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.
be an acute scalene triangle. Denote by 
the circle with diameter 
, and let 
be the contact points of the tangents from 
to , chosen so that 
and 
lie on opposite sides of 
and 
and 
lie on opposite sides of 
. Similarly, let 
be the circle with diameter 
, with tangents from touching at 
, and 
the circle with diameter , with tangents from touching at 
.
are concurrent.
were selected 
pairwise distinct points, abscissas of which are integer numbers from the segment ![$[0; 100000]$](http://latex.artofproblemsolving.com/0/5/e/05e1d2855ef6a6dfaf9b24f2713455ab1017a30d.png)
. Prove that it is possible to choose six different selected points , , , 
, 
, 
such that areas of triangles 
and 
are equals.
baskets, each containing at least one candy. The total number of candies is 
. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad takes all the candies from 
consecutive non-empty baskets, while Sevinch takes all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least 
candies, regardless of the initial distribution of candies and Sevinch's actions.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers. be a triangle. On side take points 
and 
such that 
take points 
and 
such that 
and on side take points 
and 
such that 
Let 
and 
are concurrent.
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
. cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from 
to . In a move, Madina is allowed to choose any 
cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?
and 
of the acute-angled triangle intersect at point 
, the heights of the triangle 
intersect at point 
, point is the midpoint of side . Prove that 
.
be positive real numbers. Prove that there exists a cyclic permutation of 
such that for all positive real numbers 
the following holds:![\[
\frac{a}{x\,a + y\,b + z\,c}
\;+\;
\frac{b}{x\,b + y\,c + z\,a}
\;+\;
\frac{c}{x\,c + y\,a + z\,b}
\;\ge\;
\frac{3}{x + y + z}.
\]](http://latex.artofproblemsolving.com/1/5/2/15261b3f712c2cc8ffbeb92a9711e06ad0992b5d.png)
. Dasha and Sonya play, taking turns making moves, Dasha starts. In one move, it is allowed to select any coin and move it: Dasha moves the coin to a diagonally adjacent cell, Sonya is to the side adjacent. If two coins end up in the same cell, one of them is immediately removed from the board and goes to Sonya. Sonya can stop the game at any time and take all the coins she has received. What is the biggest win she can get, no matter how she plays Dasha?
be distinct primes and let 
be nonnegative integers. Define![\[
m \;=\;
\frac12
\Bigl(\prod_{i=2}^k p_i^{a_i}\Bigr)
\Bigl(\prod_{i=1}^k(p_i+1)\;+\;\sum_{i=1}^k(p_i-1)\Bigr),
\]](http://latex.artofproblemsolving.com/7/9/a/79af85375957986298d37c00d288b6d402f40106.png)
![\[
n \;=\;
\frac12
\Bigl(\prod_{i=2}^k p_i^{a_i}\Bigr)
\Bigl(\prod_{i=1}^k(p_i+1)\;-\;\sum_{i=1}^k(p_i-1)\Bigr).
\]](http://latex.artofproblemsolving.com/e/2/3/e23799ff03400d5eade3731bb83456fcd702a5e1.png)
Prove that![\[
p^2-1 \;\bigm|\; p\,m \;-\; n.
\]](http://latex.artofproblemsolving.com/3/7/e/37e4535b59ef183908730946213429e9ebc05dd7.png)
is satisfied. Point 
is taken on the circumcircle of such that 
.A line parallel to which passes from intersects and respectively at 
and .Show that circumcircle of passes from circumcenter of 
.
is an acute angle triangle such that 
and 
. Let's denote by 
the center of the circumscribed circle of the triangle and the intersection of altitudes of this triangle. Line 
intersects in point and in point . Find the value of the ration 
.
is an acute angle triangle such that and . Let's denote by the center of the circumscribed circle of the triangle and the intersection of altitudes of this triangle. Line intersects in point and in point . Find the value of the ration .
is equlilateral. Now the nine point centre lies on the bisector of 
. Because if 
are the middle points of 
and 
the nine point centre, the quadrilateral 
is concyclic 
,So 
are symmetric points about the bisector of 
, So 
(since 
is an altitude)
(since 
is inscribed in the circle with center )
(since 
is isoceles since 
)
be the perpendicular from to . In any triangle, 
to meet the circumcircle at , then draw . 
since it's inscribed in a semicircle, so 
, and since 
. Now we have 
, and 
. Thus, 
, so 
is a parallelogram, so 
.
be the perpendicular from to . Then 
so 
is a 
right triangle, so 
. Thus, since 
and 
, 
so 
.
of the line 
(
is incenter) with the circumcircle 
of . Since 
obtain that the points , , belong to the circle 
and the quadrilateral 
is a rhombus, 
a.s.o.
satisfying
Prove
.
.
and 
are congruent, and so 
.
and so 
is cyclic and so by simple angle chasing we get that 
and 
