HANOI MATHEMATICAL SOCIETY
NGUYEN VAN MAU
HANOI OPEN MATHEMATICS COMPETITON PROBLEMS 2006 - 2013
Contents
1
Hanoi Op en Mathematics Comp etition
1.1 1.1 Hano Hanoii 1.1.1 1.1.2 1.2 1.2 Hano Hanoii 1.2.1 1.2.2 1.3 1.3 Ha i
Open Open Math Mathem emat atic icss Co Comp mpet etit itio ion n 2006 2006 Junior Section . . . . . . . . . . . . . Senior Section . . . . . . . . . . . . . Open Open Math Mathem emat atic icss Co Comp mpet etit itio ion n 2007 2007 Junior Section . . . . . . . . . . . . . Senior Section . . . . . . . . . . . . . Ope Math Math atic atic Co titi titi 2008 2008
3
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3 3 4 6 6 8 10
1.7.1 1.7.2 1.8 1.8 Hano Hanoii 1.8.1 1.8.2
Junior Section . . . . . . . . . . . . . Senior Section . . . . . . . . . . . . . Open Open Math Mathem emat atic icss Co Compe mpeti titi tion on 2013 2013 Junior Section . . . . . . . . . . . . . Senior Section . . . . . . . . . . . . .
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23 26 28 28 31
Chapter 1 Hanoi Open Mathematics Competition 1.1 1.1. 1.1.1 1
Hanoi Hanoi Open Mathem Mathemati atics cs Com Competit petition ion 200 2006 6 Juni Junior or Sect Sectio ion n
Question 1. What is the last two digits of the number
Questi Question on 5. Su Supp ppos osee n is a positive integer and 3 arbitrary
numbers are choosen from the set 1, 2, 3, . . . , 3n + 1 with their sum equal to 3 n + 1. What is the largest possible product of those 3 numbers?
{
}
Question 6. The figure ABCDEF is a regular hexagon. Find
all points M belonging to the hexagon such that Area of triangle M AC = = Area of triangle MCD. Question 7. On the circle ( O) of radius 15cm are given 2 points
of the triangle O A, B . The altitude O OH H of OAB AB intersect ( O ) at C . What is AC if AB = 16cm? Question 8. In ∆ABC , P Q
BC where P and Q are points
on AB and AC respectiv respectively ely.. The lines P C and QB intersect at G. It is also also giv given EF//BC , where G E F , E AB and
∈
∈ ∈
Find the value of clog
b
a
+ alog b ? c
Question 4. Which is larger
√ 2
2 ,
21+
1 √
2
and
3.
Question 5. The figure ABCDEF is a regular hexagon. Find
all points M belonging to the hexagon such that Area of triangle M AC = = Area of triangle MCD. Question 6. On the circle of radius 30cm are given 2 points A ,
B with AB = 16cm and C is a midpoint of AB . What What is is the the perpendicular distance from C to the circle?
1.2 1.2. 1.2.1 1
Hanoi Hanoi Open Mathem Mathemati atics cs Com Competit petition ion 200 2007 7 Juni Junior or Sect Sectio ion n
Question 1. What is the last two digits of the number
(3 + 7 + 11 +
2
· · · + 2007) ?
(A) (A) 01; 01; (B) (B) 11; 11; (C) (C) 23; 23; (D) (D) 37; 37; (E) (E) None None of the the abo abov ve. Questi Question on 2. What What is larges largestt positiv positivee integ integer er n satisfying the
following inequality: n2006 < 7 2007 ? (A) 7; (B) 8; (C) 9; (D) 10; (E) 11. Questi Question on 3. Whic Which h of the the foll folloowing wing is a possi possibl blee number umber of diagonals of a convex polygon? (A) 02; (B) 21; (C) 32; (D) 54; (E) 63
Questi Question on 7. Nine Nine points points,, no three three of wh whic ich h lie lie on the same 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.
√
a, b, c be positive integers. Prove that Question 8. Let a,
(b + c a)2 (c + a b)2 (a + b c)2 3 . + + 2 2 2 2 2 2 (b + c) + a (c + a) + b (a + b) + c 5 Questi Question on 9. A tria triang ngle le is said said to be the the Hero Heron n tria triang ngle le 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.
