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Decreasing primes
MithsApprentice 20
N
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.
be the prime numbers listed in increasing order, and let 
be a real number between 0 and 1. For positive integer 
, define![\[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} \]](http://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.
20 replies
Sequence of rational numbers
mojyla222 2
N
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 ![$[0,1]$](//latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
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
of positive integers, Ali proceed the following algorythm: In the i-th step he markes all rational numbers in the interval ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
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
2 replies
Midpoint in a weird configuration
Gimbrint 1
N
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
.
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
.
1 reply
Another FE
M11100111001Y1R 2
N
by AblonJ
Source: Iran TST 2025 Test 2 Problem 3
Find all functions 
such that for all 
we have:
such that for all 
we have:
2 replies
Shortest number theory you might've seen in your life
AlperenINAN 13
N
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.
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
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 .
-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
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
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
19 replies
Odd digit multiplication
JuanDelPan 12
N
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 .
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
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.
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
by Adywastaken
Source: JBMO 2016 shortlist
If the non-negative reals 
satisfy 
. Prove that
When does the equality occur?
Proposed by Dorlir Ahmeti, Albania
satisfy 
. Prove that
When does the equality occur?Proposed by Dorlir Ahmeti, Albania
17 replies
