Hanoi open mathematical olympiad - Problems and solutions - Nguyen Van Mau

 = Area of triangle MCD.
Q6. On the circle of radius 30cm are given 2 points A, B with AB =
16cm and C is a midpoint of AB. What is the perpendicular distance
from C to the circle?
Q7. In ∆ABC, PQ//BC where P and Q are points on AB and AC
respectively. The lines PC and QB intersect at G. It is also given
EF//BC, where G ∈ EF , E ∈ AB and F ∈ AC with PQ = a and
EF = b. Find value of BC.
Q8. Find all polynomials P (x) such that
P (x) + P
(1
x
)
= x+
1
x
, ∀x 6= 0.
Q9. Let x, y, z be real numbers such that x2 + y2 + z2 = 1. Find the
largest possible value of
|x3 + y3 + z3 − xyz|?
1.2 Hanoi Open Mathematical Olympiad 2007
1.2.1 Junior Section, Sunday, 15 April 2007
Q1. What is the last two digits of the number
(3 + 7 + 11 + · · ·+ 2007)2?
(A) 01; (B) 11; (C) 23; (D) 37; (E) None of the above.
Q2. What is largest positive integer n satisfying the following inequality:
1.2. Hanoi Open Mathematical Olympiad 2007 6
n2006 < 72007?
(A) 7; (B) 8; (C) 9; (D) 10; (E) 11.
Q3. Which of the following is a possible number of diagonals of a convex
polygon?
(A) 02; (B) 21; (C) 32; (D) 54; (E) 63.
Q4. Let m and n denote the number of digits in 22007 and 52007 when
expressed in base 10. What is the sum m+ n?
(A) 2004; (B) 2005; (C) 2006; (D) 2007; (E) 2008.
Q5. Let be given an open interval (α; β) with β−α = 1
2007
. Determine
the
maximum number of irreducible fractions
a
b
in (α; β) with 1 ≤ b ≤
2007?
(A) 1002; (B) 1003; (C) 1004; (D) 1005; (E) 1006.
Q6. In triangle ABC, ∠BAC = 600, ∠ACB = 900 and D is on BC. If
AD
bisects ∠BAC and CD = 3cm. Then DB is
(A) 3; (B) 4; (C) 5; (D) 6; (E) 7.
Q7. Nine points, no three of which lie on the same straight line, are
located
inside an equilateral triangle of side 4. Prove that some three of
these
points are vertices of a triangle whose area is not greater than
√
3.
Q8. Let a, b, c be positive integers. Prove that
(b+ c− a)2
(b+ c)2 + a2
+
(c+ a− b)2
(c+ a)2 + b2
+
(a+ b− c)2
(a+ b)2 + c2
≥ 3
5
.
1.2. Hanoi Open Mathematical Olympiad 2007 7
Q9. A triangle is said to be the Heron triangle if it has integer sides and
integer area. In a Heron triangle, the sides a, b, c satisfy the equation
b = a(a− c). Prove that the triangle is isosceles.
Q10. Let a, b, c be positive real numbers such that
1
bc
+
1
ca
+
1
ab
≥ 1.
Prove
that
a
bc
+
b
ca
+
c
ab
≥ 1.
Q11. How many possible values are there for the sum a + b + c + d if
a, b, c, d
are positive integers and abcd = 2007.
Q12. Calculate the sum
5
2.7
+
5
7.12
+ · · ·+ 5
2002.2007
.
Q13. Let be given triangle ABC. Find all points M such that
area of ∆MAB= area of ∆MAC.
Q14. How many ordered pairs of integers (x, y) satisfy the equation
2x2 + y2 + xy = 2(x+ y)?
Q15. Let p = abc be the 3-digit prime number. Prove that the equation
ax2 + bx+ c = 0
has no rational roots.
1.2.2 Senior Section, Sunday, 15 April 2007
Q1. What is the last two digits of the number(
112 + 152 + 192 + · · ·+ 20072)2?
1.2. Hanoi Open Mathematical Olympiad 2007 8
(A) 01; (B) 21; (C) 31; (D) 41; (E) None of the above.
Q2. Which is largest positive integer n satisfying the following inequal-
ity:
n2007 > (2007)n.
(A) 1; (B) 2; (C) 3; (D) 4; (E) None of the above.
Q3. Find the number of different positive integer triples (x, y, z) satsfy-
ing
the equations
x+ y − z = 1 and x2 + y2 − z2 = 1.
(A) 1; (B) 2; (C) 3; (D) 4; (E) None of the above.
Q4. List the numbers
√
2, 3
√
3, , 4
√
4, 5
√
5 and 6
√
6 in order from greatest
to
least.
Q5. Suppose that A,B,C,D are points on a circle, AB is the diameter,
CD
is perpendicular to AB and meets AB at E, AB and CD are integers
and AE − EB = √3. Find AE?
Q6. Let P (x) = x3 + ax2 + bx + 1 and |P (x)| ≤ 1 for all x such that
|x| ≤ 1.
Prove that |a|+ |b| ≤ 5.
Q7. Find all sequences of integers x1, x2, . . . , xn, . . . such that ij divides
xi + xj for any two distinct positive integers i and j.
Q8. Let ABC be an equilateral triangle. For a point M inside ∆ABC,
letD,E, F be the feet of the perpendiculars fromM ontoBC,CA,AB,
respectively. Find the locus of all such points M for which ∠FDE
is a
1.2. Hanoi Open Mathematical Olympiad 2007 9
right angle.
Q9. Let a1, a2, . . . , a2007 be real numbers such that
a1+a2+ · · ·+a2007 ≥ (2007)2 and a21+a22+ · · ·+a22007 ≤ (2007)3−1.
