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Distinct Integers with Divisibility Condition
tastymath75025 16
N
by ihategeo_1969
Source: 2017 ELMO Shortlist N3
For each integer 
decide whether there exist pairwise distinct positive integers 
such that for every 
, 
divides 
.
Proposed by Daniel Liu
decide whether there exist pairwise distinct positive integers 
such that for every 
, 
divides 
.Proposed by Daniel Liu
16 replies
GCD of a sequence
oVlad 6
N
by Rohit-2006
Source: Romania EGMO TST 2017 Day 1 P2
Determine all pairs 
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
6 replies
1 viewing
Maximum with the condition $x^2+y^2+z^2=1$
hlminh 1
N
by rchokler
Let 
be real numbers such that 
find the largest value of 
be real numbers such that 
find the largest value of 
1 reply
Mock 22nd Thailand TMO P10
korncrazy 2
N
by korncrazy
Source: own
Prove that there exists infinitely many triples of positive integers 
such that 
and 
are all perfect square.
such that 
and 
are all perfect square.
2 replies
IMO Shortlist 2014 N6
hajimbrak 26
N
by ihategeo_1969
Let 
be pairwise coprime positive integers with 
being prime and 
. On the segment ![$I = [0, a_1 a_2 \cdots a_n ]$](//latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by .
Proposed by Serbia
be pairwise coprime positive integers with 
being prime and 
. On the segment ![$I = [0, a_1 a_2 \cdots a_n ]$](http://latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by .Proposed by Serbia
26 replies
An easy FE
oVlad 2
N
by BR1F1SZ
Source: Romania EGMO TST 2017 Day 1 P3
Determine all functions 
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](//latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
2 replies
Nationalist Combo
blacksheep2003 15
N
by cj13609517288
Source: USEMO 2019 Problem 5
Let 
be a regular polygon, and let 
be its set of vertices. Each point in is colored red, white, or blue. A subset of is patriotic if it contains an equal number of points of each color, and a side of is dazzling if its endpoints are of different colors.
Suppose that is patriotic and the number of dazzling edges of is even. Prove that there exists a line, not passing through any point in , dividing into two nonempty patriotic subsets.
Ankan Bhattacharya
be a regular polygon, and let 
be its set of vertices. Each point in is colored red, white, or blue. A subset of is patriotic if it contains an equal number of points of each color, and a side of is dazzling if its endpoints are of different colors.Suppose that
is patriotic and the number of dazzling edges of is even. Prove that there exists a line, not passing through any point in , dividing into two nonempty patriotic subsets.Ankan Bhattacharya
15 replies
UIL Number Sense problem
Potato512 2
N
by buddy2007
I keep seeing a certain type of problem in UIL Number Sense, though I can't figure out how to do it (I aim to do it in my head in about 7-8 seconds).
The problem is x^((p+1)/2) mod p, where p is prime.
For example 11^15 mod 29
I know it technically doesn't work this way, but using fermats little theorem (on √x^(p+1)) always gives either the number itself, x, or the modular inverse, p-x.
By using the theorem i mean √x^28 mod 29 = 1, and then youre left with √x^2 mod 29 or x, but then its + or -.
I was wondering if there is a way to figure out whether its + or -, a slow or fast way if its slow maybe its possible to speed it up.
The problem is x^((p+1)/2) mod p, where p is prime.
For example 11^15 mod 29
I know it technically doesn't work this way, but using fermats little theorem (on √x^(p+1)) always gives either the number itself, x, or the modular inverse, p-x.
By using the theorem i mean √x^28 mod 29 = 1, and then youre left with √x^2 mod 29 or x, but then its + or -.
I was wondering if there is a way to figure out whether its + or -, a slow or fast way if its slow maybe its possible to speed it up.
2 replies
Concurrency with 10 lines
oVlad 1
N
by kokcio
Source: Romania EGMO TST 2017 Day 1 P1
Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.
1 reply
Advanced topics in Inequalities
va2010 21
N
by Novmath
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
21 replies
