[/quote]
,
such that 
.
Where 
be a point on 
,
be a point on 
be a point on 
such that 
and̸ 
.
and 
,
.
and circumcircle 
.
be the midpoints of the arcs 
of 
respectively. Let 
be the point of 
and the midpoint 
of 
lie on a line.
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
such that 
for all 
?
be a monic polynomial with complex coefficients such that there exist a polynomial 
with complex coefficients for which ![\[P(x^2-2)=P(x)Q(x).\]](http://latex.artofproblemsolving.com/a/6/2/a62b06890b6bbcc5c9634f6de8c7ae5fe9f37828.png)
Determine all complex numbers that could be the root of 
and 
Prove that
Prove that
such that for all 
,![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
.
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
of integers that has the following property:
(possibly 
)
for every integer 
.
of nonzero integers such that the only admissible set containing both 
and 
Y by Windflower
Second Entrance Mock test for grade 10 specialized in Mathematics at High School for Gifted Students, HNUE, Vietnam
13/4/2025
Problem 1:
1) Let
be positive reals. Prove that: 
2) In a small garden there are
rabbits and carrots. Each rabbit will choose randomly a carrot to eat. Find the probability of a carrot was chose by less than 
rabbit.
Problem 2:
1) Solve the equation system:
and 
2) Let
be positive rational numbers such that: 
Problem 3:
Let triangle
(
, 
, 
) with excircle 
and incircle 
. Incircle touches 
at 
. The excircle with the diameter of 
cuts excircle at 
(
). 
cuts the excircle with the diameter of at 
(
) and 
cuts 
at 
. Prove that:
1)
~ 
and is the bisector of 
2)
3)
is the tangent line of the excircle of 
Problem 4:
A positive integer is called "rich number" if the sum of all its divisors greater than two times of it. If a positive integer can't be called "rich number", it will be called "poor number". Prove that:
1) There are many "poor number".
2)
is a "rich number" with 
is a positive integer.
Problem 5:
There are
consecutive integers written on the board. In each turn, we divide the numbers to small groups (each group has number) and replace the numbers in each group with their sums and subtractions (we don't need to let the bigger numbers subtract to the smaller ones, every turn happens simultaneously). After we done a finite numbers of turn, are there consecutive integers written on the board ?
13/4/2025
Problem 1:
1) Let
be positive reals. Prove that: 
2) In a small garden there are
rabbits and carrots. Each rabbit will choose randomly a carrot to eat. Find the probability of a carrot was chose by less than 
rabbit.Problem 2:
1) Solve the equation system:
and 
2) Let
be positive rational numbers such that: 
Problem 3:
Let triangle
(
, 
, 
) with excircle 
and incircle 
. Incircle touches 
at 
. The excircle with the diameter of 
cuts excircle at 
(
). 
cuts the excircle with the diameter of at 
(
) and 
cuts 
at 
. Prove that:1)
~ 
and is the bisector of 
2)
3)
is the tangent line of the excircle of 
Problem 4:
A positive integer is called "rich number" if the sum of all its divisors greater than two times of it. If a positive integer can't be called "rich number", it will be called "poor number". Prove that:
1) There are many "poor number".
2)
is a "rich number" with 
is a positive integer.Problem 5:
There are
consecutive integers written on the board. In each turn, we divide the numbers to small groups (each group has number) and replace the numbers in each group with their sums and subtractions (we don't need to let the bigger numbers subtract to the smaller ones, every turn happens simultaneously). After we done a finite numbers of turn, are there consecutive integers written on the board ?Attachments:
This post has been edited 2 times. Last edited by imnotgoodatmathsorry, Friday at 11:48 AM
and 
?
,![\[
\angle KFB = 180^\circ - \angle KFA = 180^\circ - \angle KEA = \angle KEC,
\]](http://latex.artofproblemsolving.com/1/6/3/163454dc31ef85ed5073d975380ac35dbe5d7fd3.png)
we deduce that 
(by angle-angle similarity).![\[
\frac{KB}{KC} = \frac{BF}{CE} = \frac{BD}{CD},
\]](http://latex.artofproblemsolving.com/a/d/7/ad759b58103ad4aed4023db036e5ce421465e89b.png)
so 
is the angle bisector of 
.
be the intersection of 
with 
(where 
,
are collinear.![\[
\angle BKM = \angle BAM = \angle CAM = \angle DBM,
\]](http://latex.artofproblemsolving.com/b/c/b/bcbb4746bd9b8fe390dd4ab6dffaa46f3b132035.png)
thus 
(by angle-angle similarity), implying:![\[
MD \cdot MK = MB^2 = MI^2.
\]](http://latex.artofproblemsolving.com/8/e/c/8ec379de93a8386b811f12fa48cf88efd5966c33.png)
Therefore, 
(by side-angle-side similarity), hence:![\[
\overline{MID} = \overline{MKI} = \overline{IAP},
\]](http://latex.artofproblemsolving.com/6/b/7/6b7f4213c4b3ae16fe1aeef86c1e5a7d68399c2a.png)
so 
.
,
.
,
,
,
,
is the common external tangent to both 