−
−
− ≥
−
Questi Question on 10. Let a,b,c be positive real numbers such that
1 1 1 + + bc ca ab
≥ 1. Prove that bca + cab + abc ≥ 1.
Question 11. How many possible values are there for the sum
1.2. 1.2.2 2
Seni Senior or Sect Sectio ion n
Question 1. What is the last two digits of the number
2
2
2
11 + 15 + 19 +
· · · + 2007
2 2
?
(A) 01; (B) 21; (C) 31; (D) 41; (E) None of the above. Question 2. Which Which is largest largest positive positive integer integer n satisfying the
following inequality: n2007 > (2007)n .
(A) 1; (B) 2; (C) 3; (D) 4; (E) None of the above. Ques Questi tion on 3.
Find Find the the number umber of diffe differe ren nt posit positiv ivee inte intege gerr triples (x,y,z ) satsfying the equations x + y
− z = 1
and x2 + y 2
2
− z = 1.
Question 8.
Let ABC be an equilat equilatera erall trian triangle gle.. For a point point M inside D , E , F be ∆ABC , let D, be the feet of the perpendiculars from M onto BC, BC, CA, CA, AB , respectively. Find the locus of all such points M for which ∠F DE is a right angle. Question 9. Let a1 , a2, . . . , a2007 be real numbers such that
a1 +a2 +
2
2 1
2 2
2 2007
3
· · ·+a ≥ (2007) and a +a +· · ·+a ≤ (2007) −1. Prove that a ∈ [2006 [2006;; 2008] 2008] for all k ∈ {1, 2, . . . , 2007}. 2007
k
Question 10. What is the smallest possible value of
x2 + 2y 2
− x − 2y − xy?
Question Question 11. Find all polynomials P (x) satisfying the equation
(2x
− 1)P (x) = (x − 1)P (2x) ∀x.
Question 15. Let p = abcd be the 4-digit prime number. Prove
that the equation ax3 + bx2 + cx + d = 0
has no rational roots.
1.3 1.3. 1.3.1 1
Hanoi Hanoi Open Mathem Mathemati atics cs Com Competit petition ion 200 2008 8 Juni Junior or Sect Sectio ion n
Question 1. How How many integer integerss from 1 to 2008 2008 have have the sum
of their digits divisible by 5 ? Question 2. How many integers belong to ( a, 2008a), where a
(a > 0) is given. Question 3. Find the coefficient of x in the expansion of
Find P (x2 + 1)? Question 7. Th Thee figur figuree ABCDE is is a conve convex x pen p entago tagon. n. Find
the sum ∠DAC + AC E + + ∠E BD + ∠ACE + ∠B DA + ∠C E B ?
Question 8. The sides of a rhombus have length a and the area
is S . What is the length of the shorter diagonal? Questi Question on 9. Let Let be given given a righ right-angl t-angled ed triang triangle le ABC with
= 900 , AB = c, AC = b. Let E AC and F AB AE F = ∠ABC and ∠AF E = ∠ACB AC B . Deno such that ∠AEF Denote te by by P B C and Q BC such that E P B C and F Q BC . E F + P Q? Determine E P + EF ∠A
∈
∈
∈
⊥
a, b, c Question 10. Let a,
ditions
∈
⊥
∈ [1, 3] and satisfy the following con-
has no solutions of positive integers x, y and z . Question 4. Prove that there exists an infinite number of rela-
tively prime pairs ( m, n) of positive integers such that the equation x3 nx + mn = 0
−
has three distint integer roots. Question 5. Find all polynomials P (x) of degree 1 such that
− min P (x) = b − a, ∀a, b ∈ R where a < b. ≤≤ a, b, c ∈ [1, 3] and satisfy the following condiQuestion 6. Let a, max P (x)
≤≤
a x b
a x b
tions
max a,b,c 2, a + b + c = 5.
{
}
What is the smallest possible value of
parallelogram. parallelogram. Let O be the intersection of BN and C P . Find M B BC C such that ∠P M O = ∠OM N .
∈ ∈
Questi Question on 10. Let be given given a right-an right-angled gled triangl trianglee ABC with
AC and F AB = 900 , AB = c, AC = b. Let E AE F = ∠ABC and ∠AF E = ∠ACB AC B . Deno such that ∠AEF Denote te by by P B C and Q BC such that E P B C and F Q BC . E F + F Q? Determine E P + EF ∠A
∈
1.4 1.4. 1.4.1 1
∈
∈
∈
⊥
⊥
Hanoi Hanoi Open Mathem Mathemati atics cs Com Competit petition ion 200 2009 9 Juni Junior or Sect Sectio ion n
Ques Questi tion on 1. Let a,b,c be 3 distinct numbers numbers from 1, 2, 3, 4, 5, 6 .
