2 13b + 37 = 54, 13b 37 = 16, no solution

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1 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Individual Events SI P 6 I P 5 I P 6 I P I P I5 P Q 7 Q 8 Q 8 Q Q Q R R 7 R R 996 R R S 990 S 6 S S 666 S S 0 Group Events SG a G a G a 905 G a 9900 G a 50 G5 a b 5 b 5 b 9 b 5 b b c 80 c c 6 c c 60 c 6 d d d 0 d 8 d 8 d Sample Individual Event (999 Individual Event ) SI. For all integers m and n, m n is defined as m n = m n + n m. If P = 00, find the value of P. P + P = = = 00, P = 6 SI. If Q 6P Q 6P, where Q > 0, find the value of Q. Q 7 Q 7 Q + 7 Q 7 Q Q Q 7 (Q 7) = = Q 7 Q 7 Q 7 = Q 7 ; ( 7 Q 7 Q 7 Q ) 8 = [(Q) 69] = 69 Q Q = 9 Q = 7 Method b 7 b 7, We look for the difference of multiples of 8 b + 7 = 6, b 7 =, no solution b + 7 = 5, b 7 = 6, no solution 7 b + 7 = 8, b 7 = 5 b = 7 SI. In figure, AB = AC and KL = LM. If LC = Q 6 cm and KB = R cm, find the value of R. Draw LN // AB on BM. BN = NM intercept theorem LNC = ABC = LCN (corr. s, AB // LN, base s, isos. ) LN = LC = Q 6 cm = cm (sides opp. eq. s) R cm = KB = LN = cm (mid point theorem) SI. The sequence {a n } is defined as a = R, a n+ = a n + n (n ). If a 00 = S, find the value of S. a =, a = +, a = + +, a 00 = = = 990= S Page B K A N L C M

2 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Individual Event I. Let [x] represents the integral part of the decimal number x. Given that [.6] + [.6+ 8 ]+[.6+ 8 ]++[ ] = P, find the value of P. P = [.6] + [.6+ 8 ]+[.6+ 8 ]++[ ] = = 5 a b c I. Let a + b + c = 0. Given that = P Q, find the value of Q. a bc b ac c ab a b c a = b c + + Method a bc b ac c ab a b c b c b c + + =5 Q = + + a bc b ac c ab b 5bc c b bc c c bc b The above is an identity which holds for all a b c values of a, b and c, provided that a+b+ c = 0 = + + b cb c b cb c b cc b Let a = 0, b =, c =, then = b c b c b b c c b c = 5 Q. b cb cb c = b c b c b c bc b c Q = 8 b cb cb c = b c b bc c b bc c bc b c b cb c b 5bc c = = = 5 Q Q = 8 b cb c I. In the first quadrant of the rectangular co-ordinate plane, all integral points are numbered as follows, point (0, 0) is numbered as, point (, 0) is numbered as, point (, ) is numbered as, point (0, ) is numbered as, point (0, ) is numbered as 5, point (, ) is numbered as 6, point (, ) is numbered as 7, point (, ) is numbered as 8, Given that point (Q,Q) is numbered as R, find the value of R. point (0, ) is numbered as = point (, 0) is numbered as 9 = point (0, ) is numbered as 6 = point (, 0) is numbered as 5 = 5 point (0, 7) is numbered as 6 = 8 point (0, 8) is numbered as 65, point (, 8) is numbered as 66, point (, 8) is numbered as 67 (Q, Q) = (7, 8) is numbered as 7 I. When x + y =, the minimum value of x + y R is, find the value of S. S x + y = x + ( x) = x 8x + 6 = (x ) 7 +, min = = ; S = 6 S Page

