[/quote]
,
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
classes and 
student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than 
is not a perfect square, prove that there exist a student that has attended in at least 
classes.
,
,
,
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
and 
.
and 
Prove that
where 
be a function defined on 
taking values in 
such that
for all 
for all 
and 
Prove that
;
identical-looking coins in a circle. It is known that exactly two of them are fake. Real coins weigh the same, fake ones too, but they are lighter than the real ones. How can you determine in three weighings on a cup scale without weights whether there are fake coins lying nearby or not??
points of general position marked on the coordinate plane (i.e., points among which there are no three lying on the same straight line). Is there a polynomial of two variables 
a) of degree 
;
be triangle, inscribed in parabola. Tangents in points 
.
)
such that there exist integers 
with ![\[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]](http://latex.artofproblemsolving.com/e/f/a/efaa98e77ac0273284ee3446e4335d8cc4f681cb.png)
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 4,5: (will type tomorrow)
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 4,5: (will type tomorrow)
Attachments:
This post has been edited 1 time. Last edited by imnotgoodatmathsorry, Apr 13, 2025, 4:14 PM
and 
?
