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Tech support and questions about AoPS classes and materials
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Site Support Tech support and questions about AoPS classes and materials
Tech support and questions about AoPS classes and materials
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Decreasing primes
MithsApprentice   20
N an hour ago by Ilikeminecraft
Source: USAMO 1997
Let $p_1, p_2, p_3, \ldots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between 0 and 1. For positive integer $k$, define
\[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases}  \]
where $\{x\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \ldots$ eventually becomes 0.
20 replies
MithsApprentice
Oct 9, 2005
Ilikeminecraft
an hour ago
Sequence of rational numbers
mojyla222   2
N an hour ago by Assassino9931
Source: Iran 2024 3rd round number theory exam P1
Given a sequence $x_1,x_2,x_3,\cdots$ of positive integers, Ali proceed the following algorythm: In the i-th step he markes all rational numbers in the interval $[0,1]$ which have denominator equal to $x_i$. Then he write down the number $a_i$ equal to the length of the smallest interval in $[0,1]$ which both two ends of that is a marked number. Find all sequences $x_1,x_2,x_3,\cdots$ with $x_5=5$ and such that for all $n\in \mathbb N$ we have
$$
a_1+a_2+\cdots+a_n= 2-\dfrac{1}{x_n}.
$$
Proposed by Mojtaba Zare
2 replies
mojyla222
Aug 27, 2024
Assassino9931
an hour ago
Midpoint in a weird configuration
Gimbrint   1
N 2 hours ago by Beelzebub
Source: Own
Let $ABC$ be an acute triangle ($AB<BC$) with circumcircle $\omega$. Point $L$ is chosen on arc $AC$, not containing $B$, so that, letting $BL$ intersect $AC$ at $S$, one has $AS<CS$. Points $D$ and $E$ lie on lines $AB$ and $BC$ respectively, such that $BELD$ is a parallelogram. Point $P$ is chosen on arc $BC$, not containing $A$, such that $\angle CBP=\angle BDE$. Line $AP$ intersects $EL$ at $X$, and line $CP$ intersects $DL$ at $Y$. Line $XY$ intersects $AB$, $BC$ and $BP$ at points $M$, $N$ and $T$ respectively.

Prove that $TN=TM$.
1 reply
Gimbrint
May 23, 2025
Beelzebub
2 hours ago
Another FE
M11100111001Y1R   2
N 2 hours ago by AblonJ
Source: Iran TST 2025 Test 2 Problem 3
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all $x,y>0$ we have:
$$f(f(f(xy))+x^2)=f(y)(f(x)-f(x+y))$$
2 replies
M11100111001Y1R
Today at 8:03 AM
AblonJ
2 hours ago
Shortest number theory you might've seen in your life
AlperenINAN   13
N 2 hours ago by lksb
Source: Turkey JBMO TST 2025 P4
Let $p$ and $q$ be prime numbers. Prove that if $pq(p+1)(q+1)+1$ is a perfect square, then $pq + 1$ is also a perfect square.
13 replies
AlperenINAN
May 11, 2025
lksb
2 hours ago
n-term Sequence
MithsApprentice   15
N 2 hours ago by Ilikeminecraft
Source: USAMO 1996, Problem 4
An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a binary sequence of length $n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.
15 replies
MithsApprentice
Oct 22, 2005
Ilikeminecraft
2 hours ago
Drawing Triangles Against Your Clone
pieater314159   19
N 3 hours ago by Ilikeminecraft
Source: 2019 ELMO Shortlist C1
Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)

Proposed by Milan Haiman
19 replies
pieater314159
Jun 27, 2019
Ilikeminecraft
3 hours ago
Odd digit multiplication
JuanDelPan   12
N 3 hours ago by Ilikeminecraft
Source: Pan-American Girls' Mathematical Olympiad 2021, P4
Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.

$\textit{Proposed by Pablo Serrano, Ecuador}$
12 replies
JuanDelPan
Oct 6, 2021
Ilikeminecraft
3 hours ago
Cup of Combinatorics
M11100111001Y1R   7
N 3 hours ago by MathematicalArceus
Source: Iran TST 2025 Test 4 Problem 2
There are \( n \) cups labeled \( 1, 2, \dots, n \), where the \( i \)-th cup has capacity \( i \) liters. In total, there are \( n \) liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

$a)$ Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most \( \frac{4n}{3} \) steps.

$b)$ Prove that in at most \( \frac{5n}{3} \) steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
7 replies
M11100111001Y1R
May 27, 2025
MathematicalArceus
3 hours ago
Inequality
knm2608   17
N 3 hours ago by Adywastaken
Source: JBMO 2016 shortlist
If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that
$$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$When does the equality occur?

Proposed by Dorlir Ahmeti, Albania
17 replies
knm2608
Jun 25, 2017
Adywastaken
3 hours ago
a