Minnesota State High School Mathematics League

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1 01-1 Meet, Individual Event A 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 CALCULATORS are allowed on this event. x = 1. Determine exactly the value of x for which x ( 1 x) =.. What is the least integer value of n that satis>ies the inequality n +( n 1) >6?. Given that x + 7y = 6, if the value of x increases by, determine exactly the amount by which the value of y decreases. gallons. A hose providing a constant stream of water can >ill a 0- gallon pool in 0 minutes. But because of a steady leak, it took the hose 0 minutes to >ill the pool. How many gallons of water leaked out while the pool was being >illed? Name: Team:

2 01-1 Meet, Individual Event A SOLUTIONS NO CALCULATORS are allowed on this event. x = or Determine exactly the value of x for which x ( 1 x) =. x ( 1 x) = x 1+x = x = x =. n =. What is the least integer value of n that satis>ies the inequality n +( n 1) >6? n +( n 1) >6 1 0 n>10 n> 1 =, so the least integral solution is n = also accept 1 7. Given that x + 7y = 6, if the value of x increases by, determine exactly the amount by which the value of y decreases. Solve the given equation for y: y = 6 x. This is linear, so we can choose any two values 7 for x and expect that the corresponding change in y will be the same regardless of our choice. Choose x = 0 and x =, for which y = 6 7 and 6 7 respectively; Δy = 6 6 = gallons or 7.. A hose providing a constant stream of water can >ill a 0- gallon pool in 0 minutes. But because of a steady leak, it took the hose 0 minutes to >ill the pool. How many gallons of water leaked out while the pool was being >illed? If an empty 0- gallon pool with no leak takes 0 minutes to Gill, then it is being Gilled at a rate of gallons/minute. If the same pool has a leak in it and takes 0 minutes to Gill, then the Gill rate has become gallons/minute. Therefore, water must have been leaking out at a rate of = gallons gallons/minute. 0 min. = 7. total gallons leaked. 0 0 min.

3 01-1 Meet, Individual Event B 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. x = 1. In Figure 1, determine exactly the unknown length x. x Figure 1 CE =. Figure demonstrates what we might call the Exterior Angle Bisector Theorem : In ABC, if BD bisects interior angle ABC and BE bisects exterior angle CBF (where E lies on the extension of side AC ), then CE AE = CD. If AD = and CD =, AD use this Theorem to determine CE exactly. B F Figure A D C E. The lengths of the sides of PQR are, 1, and 1. Determine exactly the length of the triangle s shortest altitude. TZ =. Distinct points W, X, Y, and Z all lie on the same line such that WX = XY = YZ. For some point T not on that line, TW =, TX =, and TY = 7. Determine exactly the length TZ. Name: Team:

4 x = 8 or 8 or.6 Minnesota State High School Mathematics League 01-1 Meet, Individual Event B SOLUTIONS 1. In Figure 1, determine exactly the unknown length x. Using a straightforward treatment of Ceva s Theorem, we start at the segment labeled x and move clockwise x around the triangle: 8x =1 =1 x = 8. x Figure 1 CE = 1 or or.. Figure demonstrates what we might call the Exterior Angle Bisector Theorem : In ABC, if BD bisects interior angle ABC and BE bisects exterior angle CBF (where E lies on the extension of side AC ), then CE AE = CD. If AD = and CD =, AD use this Theorem to determine CE exactly. CE AE = CD AD B n n+7 = n= 1. F Figure 60 1 or 8 1 A D C n. The lengths of the sides of PQR are, 1, and 1. Determine exactly the length of the triangle s shortest altitude. E The shortest altitude must be drawn to the triangle s longest side (1). The side lengths of, 1, and 1 make this a right triangle, so we can calculate its area in two ways (Figure ): using as the base and 1 as the height, for an area of 0; or using 1 as the base and h as the height, for an area of 1h/. Set these quantities equal and solve to Gind that h = 60/ h 1 Figure TZ = 6. Distinct points W, X, Y, and Z all lie on the same line such that WX = XY = YZ. For some point T not on that line, TW =, TX =, and TY = 7. Determine exactly the length TZ. W d X d Y d Z 7 T Figure See Figure. Apply Stewart s Theorem to TWY: d +7 d = d +d d d 8 =+d, so d = 1. Now apply Stewart s again, this time to TXZ: d +TZ d =7 d +d d d. Divide through by d to yield 16+TZ = 98+d, and we can use d = 1 to Gind that TZ =108 and TZ =6.

