1) In the figure below, line L is tangent to circle G, AG GB, and m GBA=25, then what is the degree measure of the supplement of CDE?
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- Reynard Caldwell
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1 Middleton Invitational 006 THETA OPEN Written b Pedro Gomez The abbreviation NOTA denotes "None Of These Answers." Diagrams ma not be drawn to scale. ) In the figure below, line L is tangent to circle G, AG GB, and mgba=5, then what is the degree measure of the supplement of CDE? 5 B. 45 C. 55 D. 65 ) A mother is now twice as old as her daughter. Ten ears ago the mother was one-ear less than three times the daughter's age. What is the sum of the mother and daughter's current age? 4 B. 57 C. 60 D. 6 ) For x 0, x, x, x, x x = x x B. x x C. x x D. x x 0 4) Find ( i ) 0 B. C. i D. i 5. Find the longest side of the figure shown. G B. X C. B D. F 6) Find K so that the graph of x x is tangent to the x-axis. K B. C. 4 D. 4i
2 Middleton Invitational 006 THETA OPEN Written b Pedro Gomez 7) If A, B and C are distinct non-zero numbers which form a geometric progression in the order listed, find C in terms of A and B. B A A B. B C. A B D. B A 7 8) The sum of the coefficients in the expansion of ( x - ) is? 7 C B. - C. - D. 0 9) Solve for x. AB is a tangent and AC is a secant of circle P B. C. 4 D. 7 0) In the planar figure shown below, a b c... p q r x. Find x. 70 B. 440 C. 080 D. 60 ) Find twice BD in the right triangle shown. 8 B. 4 C. 5 D. 5 ) B. C. 6 D. ) Evaluate: ( ) K K - B. i C. i D.
3 Middleton Invitational 006 THETA OPEN Written b Pedro Gomez 4) Let R be the radius of the inscribed circle and Let r be the radius of the circumscribed circle. Find R. r 4 B. 7 C. 6 7 D. 7 5) Simplif: ( i ) ( - i ) ( - i ) ( 4-4 i ) B. - C. 0 D. 4 6) Find the area of the conic with equation 4x 8 4x B. 7 C. 6 D. 6 7) Point B is a mutual vertex of a regular pentagon, a square, and a third regular polgon. If of the sides of this polgon are AB and BC, then the polgon has p sides. Find p -. 9 B. 0 C. 48 D. 49 8) Find g (0), given that f ( x) x, h( x) x and g( x) f ( h( x)) for x 0. B. C. D. 9) Let p be the ratio of the radius to the circumference of circle E, and let g be the circumference of circle E. Let x 5 be the equation of circle E. Find pg. 0 5 B. 5 5 C. 0 5 D. 5 5
4 Middleton Invitational 006 THETA OPEN Written b Pedro Gomez 0) Let line P be defined b x. Find the product of the slope and the -intercept of the line perpendicular to P at the intersection point with abscissa of. 4 9 B. -6 C. D. ) Find: A - F B - I C - J D -G E - H -80 B. 50 C. -60 D. 90 p-7 ) a * b a b, for a 0, and p d, then find the greatest absolute value for ). B. 9 C. 7 D. 4 9 ) Simplif. 5-5 B. 5 C. 5 D. 5 4 x x 4 4. Evaluate for values of x such that x 4. 8 x x 8 B. 4 x x 4 C. i D. E.NOTA 5) Find the coordinates of the vertex of the parabola with equation x x-. 9, 8 B. 7, 4 8 C. 9, 4 D. 0,
