WYSE MATH REGIONAL 2014 SOLUTIONS. 1. Ans C: There are eight vertices in a cube and six in an octahedron.

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1 WYSE MATH REGIONAL 04 SOLUTIONS. Ans C: There are eight vertices in a cube and six in an octahedron.. Ans E: There are C 9 ways to select the 9 doughnuts. This is 8,57,00 ways. There are 8 C ways to select glazed, 5 C ways to select chocolate frosted, and 0 C 4 8! 5! 0! ways to select 4 jelly doughnuts. This gives us. The total ways of!5!!! 4!! making such selection is,4,800. The probability is,4, ,57,00 =.. Ans B: P PR = T QR P( R) = T QR T QR P = R 4. Ans E: When multiplying two matrices, the left must have dimensions of m x n while the right has dimensions of n x p. This is multiplying a x by a x, so the inner dimensions do not match and no product exists. 5. Ans D: From the polar coordinate we know the radius is while the angle θ is π. π π Then x = cos = and y = sin =,.. The coordinate is ( ) Ans E: i = and i =, so their sum would be Ans B: Since LR =40, then the inscribed angle LSR = 70. LSH = 80 (a straight angle). Then RSH = 0 (80 70). 8. Ans A: If we let x be the side of one of the squares, then each triangle has sides of x, and the overall perimeter is x + x + x + x + x + x = 0x. Solving for 0x = gives us x = 0.. The area of each square is x = 0.0 square inches, and the area of each triangle is x x 0.07 square inches. This makes the overall area of the region square inches. 9. Ans A: By the binomial theorem, ( ) = x y = x 5x y + 0x y 0x y + 5xy y. 0. Ans E: The median is half the length of the sum of the bases. Then the longer base side can be found by solving for L in the following equation: ( L + 7) = 0. L =. The longer base is yards or 9 feet. The shortest row is 7 yards or feet. Larger row has 9 plants while the shortest has plants. The difference is 8 plants.

2 WYSE MATH REGIONAL 04 SOLUTIONS. Ans A: Let h be the height of the pile above eye level, and d the distance from the center of the pile to the girl. Based on angles given in degrees, the two measurements give us h tan = d + 0 and tan8 = h h. Solve for d to get d = 0 and d = h d tan tan8. h h h h Substitute and solve for h. 0 = = 0 tan tan8 tan tan8 0 h = 0 h = 4.8. When we add the five for eye level, tan tan8 tan tan8 we end up with a dirt pile that is nine feet tall.. Ans D: Since it is an arithmetic sequence with a negative common difference, there is an infinite number of increasingly negative terms. So for every negative number, there s a term in the sequence for which the partial sum is less than that number. This is the very definition of a series sum going to negative infinity. Alternatively, there s an infinite number of terms which are less than negative and only a finite number of positive terms, so the sum would go to negative infinity.. Ans A: A regular hexagon is made up of equilateral triangles. Since the radius is 0 inches, each side of the hexagon is also 0 inches. Then the perimeter of the hexagon is 0 inches times the sides. The perimeter is 0 inches. 4. Ans D: There are 0 letters in iconoclasm, two of which are repeated twice each. So the 0! number of distinguishable strings is = 907,00.!! 5. Ans C: ( ) m n mn mn m n m n mn mn. Ans B: 0 have been to Africa and Europe, 5 Africa only, 0 Europe only, and 45 neither. Of the 75 who have not gone to Africa, 45 have not gone to Europe. 45 / Ans E: Using D = RT and solving for T, we get that the trip upstream takes = 5..5 hours and the return trip takes = 4 hours. The total trip thus takes 9. hours Ans D: Let x represent the dispense rate of the first machine in terms of peanut m&m s per minute and y represent the dispense rate of the second machine in terms of peanut m&m s per minute. Then 8x + 5y = 480. Since the first machine cannot dispense more than 40 peanut m&m s per minute, 0 x 40. Solving for x in our equation we have x = 0 0.5y. Then y 40. Solving for y in this inequality we find y 9 peanut m&m s per minute.

3 WYSE MATH REGIONAL 04 SOLUTIONS 9. Ans D: Using the formula mt r FV = P +, m = P + *5, P 75.. n(n ) 8(8 ) 0. Ans C: An n-gon has diagonals. If n is 8, this would be = 0.. Ans C: x x + 4 = 8 x 8 = x + 4 x x + 4 = x + 4 x 7x + 0 = 0. Then ( x )( x 5) = 0 x = or x = 5. Checking for extraneous roots we find that x 5. So x =.. Ans A: Let x be the number of hours a cookie takes to make and let y be the number of hours a cake takes to make. A single elf can make cookies and cakes or 9 cookies and cakes in a 48 hour period, so x + y = 48 while 9x + y = 48. By multiplying the first equation through by and the second one through by, we get that 8x + 9y = 44 while 8x + 4y = 9. By subtracting the two equations, 5y = 48, so y = 9.. Thus, each cake takes 9. hours to make and by dividing that into the 48 hours, the elf makes five complete cakes.. Ans B: Using Des Cartes rule and synthetic division the possible roots are ±, ± 4, ±, ±, ±, ±, ±, ±, ±. We will first test - to see if it is a root of the 4 4 polynomial by using synthetic division: Since the signs on the bottom alternate there are no negative roots less than -. Since the bottom row last entry is not 0, we know x + is not a factor of the polynomial. Next we will try. 4 4 this root is a zero of the polynomial. Therefore x + is 4 0 a factor. Next we will look for a positive root. We will try. 4 this root is also a zero of the polynomial. Then x is a factor of the polynomial. Since ( x )( x + ) = 4x. We divide the original polynomial by 4x. Here we find the following factors for the original polynomial: x + x x x +. ( )( )( )

