Minnesota State High School Mathematics League

Transcription

1 03-4 Meet, Individual Event Question # is intended to be a quickie and is worth point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have minutes for this event. NO LULTORS are allowed on this event.. Determine exactly the value of. LM 0,4 GD 0,4. The expression 4 3 can be simplified and written as a single rational number. Determine exactly that rational number. Day # 3. The epad was originally priced at $00, but the newest model of epad is coming out, so the old one is going on sale. Each day, the price is reduced by 0%, and rounded down to the nearest dollar, as necessary. (On Day, the price is reduced to $90; on Day it is $8; on Day 3 it is $7, and so on.) What is the First day on which the old epad will cost $? 4. Express as a repeating decimal. Name: Team:

2 03-4 Meet, Individual Event SOLUTIONS NO LULTORS are allowed on this event. 70. Determine exactly the value of. LM 0,4 GD 0,4 LM 0,4 GD 0,4 7 LM, 7 GD, The expression 4 3 can be simplified and written as a single rational number. Determine exactly that rational number. Work from the inside out: Day # 7 3. The epad was originally priced at $00, but the newest model of epad is coming out, so the old one is going on sale. Each day, the price is reduced by 0%, and rounded down to the nearest dollar, as necessary. (On Day, the price is reduced to $90; on Day it is $8; on Day 3 it is $7, and so on.) What is the First day on which the old epad will cost $? The basic idea is that we re subtracting the tens place of the previous day s cost, along with an additional dollar if the ones place is greater than zero. (For example, 7 7. rounds to ) Make a table: Day # ost($) Express as a repeating decimal. 9 0., 0.09, We have repetends of lengths,, and 3, so to add them, we 999 need to extend to a common multiple: 6 decimal places: ( )

3 03-4 Meet, Individual Event Question # is intended to be a quickie and is worth point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have minutes for this event.. If 3 and 4 are the lengths of the two legs of a right triangle, determine exactly the length of the triangle s hypotenuse. m PQ s r. In Figure, uu sur P D sur, PQ is a transversal, P m PQ x, and m DQP x. Determine exactly the measure of angle PQ. x x - Q D Figure m FKH 3. In isosceles triangle FGH (), the vertex angle measures 0 and its opposite side is 0 cm long. G segment from point H meets side FG at point K so that HK 0 cm. Determine the measure of angle FKH. K 0 F 0 cm H m WVZ 4. In Figure 4, XY XZ, m YXV 30, and XV XW. Determine exactly the measure of angle WVZ. X W Y V Figure 4 Z Name: Team:

4 03-4 Meet, Individual Event SOLUTIONS or If 3 and 4 are the lengths of the two legs of a right triangle, determine exactly the length of the triangle s hypotenuse. a + b c c c c 44. m PQ 4 s r. In Figure, uu sur P D sur, PQ is a transversal, m PQ x, and m DQP x. Determine exactly the measure of angle PQ. P x PQ and DQP are same- side interior angles, so 80 they are supplementary: x + x 7x 3 x 33, and since PQ and DQP Q x - Figure D are congruent (alternate interior angles), m PQ x ( 33) 6 4. G m FKH In isosceles triangle FGH (), the vertex angle measures 0 and its opposite side is 0 cm long. segment from point H meets side FG at point K so that HK 0 cm. Determine the measure of angle FKH. K 0 θ VHFK is isosceles, so label m FKH m F θ. We are told that VFGH is isosceles, so m FHG θ also. onsidering the three angles in VFGH, 0+θ +θ 80 θ 80. F θ 0 cm H m WVZ 4. In Figure 4, XY XZ, m YXV 30, and XV XW. Determine exactly the measure of angle WVZ. Let m VXW α. Then m YXZ 30 +α, and 90 α m Z α 7 α. lso, m VWX 80 α. VWX is Y X α 30 V Figure 4 W Z exterior to VVWZ, so m VWX m WVZ + m Z 90 α m WVZ + 7 α m WVZ.

