Ôn tập môn Toán 12

HANOI MATHEMATICAL SOCIETY
Hanoi Open Mathematical Olympiad 2011
Junior Section
Sunday, February 20, 2011 08h45-11h45
Important:
Answer all 12 questions.
Enter your answers on the answer sheet provided.
For the multiple choice questions, enter only the letters (A, B, C, D or E) corresponding
to the correct answers in the answer sheet.
No calculators are allowed.
Multiple Choice Questions
Question 1. Three lines are drawn in a plane. Which of the following could NOT be
the total number of points of intersections?
(A): 0; (B): 1; (C): 2; (D): 3; (E): They all could.
Question 2. The last digit of the number A = 72011 is
(A) 1; (B) 3; (C) 7; (D) 9; (E) None of the above.
Question 3. What is the largest integer less than or equal to
3
√
(2011)3 + 3× (2011)2 + 4× 2011 + 5?
(A) 2010; (B) 2011; (C) 2012; (D) 2013; (E) None of the above.
Question 4. Among the four statements on real numbers below, how many of them are
correct?
“If a < b < 0 then a < b2”;
“If 0 < a < b then a < b2”;
“If a3 < b3 then a < b”;
“If a2 < b2 then a < b”;
“If |a| < |b| then a < b”.
(A) 0; (B) 1; (C) 2; (D) 3; (E) 4
1
www.VNMATH.com
Short Questions
Question 5. Let M = 7!× 8!× 9!× 10!× 11!× 12!. How many factors of M are perfect
squares?
Question 6. Find all positive integers (m,n) such that
m2 + n2 + 3 = 4(m+ n).
Question 7. Find all pairs (x, y) of real numbers satisfying the system{
x+ y = 3
x4 − y4 = 8x− y
Question 8. Find the minimum value of
S = |x+ 1|+ |x+ 5|+ |x+ 14|+ |x+ 97|+ |x+ 1920|.
Question 9. Solve the equation
1 + x+ x2 + x3 + · · ·+ x2011 = 0.
Question 10. Consider a right-angle triangle ABC with A = 90o, AB = c and AC = b.
Let P ∈ AC and Q ∈ AB such that ∠APQ = ∠ABC and ∠AQP = ∠ACB. Calculate
PQ+ PE +QF, where E and F are the projections of P and Q onto BC, respectively.
Question 11. Given a quadrilateral ABCD with AB = BC = 3cm, CD = 4cm,
DA = 8cm and ∠DAB + ∠ABC = 180o. Calculate the area of the quadrilateral.
Question 12. Suppose that a > 0, b > 0 and a+ b 6 1. Determine the minimum value of
M =
1
ab
+
1
a2 + ab
+
1
ab+ b2
+
1
a2 + b2
.
——————————————————-
2
www.VNMATH.com
HANOI MATHEMATICAL SOCIETY
Hanoi Open Mathematical Olympiad 2011
Senior Section
Sunday, February 20, 2011 08h45-11h45
Important:
Answer all 12 questions.
Enter your answers on the answer sheet provided.
For the multiple choice questions, enter only the letters (A, B, C, D or E) corresponding
to the correct answers in the answer sheet.
No calculators are allowed.
Multiple Choice Questions
Question 1. An integer is called ”octal” if it is divisible by 8 or if at least one of its
digits is 8. How many integers between 1 and 100 are octal?
(A): 22; (B): 24; (C): 27; (D): 30; (E): 33.
Question 2. What is the smallest number
(A) 3; (B) 2
√
2; (C) 2
1+ 1√
2 ; (D) 2
1
2 + 2
2
3 ; (E) 2
5
3 .
Question 3. What is the largest integer less than to
3
√
(2011)3 + 3× (2011)2 + 4× 2011 + 5?
(A) 2010; (B) 2011; (C) 2012; (D) 2013; (E) None of the above.
Short Questions
Question 4. Prove that
1 + x + x2 + x3 + · · ·+ x2011 > 0
for every x > −1.
Question 5. Let a, b, c be positive integers such that a + 2b + 3c = 100. Find the
greatest value of M = abc.
1
www.VNMATH.com
Question 6. Find all pairs (x, y) of real numbers satisfying the system{
x + y = 2
x4 − y4 = 5x− 3y
Question 7. How many positive integers a less than 100 such that 4a2 + 3a + 5 is
divisible by 6.
Question 8. Find the minimum value of
S = |x + 1|+ |x + 5|+ |x + 14|+ |x + 97|+ |x + 1920|.
Question 9. For every pair of positive integers (x; y) we define f(x; y) as follows:
f(x, 1) = x
f(x, y) = 0 if y > x
f(x + 1, y) = y[f(x, y) + f(x, y − 1)]
Evaluate f(5; 5).
Question 10. Two bisectors BD and CE of the triangle ABC intersect at O. Suppose
that BD.CE = 2BO.OC. Denote by H the point in BC such that OH ⊥ BC. Prove
that AB.AC = 2HB.HC.
Question 11. Consider a right-angle triangle ABC with A = 90o, AB = c and
AC = b. Let P ∈ AC and Q ∈ AB such that ∠APQ = ∠ABC and ∠AQP = ∠ACB.
Calculate PQ + PE + QF, where E and F are the projections of P and Q onto BC,
respectively.
Question 12. Suppose that |ax2 + bx + c| > |x2 − 1| for all real numbers x. Prove
that |b2 − 4ac| > 4.
——————————————————-
2
www.VNMATH.com