G
Topic
First Poster
Last Poster
Find the value
sqing 11
N
by mathematical-forest
Source: 2024 China Fujian High School Mathematics Competition
Let 
and 
Find the value of 
and 
Find the value of 
11 replies
Self-evident inequality trick
Lukaluce 20
N
by ytChen
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let 
, and 
be positive real numbers, such that 
. Prove the inequality
![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](//latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
20 replies
1 viewing
three circles
barasawala 8
N
by FrancoGiosefAG
Source: Mexico 2003
are collinear with 
betweeen 
and 
. 
is the circle with diameter 
, and 
is the circle with diameter 
. Another circle touches 
at and meets again at 
and again at 
. The line 
meets again at 
and again at 
. Show that the lines 
and 
meet on the perpendicular to at .
8 replies
d(2025^{a_i}-1) divides a_{n+1}
navi_09220114 3
N
by quacksaysduck
Source: TASIMO 2025 Day 2 Problem 5
Let 
be a strictly increasing sequence of positive integers such that for all positive integers 
![\[d(2025^{a_n}-1)|a_{n+1}.\]](//latex.artofproblemsolving.com/8/f/d/8fda531266d39c7f11c58d3c1077d2cab3357bea.png)
Show that for any positive real number 
there is a positive integers 
such that 
for all 
.
Note. Here
denotes the number of positive divisors of the positive integer 
.
be a strictly increasing sequence of positive integers such that for all positive integers 
![\[d(2025^{a_n}-1)|a_{n+1}.\]](http://latex.artofproblemsolving.com/8/f/d/8fda531266d39c7f11c58d3c1077d2cab3357bea.png)
Show that for any positive real number 
there is a positive integers 
such that 
for all 
.Note. Here
denotes the number of positive divisors of the positive integer 
.
3 replies
Prove that
Trapezium with two right-angles: prove < AKB = 90° and more
Leonardo 6
N
by FrancoGiosefAG
Source: Mexico 2002
Let 
be a quadrilateral with 
. Denote by 
the midpoint of the side 
, and assume that 
. Let 
be the foot of the perpendicular from the point to the line 
. The line 
meets 
at 
, and the line 
meets 
at 
. Show that 
and 
.
[Moderator edit: The proposed solution can be found at http://erdos.fciencias.unam.mx/mexproblem3.pdf .]
be a quadrilateral with 
. Denote by 
the midpoint of the side , and assume that 
. Let 
be the foot of the perpendicular from the point to the line 
. The line 
meets 
at , and the line 
meets at . Show that 
and 
.[Moderator edit: The proposed solution can be found at http://erdos.fciencias.unam.mx/mexproblem3.pdf .]
6 replies
Prove that
Nice concurrency
navi_09220114 4
N
by quacksaysduck
Source: TASIMO 2025 Day 1 Problem 2
Four points 
, 
, 
, 
lie on a semicircle 
in this order with diameter 
, and is not parallel to 
. Points 
and 
lie on segments and respectively such that 
and 
. A circle 
passes through and is tangent to , and intersects again at 
. Prove that the lines 
, and 
are concurrent.
, , , 
lie on a semicircle 
in this order with diameter 
, and is not parallel to . Points 
and 
lie on segments and respectively such that 
and 
. A circle 
passes through and is tangent to , and intersects again at 
. Prove that the lines 
, and 
are concurrent.
4 replies
Numbers on a circle
navi_09220114 3
N
by quacksaysduck
Source: TASIMO 2025 Day 1 Problem 1
For a given positive integer 
, determine the smallest integer 
, such that it is possible to place numbers 
around a circle so that the sum of every consecutive numbers takes one of at most values.
, determine the smallest integer 
, such that it is possible to place numbers 
around a circle so that the sum of every consecutive numbers takes one of at most values.
3 replies
D1033 : A problem of probability for dominoes 3*1
Dattier 1
N
by Dattier
Source: les dattes à Dattier
Let 
a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.
What the probability of cover with 3*1 dominoes this grid removed by theses little squares (one, two or three) ?
a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.What the probability of cover with 3*1 dominoes this grid removed by theses little squares (one, two or three) ?
1 reply
