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.
vertices of a convex pentagon (but not the sides!). Let 
be the number of ways you can draw at least 
straight line segments between the vertices so that no two line segments intersect in the interior of the pentagon. What is 
? (Note what the question is asking for! You have been warned!)
, let 
be the number of numbers for which the sum of the digits is divisible by . What are the first three digits (from the left) of ?
digits. He can stop when he cannot fit the next perfect square on the board. (At the end, there might be some space left on the board - he does not write only part of the next perfect square.) If 
is the largest perfect square he writes, what is ? are there such that 
, and the greatest common divisor of and is a prime number?
and 
, inclusive, with equal probability. However, right now, I want to produce a random number between and , inclusive, so I do the following:
I use my machine to generate a number between and . Call this 
. I take a and divide it by to get remainder 
. If 
, then I record as the randomly generated number. If 
, then I record instead. and 
. Let 
be the probability that the two numbers are equal. What is 
?
on the plane, where 
and 
are nonnegative integers.
's turn, A tells his opponent 
to move to another point on the plane. Then waits for a while. If is not eaten by a tiger, then moves to that point as well. there are three places can tell to walk to: leftwards to 
, downwards to 
, and simultaneously downwards and leftwards to 
. However, you cannot move to a point with a negative coordinate. with 
and 
, and and are not both zero. Also suppose that you two play strategically, and you go first (i.e., by telling your friend where to go). For how many of the starting points do you win?
, 
degrees, and 
degrees. Find the area of triangle 
. Your answer in its simplest form can be written as 
, where where 
are integers and 
is square-free. Find 
. and are integers which satisfy 
What is the maximum possible value of 
? The square has side length 
. The boundary of the square intersects the graph of 
at at least 
points. Mom: Tell me the number of zeros at the end of 
PT: Huh? ends in 
, so there aren't any zeros. Mom: No, the exclamation point at the end was not to signify me yelling. I was not asking about , I was asking about . 
. Exactly one of the four is a rational number. Let that number be 
, where and 
are nonnegative integers and 
. Concatenate 
.
.)
, 
, 
, 
. You remove a section of the bottom of the paper by cutting along the function 
, where 
satisfies 
. (In other words, you keep the bottom two vertices.)
-shaped elbow tube.
, where are integers and is square-free. 
.
find the area of the region in the 
-plane satisfying:
, and the other cone has a radius of . The two cones with radii have height 
, and the other cone has height 
. Let 
be the volume of the tetrahedron with three of its vertices as the three vertices of the cones and the fourth vertex as the center of the base of the cone with height . Find 
.
, where r and s are integers and . What is 
?
and 
, and for 
, 
and 
. The minimum value of 
can be written in the form 
, where are integers and is square-free. Concatenate 
(in that order!).
, where are integers and 
,
, 
are all less than ?
with the property that for every real numbers and it holds that
coins.
with center at point 
inscribed in an triangle 
(
) touches the sides 
, 
and 
at points 
, 
and 
, respectively. The circumcircles of triangles and 
intersect secondary at point 
The lines 
and 
intersect at point 
and intersects the line at points 
and 
, respectively. The tangent lines to , other than , passing through points and touch at points 
and 
, respectively. Let the lines 
and 
intersect at the point 
. Prove that if 
is a midpoint of segment 
then 
.
, where 
, such that![\[ \frac{2^n+2}{n} \]](http://latex.artofproblemsolving.com/7/1/d/71d8537a6e5a489ab3ec382fee21d638ab65e213.png)
, let
denote the number of positive integer divisors of ,
denote the sum of the positive integer divisors of , and
denote the number of positive integers less than or equal to that are relatively prime to .
be integers. Brandon has a calculator with three buttons that replace the integer currently displayed with , , or , respectively. Prove that if the calculator currently displays , then Brandon can make the calculator display after a finite (possibly empty) sequence of button presses.
be a point inside the triangle such that 
. Let 
and 
be the midpoints of the arcs 
and 
, respectively. Prove that the circumcircles of triangles 
and 
are tangent to each other.
be an interior point on the side of an acute-angled triangle . Let the circumcircle of triangle 
intersect 
again at 
and the circumcircle of triangle 
intersect again at 
. Let 
, 
, and 
intersect the circumcircle of triangle again at 
, 
and 
, respectively. Let and 
be the incentres of triangles 
and 
, respectively. Prove that 
are concyclic.
.Find all functions 
so that![\[
f(x + y) = f(x) + f(y) + a
\]](http://latex.artofproblemsolving.com/4/5/1/4517c91563805f38690b59828e76164754da340c.png)
holds for all 
.
be the assertion for the above mentioned function.Note that this function might look very similar to an additive equation but the only thing that is annoying us is that constant a,so what we do is simple,we seek to transform the equation,and such add a to both sides and the function changes to,![\[
f(x+y)+a=f(x)+a+f(y)+a
\]](http://latex.artofproblemsolving.com/4/3/f/43f49a8d2bab200b2fea6f38aa1100117ae50c5a.png)
,and thus the function becomes,and we define the domain on positive reals and the codomain on reals![\[
g(x+y)=g(x)+g(y)
\]](http://latex.artofproblemsolving.com/9/6/0/96066d6278f53c8e3aae75bef9ff0f8dda7b2b3d.png)
,which makes g bounded below,and such g is linear.Now maybe I am wrong and if so please tell me.So 
,for some real c. so it means that the solution is 
,for all real x.
, but he only wants the sets wherein the sum of the elements is divisible by . How many subsets can he pick?
is the amount of bananas Kyle currently has in his garage (where 
is in hours), then what does 
mean in this context with units or something.
hours, the rate at which the amount of Bananas Kyle has is decreasing by 
bananas per hour per hour)
across, start at the hexagon labeled 
.
is not in the set, and neither is 
,
,
,
.
.![$\sqrt[3]{\dfrac{b^2+c^2}{a^2+bc}}+\sqrt[3]{\dfrac{a^2+c^2}{b^2+ac}}+\sqrt[3]{\dfrac{a^2+b^2}{c^2+ab}}\le \dfrac{a+b+c}{\sqrt[3]{abc}}$](http://latex.artofproblemsolving.com/b/5/8/b58b5566787bd32e2dfda5c47c7a404902b57e5a.png)
other than 
such that 
and 
are both perfect squares.