Show that 7 divides abc + (7
− a)(7 − b)(7 − c).
{
}
Question 6. Sup Suppose pose that that 4 real numbers numbers a,b,c,d satisfy the
conditions
a2 + b2 = 4 c2 + d2 = 4 ac + bd = 2
Find the set of all possible values the number M = ab + cd can take. a, b, c, d be positive integers such that a + b + Question 7. Let a, c + d = 99. Find the smallest and the greatest values of the following product P = abcd. Question 8. 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 1004 .
1.4. 1.4.2 2
Seni Senior or Sect Sectio ion n
Ques Questi tion on 1. Let a,b,c be 3 distinct numbers numbers from 1, 2, 3, 4, 5, 6 .
Show that 7 divides abc + (7
− a)(7 − b)(7 − c).
{
}
Question 2. Show that there is a natural number n such that
the number a = n ! ends exacly in 2009 zeros. a, b, c be positive integers with no common Questi Question on 3. Let a,
factor and satisfy the conditions 1 1 1 + = . a b c Prove that a + b is a square. 10n n+1 . Prove that Question 4. Suppose that a = 2b , where b = 210
a is divisible by 23 for any positive integer n.
Question 8. 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 1004 . Question 9.Given an acute-angled triangle ABC with area S ,
let points A , B , C be located as follows: A is the point where altitud altitudee from A on B C meets the outwa outwards rds facing facing semicirle semicirle drawn on B C as diameter diameter.. Po Poin ints ts B , C are located similarly. Evaluate the sum = (area ∆BC A)2 + (area ∆C AB )2 + (area ∆ABC )2 . T = Question 10. Prove that d 2 + (a
of the inscribed circle of ∆ ABC.
2
− b)
< c2 , where d is diameter
(A): (A): 0; (B): (B): 1; (C): (C): 2; (D): (D): 3; (E) (E) None None of the the abo abov ve. Question 3. 5 last digits of the number M = 52010 are
(A): (A): 65625 65625;; (B): (B): 45625 45625;; (C): (C): 25625 25625;; (D): (D): 15625 15625;; (E) None None of the the above. Question 4. How How many real num numbers bers a
corresponding number a
∈ (1, 9) such that the
− a1 is an integer.
(A): (A): 0; (B): (B): 1; (C): (C): 8; (D): (D): 9; (E) (E) None None of the the abo abov ve. Question 5. Each box in a 2
× 2 table can be colored black or
white. How many different colorings of the table are there? (A): (A): 4; (B): (B): 8; (C): (C): 16; 16; (D): (D): 32; 32; (E) (E) None None of the the abo abov ve.
√ Question 6. The greatest integer less than (2 + 3)
5
are
Find the maximum maximum value value of x y z M = , x, y, z > 0 . + + 2x + y 2y + z 2z + + x
Question 10.
1.5. 1.5.2 2
Seni Senior or Sect Sectio ion n
Question 1. The number of integers n
22n + 2n + 5 is divisible by 7 is
∈ [2000, 2010] such that
(A): (A): 0; (B): (B): 1; (C): (C): 2; (D): (D): 3; (E) (E) None None of the the abo abov ve. Question 2. 5 last digits of the number 5 2010 are
(A): (A): 65625 65625;; (B): (B): 45625 45625;; (C): (C): 25625 25625;; (D): (D): 15625 15625;; (E) None None of the the above. Question 3. How How many real num numbers bers a
corresponding number a
∈ (1, 9) such that the
− a1 is an integer.
a, b, c, d. is an integer. Determine a, Question 7. Let P be the common point of 3 internal bisectors
of a given ABC. The line passing through P and perpendicular to C P intersects AC and BC at M and N , respecti respectiv vely. ely. If AM ? AP = 3cm, BP = 4cm, compute the value of B N n and n 3 + 2n2 + 2n +4 are both perfect squares, Question 8. If n
find n. Let x, y be the positive integers such that 3 x2 + x = 4y 2 + y. Prove that x y is a perfect integer.