3 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Individual Event I. If log (log P) = log (log P) and P, find the value of P. loglog P loglog P log log log loglog log P P log log log(log P) = log(log P) log(log P) = log(log P) (log P) = log P log P log log P log log P P, log P 0 log log log P = log = log 6 P = 6 I. In the trapezium ABCD, AB // DC. AC and BD intersect at O. The areas of triangles AOB and COD are P and 5 respectively. Given that the area of the trapezium is Q, find the value of Q. Reference 99 HI, 997 HG, 00 FI., 00 HG7, 00HG, 0 HG AOB ~ COD (equiangular) OA OA ; area of AOB 6 area of COD OC 5 OC OA : OC = : 5 area of AOB (the two triangles have the same height, but different bases.) area of BOC 5 5 Area of BOC = 6 = 0 D A O B C Similarly, area of AOD = 0 Q = the area of the trapezium = = 8 I. When 999 Q is divided by 7, the remainder is R. Find the value of R = (785+) 8 = 7m + 8 = 7m + ( ) 7 = 7m + (79+) 7 = 7m + 7n +, where m and n are integers R = I. If = (R + S) and S > 0, find the value of S. Reference: 995 FG7. = ( + S) (00000 ) = ( + S) = ( + S) = ( + S) + S = S = Page

4 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Individual Event I. Given that the unit digit of is P, find the value of P = ( ) = ( + 998) P = unit digit = I. Given that x + = P. If x 6 + x x 6 = Q, find the value of Q. x + = x x = x x + x = x x 6 x 6 x x x 6 x x 6 Q = Q Q Q R I. Given that Q Q Q Q 998Q 999Q Q 999Q value of R. R R R R = = 996 I. Let f (0) = 0; f (n) = f (n ) + when n =,,,,. If f (S) = R, find the value of S. f () = 0 + =, f () = + =, f () =,, f (n) = n R = 996 = f (S) = S S = 666, find the Page

5 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Individual Event I. Suppose a b, where a, b, and a b + 0. a b Given that ab a + b = P, find the value of P. a b + + = 0 a b ] = 0 a b (a b + )[ ab + b a = 0 P = I. In the following figure, AB is a diameter of the circle. C and D divide the arc AB into three equal parts. The shaded area is P. If the area of the circle is Q, find the value of Q. Let O be the centre. Area of ACD = area of OCD (same base, same height) and COD = 60 Shaded area = area of sector COD = area of the circle = 6 = I. Given that there are R odd numbers in the digits of the product of the two Q-digit numbers and , find the value of R. Reference: 05 FI. Note that 99 = 089; 999 = Deductively, = R = odd numbers in the digits. I. Let a, a,, a R be positive integers such that a < a < a < < a R- < a R. Given that the sum of these R integers is 90 and the maximum value of a is S, find the value of S. a + a + + a = 90 a + (a + ) + (a + ) + + (a + ) 90 a a.967 S = maximum value of a = Page 5

6 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Individual Event I5. If = P, find the value of P. 999 Reference: 05 FG P = = = 8 = I5. If (x P)(x Q) = 0 has two integral roots, find the value of Q. Reference: 00 FI., 00 FI., 0 FI., 0 HG (x )(x Q) = 0 x ( + Q)x + Q = 0 Two integral roots is perfect square = [( + Q) (Q )] = (Q Q + ) = (Q ) + It is a perfect square Q = 0, Q = Method (x )(x Q) = (x = and x Q = ) or (x = and x Q = ) (x = and Q = ) or (x = and Q = ) Q = I5. Given that the area of the ABC is Q; D, E and F are the points on AB, BC and CA respectively such that AD = AB, BE = BC, CF = CA. If the area of DEF is R, find the value of R. Reference: 99 FG9. R = area ADF area BDE area CEF AD AF sin A BE BDsin B CE CF c b sin A c a sin B a b sin C = = bc sin A ac sin B ab sin C 9 = area of ABC 9 sinc = 9 = 9 I5. Given that (Rx x + ) 999 a 0 + a x + a x + + a 998 x 998. If S = a 0 + a + a + + a 997, find the value of S. (x x + ) 999 a 0 + a x + a x + + a 998 x 998 compare coefficients of x 998 on both sides, a 998 = Put x =, 999 = a 0 + a + a + + a 998 S = a 0 + a + a + + a 997 = (a 0 + a + a + + a 998 ) a 998 = = 0 Page 6

