and 
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Post any solutions you have, with explanation or proof if possible, good luck!
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Concurrence, Isogonality
Wictro 40
N
by CatinoBarbaraCombinatoric
Source: BMO 2019, Problem 3
Let 
be an acute scalene triangle. Let 
and 
be two distinct interior points of the segment 
such that 
. Suppose that:
and 
are the feet of the perpendiculars from from 
to the lines 
and 
respectively.
and 
are the feet of the perpendiculars from 
to the lines and respectively.
Prove that
and 
intersect on the line .
be an acute scalene triangle. Let 
and 
be two distinct interior points of the segment 
such that 
. Suppose that:
and 
are the feet of the perpendiculars from from 
to the lines 
and 
respectively.
and 
are the feet of the perpendiculars from 
to the lines and respectively.Prove that
and 
intersect on the line .
40 replies
Tango course
oVlad 1
N
by kokcio
Source: Romania EGMO TST 2019 Day 1 P4
Six boys and six girls are participating at a tango course. They meet every evening for three weeks (a total of 21 times). Each evening, at least one boy-girl pair is selected to dance in front of the others. At the end of the three weeks, every boy-girl pair has been selected at least once. Prove that there exists a person who has been selected on at least 5 distinct evenings.
Note: a person can be selected twice on the same evening.
Note: a person can be selected twice on the same evening.
1 reply
1 viewing
Easy Number Theory
math_comb01 36
N
by anudeep
Source: INMO 2024/3
Let 
be an odd prime and 
be integers so that the integers 
are divisible by .
Prove that divides each of .
Proposed by Navilarekallu Tejaswi
be an odd prime and 
be integers so that the integers 
are divisible by .Prove that
divides each of .
Proposed by Navilarekallu Tejaswi
36 replies
A cyclic inequality
KhuongTrang 0
Source: own-CRUX
IMAGE
https://cms.math.ca/.../uploads/2025/04/Wholeissue_51_4.pdf
https://cms.math.ca/.../uploads/2025/04/Wholeissue_51_4.pdf
0 replies
Prove excircle is tangent to circumcircle
sarjinius 7
N
by markam
Source: Philippine Mathematical Olympiad 2025 P4
Let be a triangle with incenter 
, and let 
be a point on side . Points and are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
are chosen on side such that and are the angle bisectors of angles 
and 
respectively. Let 
be the circle tangent to segment 
, the extension of 
past , and the extension of 
past . Prove that is tangent to the circumcircle of triangle .
be a triangle with incenter 
, and let 
be a point on side . Points and are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
are chosen on side such that and are the angle bisectors of angles 
and 
respectively. Let 
be the circle tangent to segment 
, the extension of 
past , and the extension of 
past . Prove that is tangent to the circumcircle of triangle .
7 replies
Distinct Integers with Divisibility Condition
tastymath75025 15
N
by cursed_tangent1434
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
15 replies
An easy FE
oVlad 1
N
by pco
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 
1 reply
Fractions and reciprocals
adihaya 34
N
by de-Kirschbaum
Source: 2013 BAMO-8 #4
For a positive integer 
, consider the 
fractions 
The product of these fractions equals 
, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of for which this is possible and prove that you have found them all.
, consider the 
fractions 
The product of these fractions equals 
, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of for which this is possible and prove that you have found them all.
34 replies
GCD Functional Equation
pinetree1 60
N
by cursed_tangent1434
Source: USA TSTST 2019 Problem 7
Let 
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
and such that 
for all integers .
Ankan Bhattacharya
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
and such that 
for all integers .Ankan Bhattacharya
60 replies
New geometry problem
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is 
so 
and thus 
Thus, 
The power of 
so the tangent length is 
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#4
Apr 4, 2025, 11:16 PM
Y by
fruitmonster97 wrote:
w problem 
@above is correct.
@above is correct.bro this is so smart
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#5
Apr 13, 2025, 4:40 PM
Y by
fruitmonster97 wrote:
w problem @above is correct.
@above is correct.Thanks! This was my intended solution.
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