Question 9.
−
Question 10.
M =
Find the maximum maximum value value of x y z , x, y, z > 0 . + + 2x + y 2y + z 2z + + x
3
(2011)3 + 3
× (2011)
2
+4
× 2011 + 5?
(A) (A) 2010 2010;; (B) (B) 2011 2011;; (C) (C) 2012 2012;; (D) (D) 2013 2013;; (E) (E) None None of the the above. Question 4. Among the four statements on real numbers be-
low, how many of them are correct? a < b < 0 then a < b2”; “If a “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 Question 5. Let M = 7!
× 8! × 9! × 10! × 11! × 12!. How many
factors of M are perfect squares?
Question 9. Solve the equation
1 + x + x2 + x3 +
· · · + x
2011
= 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 ∠AP Q = ∠ABC and ∠AQP = ∠ACB. Calculate P Q + P E + + QF, where E and F are the projections of P and Q onto BC , respectively.
∈
∈
Question 11. Given a quadrilateral ABCD with AB = BC B C =
3cm, C D = 4cm, DA = 8cm and ∠DAB + ∠ABC = 180o . Calculate the area of the quadrilateral. Questi Question on 12. Suppo Suppose se that a > 0, b > 0 and a + b 1.
Determine the minimum value of 1
1
1
1
(A) (A) 2010 2010;; (B) (B) 2011 2011;; (C) (C) 2012 2012;; (D) (D) 2013 2013;; (E) (E) None None of the the above. Question 4. Prove that
1 + x + x2 + x3 + for every x
· · · + x
2011
0
−1.
a, b, c be positive integers such that a + 2 b + Question 5. Let a,
3c = 100. Find the greatest value of M = abc. Question 6. Find all pairs ( x, y ) of real numbers satisfying the
system
x + y = 2 x4 y 4 = 5x
−
− 3y
Question 7. How many positive integers a less than 100 such
Question 10. Two bisectors BD and C E of of the triangle ABC
intersect intersect at O . Suppose that BD.CE = 2BO.OC. Denote by H the point in BC such that OH B C . Prove Prove that 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 ∠AP Q = ∠ABC and ∠AQP = ∠ACB. Calculate P Q + P E + + QF, where E and F are the projections of P and Q onto BC , respectively.
∈
∈
Question 12. Suppose that ax2 + bx + c x2
numbers x. Prove that b2
|
| − 4ac| 4.
1.7 1.7. 1.7.1 1
| | − 1| for all real
Hanoi Hanoi Open Mathem Mathemati atics cs Com Competit petition ion 201 2012 2 Juni Junior or Sect Sectio ion n
Questi Question on 4. A man trav travels from from town town A to town E through
towns B, C and D with uniform speeds 3km/h, 2km/h, 6km/h and 3km/h on the horizontal, up slope, down slope and horizontal road, respectively. If the road between town A and town can be classified as horizontal, up slope, down slope and horE can izontal and total length of each type of road is the same, what is the average speed of his journey? (A) (A) 2km/ 2km/h; h; (B) (B) 2,5k 2,5km/ m/h; h; (C) (C) 3km/ 3km/h; h; (D) (D) 3,5k 3,5km/ m/h; h; (E) (E) 4km/h.
Question 5. How How many many different different 4-digit even even integers integers can be
form from the elements of the set 1, 2, 3, 4, 5 .
{
}
(A): (A): 4; (B): (B): 5; (C): (C): 8; (D): (D): 9; (E) (E) None None of the the abo abov ve. Question 6. At 3:00 A.M. A.M. the temperature temperature was 13 o below zero.
By noon it had risen to 32 o. What is the average hourly increase in teparature? Questi Question on 7. Find Find all inte integer gerss n such that 60 + 2 n
perfect square.
