AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
take on exactly 
distinct values?
be one of the numbers 
. Suppose that for all positive integers 
, the number 
never has remainder upon division by 
. List all possible values of .
where each 
is chosen from 
. A line is an (unordered) set of three (distinct) cards 
,
,
such that for each , the numbers 
are either all the same or all different. How many different lines are there?
, where 
, is a 
-tangent pair if we have
. Compute the second largest integer that appears in a 
-tangent pair.
has side length 
. For 
, choose 
to be a point on the segment 
uniformly at random, assuming the convention that 
for all integers 
. What is the expected value of the area of hexagon 
?
.-dimensional space passes through the point 
, with 
, 
, and 
all positive. The plane also intersects all three coordinate axes with intercepts greater than zero (i.e. there exist positive numbers 
, 
, 
such that 
, 
, and 
all lie on this plane). Find, in terms of , , , the minimum possible volume of the tetrahedron formed by the origin and these three intercepts.
, when it begins walking to the right along the rubber band as the child begins stretching it. The increasingly tired ant walks at a rate of 
centimeters per second, while the child uniformly stretches the rubber band at a rate of one meter per second. The rubber band is infinitely stretchable and the ant and child are immortal. Compute the time in seconds, if it exists, at which the ant reaches the right end of the rubber band. If the ant never reaches the right end, answer 
.. We say that a point lies at distance d from a line segment if 
is the minimum distance between the point and any point on the line segment. Finally, we say that a lattice point 
is nearby a line segment if the distance between and the line segment is no greater than the distance between the line segment and any neighbor of . Find the number of lattice points that are nearby the line segment connecting the origin and the point 
.
, comes after 
in the permutation. What is the probability that a random permutation of the first 
integers is valid?
and 
, find the range of all possible values of
.
, 
, and 
. Compute the area of the smallest regular polygon that has three vertices coinciding with the vertices of the given triangle. are there such that for any natural numbers 
, we have 
implies 
?
and 
such that the following limit is finite and nonzero: 
.
.
, and let 
and be independently and uniformly distributed random variables in the interval 
. For the ordered 
-tuple 
, let 
denote the ratio between the area of a sheet with dimension a×b and the area of the horizontal cross-section of the box with dimension 
after the sheet has been folded in halves along each dimension until it occupies the largest possible area that will still fit in the box (because Sean is picky, the sheet must be placed with sides parallel to the box’s sides). Compute the smallest value of b/a that maximizes the expectation 
.
If 
for 
and 
, then find the value of 
.
If the 
th term of an A.P. is 
and the th term is , then find its 
th term.
If 
is a positive real number different from , then prove that the numbers 
are in A.P. Also find their common difference.
be an acute triangle with 
. Let 
be its circumcircle, 
its orthocenter, and 
the foot of the altitude from . Let 
be the midpoint of 
. Let 
be the point on 
such that 
and let 
be the point on such that 
. Assume that the points , 
, 
, and are all different and lie on in this order.
and 
are tangent to each other.
+1 w
touch sides , 
, and 
at points 
, 
, and , respectively. Let 
be the diametrically opposite point of with respect to the incircle. Let lines 
and 
intersect the incircle again at 
and 
, respectively. Prove that the lines 
, 
, and 
are concurrent, i.e., the lines intersect at the same point.
and 
be two similar right-angled triangles (in the same plane) with 
such that and lie on opposite sides of the line 
. If 
, calculate the ratio 
.
is written on the blackboard. Every time, Carmela can erase the number on the black board and replace it with a new number , if and only if 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
and 
touch externally at 
and the radius of is larger than that of . 
is any point on which does not lie on the line joining the centers of the circles. 
and 
are points on such that 
and 
are tangent to . The lines 
, 
intersect again at 
, 
respectively. 
is the intersection of the tangent at and the line 
. Show that the locus of as varies is a straight line.
be real numbers such that 
Prove that
Let be real numbers such that 
Prove thatLet be real numbers such that 
Prove that
be positive real numbers . Prove that :
with 
, 
and 
. Let 
and 
such that 
and 
. Calculate 
, where and are the projections of and onto , respectively.
?
such that 
and : two times the same angle 
? and are the projections of and onto , respectively: I presume, the projections of 
and .
lies on, sorry for the many typos,
and then use that for find 
but I must have gone wrong somewhere.
in the end indicate, as in what am I supposed to do for the later terms in the sequence?
be positive real numbers. Find the minimum value of![\[ T = (x+y+z+t)^2.\left[\dfrac{1}{x(y+z+t)}+\dfrac{1}{y(z+t+x)}+\dfrac{1}{z(t+x+y)}+\dfrac{1}{t(x+y+z)}\right] \]](http://latex.artofproblemsolving.com/c/a/c/cac0967ebdf30c1f79a1ec3e468cc035402333a7.png)
,
such that 
lie on 
be its center. Let 
be some point lying on the circumcircle of 
.
is tangent to 
is tangent to 
and 
,
meet the circumcircle of 
and 
.
,
.
and 
exist on 
and 
are tangent to 
,
,
,
.
at the orthocenter of 
,
be a point lying on 
such that 
.
.
be its incircle, and let 
be the midpoint of minor arc 
on the circumcircle of 
at points 
,
the intersection of 
and 
.
and 
denote the 
,
,
and 
,
and 
.
and 
are tangent to each other.
and 
be circles tangent to 
meets 
.
similarly.
and 
.
and 
.
be non-negative real numbers.Prove that :![\[ \sqrt{x^4+y^4 } +(2+\sqrt{2})xy \geq x^2+y^2 \]](http://latex.artofproblemsolving.com/6/2/3/623d50afffd81b2783ff98568791164b53747567.png)
It's easy to find that 
