Grade 11 - Fifth Exam
1. Given positive reals 
show that we have:
2. Let
be the angles of a triangle 
of perimeter 
and 
is the radius of its circumscribed circle.
Prove that:
show that we have: 2. Let
be the angles of a triangle of perimeter and is the radius of its circumscribed circle. Prove that: 
When do we have equality? 3. Two circles are tangent to each other internally at a point
. Let the chord 
of the larger circle be tangent to the smaller circle at a point 
. Prove that the line 
bisects 
. 4. The diagonals of a trapezoid
whose bases are 
and 
intersect at 
Prove that
Where 
denotes the area of 
.Grade 11 - First Exam
1. Solve the following equation in 
:
2. Solve in
the equation :
3. Let
and 
be two real numbers and let
. Find the values of and for which 
is minimal.4. Let
be a triangle. 
and 
are two points on the side 
such that 
.Find the mesure of the angle 
knowing that 
.
2. One integer was removed from the set
of the integers from 
to 
. The arithmetic mean of the other integers of 
is equal to 
.What integer was removed ?
3. When dividing an integer
by a positive integer , 
, a student finds 
.
Prove that the student made a mistake while computing.
4.
and 
are two circles which intersect in 
and 
. 
is a line that moves and passes through , intersecting in P and in P'. Prove that the bisector of 
passes through a non-moving point.
2. Compute the sum
where every three consecutive
are followed by two 
.
3. Solve in
the following system
4. Let be a triangle with area and 
the middle of the side 
. 
and 
are two points of 
and 
respectively such that 
and
. The two lines 
and 
intersect in a point 
. Find the area of the triangle 
.
such that 
, and 
for all reals 
.
2. Prove that the equation
has 4 distinct real solutions if and only if 
(
and 
are two real parameters).
3. Find all functions
which verify the relation
4. Let be a convex quadrilateral with angles 
and 
not less than 
. Prove that
.
knowing that . Grade 11 - Fourth Exam
1. Find all positive integers n such that : 2. One integer was removed from the set
of the integers from to . The arithmetic mean of the other integers of is equal to .What integer was removed ? 3. When dividing an integer
by a positive integer , , a student finds .Prove that the student made a mistake while computing.
4.
and are two circles which intersect in and . is a line that moves and passes through , intersecting in P and in P'. Prove that the bisector of passes through a non-moving point. Grade 11 - Second Exam
1. Prove that 2. Compute the sum
where every three consecutive
are followed by two . 3. Solve in
the following system 4. Let
be a triangle with area and the middle of the side . and are two points of and respectively such that and. The two lines and intersect in a point . Find the area of the triangle . Grade 11 - Third Exam
1. Find the maximum value of the real constant such that , and for all reals . 2. Prove that the equation
has 4 distinct real solutions if and only if (
and are two real parameters). 3. Find all functions
which verify the relation 4. Let
be a convex quadrilateral with angles and not less than . Prove that .
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