MASSACHUSETTS ASSOCIATION OF MATHEMATICS LEAGUES STATE PLAYOFFS Arithmetic and Number Theory 1.

Transcription

1 STTE PLYOFFS 004 Round 1 rithmetic and Number Theory How many integers have a reciprocal that is greater than 1 and less than π?. Let 9 b,10 b, and 11 b be numbers in base b. In what positive base b do the numbers form a Pythagorean Triple? 3. Let P j be the j th prime number. P n 103 is the n th prime number. k Pn+ 3 + Pn+ 4. How many different positive integral factors does k have?

2 STTE PLYOFFS 004 Round lgebra Factor completely: 4 + 6y 135y. Find the largest integer value of such that < Given the system a + by a b + ay b with a ± b, find y in simplest form.

3 STTE PLYOFFS 004 Round 3 Geometry Each side of square D is of length 1 cm. E is the midpoint of, F is the trisection point of closer to and G is on D such that G 1 D. How many square centimeters are in 4 the area of EFG?. Given 0 < θ < 90, if the ratio of the complement of θ to the supplement of θ is less than one-tenth, determine the number of integer values of θ and E 7., D, and E D DE. lso,m m D m DE. Find the number of square units in the area of DE. D

4 STTE PLYOFFS 004 Round 4 lgebra / sum of a and b?. 3. can be written in the form a, where a and b are relatively prime. What is the b. If log ( 9 log b) 5 +, determine the value of b Let an be the th n term of an arithmetic progression. Let S n be the sum of the first n terms of the arithmetic progression with a 1 1 and a3 3a8. Determine the largest possible value of S n.

5 STTE PLYOFFS 004 Round 5 nalytic Geometry Points (-10, 0), (0, 5), and (10, 0) are the vertices of a triangle. How many points P(, y) are inside the triangle given that and y are integers Find the endpoint in the first quadrant of the 4 16 chord of the ellipse which passes through its center and is perpendicular to the line whose equation is y. Epress your answer as an ordered pair with the coordinates in eact simplified form.. n ellipse has the equation ( ) ( y) 3. Starting at P(, 5), a bug walks a straight line path, make a 90 turn, and then walks another straight line path until it reaches the point Q(8, 13). Determine the greatest possible distance from the -ais that the bug can attain.

6 STTE PLYOFFS 004 Round 6 Trig and omple Numbers Simplify ( 3+ 4 ) ( 1 ) i i into the form a + bi.. If sec tan 3, determine the numerical value of sec + tan. 3. Regular heagon DEF is inscribed in a circle. F X is the midpoint of arc D. Determine the numerical value of X DX. E D X

7 STTE PLYOFFS 004 Team Round Let f be a periodic function with a period of 3 defined for all real numbers. If, on the interval (1,4], f y is 3y 11 f y is a + by c where a, b, and c are relatively prime positive integers, determine the ordered triple (a, b,c). an equation for ( ) + and on the interval (00, 005] an equation for ( ). In kite D, D and D. If m 108 and m 36 then the ratio of the area of D to the area of D can be written in the form ( ) a btan 36 c are relatively prime positive integers. Determine the ordered triple ( abc.,, ) where a, b, and c 3. n infinite number of squares are inscribed in right as indicated in the diagram. If the sum of the areas of the squares is one-fifth the area of determine the ratio of to., 4. D is an isosceles trapezoid with bases and D with < D. P is the point of intersection of and D. Point X is chosen at random from the interior of the trapezoid. If the probability that X lies in PD is 1 D, find Determine all values of a such that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ln a + ln a + ln a + ln a ln a + ln a + ln a + ln a For { 1,,3,,100}, determine the number of positive integer values of y such that y ( + 1)( + )( + 3) + 1.

8 STTE PLYOFFS 004 nswer Sheet Round Round 1. 3(4 9y)( + 5y) a + b Round Round Round (, ) Round i Team 1. (, 3, 4013). ( 1,1,) , e 14 /