− n
AB C and 2 points K Questi Question on 8. Giv Given a trian triangl glee ABC
2
is a
∈ AB,
∈ BC such that BK = 2AK , C N = 2B N and Q is the S · common point of AN and C K . Compute S N
∆ABC
∆BC Q
Question 9. Evaluate the integer part of the number
20112
2011
Questi Question on 14. Let Let be given given a tria triang ngle le ABC with ∠A = 900
and the bisectrices of angles B and C meet at I . Suppose that I H is is perpendicular to B C (H belongs to BC ). If H B = 5cm, = 8cm, compute the area of ABC . H C =
Questi Question on 15. Determ Determin inee the great greatest est valu valuee of the sum M =
xy + yz y z + z x, where x, x, y, z are + zx, are real numbers satisfying the following condition x2 + 2y 2 + 5z 2 = 22. 1.7. 1.7.2 2
Seni Senior or Sect Sectio ion n
Questi Question on 1. Let x =
H = = (1 + x5
7 20123
−x )
11
√ √ 6+2 5+ 6−2 5 √ 20 · The value of
is
(A): A): 1; (B): B): 11; (C): C): 21; (D) (D): 101; (E) (E) Non Nonee of the abo bov ve. Question 2. Compare the numbers:
(A): 1; (B): 2; (C): 3; (D): 4; (E) No None of the abo bov ve.
x + 2010 Ques Questi tion on 5. Let f (x) be a function such that that f (x)+2f = x 1 4020 x for all x = 1. Then the value of f (2012) is
−
−
(A): A): 2010; (B): 2011 011; (C): C): 2012; 12; (D): 2014 014; (E) Non onee of the above. Question 6. For every n = 2, 3, . . . , we put
− × −
1 An = 1 1+2
1 1 1+2+3
×···× −
Determine all positive integer n (n integer.
≥
···
Questi Question on 7. Pro Prove that that the num number a = 1 . . . 1 5 . . . 5 6 is a
perfect square.
1 . 1 1+2+3+ + n 1 2) such that is an An
2012
2011
Question 11. Suppose that the equation x 3 + px2 + qx + r = 0
p, q , r are integer numbers. Put has 3 real roots x 1, x2 , x3, where p, S n = x n1 + xn2 + xn3 , n = 1, 2, . . . Prove that S 2012 2012 is an integer. AB C with In an an isosceles isosceles triangle ABC with the base AB BC. C. Let O be the center of its circumscribed given a point M B circle and S be be the center of the inscribed circle in ∆ ABC and S M AC. AC . Prove that OM B BS. S. Question 12.
∈ ∈
⊥ ⊥
A cube with sides sides of length length 3cm is paint painted ed red and then cut into 3 3 3 = 27 cubes with sides of length 1cm. a denotes the number of small cubes (of 1cm 1cm 1cm) that If a are not painted at all, b the number painted on one sides, c the number painted on two sides, and d the number painted on three sides, determine the value a b c + d? Question 13.
× ×
×
×
− −
Question 14. Solve, in integers, the equation 16 + 1 = (
2
Questi Question on 2. How How many many natur natural al numbers numbers n are there so that
n2 + 2014 is a perfect square.
(A): 1; (B): 2; (C): 3; (D): 4; (E) No None of the abo bov ve.
Question 3. The largest integer not exceeding [( n + 1)α]
√ 2013 where n is a natural number, α = √ , is: 2014
− [nα],
(A): 1; (B): 2; (C): 3; (D): 4; (E) No None of the abo bov ve.
Question 4. Let A be an even number but not divisible by 10.
The last two digits of A20 are: (A): 46; (B): 56; (C): 66; above.
(D): 76; (E): None of the
AB C be Question 7. Let ABC be a triangle with A = 900, B = 600 and BC = 1cm. Draw Draw outside outside of ∆ ABC three equilateral triangles ABD,ACE and BCF. Determine the area of ∆ DEF. Question 8. Let ABCDE be be a convex pentagon. Given that
area of ∆ABC = area of ∆BC D = area of ∆ C DE = area of ∆ DEA DE A = area of ∆E AB = 2cm2, Find the area of the pentagon. Question 9. Solve the following system in positive numbers
Questi Question on 10
x + y 1 2 1 + 2 = 10. xy x + y 2
≤
Consid Con sider er the set of all rectan rectangle gless with with a given given
prove that the equation f (x) = 2x2
− 1 has two real roots.
Question 13. Solve the system of equations
1 1 1 + = x y 6 3 2 5 + = x y 6
Question 14. Solve the system of equations
x3 + y = x 2 + 1 2y 3 + z = 2y 2 + 1 3z 3 + x = 3z 2 + 1
Question 15. Denot Denotee by Q and N∗ the set of all rational and
ax + b Q positive integer integer numbers, numbers, respectively respectively.. Suppose that x A, B , C such for every x N∗ . Prove that there exist integers A,
∈
∈
1 1 1 (A): ; (B): ; (C): ; 2012 2013 2014 of the above.