5 Middleton Invitational 006 THETA OPEN Written b Pedro Gomez 6) Given that OZ, OR and OM are perpendicular to EG, GP and PE, respectivel. Find moz mor mom, if PEG is equilateral with side length. B. C. D. 7) Find the product of the roots of x - 7x 6 x B. C. 5 D ) Given: F( x) x and G( x) 4x x 6x 54x 78x x x x, and k the 6th prime number. Find G( F( G( F( G( F( k)))))) to the nearest thousandth B. C. 0 D i 9) Find the sum of the first 40 natural numbers. 80 B. 780 C. 640 D ) Find the length of the major arc formed b the hands of a clock with a diameter of units and the time is :47 pm B. C. D
6 Middleton Invitational 006 THETA OPEN Written b Pedro Gomez Solutions. A. D. D 4. C 5. D CE = 0 degrees b definition of central angle, since AGB is an isosceles triangle. CDE = 65 b definition of inscribed angle. So the supplement is 80-65=5. M D M -0 ( D -0) - D 0 D D, M 4. D M 4 6 x x x x x x x x x x x x x x x Which is equivalent to. x ( ) i i i i x is the largest side of the ; however, the smallest side of the B is the largest side of the ; however, the smallest one of the So the largest side is F, since it is across from the largest angle (95). 6. C 7. A 8. C 9. C 0. B K 4 K 6 K 4 The ratio in a geometric progression is the quotient of a and a. B B So rb B A A x 7 ( ) ( ) ( x ) 8 9 x 7 x x 4 The sum of ever three-pair angle and the unmarked angle is equal to 60. So the sum of all the letters is equal to 60(6) 80(6 ) 440 n n
7 Middleton Invitational 006 THETA OPEN Written b Pedro Gomez NO CALCULATOR ALLOWED. D B the definition of a median in a right-triangle, its measure is half of the 5 5 length of the hpotenuse. The 9--5 triangle has a median of. Twice is 5.. D. A x 6 x x x x x x x k 6 0 ( )( ) 0., becasue a... cannot be negative. k i ever 8 terms the sum goes to zero. The pattern is i,-,- i,. So the sum of the first eleven terms would be B (This is a MEAN one!!) R = inscribed, r = Circumscribed. A R (where a, b and c are the sides and A is the area of the triangle) a b c abc A abc abc r = R Rr 4A r a b c 4A ( a b c ) Rr 7 (5 7 8) 5. C ( i ) ( - i ) ( - i ) ( 4-4 i ) ( 4 4 i ) ( 4 4) 0. The product is zero. 6. C 7. A 8. B 9. B x 4x x = 0 6 The area of the ellipse is ab (90 08) 6 80( n ) n n 0. n - 9 n x g( x) f ( h( x)) 4 x x 4 4 x g ( x) g (0) r p g r r r pg r 5 5 5
8 Middleton Invitational 006 THETA OPEN Written b Pedro Gomez x, when x=, =. NO CALCULATOR ALLOWED 4 - ( x x ) x. The product of the slope and the 0. A 4 4 -intercept is 9 ( A B C D E ) ( F G H I J ) (80) (540) 60. C. A 79@ 4*, ( ) *(-) ( ) ( 7) so or @=79 ( ) 79@ 79 ( ) = gives a greater absolute value.. C 4. C 5. B 5 5 (5) 5 5 ( x 4) i ( x 4) - b - b 7 x x has a vertex at, f, a a 4 8 moz mor mom The altitude of the equilateral triangle. so 6. C 7. C The product of the roots is -5 x 7x 6x 5 -z, where z is the constant and a is the leading coeffiecient. a 8. B F( x ) x - and G( x ) 4 x x - 6x 54 x -78x x -x x, k G( F( G( F( G( F()))))) G ( F( G( F( G(0))))) G ( F( G( F()))) G ( F( G(0))) G( F()) G (0) 9. A 0. C n( a a ) 40(4) 40 n p p 80 (60 47) 457 7(0) (0)
9 Middleton Invitational 006 THETA OPEN Written b Pedro Gomez NO CALCULATOR ALLOWED Answer to Theta Open. A. D. D 4. C 5. D 6. C 7. A 8. C 9. C 0. B. D. D. A 4. B 5. C 6. C 7. A 8. B 9. B 0. A. C. A. C 4. C 5. B 6. C 7. C 8. B 9. A 0. C
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