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 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 
.
, find the value of x that minimizes 
.
and 
such that the only 
-intercept of 
equals its 
-intercept. Compute 
.
digit numbers 
(with 
) such that 
, 
, and 
. What’s the size of this set? be the midpoint of in 
. A line perpendicular to D intersects at . If the area of is four times that of the area of 
, what is 
in degrees?
with 
and 
for 
. Find 
.
.
. Let 
(x) be the 
th derivative of 
. What is the largest integer 
such that 
divides 
?
be the set of vectors 
where 
are all real numbers. Let 
denote 
. Let 
be the set in 
given by 
If a point 
is uniformly at random from , what is ![$E[||z||^2]$](http://latex.artofproblemsolving.com/e/1/a/e1aa966627c05d1f317981d82094e07d1b1ce0e1.png)
?
be the unique integer between 
and 
, inclusive, that is equivalent modulo to 
. Let be the set of primes between and 
, inclusive. Find 
.
,
,
,
. We pick an integer between and 
, inclusive, uniformly at random. We shoot a puck from in the direction of 
and the puck bounces perfectly around the box (angle in equals angle out, no friction) until it hits one of the four vertices of the box. What is the expected number of times it will hit an edge or vertex of the box, including both when it starts at and when it ends at some vertex of the box?
. She can only buy full packs of each of the following items. A pack of pens is 
, a pack of pencils is 
, and any type of notebook or stapler is 
. Sarah buys at least pack of pencils. She will either buy stapler or no stapler. She will buy at most college-ruled notebooks and at most 
graph paper notebooks. How many ways can she buy school supplies?
be the center of the circumcircle of right triangle with 
. Let be the midpoint of minor arc and let be a point on line such that 
. Let be the intersection of line 
and the Circle and let be the intersection of line 
and 
. If 
and 
, compute the radius of the Circle .
.
to be the maximum possible least-common-multiple of any sequence of positive integers which sum to . Find the sum of all possible odd 
of the pyramid 
are equal to 
, and , respectively the base of the ABC is a regular triangle. It is known that the area of the lateral faces of the pyramid are equal to each other. Find the volume of the pyramid .
of the pyramid are equal to , and , respectively the base of the ABC is a regular triangle. It is known that the area of the lateral faces of the pyramid are equal to each other. Find the volume of the pyramid .
satisfying
Prove
be positive reals such that 
.
![\[
M = \frac{1 - x^2}{x^2} + 2y^2 + 3x + \frac{24}{y} + 2025.
\]](http://latex.artofproblemsolving.com/6/b/d/6bd846ab0f95b1cf4766c7bf0356a439f3bfc93a.png)
.
has sides 
,
has sides 
,
for convenience), we have 
,
,
.
,
.
times the square of the volume) of this purported tetrahedron to check if it positive. However, 
)