(D):
1 ; (E): None 2015
Question 3. What What is the the larges largestt inte integer ger not exceedi exceeding ng 8 x3 +
6x
−
1 1, where x = 2
√ − √ 3
2+
5+
3
2
5 ?
(A): 1; (B): 2; (C): 3; (D): 4; (E) No None of the abo bov ve. Questi Question on 4. Let x0 = [α], x1 = [2α]
x4 = [5α]
− [4α], x
value of x9 is
5
− [α], x = √ [3α] − [2α], 2013 = [6α] − [5α], ..., where α = √ . The 2014
(A): 2; 2; (B): 3; 3; (C): 4; 4;
2
(D): 5; 5; (E): Non None of the abo bov ve.
Question 5. The num number ber n is called a composite number if it
can be written in the form n = a
× b, where a, b are positive
AC D such that AM N is an equilatera ∆ACD equilaterall triangle. triangle. Determin Determinee BMC. Question 8. Let ABCDE be be a convex pentagon and
area of ∆ABC = area of ∆BC D = area of ∆ C DE DE A = area of ∆EAB. = area of ∆ DEA
Given that area of ∆ ABCDE = 2. Evaluate the area of area of ∆ ABC.
Question 9. A given polynomial P (t) = t 3 + at2 + bt + c has 3
distinct real roots. If the equation ( x2 + x + 2013 2013))3 + a(x2 + x + 2013)2 + b(x2 + x + 2013) + c = 0 has no real roots, prove that 1 P (2013) > . 64 Questi Question on 10
Consid Con sider er the set of all rectan rectangle gless with with a given given
√ Evaluate the value of f ( 2013) . Question 13. Solve the system of equations
xy = 1 x y + =1 x4 + y 2 x2 + y 4
Question 14. Solve the system of equations:
1 4 x3 + y = x 2 + x 3 3 1 5 y3 + z = y 2 + y 4 4 1 6 z 3 + x = z 2 + z 5 5
− − −
Question 15. Denot Denotee by Q and N∗ the set of all rational and
b
Hanoi Open Mathematical Competition 2016 Senior Section Saturday, 12 March 2016
08h30-11h30
How many many are there 10-digit 10-digit numbers numbers composed from the digits digits 1, 2, 3 only and in which, two neighbouring digits differ by 1.
Question 1.
(A): (A): 48 48 (B): (B): 64 64 (C): (C): 72 72 (D) (D): 128 128 (E): E): None None of the abo above. Given Given an arra array y of num numbers bers A = (672, 673, 674, . . . , 2016) on table. ble. Three Three arbitra arbitrary ry numbers numbers a,b,c A are step by step replaced by number 1 min(a,b,c). After 672 times, on the table there is only one number m, such that 3
Question Question 2.
∈
(A): 0 < m < 1 (B): m = 1 above.
(C): 1 < m < 2 (D): m = 2 (E): E): None one of of the the
two positive numbers a, b such that the condition a3 + b 3 = Question 3. Given two a5 + b5 , then the greatest value of M = a 2 + b2 ab is
−
(A):
1 1 (B): 4 2
Question Question 4.
(C): (C): 2 (D): (D): 1 (E): (E): None None of the the abo above. In Zoo, a monkey monkey becomes becomes lucky lucky if he eats eats three differe different nt fruits fruits
a, b, c satisfy the conditions Question 9. Let rational numbers a, a + b + c = a 2 + b2 + c2
∈ Z.
m2 Prove that there exist two relative prime numbers m, n such that abc = 3 . n Question 10. Given natural numbers a, b such that 2015 a2 + a = 2016b2 + b. Prove that a b is a natural number.
√ −
Question 11. Let I be the incenter of triangle AB C and ω be its circumcircle. Let the line AI intersect ω at point D = A. Let F and E be points on side BC and
1 arc B DC respectively such that ∠BAF = ∠CAE < ∠BAC . Let X be the second 2 point of intersection of line E I with with ω and T be the point of intersection of segment DX with line AF . Prove that TF.AD = ID.AT .
