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Decreasing primes
MithsApprentice 20
N
an hour ago
by Ilikeminecraft
Source: USAMO 1997
Let
be the prime numbers listed in increasing order, and let
be a real number between 0 and 1. For positive integer
, define
![\[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} \]](//latex.artofproblemsolving.com/e/1/7/e17b3fc5962fba938dee877dac266c3cd024a6e3.png)
where
denotes the fractional part of
. (The fractional part of
is given by
where
is the greatest integer less than or equal to
.) Find, with proof, all
satisfying
for which the sequence
eventually becomes 0.



![\[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} \]](http://latex.artofproblemsolving.com/e/1/7/e17b3fc5962fba938dee877dac266c3cd024a6e3.png)
where









20 replies
Sequence of rational numbers
mojyla222 2
N
an hour ago
by Assassino9931
Source: Iran 2024 3rd round number theory exam P1
Given a sequence
of positive integers, Ali proceed the following algorythm: In the i-th step he markes all rational numbers in the interval
which have denominator equal to
. Then he write down the number
equal to the length of the smallest interval in
which both two ends of that is a marked number. Find all sequences
with
and such that for all
we have

Proposed by Mojtaba Zare

![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)


![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)




Proposed by Mojtaba Zare
2 replies
Midpoint in a weird configuration
Gimbrint 1
N
2 hours ago
by Beelzebub
Source: Own
Let
be an acute triangle (
) with circumcircle
. Point
is chosen on arc
, not containing
, so that, letting
intersect
at
, one has
. Points
and
lie on lines
and
respectively, such that
is a parallelogram. Point
is chosen on arc
, not containing
, such that
. Line
intersects
at
, and line
intersects
at
. Line
intersects
,
and
at points
,
and
respectively.
Prove that
.
































Prove that

1 reply
Another FE
M11100111001Y1R 2
N
2 hours ago
by AblonJ
Source: Iran TST 2025 Test 2 Problem 3
Find all functions
such that for all
we have:



2 replies


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
and
be prime numbers. Prove that if
is a perfect square, then
is also a perfect square.




13 replies
n-term Sequence
MithsApprentice 15
N
2 hours ago
by Ilikeminecraft
Source: USAMO 1996, Problem 4
An
-term sequence
in which each term is either 0 or 1 is called a binary sequence of length
. Let
be the number of binary sequences of length
containing no three consecutive terms equal to 0, 1, 0 in that order. Let
be the number of binary sequences of length
that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that
for all positive integers
.









15 replies
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,
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
.)
Proposed by Milan Haiman


Proposed by Milan Haiman
19 replies
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
greater than 10. Then, she multiplies all the digits of
and obtains an odd number. Find all possible values of the units digit of
.




12 replies
Cup of Combinatorics
M11100111001Y1R 7
N
3 hours ago
by MathematicalArceus
Source: Iran TST 2025 Test 4 Problem 2
There are
cups labeled
, where the
-th cup has capacity
liters. In total, there are
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.
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
steps.
Prove that in at most
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.









7 replies
Inequality
knm2608 17
N
3 hours ago
by Adywastaken
Source: JBMO 2016 shortlist
If the non-negative reals
satisfy
. Prove that
When does the equality occur?
Proposed by Dorlir Ahmeti, Albania



Proposed by Dorlir Ahmeti, Albania
17 replies
