Minnesota State High School Mathematics League

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1 011-1 Meet 1, Individual Event Question #1 is intended to be a quickie and is worth 1 point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have 1 minutes for this event. NO LULTORS are allowed on this event. 1. There were 7 boys and 13 girls at a party. What percentage of the group were boys? x =. If 11 x mod 13, and x is a single positive digit, determine the value of x. 3. onvert r = into a fraction expressed as the quotient of two relatively prime integers. 4. n integer, increased by the value of its cube, becomes 9,788. What is the original integer? Name: Team:

2 011-1 Meet 1, Individual Event SOLUTIONS NO LULTORS are allowed on this event. 3% (lso accept 3 ) 1. There were 7 boys and 13 girls at a party. What percentage of the group were boys? 7 boys 0 total in group = 3%. x =. If 11 x mod 13, and x is a single positive digit, determine the value of x. Since 19 = 13 13, we know that 11 is 8 short of a multiple of i.e., x mod 13. Subtracting 8 from both sides, x = onvert r = into a fraction expressed as the quotient of two relatively prime integers. Multiply both sides of the given equation by 10, so that 10r = 14 + x where x = Now note that 100x = 7+ x x = Thus, r = 11 = 1 11 r = n integer, increased by the value of its cube, becomes 9,788. What is the original integer? Since = 1,080 and = 79,090, the integer must be between 80 and 90, and closer to 80. Use modular arithmetic based on the last digit of x + x 3 : x x x + x Thus our only two options are 84 and 89, and 84 is closer to > Only two values yield a last digit of 8.

3 011-1 Meet 1, Individual Event Question #1 is intended to be a quickie and is worth 1 point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have 1 minutes for this event. m = 1. In Figure 1, m = 90, and point is located on such that = =. What is the measure of? Figure 1 S Q m SRQ =. Figure shows an equilateral triangle sharing a common side with a square. etermine the measure of SRQ. R U Figure In regular 9- sided polygon 1 9 (Figure 3), what is the measure of the obtuse angle between diagonals 1 and 4 7? Figure 3 4. The distances from a point to three vertices of a rectangle are,, and 10, as shown in Figure 4. What is the distance from the point to the fourth vertex? 10 Figure 4 Name: Team:

4 m = 10 m SRQ = Meet 1, Individual Event SOLUTIONS 1. In Figure 1, m = 90, and point is located on such that = =. What is the measure of? In, α + β + 90 = 180, so α + β = 90. lso, m = 180 α = 180 β, so α = β. This means β = 30, so m = ( ) = 10.. Figure shows an equilateral triangle sharing a common side with a square. etermine the measure of SRQ. Note that SUR and RQ are isosceles, allowing us to label the angles as shown in the Sigure. Focus on point R: m SRQ = 30 m SRQ = 10.! S U " 30! R Figure " Figure Q In regular 9- sided polygon 1 9 (Figure 3), 9 what is the measure of the obtuse angle between diagonals 1 and 4 7? is equilateral, because each of its sides cut off an identical section of the nonagon. So m = 0, and by similar reasoning, m 7 1 = m 1 = m 1 4 = Using the shaded isosceles triangle, the desired obtuse angle has measure = 100. Figure The distances from a point to three vertices of a rectangle are,, and 10, as shown in Figure 4. What is the distance from the point to the fourth vertex? X b 10 Through the interior point, draw line segments perpendicular to each side of the rectangle. Using Pythagorean relationships, a + c = a + b + c + d = 4 + X = 100+ b + c = 10 X = 11 b + d = X X = 11 d + a = d c a Figure 4 Note the theorem proven here: The sum of the squares of opposing distances from the interior point must be equal!

