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Cute NT Problem
M11100111001Y1R 4
N
by RANDOM__USER
Source: Iran TST 2025 Test 4 Problem 1
A number 
is called lucky if it has at least two distinct prime divisors and can be written in the form:
![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](//latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
where 
are distinct prime numbers that divide . (Note: it is possible that has other prime divisors not among .) Prove that for every prime number 
, there exists a lucky number such that 
.
is called lucky if it has at least two distinct prime divisors and can be written in the form:![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](http://latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
where 
are distinct prime numbers that divide . (Note: it is possible that has other prime divisors not among .) Prove that for every prime number 
, there exists a lucky number such that 
.
4 replies
USAMO 2003 Problem 4
MithsApprentice 72
N
by endless_abyss
Let 
be a triangle. A circle passing through 
and 
intersects segments 
and 
at 
and 
, respectively. Lines 
and 
intersect at 
, while lines 
and 
intersect at 
. Prove that 
if and only if 
.
be a triangle. A circle passing through 
and 
intersects segments 
and 
at 
and 
, respectively. Lines 
and 
intersect at 
, while lines 
and 
intersect at 
. Prove that 
if and only if 
.
72 replies
Easy but unusual junior ineq
Maths_VC 1
N
by blug
Source: Serbia JBMO TST 2025, Problem 2
Real numbers 
satisfy 
. Determine the minimal and the maximal value of the expression 
satisfy 
. Determine the minimal and the maximal value of the expression 
1 reply
Bosnia and Herzegovina JBMO TST 2009 Problem 1
gobathegreat 1
N
by FishkoBiH
Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2009
Lengths of sides of triangle are positive integers, and smallest side is equal to 
. Determine the area of triangle 
if 
, where 
, 
and 
are lengths of altitudes in triangle from vertices , and 
, respectively.
are positive integers, and smallest side is equal to 
. Determine the area of triangle 
if 
, where 
, 
and 
are lengths of altitudes in triangle from vertices , and 
, respectively.
1 reply
USAMO 2001 Problem 2
MithsApprentice 53
N
by lksb
Let be a triangle and let 
be its incircle. Denote by 
and 
the points where is tangent to sides and , respectively. Denote by 
and 
the points on sides and , respectively, such that 
and 
, and denote by the point of intersection of segments 
and 
. Circle intersects segment at two points, the closer of which to the vertex is denoted by 
. Prove that 
.
be a triangle and let 
be its incircle. Denote by 
and 
the points where is tangent to sides and , respectively. Denote by 
and 
the points on sides and , respectively, such that 
and 
, and denote by the point of intersection of segments 
and 
. Circle intersects segment at two points, the closer of which to the vertex is denoted by 
. Prove that 
.
53 replies
A=b
k2c901_1 89
N
by reni_wee
Source: Taiwan 1st TST 2006, 1st day, problem 3
Let 
, 
be positive integers such that 
is a multiple of 
for all positive integers 
. Prove that 
.
Proposed by Mohsen Jamali, Iran
, 
be positive integers such that 
is a multiple of 
for all positive integers 
. Prove that 
.Proposed by Mohsen Jamali, Iran
89 replies
Strange angle condition and concyclic points
lminsl 129
N
by Aiden-1089
Source: IMO 2019 Problem 2
In triangle , point 
lies on side and point 
lies on side . Let and be points on segments 
and 
, respectively, such that 
is parallel to . Let 
be a point on line 
, such that lies strictly between and , and 
. Similarly, let 
be the point on line 
, such that lies strictly between and , and 
.
Prove that points
, and are concyclic.
Proposed by Anton Trygub, Ukraine
, point 
lies on side and point 
lies on side . Let and be points on segments 
and 
, respectively, such that 
is parallel to . Let 
be a point on line 
, such that lies strictly between and , and 
. Similarly, let 
be the point on line 
, such that lies strictly between and , and 
.Prove that points
, and are concyclic.Proposed by Anton Trygub, Ukraine
129 replies
Simple inequality
sqing 12
N
by Rayvhs
Source: MEMO 2018 T1
Let 
and 
be positive real numbers satisfying 
Prove that
and 
be positive real numbers satisfying 
Prove that
12 replies
Random concyclicity in a square config
Maths_VC 2
N
by Maths_VC
Source: Serbia JBMO TST 2025, Problem 1
Let be a random point on the smaller arc of the circumcircle of square 
, and let 
be the intersection point of segments and 
. The feet of the tangents from point to the circumcircle of the triangle 
are and , where 
is the center of the square. Prove that points , , and lie on a single circle.
be a random point on the smaller arc of the circumcircle of square 
, and let 
be the intersection point of segments and 
. The feet of the tangents from point to the circumcircle of the triangle 
are and , where 
is the center of the square. Prove that points , , and lie on a single circle.
2 replies
Serbian selection contest for the IMO 2025 - P3
OgnjenTesic 3
N
by atdaotlohbh
Source: Serbian selection contest for the IMO 2025
Find all functions 
such that:
-
is strictly increasing,
- there exists
such that 
for all 
,
- for every
, there exists 
such that
![\[
f(y) = \frac{f(x) + f(x + 2024)}{2}.
\]](//latex.artofproblemsolving.com/b/e/2/be26213154bb74bd5a35b8d160011351871bfa9b.png)
Proposed by Pavle Martinović
such that:-
is strictly increasing,- there exists
such that 
for all 
,- for every
, there exists 
such that![\[
f(y) = \frac{f(x) + f(x + 2024)}{2}.
\]](http://latex.artofproblemsolving.com/b/e/2/be26213154bb74bd5a35b8d160011351871bfa9b.png)
Proposed by Pavle Martinović
3 replies