Question Question 12. Let A be point inside the acute angle xOy. An arbitrary circle ω passes through O, A; intersecting Ox and Oy at the second intersection B and C, respectively. Let M be the midpoint of BC. Prove that M is always on a fixed line (when ω changes, but always goes through O and A). Question 13.
Find all triples triples (a,b,c) of real numbers such that 2a + b
|
2
|ax
+ bx + c
| ≤ 1 ∀x ∈ [−1 1]
| ≥ 4 and
Digit
1 1 2 3
2 12 21 23 32
3 12 1 12 3 212 23 2 32 1 32 3
4
6
3
4 1 212 1 232 2 121 2 12 3 2 321 2 323 3212 3232 8
.
We can see that a number ending by 2 in previous column generates 2 numbers for next column (we can add 1 or 3 at the end), but a number ending by 1 or 3 generate 1 number for next column (we can add only 2 at the end). From this, we can make a table. The first row is number of digits, the second row is the number of k-digit numbers satisfying the condition and ending with 1, 3, the third row is the number of k-digit numbers satisfying the condition and ending with 2. 1 2 1
2 2 2
Question 2. (A).
3 4 2
4 4 4
5 8 4
6 8 8
7 16 8
8 16 16
9 32 16
10 32 . 32
We have ab(a2
2 2
3 3
5
− b ) ≥ 0 ⇔ 2a b ≤ ab
+ a5 b
⇔ (a
3
+ b3 ) 2
5
≤ (a + b)(a
+ b5 ).
(1)
Combining a3 + b3 = a 5 + b5 and (1), we find a3 + b3
2
≤ a+b ⇔ a
+ b2
− ab ≤ 1.
The equality holds if a = 1, b = 1. Question 4. (D).
First we leave tangerines on a side. We have 20 + 30 + 40 = 90 fruites. As we feed the happy monkey is not more than one tangerine, each monkey eats fruits of these 90 at least 2. Hence, Hence, the monkeys monkeys are not more than 90/2 = 45. We will show how you can bring happiness to 45 monkeys: 5 monkeys eat: orange, banana, tangerine; 15 monkeys eat: orange, peach, tangerine; 25 monkeys eat peach, banana, tangerine. At all 45 lucky monkeys - and left five unused tangerines! Question 5. (E).
We have 3x2 + x = 4y 2 + y (x y )(3x + 3y + 1) = y 2 . We prove that ( x y ; 3x + 3y + 1) = 1.
−
⇔ −
Type 2: Three vertices lie in distinct horizontal lines. We have 3 3 3 triangles of these type. But we should remove remove degenerated degenerated triangles triangles from them. There There are 5 of those (3 vertical vertical lines and two two diagonals). diagonals). So, we have 27 - 5 = 22 triangles of this type. Total, we have 54 + 22 = 76 triangles. For those students who know about C nk this problem problem can be b e also solved as C 93 8 where 8 is the number of degenerated triangles.
× ×
−
a,b, c Let abc, where a,b,
Question Question 8. (a + b + c)3 .
∈ {1, 2, 3, 4, 5, 6, 7, 8, 9}, a = 0 and abc = ≤ 999 and √ 100 ≤ a + b + c ≤ √ 999. Hence
Note that 100 (a + b + c)3 5 a + b + c 9. 3 a + b + c = 5 then abc = (a + b + c) =53 = 125 and a + b + c = 8 (not suitable). If a 3 If a a + b + c = 6 then abc = (a + b + c) =63 = 216 and a + b + c = 9 (not suitable). If a + b + c = 7 then abc = (a + b + c)3 =73 = 343 and a + b + c = 10 (not suitable). If a + b + c = 8 then abc = (a + b + c)3 =83 = 512 and a + b + c = 8 (suitable). If a + b + c = 9 then abc = (a + b + c)3 =93 = 729 and a + b + c = 18 (not suitable). Conclusion: abc = 512.
≤
≤
3
≤
Put a + b + c = a 2 + b2 + c2 = t. We have have 3 (a2 + b2 + c2 ) (a + b + c)2 , then t Since t Z then t 0;1;2;3 . 0 If = 0 then = 0 and abc 0 b
3
Question 9.