5 01-1 Meet, Individual Event C 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 CALCULATORS are allowed on this event. sin θ = 1. If π <θ < π and sinθ =, determine exactly the value of sin θ. cos (A C) =. In right triangle ABC, AB =, BC = 1, and AC = 1. Determine cos (A C) exactly. α =. What is the smallest positive angle α, in degrees, for which sinα = 1 cos 18? β =. What is the smallest positive angle β, in degrees, for which sinβ tanβ = tanβ sin8? Name: Team:

6 01-1 Meet, Individual Event C SOLUTIONS NO CALCULATORS are allowed on this event. sin θ = or If π <θ < π and sinθ =, determine exactly the value of sin θ. See Figure 1. sinθ =sinθ cosθ = =.! θ Figure 1 cos (A C) = In right triangle ABC, AB =, BC = 1, and AC = 1. Determine cos (A C) exactly. ( ) = cos AcosC +sin AsinC = 1 cos A C = C 1 1 B A α = 71. What is the smallest positive angle α, in degrees, for which sinα = 1 cos 18? 1 cos 18 Divide both sides of the equation by to obtain sinα =. The right side of this equation is the half- angle formula for sine, so we can rewrite: sinα = sin109. ReGlect 109 across the y- axis to Gind the smallest α with this sine value, 71. β = 16. What is the smallest positive angle β, in degrees, for which sinβ tanβ = tanβ sin8? Rewrite the equation as sinβ cosβ sinβ cosβ = sinβ sin8. Then multiply both sides by cosβ cosβ sinβ : cos β 1= sin8 cosβ = sin8. But using the sine/cosine cofunction identity, sin 8 = cos (90 8 ) = cos. So cosβ = cos, and β =16.

7 01-1 Meet, Individual Event D 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. slope = 1. Determine exactly the slope of the line x y =6. (x, y) =. Determine exactly the coordinates of the intersection point of x + y = 8 and x + y =1.. A lattice point is a point on the xy- plane whose coordinates are both integers. How many lattice points lie on the line x y = 10, are within the >irst quadrant, and have a y- coordinate of at most 01?. Determine exactly the shortest distance between the lines x y = 6 and y = x + 1. Name: Team:

Multiply the second equation through by the LCD (6) so that both equations are in standard form: x + y =8 x +y =6 Using elimination, y = x +( ) =8 x = 0, so the lines intersect at the point ( 0, ).

8 01-1 Meet, Individual Event D SOLUTIONS slope = 1. Determine exactly the slope of the line x y =6. Place in slope- intercept form: x y =6 y = x 6 y = x, so the slope =. (x, y) = ( 0, ) Graders: Award 1 point per correct coordinate.. Determine exactly the coordinates of the intersection point of x + y = 8 and x + y =1. Multiply the second equation through by the LCD (6) so that both equations are in standard form: x + y =8 x +y =6 Using elimination, y = x +( ) =8 x = 0, so the lines intersect at the point ( 0, ) A lattice point is a point on the xy- plane whose coordinates are both integers. How many lattice points lie on the line x y = 10, are within the >irst quadrant, and have a y- coordinate of at most 01? Figure shows the graph of the line, which has x- intercept.. As the line enters the Girst quadrant, it passes through the lattice points (, 1), (, ), (, ), and so on. The y- coordinates are simply the set of odd numbers! So: we simply count the odd numbers 01. This is the same as the # of even numbers 01, which is 01 = Figure 9. Determine exactly the shortest distance between the lines x y = 6 and y = x + 1. Rewriting the equation of the Girst line (y = x + 6), we can see that the lines have the same slope, meaning that they are parallel. Thus all distances between the two lines are equivalent (they are all shortest ), and any distance between the two lines will sufgice. y = x + 1 passes through (0, 1), so use the formula for the distance from a point to a line to Gind the distance from (0, 1) to x y = 6 (Figure ): ( )+ 1( 1) ( 6) 9 d = Ax +By C 0 = A +B + 1 ( ) = = 9. Figure

