. Prove that:
and 
Show that
Let 
Show that
Prove that
. Prove that
Y by PikaPika999, RainbowJessa
Prove that 
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trigonometric functions
VivaanKam 16
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by Shan3t
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
Thank you!
16 replies
Minimum number of points
Ecrin_eren 2
N
by Shan3t
There are 18 teams in a football league. Each team plays against every other team twice in a season—once at home and once away. A win gives 3 points, a draw gives 1 point, and a loss gives 0 points. One team became the champion by earning more points than every other team. What is the minimum number of points this team could have?
2 replies
Weird locus problem
Sedro 7
N
by ReticulatedPython
Points 
and 
are in the coordinate plane such that 
. Let 
denote the locus of all points 
in the coordinate plane satisfying 
, and let 
be the midpoint of 
. Points 
and 
are on such that 
and 
. The value of 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
and 
are in the coordinate plane such that 
. Let 
denote the locus of all points 
in the coordinate plane satisfying 
, and let 
be the midpoint of 
. Points 
and 
are on such that 
and 
. The value of 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
7 replies
IOQM P23 2024
SomeonecoolLovesMaths 3
N
by lakshya2009
Consider the fourteen numbers, 
. The smallest natural numebr such that they leave distinct remainders when divided by is:
. The smallest natural numebr such that they leave distinct remainders when divided by is:
3 replies
Pells equation
Entrepreneur 0
A Pells Equation is defined as follows 
Where 
are positive integers and 
is a non-square positive integer. If 
denotes the n-th set of solution to the equation with 
Then, prove that 
Where 
are positive integers and 
is a non-square positive integer. If 
denotes the n-th set of solution to the equation with 
Then, prove that 
0 replies
Incircle concurrency
niwobin 1
N
by niwobin
Triangle ABC with incenter I, incircle is tangent to BC, AC, and AB at D, E and F respectively.
DT is a diameter for the incircle, and AT meets the incircle again at point H.
Let DH and EF intersect at point J. Prove: AJ//BC.
DT is a diameter for the incircle, and AT meets the incircle again at point H.
Let DH and EF intersect at point J. Prove: AJ//BC.
1 reply
The centroid of ABC lies on ME [2023 Abel, Problem 1b]
Amir Hossein 3
N
by Captainscrubz
In the triangle 
, points 
and 
lie on the side 
, with 
. Also, is the midpoint of 
. Show that the centroid of lies on 
.
, points 
and 
lie on the side 
, with 
. Also, is the midpoint of 
. Show that the centroid of lies on 
.
3 replies
ALGEBRA INEQUALITY
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Y by PikaPika999, RainbowJessa
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Share some better sols.
Now expand and finish by AM-GM or by MuriheadShare some better sols.
This post has been edited 3 times. Last edited by Tony_stark0094, Apr 23, 2025, 12:53 AM
Reason: fjajja
Reason: fjajja
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to both sides to obtain ![\[\sum_{cyc}\frac{(a+b)(c+a)}{(b+c)} \ge 2(a+b+c).\]](http://latex.artofproblemsolving.com/b/1/e/b1e1674cee0e7445e5e2a8e8d3b201fd49247a65.png)
Substitute 
,
,
to turn this into ![\[\sum_{cyc} \frac{xz}{y} \ge \sum_{cyc} x,\]](http://latex.artofproblemsolving.com/2/e/e/2ee5eb58e69757f7815b75e979304b5f50d28c47.png)
which is clear by Muirhead.
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