5 011-1 Meet 1, Individual Event Question #1 is intended to be a quickie and is worth 1 point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have 1 minutes for this event. NO LULTORS are allowed on this event. 1. etermine exactly the value of cos π 1 + cos π + cos π 3. m =. Find the radian measure of the smallest positive angle for which cos has the same value as sin In the xy- plane, the graphs of y = sin x and y = 1 intersect at points P and Q, where P is in the Xirst quadrant and Q is in the second quadrant. In terms of π, determine exactly the smallest possible distance between P and Q. cos Q = 4. In rectangle (Figure 4), = 8 and = 1. PQ is a right triangle, as shown. If cos PQ = 4, determine exactly the value of cos Q. P 8 Q Figure 4 1 Name: Team:

6 011-1 Meet 1, Individual Event SOLUTIONS NO LULTORS are allowed on this event etermine exactly the value of cos π 1 + cos π + cos π 3. cos π + cos π + cos π 3 = ( 1 )+0+ 1 = 1. m = π 3. Find the radian measure of the smallest positive angle for which cos has the same value as sin 10. sin 10 = sin 30 = 1. The earliest that cosine takes on negative values is in quadrant II, so we seek π < < π for which cos = 1. Since cos π 3 = 1, reslect across the y- axis to = π 3. 4π 3 3. In the xy- plane, the graphs of y = sin x and y = 1 intersect at points P and Q, where P is in the Xirst quadrant and Q is in the second quadrant. In terms of π, determine exactly the smallest possible distance between P and Q. 0. 7π π π π 3 π π 3 π π cos Q = 4 4. In rectangle (Figure 4), = 8 and = 1. PQ is a right triangle, as shown. If cos PQ = 4, determine exactly the value of cos Q. ngle chasing shows us that P PQ, so cos P = 4 = 8, and P = 10. Thus P is P This means P =, so cos PQ = 4 = PQ PQ = 1. onsidering P s length as 0, PQ is 1, 0,. 1/ 10 8 Q /! 90! P! cos Q = 1 = 1 = 4. 1 Figure 4

7 011-1 Meet 1, Individual Event Question #1 is intended to be a quickie and is worth 1 point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have 1 minutes for this event. 1. Write, in x + bx + c = 0 form, the quadratic equation whose roots are 3 and 1.. Find the remainder when x is divided by x alculate the difference between the two roots of the equation x px + p 1 4 = 0. p = q = 4. If the solutions (for x) of x + px + q = 0 are the cubes of the solutions for x + mx + n = 0, express p and q as polynomials in terms of m and n. Name: Team:

8 011-1 Meet 1, Individual Event SOLUTIONS x +x 3= 0 1. Write, in x + bx + c = 0 form, the quadratic equation whose roots are 3 and 1. ( x +3) ( x 1) = x 1x +3x 3= x +x 3= 0.. Find the remainder when x is divided by x 1. y the Remainder Theorem, if f x simply f 1 ( ) = ( 1) 13 +1=1+1=. ( ) = x 13 +1, then the remainder when f(x) is divided by x 1 is 1 (lso accept 1 ) 3. alculate the difference between the two roots of the equation x px + p 1 4 = 0. [Mathematics Teacher] p 1 4 is the product of the two roots, and suspiciously factors into p 1 p 1 + p+1 = p heck the sum of the roots: So the two roots are p 1 and p+1 and p+1. = p, which checks vs. the coefsicient of the x term., and their difference is p+1 p 1 = = 1. p = q = m 3 3mn n 3 4. If the solutions (for x) of x + px + q = 0 are the cubes of the solutions for x + mx + n = 0, express p and q as polynomials in terms of m and n. [Mathematics Teacher] Graders: 1 point for each correct value Let the two roots of x + mx + n = 0 be R 1 and R. Using sum and product of roots, R 1 + R = m, and R 1 R = n. lso, the roots of x + px + q = 0 are ( R 1 ) 3 and ( R ) 3, so ( R 1 ) 3 + ( R ) 3 = p, and ( R 1 ) 3 ( R ) 3 = q. So we have q = n 3. Factoring: p = ( R 1 ) 3 + ( R ) 3 = R 1 + R ( ) ( R 1 ) R 1 R + ( R ) ( ) ( R 1 + R ) 3R 1 R = m ( )( m 3n) = m 3 +3mn, so p = m 3 3mn. = m

