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Topic
First Poster
Last Poster
polynomial
hanzo.ei 1
N
by alexheinis
Source: from a book
Given a polynomial 
satisfying 
, 
, and for 
(
) positive real numbers 
. Prove that there exists a strictly increasing sequence of real numbers 
such that
![\[
\sum_{i=1}^{n} \frac{k_i}{P'(a_i)} = \sum_{i=1}^{n} k_i.
\]](//latex.artofproblemsolving.com/4/4/e/44e5eb17b17531e36c362e062471d9bdffc8144c.png)
satisfying 
, 
, and for 
(
) positive real numbers 
. Prove that there exists a strictly increasing sequence of real numbers 
such that![\[
\sum_{i=1}^{n} \frac{k_i}{P'(a_i)} = \sum_{i=1}^{n} k_i.
\]](http://latex.artofproblemsolving.com/4/4/e/44e5eb17b17531e36c362e062471d9bdffc8144c.png)
1 reply
Prove that AY is tangent to (AEF)
geometry6 4
N
by V-217
Source: IMOC 2021 G8
Let 
be an arbitrary interior point of 
, and 
, 
, 
intersect 
, 
, 
at 
, 
, 
, respectively. Suppose that 
be the midpoint of , 
and 
intersect at 
, 
intersects at 
, and 
intersects at 
. Show that 
is tangent to .
be an arbitrary interior point of 
, and 
, 
, 
intersect 
, 
, 
at 
, 
, 
, respectively. Suppose that 
be the midpoint of , 
and 
intersect at 
, 
intersects at 
, and 
intersects at 
. Show that 
is tangent to .
4 replies
A positive integer changes every second and becomes a power of two
nAalniaOMliO 5
N
by RagvaloD
Source: Belarusian National Olympiad 2025
A positive integer with three digits is written on the board. Each second the number 
on the board gets replaced by 
, where 
is the largest prime divisor of .
Prove that either after 999 seconds or 1000 second the number on the board will be a power of two.
on the board gets replaced by 
, where 
is the largest prime divisor of .Prove that either after 999 seconds or 1000 second the number on the board will be a power of two.
5 replies
possible triangle inequality
sunshine_12 1
N
by kiyoras_2001
a, b, c are real numbers
|a| + |b| + |c| − |a + b| − |b + c| − |c + a| + |a + b + c| ≥ 0
hey everyone, so I came across this inequality, and I did make some progress:
Let (a+b), (b+c), (c+a) be three sums T1, T2 and T3. As there are 3 sums, but they can be of only 2 signs, by pigeon hole principle, atleast 2 of the 3 sums must be of the same sign.
But I'm getting stuck on the case work. Can anyone help?
Thnx a lot
|a| + |b| + |c| − |a + b| − |b + c| − |c + a| + |a + b + c| ≥ 0
hey everyone, so I came across this inequality, and I did make some progress:
Let (a+b), (b+c), (c+a) be three sums T1, T2 and T3. As there are 3 sums, but they can be of only 2 signs, by pigeon hole principle, atleast 2 of the 3 sums must be of the same sign.
But I'm getting stuck on the case work. Can anyone help?
Thnx a lot
1 reply
Equal radius
FabrizioFelen 9
N
by ihategeo_1969
Source: Centroamerican Olympiad 2016, Problem 6
Let be triangle with incenter 
and circumcircle 
. Let 
and 
, the line parallel to 
through cuts , 
in and 
. Prove that the circumradius of 
and 
are equal.
be triangle with incenter 
and circumcircle 
. Let 
and 
, the line parallel to 
through cuts , 
in and 
. Prove that the circumradius of 
and 
are equal.
9 replies
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
Geo challenge on finding simple ways to solve it
Assassino9931 3
N
by africanboy
Source: Bulgaria Spring Mathematical Competition 2025 9.2
Let 
be an acute scalene triangle inscribed in a circle 
. The angle bisector of 
intersects 
at 
and at 
. The point 
is the midpoint of 
. Let 
be the altitude in 
, and the circumcircle of 
intersects again at 
. Let 
be the midpoint of , and let 
be the reflection of 
with respect to . Prove that the triangles 
and 
are similar.
be an acute scalene triangle inscribed in a circle 
. The angle bisector of 
intersects 
at 
and at 
. The point 
is the midpoint of 
. Let 
be the altitude in 
, and the circumcircle of 
intersects again at 
. Let 
be the midpoint of , and let 
be the reflection of 
with respect to . Prove that the triangles 
and 
are similar.
3 replies
Special students
BR1F1SZ 1
N
by Lil_flip38
Source: 2013 Argentina L2 P4
In a school with double schooling, in the morning the language teacher divided the students into 
groups for an activity. In the afternoon, the math teacher divided the same students into 
groups for another activity. A student is considered special if the group they belonged to in the afternoon is smaller than the group they belonged to in the morning. Find the minimum number of special students that can exist in the school.
Note: Each group has at least one student.
groups for an activity. In the afternoon, the math teacher divided the same students into 
groups for another activity. A student is considered special if the group they belonged to in the afternoon is smaller than the group they belonged to in the morning. Find the minimum number of special students that can exist in the school.Note: Each group has at least one student.
1 reply
Finding big a_i a_i+1
nAalniaOMliO 1
N
by RagvaloD
Source: Belarusian National Olympiad 2025
Positive real numbers 
with sum 
are such that the equation 
has a positive root 
smaller than 
.
Prove that there exists a positive integer
such that the inequality 
holds.
with sum 
are such that the equation 
has a positive root 
smaller than 
.Prove that there exists a positive integer
such that the inequality 
holds.
1 reply
nice problem
hanzo.ei 2
N
by Lil_flip38
Source: I forgot
Let triangle be inscribed in the circumcircle 
and circumscribed about the incircle 
, with 
. The incircle touches the sides , , and at , , and , respectively. A line through , perpendicular to 
, intersects , , and at , , and 
, respectively. The line meets at (distinct from 
). The circumcircle of triangle 
intersects at 
(distinct from ). Let be the midpoint of the arc 
of . The line cuts segments 
and 
at 
and 
, respectively, and the tangents to the circle 
at and intersect at 
. Prove that 
.
be inscribed in the circumcircle 
and circumscribed about the incircle 
, with 
. The incircle touches the sides , , and at , , and , respectively. A line through , perpendicular to 
, intersects , , and at , , and 
, respectively. The line meets at (distinct from 
). The circumcircle of triangle 
intersects at 
(distinct from ). Let be the midpoint of the arc 
of . The line cuts segments 
and 
at 
and 
, respectively, and the tangents to the circle 
at and intersect at 
. Prove that 
.
2 replies
