angle EAC \).

Prove that \( 2\angle FAB - \angle BDF = 180^\circ \).

Created by Hongdao Chen

Easy Sequence Problem

by zqy648, Jul 12, 2025, 8:57 AM

Let \(\{a_n\}_{n\geq 1}\) be a strictly increasing sequence of positive integers such that there exists a positive number \( M \) satisfying, for any positive integer \( k \),
\[\sum_{i=1}^k a_i^3 \leq \left( M + \sum_{i=1}^k a_i \right)^2.\]Prove that the sequence \(\{a_n - n\}_{n\geq 1}\) is eventually constant.

Note: A sequence \(\{b_n\}_{n\geq 1}\) is called strictly increasing if \( b_1 < b_2 < \cdots \); it is called eventually constant if there exists a positive integer \( N \) such that \( b_{k+1} = b_k \) holds for all positive integers \( k \geq N \).

Created by Zhou Yang

Existence of m|a(k) and a(k)≤m¹⁹⁹

by EthanWYX2009, Jul 12, 2025, 8:41 AM

A sequence \(\{a_n\}\) is defined as follows: \( a_1 = 1 \); for any positive integer \( k \), \( a_{k+1} \) is the smallest positive integer not equal to \( a_1, a_2, \cdots, a_n \) that satisfies \( \gcd(a_{k+1}, a_k) \geq a_k^{0.99} \).

Prove that for any positive integer \( m \), there exists a positive integer \( k \) such that \( m | a_k \) and \( a_k \leq m^{199} \).

Created by Zhenyu Dong

Great Inequality

by EthanWYX2009, Jul 12, 2025, 8:38 AM

Given positive integer \( n \geq 2 \) and positive real \( t \). Let positive real numbers \( a_1, a_2, \ldots, a_n \) satisfy \( \sum_{i=1}^n a_i = t \). Denote \( S = \{1, 2, \cdots, n\} \). A non-empty subset \( I \) of \( S \) is called good if
\[\sum_{i\in I} a_i^3 \geq \left( \sum_{i\in I} a_i \right)^2.\]Determine the maximum possible number of good subsets of \( S \).

Created by Yuxing Ye

Parallel lines (extension of previous problem)

by RANDOM__USER, Jul 12, 2025, 7:28 AM

Let \(D\) be an arbitrary point on the side \(BC\) in a triangle \(\triangle{ABC}\). Let \(E\) and \(F\) be the intersection of the lines parallel to \(AC\) and \(AB\) through \(D\) with \(AB\) and \(AC\). Let \(G\) be the intersection of \((AFE)\) with \((ABC)\). Let \(M\) be the midpoint of \(BC\) and \(X\) the intersection of \(AM\) with \((ABC)\). Let \(H\) be the intersection of \((XMG)\) with \(BC\). Prove that \(EF\) is parallel to \(AH\).

[asy]
pair A = (5.48934,3.62066);
pair B = (7.08963,-0.46477);
pair C = (2.10049,-0.44595);
pair M = (4.59506,-0.45536);
pair X = (4.27548,-1.91196);
pair D = (6.07675,-0.46095);
pair E = (6.76474,0.36463);
pair F = (4.80135,2.79507);
pair G = (7.43276,0.63137);
pair H = (8.79490,-0.47121);

import graph;
size(12.00462cm);
pen zzttqq = rgb(0.6,0.2,0.);
pen qqwuqq = rgb(0.,0.39215,0.);
draw(A--B--C--cycle, linewidth(0.6) + zzttqq);
draw(A--B, linewidth(0.6) + zzttqq);
draw(B--C, linewidth(0.6) + zzttqq);
draw(C--A, linewidth(0.6) + zzttqq);
draw(circle((4.60023,0.91625), 2.84681), linewidth(0.6));
draw(D--E, linewidth(0.6));
draw(F--D, linewidth(0.6));
draw(circle((6.45122,2.11963), 1.78278), linewidth(0.6));
draw(A--X, linewidth(0.6));
draw(circle((6.69039,-1.67843), 2.42618), linewidth(0.6));
draw(E--F, linewidth(0.6) + qqwuqq);
draw((5.70801,1.67274)--(5.80126,1.67132), linewidth(0.6) + qqwuqq);
draw((5.70801,1.67274)--(5.68979,1.58127), linewidth(0.6) + qqwuqq);
draw((5.78304,1.57985)--(5.87629,1.57843), linewidth(0.6) + qqwuqq);
draw((5.78304,1.57985)--(5.76483,1.48839), linewidth(0.6) + qqwuqq);
draw(H--A, linewidth(0.6) + qqwuqq);
draw((7.06708,1.66761)--(7.16033,1.66619), linewidth(0.6) + qqwuqq);
draw((7.06708,1.66761)--(7.04887,1.57614), linewidth(0.6) + qqwuqq);
draw((7.14212,1.57472)--(7.23537,1.57330), linewidth(0.6) + qqwuqq);
draw((7.14212,1.57472)--(7.12391,1.48326), linewidth(0.6) + qqwuqq);
draw(B--H, linewidth(0.6));
draw(X--G, linewidth(0.6));

dot("$A$", A, dir(99));
dot("$B$", B, dir(286));
dot("$C$", C, dir(223));
dot("$M$", M, dir(120));
dot("$X$", X, dir(233));
dot("$D$", D, dir(-60));
dot("$E$", E, dir(82));
dot("$F$", F, dir(160));
dot("$G$", G, dir(280));
dot("$H$", H, dir(66));
[/asy]

Note: This is another property of a configuration I posted before where one needed to prove that \(X, D\) and \(G\) are collinear. There are surprisingly many properties in the configuration posted earlier :P

AOPS MO Introduce

by MathMaxGreat, Jul 12, 2025, 1:04 AM

$AOPS MO$

Problems: post it as a private message to me or @jerryZYang, please post it in $LATEX$ and have answers

6 Problems for two rounds, easier than $IMO$

If you want to do the problems or be interested, reply ’+1’
Want to post a problem reply’+2’ and message me
Want to be in the problem selection committee, reply’+3’

Cubic sequence

by huricane, Feb 27, 2016, 9:27 AM

A $\textit{cubic sequence}$ is a sequence of integers given by $a_n =n^3 + bn^2 + cn + d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers.
$\textbf{(a)}$ Show that there exists a cubic sequence such that the only terms
of the sequence which are squares of integers are $a_{2015}$ and $a_{2016}$.
$\textbf{(b)}$ Determine the possible values of $a_{2015} \cdot a_{2016}$ for a cubic sequence
satisfying the condition in part $\textbf{(a)}$.

Asian Pacific Mathematical Olympiad 2010 Problem 4

by Goutham, May 7, 2010, 6:50 PM

Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.

The oldest, shortest words — "yes" and "no" — are those which require the most thought.

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