Prove that ak ∈ [2006; 2008] for all k ∈ {1, 2, . . . , 2007}.
Q10. What is the smallest possible value of
x2 + 2y2 − x− 2y − xy?
Q11. Find all polynomials P (x) satisfying the equation
(2x− 1)P (x) = (x− 1)P (2x), ∀x.
Q12. Calculate the sum
1
2.7.12
+
1
7.12.17
+ · · ·+ 1
1997.2002.2007
.
Q13. Let ABC be an acute-angle triangle with BC > CA. Let O, H
and F
be the circumcenter, orthocentre and the foot of its altitude CH,
respectively. Suppose that the perpendicular to OF at F meet the
side
CA at P . Prove ∠FHP = ∠BAC.
Q14. How many ordered pairs of integers (x, y) satisfy the equation
x2 + y2 + xy = 4(x+ y)?
Q15. Let p = abcd be the 4-digit prime number. Prove that the equation
ax3 + bx2 + cx+ d = 0
has no rational roots.
1.3. Hanoi Open Mathematical Olympiad 2008 10
1.3 Hanoi Open Mathematical Olympiad 2008
1.3.1 Junior Section, Sunday, 30 March 2008
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
m2 + n2 = 3(m+ n).
Q5. Suppose x, y, z, t are real numbers such that
|x+ y + z − t| 6 1
|y + z + t− x| 6 1
|z + t+ x− y| 6 1
|t+ x+ y − z| 6 1
Prove that x2 + y2 + z2 + t2 6 1.
Q6. Let P (x) be a polynomial such that
P (x2 − 1) = x4 − 3x2 + 3.
Find P (x2 + 1)?
Q7. The figure ABCDE is a convex pentagon. Find the sum
∠DAC + ∠EBD + ∠ACE + ∠BDA+ ∠CEB?
Q8. The sides of a rhombus have length a and the area is S. What is
the length of the shorter diagonal?
1.3. Hanoi Open Mathematical Olympiad 2008 11
Q9. Let be given a right-angled triangle ABC with ∠A = 900, AB = c,
AC = b. Let E ∈ AC and F ∈ AB such that ∠AEF = ∠ABC and
∠AFE = ∠ACB. Denote by P ∈ BC and Q ∈ BC such that EP ⊥ BC
and FQ ⊥ BC. Determine EP + EF + PQ?
Q10. Let a, b, c ∈ [1, 3] and satisfy the following conditions
max{a, b, c} > 2, a+ b+ c = 5.
What is the smallest possible value of
a2 + b2 + c2?
1.3.2 Senior Section, Sunday, 30 March 2008
Q1. How many integers are there in (b, 2008b], where b (b > 0) is given.
Q2. Find all pairs (m,n) of positive integers such that
m2 + 2n2 = 3(m+ 2n).
Q3. Show that the equation
x2 + 8z = 3 + 2y2
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
x3 − nx+mn = 0
has three distint integer roots.
Q5. Find all polynomials P (x) of degree 1 such that
max
a≤x≤b
P (x)− min
a≤x≤b
P (x) = b− a, ∀a, b ∈ R where a < b.
1.4. Hanoi Open Mathematical Olympiad 2009 12
Q6. Let a, b, c ∈ [1, 3] and satisfy the following conditions
max{a, b, c} > 2, a+ b+ c = 5.
What is the smallest possible value of
a2 + b2 + c2?
Q7. Find all triples (a, b, c) of consecutive odd positive integers such
that a < b < c and a2 + b2 + c2 is a four digit number with all digits
equal.
Q8. Consider a convex quadrilateral ABCD. Let O be the intersection
of AC and BD; M,N be the centroid of4AOB and4COD and P,Q be
orthocenter of4BOC and4DOA, respectively. Prove that MN ⊥ PQ.
Q9. Consider a triangle ABC. For every point M ∈ BC we difine
N ∈ CA and P ∈ AB such that APMN is a parallelogram. Let O be
the intersection of BN and CP . Find M ∈ BC such that ∠PMO =
∠OMN .
Q10. Let be given a right-angled triangle ABC with ∠A = 900, AB = c,
AC = b. Let E ∈ AC and F ∈ AB such that ∠AEF = ∠ABC and
∠AFE = ∠ACB. Denote by P ∈ BC and Q ∈ BC such that EP ⊥ BC
and FQ ⊥ BC. Determine EP + EF + FQ?
1.4 Hanoi Open Mathematical Olympiad 2009
1.4.1 Junior Section, Sunday, 29 March 2009
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.
1.4. Hanoi Open Mathematical Olympiad 2009 13
Q2. Which is largest positive integer n satisfying the inequality
1
1.2
+
1
2.3
+
1
3.4
+ · · ·+ 1
n(n+ 1)
<
6
7
.
(A) 3; (B) 4; (C) 5; (D) 6; (E) None of the above.
Q3. How many positive integer roots of the inequality
−1 < x− 1
x+ 1
< 2
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 a, b, c ∈ {1, 2, 3, 4, 5, 6} 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
1
a
+
1
b
=
1
c
.
Prove that a+ b is a square.
Q7. Suppose that a = 2b+19, where b = 210n+1. Prove that a is divisible
by 23 for any positive integer n.
Q8. Prove that m7 −m is divisible by 42 for any positive integer m.
Q9. Suppose that 4 real numbers a, b, c, d satisfy the conditions{
a2 + b2 = c2 + d2 = 4
ac+ bd = 2
Find the set of all possible values the number M = ab+ cd can take.
1.4. Hanoi Open Mathematical Olympiad 2009 14
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 x2 + y2 = (2xy + 1)2.
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 ∆ABC with area (∆ABC) = 60cm2. Let R, S
l