9 01-1 Meet, Team Event Each question is worth 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. F S = 1. On a father/son road trip, the father drove 60% of the time, and the son covered 60% of the distance while driving. Each drove at a constant rate (though not necessarily the same rate). If the father s rate was F mph and the son s rate was S mph, determine F S exactly.. If the lengths of a triangle s sides are,, and 6, determine exactly the length of the triangle s shortest median. r =. The line x y = 6 is tangent to a circle whose center lies on the positive x- axis. If the point of tangency is (6, ), determine exactly the length of the circle s radius. x = y =. In Figure, the segment between x and y is an angle bisector. Determine exactly the unknown lengths x and y. y x Figure k =. Determine exactly the value of k such that the y- intercept of ky x = 9 is 0 greater than the y- intercept of y x = 9. sec BFE = 6. In right triangle ABC, the legs measures are AB = 6 and AC =, and medians BD and CE intersect at F. Determine exactly sec BFE. Team:

01-1 Meet, Team Event SOLUTIONS (page 1) F S = 9 1. On a father/son road trip, the father drove 60% of the time, and the son covered 60% of the distance while driving.

10 01-1 Meet, Team Event SOLUTIONS (page 1) F S = 9 1. On a father/son road trip, the father drove 60% of the time, and the son covered 60% of the distance while driving. Each drove at a constant rate (though not necessarily the same rate). If the father s rate was F mph and the son s rate was S mph, determine F S exactly. 6. If the lengths of a triangle s sides are,, and 6, determine exactly the length of the triangle s shortest median. m Figure 1. The line x y = 6 is tangent to a circle whose center lies on the positive x- axis. If the point of tangency is (6, ), determine exactly the length of the circle s radius. Figure 8 x = y = 9. In Figure, the segment between x and y is an angle bisector. Determine exactly the unknown lengths x and y. y x Graders: award points per correct value. Figure k = 9. Determine exactly the value of k such that the y- intercept of ky x = 9 is 0 greater than the y- intercept of y x = 9. sec BFE = In right triangle ABC, the legs measures are AB = 6 and AC =, and medians BD and CE intersect at F. Determine exactly sec BFE.

11 01-1 Meet, Team Event SOLUTIONS (page ) 1. Suppose the trip covered a distance of d miles in a total time of t hours. Then the father drove a distance of 0.d in 0.6t hours, for a rate of F = 0.d 0.6t = d 0.6d, while the son drove a distance of 0.6d in 0.t hours, for a rate of S = t 0.t = d t. Therefore, F d S = t = d t d t t d = 9. (Notice that the total distance and total time of the road trip were irrelevant!). See Figure. Similar to what we saw in Event B, the shortest median must be drawn to the triangle s longest side (6). By Stewart s Theorem, + = m = 6m + m = 69 6 = m = = 6.. See Figure. Rewrite the equation of the tangent line to Kind that it has slope. Then the radius that passes through the point of tangency has slope, and is part of a line with equation y = ( x 6 ). This line s x- intercept is the circle s center:, 0. r = the distance from this point to (6, ): 6 + = + = = 1.. First, apply the Angle Bisector Theorem: + y + x = 9+y = 8+x x y =1. Then, by Ceva s Theorem, x 9x =1 y 8 y =1 x = 8 9 y. Substituting, 8 9 y y =1 9 9 y =1 y =, x = 8 9 y = =.. Solve both equations for y: y = k x + 9 k and y = x + 9. Thus 9 k = = 60k +9k k =. 6. Focus on BFE in Figure 6. BD = +6 = 0 = 10, so sin EBF = sin ABD = AD BD = 10 = 10 10, and cos EBF = Also, CE = + = and BEF and AEC are supplementary, so sin BEF = sin AEC = AC CE = and cos BEF = cos AEC =. Finally: m BFE = 180 m BEF + m EBF ( ) cos BFE = cos( BEF + EBF ) ( ) = 10 cos BEF cos EBF sin BEF sin EBF = = 0 0 0, and sec BFE = = 10 = B E A F D Figure 6 C

Minnesota State High School Mathematics League

Minnesota State High School Mathematics League 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