7 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Sample Group Event (999 Final Group Event ) SG. Let x * y = x + y xy, where x, y are real numbers. If a = * (0 * ), find the value of a. 0 * = = a = * (0 * ) = * = + = SG. In figure, AB is parallel to DC, ACB is a right angle, AC = CB and AB = BD. If CBD = b, find the value of b. ABC is a right angled isosceles triangle. D C BAC = 5 (s sum of, base s isos. ) b ACD = 5 (alt. s, AB // DC) BCD = 5 A Apply sine law on BCD, BD BC sin5 sin D AB sin 5 AB sin D sin D = ; D = 0, given that AB = BD B CBD = = 5 (s sum of BCD) b = 5 SG. Let x, y be non-zero real numbers. If x is 50% of y and y is c% of x, find the value of c. x =.5y...() c y = x...() 00 c sub. () into (): y =.5y 00 c = 80 SG. If log p x =, log q x =, log r x = 6 and log pqr x = d, find the value of d. log x log x log x ; ; 6 log p log q log r log p log q log r ; ; log x log x log x 6 log p log q log r log x log x log x 6 log pqr log x log x log pqr d = log pqr x = Page 7

8 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Group Event G. Given that when 889, 069 and 7 are divided by an integer n, the remainders are equal. If a is the maximum value of n, find a. Reference: 06 FI. 889 = pn + k () 069 = qn + k () 7 = rn + k () () (): 5 = (q p)n () () (): 798 = (r q)n (5) (): 0 = (q p)n (5): 7 = (r q)n a = maximum value of n = G. Let x and y. If b = x xy + y, find the value of b. b = x xy + y = x xy + y + xy = (x y) + xy = = = = 5 G. Given that c is a positive number. If there is only one straight line which passes through point A(, c) and meets the curve C: x + y x y 7 = 0 at only one point, find the value of c. The curve is a circle. There is only one straight line which passes through point A and meets the curve at only one point the straight line is a tangent and the point A(, c) lies on the circle. (otherwise two tangents can be drawn if A lies outside the circle) Put x =, y = c into the circle. + c c 7 = 0 c c 8 = 0 (c )(c + ) = 0 c = or c = (rejected) G. In Figure, PA touches the circle with centre O at A. If PA = 6, BC = 9, PB = d, find the value of d. It is easy to show that PAB ~ PCA PA PC (ratio of sides, ~'s) PB PA 6 9 d d 6 6 = 9d + d d + 9d 6 = 0 (d )(d + ) = 0 d = or (rejected) Page 8

9 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Group Event G. If 9 is the difference of two consecutive perfect squares, find the value of the smallest square number, a. Let a = t, the larger perfect square is (t +) (t + ) t = 9 t + = 9 t = 95 a = 95 = 905 G. In Figure (a), ABCD is a rectangle. DE:EC = : 5, and DE. BCE is folded along the side BE. If b is the area of the shaded part as shown in Figure (b), find the value of b. A D C B 0 60 C E Let DE = t, then CE = 5t. Suppose BC intersects AD at F, C E intersects AD at G. BC = BC' = AD = 5t tan 60 = 5 t C ED = 60, ABC = 0, C FG = 60, C GF = 0 AF = 6t tan 0 = t, DG = t tan 60 = t FG = 5 t t t = t C F = t cos 60 = t, C G = t cos 0 = t Area of C FG = t t = t = 9 G. Let the curve y = x 7x + intersect the x-axis at points A and B, and intersect the y-axis at C. If c is the area of ABC, find the value of c. x 7x + = (x )(x ) The x-intercepts of,. Let x = 0, y = c = = 6 sq. units G. Let f (x) = x x + and g(x) = x + x. If d is the smallest value of k such that f (x) + kg(x) = 0 has a single root, find d. x x + + k( x + x) = 0 ( k)x + (k )x + = 0 It has a single root = 0 or k = 0 (k ) ( k)() = 0 or k = k k = 0 or k = k + k 60 = 0 or k = A B F G D E k = 6 or 0 or, d = the smallest value of k = 0 Page 9