5 03-4 Meet, Individual Event Question # is intended to be a quickie and is worth point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have minutes for this event. a. In Figure, point P(a, b) is located in the second quadrant on the circumference of a circle of radius r. Express a in terms of b and r. P (a,b) r Figure sinθ. line segment drawn from the origin to a point (m, n) in the fourth quadrant makes an angle of \ with the positive x- axis. Express sin \ in terms of m and n. tan α 3. The point (^, k) lies on the graph of y cos x (), beneath the y- axis and with ^ between 4 and radians. Express tan ^ in terms of k. 4 6 (α, k) D 4. Semicircle O, pictured in Figure 4, has diameter 0. rc also has length 0. From, a perpendicular is dropped to meet segment at D. alculate the length D. O Figure 4 Name: Team:

6 a r b Minnesota State High School Mathematics League 03-4 Meet, Individual Event SOLUTIONS. In Figure, point P(a, b) is located in the second quadrant on the circumference of a circle of radius r. Express a in terms of b and r. P (a,b) b a r Drop a perpendicular from P to the x- axis as shown, creating a right triangle with legs a, b and hypotenuse r. a + b r a r b a ± r b, and since a lies on the negative x- axis, a r b. Figure sinθ n m + n. line segment drawn from the origin to a point (m, n) in the fourth quadrant makes an angle of \ with the positive x- axis. Express sin \ in terms of m and n. n m + n or m + n or equivalent See Figure. sin θ opp hyp n c, and since m + n c, we have c m + n sinθ n n m + n. m + n m + n c m θ n tan α k k Figure (m,n) 3. The point (^, k) lies on the graph of y cos x (), beneath the y- axis and with ^ between 4 and radians. Express tan ^ in terms of k. cos α k, which is negative. π <α < 3π, so sin α is also negative. sin α + cos α sinα ± cos α ± k. tan α sinα k cosα k. 4 6 (α, k) D +cos( ) or.99 Graders: if students use 360 as an π angle equivalent, the degree sign must be included. 4. Semicircle O, pictured in Figure 4, has diameter 0. rc also has length 0. From, a perpendicular is dropped to meet segment at D. alculate the length D. Since has a length equal to twice the semicircle s radius, by de`inition of a radian, angle O measures exactly, but also, radians. cos OD OD and cos O cos since OD and O are supplementary, their cosines are opposites. OD cos. D O OD cos( ) 0 O Figure 4 D

7 03-4 Meet, Individual Event D Question # is intended to be a quickie and is worth point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have minutes for this event. NO LULTORS are allowed on this event. r. Let r and r be the distinct roots of r r 0, with r < r. Determine r exactly. x + x. Let x and x be the solutions of x 0x Determine x + x exactly. f ( ) 3. Let f x be a quadratic polynomial. If it is known that f ( 0), f, and f ( ) 0, exactly. determine f Let ( X ) ( X ) ( X ) ( X + 3)+ ( X ) ( 3X )+ ( X + 3) ( X ), where,, and are all real numbers. Determine the sum + + exactly. Name: Team:

8 03-4 Meet, Individual Event D SOLUTIONS NO LULTORS are allowed on this event. r. Let r and r be the distinct roots of r r 0, with r < r. Determine r exactly. r r 0 ( r ) ( r + 4) 0 r 4 or. Since 4 <, r 4 and r. x + x 0 3. Let x and x be the solutions of x 0x Determine x + x exactly. or 7 3 Of course we could use the quadratic formula to `ind both roots, but it s more elegant to use the formulas for the sum & product of roots: + x + x x + x sum of roots x x x x x x x x product of roots 0 3. f ( ) 3 3. Let f x be a quadratic polynomial. If it is known that f ( 0), f, and f ( ) 0, exactly. determine f Let f ( x) ax + bx + c. Then f ( 0) a( 0) + b( 0)+ c c, f a + b a+ b+ a+ b, and f ( ) a( ) + b + c + c 4a+b+ c a+( a+b)+ c 0 a++ 0 a 3, b. Thus our quadratic polynomial is f ( x) 3 x + x +, and we have f ( ) 3 ( ) + ( ) Let ( X ) ( X ) ( X ) ( X + 3)+ ( X ) ( 3X )+ ( X + 3) ( X ), where,, and are all real numbers. Determine the sum + + exactly. or 6 6 Rearrange the right side of the equation, then factor out X + 3 ( X + 3) ( X )+ ( X ) + ( X )( 3X ) ( X + 3) ( 3X )+ X from two of the terms: ( 3X ) ( X +) ( 3X ). Since the coef`icient of the X term in each factor must be, we factor out of the `irst factor and 3 out of the second factor to yield: X + 3 X 3 6 X + X 3 X So 6,, ( X ).

