Hanoi Open Mathematical Olympiad 2008

What we love to do we find time to do! Tuan Nguyen Anh
Hanoi Mathematical Society
Hanoi Open Mathematical Olympiad 2008
Junior Section
Sunday, 30 March 2008 08h45 - 11h45
Important:
Answer all 10 questions.
Enter your answers on the answer sheet provided.
No calcultoes are allowed.
Q1. How many integers from 1 to 2008 have the sum of their digits divisible by 5?
Q2. How many integers belong to (a;2008a), where a (a > 0) is given.
Q3. Find the coefficient of x in the expansion of
(1 + x)(1 - 2x)(1 + 3x)(1 - 4x)…(1 - 2008x).
Q4. Find all pairs (m;n) of positive integers such that
.
Q5. Suppose x, y, z, t are real numbers such that

. Prove that
.
Q6. Let P(x) be a polynomial such that
. Find
?
Q7. The figure ABCDE is a convex pentagon. Find the sum


Q8. The sides of a rhombus have length a and the area is S. What is the length of the shorter diagonal?
Q9. Let be given a right-angled triangle ABC with
, AB = c, CA = b. Let
and
such that
and
. Denote by
and
such that
and
. Determine EP + EF + PQ?
Q10. Let
and satisfy the following conditions

, a + b + c = 5.
What is the smallest possible value of
?
What we love to do we find time to do! Tuan Nguyen Anh
Hanoi Mathematical Society
Hanoi Open Mathematical Olympiad 2008
Senior Section
Sunday, 30 March 2008 08h45 - 11h45
Important:
Answer all 10 questions.
Enter your answers on the answer sheet provided.
No calcultoes are allowed.
Q1. How many integers are there in
, where b ( b > 0) is given.
Q2. Find all pairs (m;n) of positive integers such that
.
Q3. Show that the equation
, has no solutions of positive integers x, y and z.
Q4. Prove that there exists an infinite number of relatively prime pairs (m;n) of positive integers such that the equation
, has three distintcinteger roots.
Q5. Find all polynomials P(x) of degree 1 such that
,
where a < b.
Q6. Let
and satisfy the following conditions

, a + b + c = 5.
What is the smallest possible value of
?
Q7. Fin all triples (a, b, c) of consecutive odd positive integers such that a < b < c and
is a four digit number with all digit equal.
Q8. Consider a covex quadrilateral ABCD. Let O be the intersection of AC and BD; M, N be the centroid of
and
and P, Q be orthocenter of
and
, respectively. Prove that
.
Q9. Consider a triangle ABC. For every point
we difine
and
such that APMN is a parallelogram. Let O be the intersection of BN and CP. Find
such that
.
What we love to do we find time to do! Tuan Nguyen Anh
Q10. Let be given a right-angled triangle ABC with
, AB = c, CA = b. Let
and
such that
and
. Denote by
and
such that
and
. Determine EP + EF + PQ?


(Sưu tầm và giới thiệu)