10 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Group Event G. Let a = , find the value of a. Reference: 99 HG6, 995 FI., 996 FG0., 00 FG., 0 FI. Let t = 998.5, then 997 = t.5, 998 = t 0.5, 999 = t + 0.5, 000 = t = t.5 t 0.5 t 0.5 t.5 t = 9 t t =.5 t 0.5 = t t = t 6 = t.5 = = (000.5).5 = = 9900 G. In Figure, A and B are two cones inside a 0 cylindrical tube with length of 0 and diameter of 6. If the volumes of A and B are in the ratio 6 : and b is the height of the cone B, find the A B value of b. 0 b : b : 0 b = b b = 5 G. If c is the largest slope of the tangents from the point A find the value of c. 0 0 Let the equation of tangent be y c x 0 cx y c= 0 Distance form centre (0, 0) to the straight line = radius c c 5 c c 5 0c + 5c = c + c 0c + = 0 (c )(c ) = 0 c = or. The largest slope =. Page 0 0, 0 to the circle C: x +y =, G. P is a point located at the origin of the coordinate plane. When a dice is thrown and the number n shown is even, P moves to the right by n. If n is odd, P moves upward by n. Find the value of d, the total number of tossing sequences for P to move to the point (, ). Possible combinations of the die:,,,,,. There are 6 C permutations, i.e. 5.,,,,. There are 5 C permutations, i.e. 5.,,,. There are C permutations, i.e..,,. There are! permutations, i.e. 6. Total number of possible ways = = 8.

11 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Group Event G. Let a be a -digit number. If the 6-digit number formed by putting a at the end of the number 50 is divisible by 7, 9, and, find the value of a. Reference: 00 HG Note that 50 is divisible by 7 and 9. We look for a -digit number which is a multiple of 6 and that a is divisible by satisfied the condition G. In Figure, ABCD is a rectangle with AB and BC. BE and BF are the arcs of circles with centres at C and A respectively. If b is the total area of the shaded parts, find the value of b. AB = AF, BC = CE Shaded area = sector ABF rectangle ABCD + sector BCE = AB AB BC BC = = 6 6 = = = = b G. In Figure 5, O is the centre of the circle and c = y. Find the value of c. BOC = c ( at centre twice at ce ) B y + y + c = 80 (s sum of OBC) y + c = 80 c + c = 80 c = 60 A c O y C G. A, B, C, D, E, F, G are seven people sitting around a circular table. If d is the total number of ways that B and G must sit next to C, find the value of d. Reference: 998 FI5., 0 FI. If B, C, G are neighbours, we can consider these persons bound together as one person. So, there are 5 persons sitting around a round table. The number of ways should be 5!. Since it is a round table, every seat can be counted as the first one. That is, ABCDE is the same as BCDEA, CDEAB, DEABC, EABCD. Therefore every 5 arrangements are the same. The number of arrangement should be 5! 5 =! =. But B and G can exchange their seats. Total number of arrangements = = 8. Page

12 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Group Event 5 G5. If a is the smallest cubic number divisible by 80, find the value of a. Reference: 00 HI 80 = 5 a = 6 5 = G5. Let b be the maximum of the function y = x 6x (where x 5), find the value of b. When x, y = x 6x = (x + ) + Maximum value occurs at x =, y = ( + ) + = When x 5, y = x 6x = (x ) Maximum value occurs at x = 5, y = 9 Combing the two cases, b = G5. In Figure 6, a square-based pyramid is cut into two shapes by a cut running parallel to the base and made of the way up. Let : c be the x ratio of the volume of the small pyramid to that of the truncated base, find the value of c. Reference: 00 HG5 The two pyramids are similar. x volume of the small pyramid x volume of the big pyramid x c = 7 = 6 7 G5. If cos 6 + sin 6 = 0. and d = + 5 cos sin, find the value of d. (cos + sin )(cos sin cos + cos ) = 0. cos + sin cos + cos sin cos = 0. (cos + sin ) sin cos = = sin cos sin cos = 0. d = + 5 cos sin = = Page

11 is the same as their sum, find the value of S.

11 is the same as their sum, find the value of S. Answers: (998-99 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Events I P 4 I a 8 I a 6 I4 a I5 a IS a Q 8 b 0 b 7 b b spare b 770 R c c c c 0 c 57 S 0 d 000 d 990 d