9 03-4 Meet, Team Event Each question is worth 4 points. Team members may cooperate in any way, but at the end of 0 minutes, submit only one set of answers. Place your answer to each question on the line provided. sum. Find the sum of all positive integers n for which LM(, n) GD(n, 0). m. On isosceles V, points P and Q are on sides and respectively, distinct from,, and, so that P PQ Q. Determine exactly the measure of angle. 3. shows a semicircle with radius. If arc ª also has length, calculate the length of chord. a 4. Let f t t + at. Given that at least one of the coefficients of the degree- four f ( t ) polynomial f is zero, list all possible values of a. y k. Figure shows rays drawn from the origin at angular intervals of 0, intersecting the line x at y, y, etc. Find the smallest positive integer k for which y k y k will equal or exceed y. y 6 y y 4 y 3 y x x"" Figure b 6. Let N be a number in base b such that N b 4 b 7 b. What is the greatest base b for which N b would be written with as its left- most digit? Team:

10 03-4 Meet, Team Event SOLUTIONS (page ) sum 384. Find the sum of all positive integers n for which LM(, n) GD(n, 0). m 7 80 or 7. On isosceles V, points P and Q are on sides and respectively, distinct from,, and, so that P PQ Q. Determine exactly the measure of angle. Q m m m Figure shows a semicircle with radius. If arc m m 80-4m 3m 3m P or 0.98 or sin( 0.) or cos ª also has length, calculate the length of chord. D a, 0,, Graders: award point per correct value 4. Let f t t + at. Given that at least one of the coefficients of the degree- four f ( t ) polynomial f is zero, list all possible values of a. y y 6 k. Figure shows rays drawn from the origin at angular intervals of 0, intersecting the line x at y, y, etc. Find the smallest positive integer k for which y k y k will equal or exceed y. y y 4 y 3 y x"" x b 3 6. Let N be a number in base b such that N b 4 b 7 b. What is the greatest base b for which N b would be written with as its left- most digit?

11 03-4 Meet, Team Event SOLUTIONS (page ). For LM(, n) GD(n, 0), both quantities must be equal to n. This means n is both a multiple of and a factor of 0. Examining the prime factorization of 0 ( 3 7 ), the acceptable values of n must be of the form 3 a b 7 c, where a, b, and c are all either 0 or. This yields eight values:, 6, 0, 4, 30, 4, 70, and 0. Their sum is See Figure. egin by labeling m as m. Then in isosceles VPQ, m P m also. PQ is an exterior angle of this triangle, and so its measure is m. Using isosceles VQP, m m, and m QP 80 4m. Subtracting along straight angle P, m P 3m, and in isosceles VP, m 3m also. Now we must remember that the original triangle, V, was given as isosceles, so m Q 3m m+ m P m P m. We now have expressions for all 80 angle measures in VP : m+ 3m+ 3m 80 m Drawing segments D and creates VD, and because it is inscribed in a semicircle, it is a right triangle with right angle at. ecause arc has the same length as its circle s radius, by the de`inition of radian, m radian, and m D 0. radians. Using trigonometry, sin( 0.) sin( 0.) f ( t + at ) t ( + at ) + a ( t + at ) t 4 + at 3 + ( a )t at + + ( at + a t a) t 4 + at 3 + ( a + a )t + ( a a)t a. Since one of the coef`icients is zero, our possibilities are a 0 a 0 ; a + a 0 a + ( a ) 0 a or ; a a 0 a( a ) 0 a 0 or (only is unique), or a 0 (repeat value again). The unique values of a are, 0,,.. y k y k y + y + + y k y + y + + y k tan 0k y Therefore, we want tan 0k tan( 0( k ) ). y tan , so tan( 0( k ) ) Generate a table of possible values for k: tan 0k k tan( 0( k ) ) The `irst value that satis`ies the inequality is k. (It turns out, in fact, that tan0 tan40 tan0. an you prove why this is true?) 6. N b 4 b 7 b N ( b + 4) ( b + 7) b +b + 8. ecause we want a quantity of in the b column, we need b + 8 b (and also b + 8 < b ). Either solve the inequality or try values of b > 0 to `ind the greatest b 3.