9 STTE PLYOFFS 004 Solutions Round 1 1. The numbers from 4 through 50, 47 altogether. b b 81 b+ 1 b ( 0) ( + 1) k which has 5 0factors. Round 1. 3(4 9y)( + 5y) < > > 1105 > Multiplying the top equation by b and the bottom by a and subtracting eliminates ab( a b) ab and gives y. Multiplying the top by a and the bottom by b and b a a+ b 3 3 a b a + ab+ b subtracting eliminates y and gives. a b a + b Thus, y a ab b + + a + b. a+ b Round 3 1. rea of square 144 cm. Subtract areas of trapezoid EGD, EF, and FG.. Since 90 θ < 1 then θ <180 θ θ > 80. ut since θ < 90, 180 θ 10 θ 81, 8,..., 89, making for a total of 9 values of θ. 3. Since the three triangles are similar, 8 y 7 y y 4 y 7. Thus, y y 87 and y y 9 y 18 and 1. Then so 4 5 and since the scale factor is 3/, D 6 5 and DE 9 5. The sum of the areas is y 7 D E

10 Round ; /4. log16 ( 9 + log16 b) 9 + log16 b 16 3 log 16 b 3 b Thus, b From a3 3a8 we obtain 1+ d 3( 1+ 7d) d. Then 19 Round 5 + ( n 1) n Sn 19 19n n( n 1 ) n 0n +. The maimum occurs at n The maimum of S n The points are on the horizontal lines y 1,,3,4. onsidering the slopes of the segments from (0, 5) to (10, 0) and (-10, 0) are 1/ and 1/, the points lie between (7, 1) and (-7, 1); (5, 1) and (-5, 1); (3, 3) and 3, 3); and (1, 4) and (-1, 4).. The center of the ellipse is (, 0) and the slope of the chord is. The equation of the chord is y + 4. Substituting into the equation of the ellipse gives y 8, leading to and, y. The ordered pair is ( ) 3. Let be the point at which the bug makes a right hand turn. Then PQ is a right triangle and can be inscribed in a circle. Thus, all points lie on a circle with diameter PQ. The circle's diameter equals (8 ) + (13 5) 10, the circle's center is the midpoint of PQ, namely (5, 9), making the circle's equation equal ( 5) + ( y 9) 5. The high point occurs directly above the center where 5 making y 14. Round 6 1. (3 + 4i) i; (1 i) i; 7+ 4i+ 11 i 4+ i.. (sec tan )(sec + tan ) 3(sec + tan ) giving 1 3(sec + tan ) so sec + tan 1 3. sec tan 3(sec + tan ) 3. Since D is a diameter, XD is a right triangle and F since mdx 30, then m DX 15. Set D 1, then X DX sin75 sin E 1 15 D X

11 STTE PLYOFFS 001 Solutions Team Round 1. On [1, 4 ] the linear function joins (1, 3) and (4, 1), so on [ 00, 005 ] the linear function will connect (00, 3) and (005, 1). The equation is y 1 ( 005 ) giving + 3y 4013 so the answer is (, 3, 4013). 3. Since the triangles have a common base, the ratio of their areas equals the ratio of their heights. Since tan36 h, then h tan 36 Since tan7 k then k tan 7. Hence, h k tan 36 tan 7 The answer is ( 1,1,) tan36 tan36 1 tan 36 1 tan h k 7 D 3. Let be the side of the first square and y be the side of the second square. Since a a ab then b a + b. pplying the same approach to the square with side y inscribed in a triangle with legs and b, then b ( ) b ( ) ab y + b b a b b ab 1 + a + b b ( ) ab b b. Hence is the common ratio of the a + b a + b a + b sides. The sum of the areas equals ab ab ab b ab b 4 a+ b K a+ b a+ b a+ b a+ b a+ b b area ab 5 5 ab a+ b b 1 a+ b 10 gives a b 8. a ( a + b) y y y y b ab a + b. Thus,

12 4. It is helpful to know that the areas of P and PD are equal and that the areas of P, P, and PD form an increasing geometric progression. Let the area of P, then the other areas are as marked on the diagram. Thus, 1 gives D 3 D Since, we solve the equation and obtain 3+. P 3 5. lna 3lna lna 3lna lna lna 3ln a 6 lna 1 lna 1 lna 1 lna 1 lna 1 1/4 4( lna) lna 0 lna 0 or. Thus, a 1, e. 4 ( ) ( ) ( ) 6. Subtract 1 from both sides obtaining: ( y )( y+ ) ( + )( + )( + ) ( ( + ))(( + )( + )) Since y 1 and y ( )( ) are two integers differing by and + 3 and are two integers differing by, then for each value of in {1,,..., 100} there is a value of y for which ( y 1)( y+ 1) ( + 1)( + )( + 3). For eample, when 1, y 5; when 100, y 10,301. The answer is 100.