9 011-1 Meet 1, 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. 1. The diagonals of the quadrilateral shown in Figure 1 are perpendicular. Lengths of three of the quadrilateral s sides are, 17, and. What is the length of the fourth side? Figure polynomial has a remainder of 3 when divided by (x 1) and a remainder of when divided by (x 3). What is the remainder when the polynomial is divided by (x 1) (x 3)? 3. In Figure 3, each small grid square measures 1 square unit. etermine exactly the sum of the measures of angles MN, MN, MN, and MN. Figure 3 M N 4. store advertised a one- day sale during which it would be selling appliances for 0% off the regular price. In addition, customers using the store s credit card could, after the price reduction, still get their regular 10% discount. mathematically- ignorant salesman sold items to these customers at 0% off the regular price. If he accounted for $1,000 of store credit card sales, how much did his error cost the store in losses?. In right triangle (Figure ), =. If tan E = tan E = 3 tan, determine exactly the value of tan E + tan. Figure E. List all integers n > 3 such that n 3 divides evenly into n n. Team:

10 011-1 Meet 1, Team Event SOLUTIONS (page 1) The diagonals of the quadrilateral shown in Figure 1 are perpendicular. Lengths of three of the quadrilateral s sides are, 17, and. What is the length of the fourth side? Figure 1 17 x +. polynomial has a remainder of 3 when divided by (x 1) and a remainder of when divided by (x 3). What is the remainder when the polynomial is divided by (x 1) (x 3)? [Mathematics Teacher] 4 tan 1 (lso accept any equivalent inverse trig expressions) 3. In Figure 3, each small grid square measures 1 square unit. etermine exactly the sum of the measures of angles MN, MN, MN, and MN. Figure 3 M N $ store advertised a one- day sale during which it would be selling appliances for 0% off the regular price. In addition, customers using the store s credit card could, after the price reduction, still get their regular 10% discount. mathematically- ignorant salesman sold items to these customers at 0% off the regular price. If he accounted for $1,000 of store credit card sales, how much did his error cost the store in losses? Q 1 y In right triangle (Figure ), =. If tan E = tan E = 3 tan, Figure E determine exactly the value of tan E + tan. P y 1 x x 4,,, 9. List all integers n > 3 such that n 3 divides evenly into n n. [Mathematics Teacher] Graders: 1 point for each correct value

11 011-1 Meet 1, Team Event SOLUTIONS (page ) 1. This is like an inside- out version of the rectangle from Event, problem #4. If you were wise to check the Event solutions page before entering the Team Event, you would have noticed a useful theorem that applies here: + = 17 + X + = 89+ X X = 31 X = 19.. Let the original polynomial be P(x). Express P( x) = Q( x) ( x 1) ( x 3)+ R( x), where Q(x) is the quotient polynomial and R(x) is the remainder polynomial we re looking for. Since R(x) is the remainder after dividing by a quadratic, R(x) can be no more than a linear polynomial. Set R x a = a = 1 b =, and R x ( ) = ax + b. P(1) = 3, so a + b = 3. lso, P(3) =, so 3a + b =. y elimination, ( ) = x Translate triangles MN, MN, and MN so that the images of points,, and align with point. The sum of the four angles can now be seen as a single large acute angle at point, inside a right triangle with legs of lengths 4 and. 4 The easiest way to express this angle exactly is to use the inverse tangent function: tan 1 4. The salesman took in 40% of the regular price p. In other words,.40p = 1000 p = $37,00. He should have Sirst taken 0% off:.0( $37,00) = $18,00, and then taken another 10% off, leaving 90%:.90( $18,00) = $1,87. Therefore, his error cost the store $1,87 $1,000 = $187.. Let = = 1, and set = x, E = y. ropping perpendiculars from and E to creates triangles P and EQ. = 1 x, so P = P = 1 x ( ) and P = P = 1 x 1 x = = 1+ x. Thus, tan = P P = 1 x 1+ x. y similar reasoning, tan E = EQ Q = 1 y. Using the given fact that tan E = tan E = 3 tan, we have 1+ y y x = y = 3x x = 1, y = 3, and so 4 1 x tan + tan E = 1+ x + 1 y 1+ y = = = = Synthetically divide: So n n n 3 = n++ n 3. We need n 3 to evenly divide so that n 3 becomes part of the quotient. has four divisors: {1,, 3, }, so set n 3= { 1,, 3, } so that n { 4,,, 9}.