[/quote]
,
,
.
and 
such that the only 
-
equals its 
-
.
digit numbers 
(with 
)
,
,
.
be the midpoint of 
in 
.
at 
.
,
in degrees?
with 
and 
for 
.
.
.
.
(x) be the 
t
.
such that 
divides 
?
be the set of vectors 
where 
are all real numbers. Let 
denote 
.
be the set in 
given by 
If a point 
is uniformly at random from ![$E[||z||^2]$](http://latex.artofproblemsolving.com/e/1/a/e1aa966627c05d1f317981d82094e07d1b1ce0e1.png)
?
be the unique integer between 
and 
,
.
and 
,
.
,
,
,
.
and 
,
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 
.
,
,
.
graph paper notebooks. How many ways can she buy school supplies?
be the center of the circumcircle of right triangle 
with 
.
be the midpoint of minor arc 
and let 
be a point on line 
.
be the intersection of line 
and the Circle 
be the intersection of line 
and 
.
and 
,
.
to be the maximum possible least-common-multiple of any sequence of positive integers which sum to 
be an acute triangle with 
,
be the center of its circumcircle. Let 
be the reflection of 
with respect to 
.
at 
,
intersects the line through 
.
be a point on the ray 
,
.
.
be a triangle with side‐lengths 
,
,
.
be the areas of triangles 
,
,
,![\[
\frac{abc}{12R}
\;\le\;
\frac{P_1^2 + P_2^2 + P_3^2}{P}
\;\le\;
\frac{3R^3}{4\sqrt[3]{abc}}.
\]](http://latex.artofproblemsolving.com/1/3/9/139a3e15f08afb78ca0b8301f24d30db8f202207.png)
lie the following points: 
and 
on 
,
on 
,
on 
.![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](http://latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet 
.
,
are collinear.
and 
be two fixed points in the plane. For each point 
of the plane, outside of the line 
be the barycenter of the triangle 
.
with 
and 
be points on segments 
and 
,
.
be the circumcircle of 
,
the circumcircle of 
.
to 
and 
,
intersect 
,
intersect 
.
is tangent to both 
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)
.
.
where 
and 
.
intersects 
,
intersects 
.
intersects 
and 
intersect at 
and 
intersect at 
.
,
is circumcircle. Let 
.
and 
,
(opposite to 
and tangents to 
)
.
be two triangles inscribed in the same circle, centered at 
.
with respect to triangle 
.
.
lie in this order on the circle and form a convex, non-degenerate hexagon.
Y by
Let 
be a prime number. Find the number of elements of the set
![\[
M_p = \left\{ x \in \mathbb{Z}^* \,\middle|\, \sqrt{x^2 + 2p^{n} x} \in \mathbb{Q} \right\}.
\]](//latex.artofproblemsolving.com/5/4/1/54152836e072da8d4ec5984fcb7fea428cda8e48.png)
Edit: I edited with a variation of the same problem because I intend to give this one to an exam for my pupils.
be a prime number. Find the number of elements of the set![\[
M_p = \left\{ x \in \mathbb{Z}^* \,\middle|\, \sqrt{x^2 + 2p^{n} x} \in \mathbb{Q} \right\}.
\]](http://latex.artofproblemsolving.com/5/4/1/54152836e072da8d4ec5984fcb7fea428cda8e48.png)
Edit: I edited with a variation of the same problem because I intend to give this one to an exam for my pupils.
This post has been edited 1 time. Last edited by td12345, Apr 14, 2025, 6:17 PM
Reason: variation of the problem
Reason: variation of the problem
,
comes from but I think when 
we get 2 extra: 
.
not the set of the square root expression no? I got a different number from yours and I don't know where I went wrong, I will lay down the 
:
where 
. This leads to ![\( [\frac{2025+1}{2}]\)](http://latex.artofproblemsolving.com/d/9/3/d93cfced933adf97b0b51a17da9558e88d0838fc.png)
values of 
where 
.
extra values for 
:
where 
. This leads to 
distinct non-zero values of 
where 
distinct non-zero values of 
?
it leads to 
for the very first family and 
we shall have 
.
,
goes from 1 to 2024. This leads to 
extra values for 
,
.
.
is natural numbers (positive integers) and 
is natural numbers without 
?