∈
∈{
≥
}
∈ [0; [0; 3] .
y = rq r q (q p) pq ( p q )] [ pq )]2 r p ( p q ) Hence x = rp abc = . 3 2 2 ( + ) p q pq z = r pq = rpq We prove that ( pq ( p q ); ); p2 + q 2 pq ) = 1. Suppose that s = ( pq ( p q ) ; p2 + q 2 pq ) ; s > 1 then s pq pq ( p q ) . 2 2 p. Since s ( p + q pq ) then s q and s = 1 (not suitable). Case 1. Let s p. Case 2. Let s q. Similarly, we find s = 1 (not suitable). s p p Case 3. If s ( p q ) then s ( p q )2 ( p2 + q 2 pq ) (not suitable). s pq pq s q If t = 2 then a + b + c = a 2 + b2 + c2 = 2. We reduce it to the case where t = 1, which was to be proved.
| |
− − ⇒ − − |
| −
− − − − − − |
| − −
−
|
⇒ |
−
⇒ |
|
Question 10. From equality
2015a2 + a = 2016b2 + b, we find a b. If a = b then from (1) we have a = b = 0 and a > b, we write (1) as If a
≥
b2 = 2015(a2
2
− b ) + (a − b) ⇔ b
2
= (a
(1)
√ a − b = 0. − b)(2015a + 2015b + 1).
(2)
Let (a, b) = d then a = md ; b = nd, where (m, n) = 1. Since a > b then m > n; and put m n = t > 0 . Let (t, n) = u then n is divisible by u; t is divisible by u and m is divisible by u. That follows = 1 and then ( ) 1
−
IL
CL
Since C I is bisector of ∠ACL = . Furthermore urthermore,, ∠DC L = AC L, we get AI AC 1 ∠DC B = ∠DAB = ∠C AD = ∠BAC. Hence, the triangles DC L and DC A are 2 CL DC similar. Therefore, = . AC AD LC D = ∠I C D. It Finally, Finally, we have ∠DI C = ∠I AC + ∠I C A = ∠I C L + ∠LCD DC ID = . AD AD T F IL CL DC ID Summarizing all these equalities, we get = = = = AT AI AC AD AD T F ID = TF.AD = ID.AT as desired. AT AD
follows DI C is a isosceles triangle at D. Hence
⇒
⇒
Since (Ox ) is tangent to Ox, ∠ADC = ∠AOB. Since OBAC is is cyclic, ∠ACD. So
Similarly,
ADC are triangles AOB, ADC are similar. Therefore
AB ABE ACO, so BE = CO AC
AB AC
=
∠ABO =
OB DC
(1) (2)
From (1) and (2), we deduce that
OB BE = CD OC
OB CD ⇒ BE = OC
Hence OE OD = BE OC
OP ON ON OP OP ⇒ ON ⇒ = = = = BE OC NB BE − N O OC − OP C P
It follows, if N P intersects BC at M, then MB
MB M C
· PP OC · NN BO = 1 (by Menelaus’
O BC ) conclusion Theorem in triangle OBC = 1, it follows N P passes through M is M C midpoint midpoint of BC.
Question 13. From the assumptions, we have f ( 1)
| ± | ≤ 1, |f (0)| ≤ 1 and
f (1) = a + b + c f ( 1) = a
−
−b+c
1 [f (1) + f ( 1)] 2 1 b = [f (1) f ( 1)] 2
⇔
a =
− − f (0) − −
Indeed, we have = f 2 (x) + 2f (x).x + x2 + pf (x) + px + q f [f (x) + x] = [f (x) + x]2 + p[f (x) + x] + q = = f (x)[f (x) + 2 x + p] + x2 + px + q = = f (x)[f (x) + 2 x + p] + f (x) = f (x)[f (x) + 2x + p + 1] = f (x)[x2 + px + q + + 2 x + p + 1] = f (x)[(x + 1)2 + p(x + 1) + q ] = f (x)f (x + 1), which proves (1). Putting m := f (2015) + 2015 gives f (m) = f [f (2015) + 2015] = f (2015)f (2015 + 1) = f (2015)f (2016),
as desired. We have
Question 15.
T 2(18ab+9ca+29bc) = (5a 3b)2 +(4a 3c)2 +(4b 5c)2 +(a 3b+3c)2
−
−
−
−
−
That follows T
≥ 2. The equality occures if and only if ≥ 5a − 3b = 0 5a − 3b = 0 4a − 3c = 0 ⇔ 4a − 3c = 0 4b − 5c = 0
4b
5
0
≥ 0, ∀a,b,c ∈ R.
