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<AEC+<BKD=90^o, <ACE=<BCF, <CDF=<BDK 2016 Armenia NMO 8.2
parmenides51 1
N
by vanstraelen
Points 
are taken on the semicircle with diameter 
(points 
are in the specified order), and points 
on the diameter (
is on the segment 
) such that 
and 
. Prove that 
.
are taken on the semicircle with diameter 
(points 
are in the specified order), and points 
on the diameter (
is on the segment 
) such that 
and 
. Prove that 
.
1 reply
Tetrahedrons and spheres
ReticulatedPython 1
N
by jb2015007
Let 
be a non-degenerate tetrahedron with 
, 
, 
, and 
Let a sphere of radius 
be circumscribed about this tetrahedron. Prove that ![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 9\sqrt[3]{16}.$$](//latex.artofproblemsolving.com/8/a/4/8a4a6595e2e5f20222f6288f89d6938301c9d7d2.png)
be a non-degenerate tetrahedron with 
, 
, 
, and 
Let a sphere of radius 
be circumscribed about this tetrahedron. Prove that ![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 9\sqrt[3]{16}.$$](http://latex.artofproblemsolving.com/8/a/4/8a4a6595e2e5f20222f6288f89d6938301c9d7d2.png)
1 reply
weird permutation problem
Sedro 3
N
by Sedro
Let 
be a permutation of 
such that there are exactly 
ordered pairs of integers 
satisfying 
and 
. How many possible exist?
be a permutation of 
such that there are exactly 
ordered pairs of integers 
satisfying 
and 
. How many possible exist?
3 replies
A problem involving modulus from JEE coaching
AshAuktober 6
N
by no_room_for_error
Solve over 
:
(There are two ways to do this, one being bashing out cases. Try to find the other.)
:
(There are two ways to do this, one being bashing out cases. Try to find the other.)
6 replies
Combinatorial proof
MathBot101101 9
N
by MathBot101101
Is there a way to prove
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}=1-\frac{1}{{n+1)!}
without induction and using only combinatorial arguments?
Induction proof wasn't quite as pleasing for me.
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}=1-\frac{1}{{n+1)!}
without induction and using only combinatorial arguments?
Induction proof wasn't quite as pleasing for me.
9 replies
such that 
.
Where 
Inscribed Semi-Circle!!!
ehz2701 2
N
by mathafou
A right triangle 
with legs 
and 
is drawn with a semicircle inscribed into the triangle. What is the smallest possible radius of the semi-circle and the largest possible radius?
with legs 
and 
is drawn with a semicircle inscribed into the triangle. What is the smallest possible radius of the semi-circle and the largest possible radius?2 replies
geometry
carvaan 1
N
by vanstraelen
OABC is a trapezium with OC // AB and ∠AOB = 37°. Furthermore, A, B, C all lie on the circumference of a circle centred at O. The perpendicular bisector of OC meets AC at D. If ∠ABD = x°, find last 2 digit of 100x.
1 reply
Inequalities
nhathhuyyp5c 1
N
by Mathzeus1024
Let 
be non-negative real numbers such that 
. Find the maximum and minimum values of the expression
![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](//latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
be non-negative real numbers such that 
. Find the maximum and minimum values of the expression![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
1 reply
