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Topic
First Poster
Last Poster
2023 Official Mock NAIME #15 f(f(f(x))) = f(f(x))
parmenides51 3
N
by jasperE3
How many non-bijective functions 
exist that satisfy 
for all real 
and the domain of f is strictly within the set of 
, the range being 
?
Even though this is an AIME problem, a proof is mandatory for full credit. Constants must be ignored as we dont want an infinite number of solutions.
exist that satisfy 
for all real 
and the domain of f is strictly within the set of 
, the range being 
?Even though this is an AIME problem, a proof is mandatory for full credit. Constants must be ignored as we dont want an infinite number of solutions.
3 replies
Geometry
AlexCenteno2007 3
N
by AlexCenteno2007
Let ABC be an acute triangle and let D, E and F be the feet of the altitudes from A, B and C respectively. The straight line EF and the circumcircle of ABC intersect at P such that F is between E and P, the straight lines BP and DF intersect at Q. Show that if ED = EP then CQ and DP are parallel.
3 replies
Cube Sphere
vanstraelen 4
N
by pieMax2713
Given the cube
with edge 
cm.Find the volume of the sphere passing through
and tangent to the plane 
.4 replies
Square number
linkxink0603 3
N
by Zok_G8D
Find m is positive interger such that m^4+3^m is square number
3 replies
Combinatorics
AlexCenteno2007 0
Adrian and Bertrand take turns as follows: Adrian starts with a pile of (
) stones. On their turn, each player must divide a pile. The player who can make all piles have at most 2 stones wins. Depending on n, determine which player has a winning strategy.
) stones. On their turn, each player must divide a pile. The player who can make all piles have at most 2 stones wins. Depending on n, determine which player has a winning strategy.
0 replies
How many pairs
Ecrin_eren 6
N
by Ecrin_eren
Let n be a natural number and p be a prime number. How many different pairs (n, p) satisfy the equation:
p + 2^p + 3 = n^2 ?
6 replies
parallelogram in a tetrahedron
vanstraelen 1
N
by vanstraelen
Given a tetrahedron 
and a plane 
, parallel with the edges 
and 
.
.
a) Prove: the intersection of the tetrahedron with the plane is a parallelogram.
b) If
and 
,
calculates the lenghts of the sides of this parallelogram.
and a plane 
, parallel with the edges 
and 
.
.a) Prove: the intersection of the tetrahedron with the plane is a parallelogram.
b) If
and 
,calculates the lenghts of the sides of this parallelogram.
1 reply
and 
.
.
triangle ABC, # BCDE, BE//AM, BE=AM/2, midpoint (Greece Junior 2014)
parmenides51 8
N
by AylyGayypow009
Let 
be a triangle and let 
be the midpoint 
. On the exterior of the triangle, consider the parallelogram 
such that 
and 
. Prove that line 
passes through the midpoint of segment 
.
be a triangle and let 
be the midpoint 
. On the exterior of the triangle, consider the parallelogram 
such that 
and 
. Prove that line 
passes through the midpoint of segment 
.
8 replies
a,b,c irrational, f(x)=ax^2+bx+c : [-1,1] to [-1,1] surjective
tom-nowy 1
N
by alexheinis
Consider a quadratic function 
, where the coefficients 
and 
are all irrational numbers.
Is it possible for this function to have a maximum value of
and a minimum value of 
over the interval ![$[-1, 1]$](//latex.artofproblemsolving.com/5/c/3/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png)
?
, where the coefficients 
and 
are all irrational numbers.Is it possible for this function to have a maximum value of
and a minimum value of 
over the interval ![$[-1, 1]$](http://latex.artofproblemsolving.com/5/c/3/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png)
?
1 reply
