Hanoi Open Mathematical Olympiad 2009

What we love to do we find time to do! Tuan Nguyen Anh
Hanoi Mathematical Society
Hanoi Open Mathematical Olympiad 2009
Junior Section
Sunday, 29 March 2009 08h45 - 11h45
Important:
Answer all 14 questions.
Enter your answers on the answer sheet provided.
No calcultoes are allowed.
Q1. What is the last two digits of the number
1000.1001 + 1001.1002 + 1002.1003 + … + 2008.2009?
(A) 25; (B) 41; (C) 36; (D) 54; (E) None of the above.
Q2. Which is largest positive integer n satisfying the inequality

.
(A) 3; (B) 4; (C) 5; (D) 6; (E) None of the above.
Q3. How many positive integer roots of the intequality
,
are there in (-10;10).
(A) 15; (B) 16; (C) 17; (D) 18; (E) None of the above.
Q4. How many triples (a;b;c) where
and a < b < c such that the number abc + (7 - a)(7 - b)(7 - c) is divisible by 7.
(A) 15; (B) 17; (C) 19; (D) 21; (E) None of the above.
Q5. Show that there is a natural number n such that the number a = n! ends exacly in 2009 zeros.
Q6. Let a, b, c be positive integers with no common factor and satisfy the conditions
. Prove that a + b is a square.
Q7. Suppose that
, where
. Prove that a is divisible by 23 for any positive integer n.
Q8. Prove that
is divisible by 42 for any positive integer m.
What we love to do we find time to do! Tuan Nguyen Anh
Q9. Suppose that 4 real numbers a, b, c, d satisfy the conditions

.
Find the set of all possible values the number M = ab + cd can take.
Q10. Let a, b be positive integers such that a + b = 99. Find the smallest and the greatest values of the following product P = ab.
Q11. Find all integers x, y such that
.
Q12. Find all the pairs of the positive integers such that the product of the numbers of any pair plus the half of one of the numbers plus one third of the other number is three times less than 15.
Q13. Let be given
with area (
) =
. Let R, S lie in BC such that BR = RS = SC and P, Q be midpoints of AB and AC, respectively. Suppose that PS intersects QR at T. Evaluate area (
).
Q14. Let
be an acute-angled triangle with AB = 4 and CD be the altitude through C with CD = 3. Find the distance between the midpoints of AD and BC.














What we love to do we find time to do! Tuan Nguyen Anh
Hanoi Mathematical Society
Hanoi Open Mathematical Olympiad 2009
Senior Section
Sunday, 29 March 2009 08h45 - 11h45
Important:
Answer all 14 questions.
Enter your answers on the answer sheet provided.
No calcultoes are allowed.
Q1. What is the last two digits of the number
1000.1001 + 1001.1002 + 1002.1003 + … + 2008.2009?
(A) 25; (B) 41; (C) 36; (D) 54; (E) None of the above.
Q2. Which is largest positive integer n satisfying the inequality

.
(A) 3; (B) 4; (C) 5; (D) 6; (E) None of the above.
Q3. How many positive integer roots of the intequality
,
are there in (-10;10).
(A) 15; (B) 16; (C) 17; (D) 18; (E) None of the above.
Q4. How many triples (a;b;c) where
and a < b < c such that the number abc + (7 - a)(7 - b)(7 - c) is divisible by 7.
(A) 15; (B) 17; (C) 19; (D) 21; (E) None of the above.
Q5. Suppose that
, where
. Prove that a is divisible by 23 for any positive integer n.
Q6. Determine all positive integral pairs (u;v) for which
.
Q7. Prove that for every positive integer n there exists a positive integer m such that the last n digists in deciman representation of
are equal to 8.
Q8. Give an example of a triangle whose all sides and altitudes are positive integers.