4 WYSE MATH REGIONAL 04 SOLUTIONS 4. Ans E: Denominators need to be non-zero and radical terms need to be non-negative. The left term gives us x > 0 or x >, the right 5 x > 0 or x < 5. We need both restrictions in effect, giving a resulting domain of (, 5). 5. Ans A: Quadrilaterals with diagonals of equal length are classified as either squares, rectangles or isosceles trapezoids. Of those, only squares must have congruent adjacent sides.. Ans A: Since the det is 84; = 84. Then 4 K K 4 4K 0 4( 5k 0) + ( 0 8) = 84. So 4K 0 + 0K = 84. 4K = 0 K = Ans D: For a cylinder, V = πr h. A square cross section means h = r, so V = πr. We were given V = 00, so r = Since SA = πr + πrh = πr, SA Ans D: a and b are equivalent as we used a double angle formula for sine. We used it again to get c. We used angle addition on a to get e. d is, of course, -cos 4x, using a double angle formula, so it is not equivalent to the other four. x 9. Ans C: ( 5 ) ( ) 5 x+ = 5 = 5 x + = x 8 = x x = 4. x ( ) 5 5 x x+ x 5 = 5. So 0. Ans B: First, note that 85 is the product of the prime numbers and 7. Second, the rational root theorem states that, for a polynomial with integral coefficients, all rational roots have numerators which are factors of the constant term (in this case, they would have numerators of either,, 7 or 85 or their opposites) and their denominators are factors of the leading coefficient (so they could have denominators of, or 4 or their opposites). is not possible as none of the possible numerators is a multiple of.. Ans E: Choose generic point (x, y) on the parabola and equate the distance from point to focus with distance from point to directrix. Then solve for y to find the equation of the parabola. ( ) ( ) 0 ( ) ( ) x + y = y x + y = y. Then x x + + y 4y + 4 y = 0 y = x x Ans E: Since A and B are supplementary, their measure adds up to 80 degrees. Since B and C are congruent, C must have the same measure as B, so A and C also add up to 80. However, the statement about C and D being acute simply means both have measure less than 90 degrees. This forces A to have a measure between 90 and 80 degrees, so that immediately eliminates a), c), and d). It is possible for b) to be true, but only if C and D have the same measure, which was not required.

5 WYSE MATH REGIONAL 04 SOLUTIONS. Ans C: To find the angle of elevation, draw an imaginary right triangle whose base (which extends from the crest of the head to a place that is up the Eiffel Tower) has a length of 00 feet (00 yards) and whose height (which runs up the Eiffel Tower) is 05 feet, inches (the height of the Eiffel Tower less the height of the human). Thus 05'" 8" the tangent of the angle in question would be =, and by using the 00' 00" degree version of the arctangent function, we get C. 4. Ans D: Using the diagram below, ( ) Plane = cos.5 d = d = 70 d cos.5 ( ) ft..5 goal line 5. Ans C: Let x be the gallons of pure syrup, then create an equation that keeps track of the syrup in the mix: *0. + x* + (4 x)*0 = 0*0.5. Solve to get x = Ans C: Note that the x-value oscillates between - and, as does the y-value. Thus if we can get x- and y-coordinates whose absolute values are, that would give the π maximum possible distance of. Thankfully, in t =, we have such a value. 7. Ans B: =, = x y. hypotenuse = ( ) ( ) ( ) hyp + =. secθ = = =. adj ( ) log 4x 4x + log x log(x ) 8. Ans A: Note that =. For x >, this is. For log( x ) log( x ) log(x ) any x other than, that result is equivalent to. So the limit must be. 9. Ans A: Numbers in ascending order are,,,,4,4,4,5,,7,7,8,9,9,0. The middle score is the 8 th score (found by calculating 5 + ). The eighth score is Ans C: First consider h-t-h matches. C beat A and D, and tied B for a point total of 5. For A to have two ties, A must have lost to C and tied B and D for a point total of. D s only tie came against A, so the B-D match didn t end in a tie. Since B can t have a point total of, B must have won the match. This means B tied A and C and beat D for a point total of 4, and D lost to B and C and tied A for a point total of. In individual, D got points from OC and A got points from EC. Since the two who tied BT must be A and B (wasn t C, must include B by IV) that means A and B each get more points from PE. This brings A s total up to, B s total up to, C s total up to 5, and D s.

2014 Academic Challenge

2014 Academic Challenge 014 Academic Challenge MATHEMATICS TEST - EGIONAL This Test Consists of 40 Questions Mathematics Test Production Team Kevin Boyer, Illinois State University Author/Team Leader Linda Wiggins, Illinois State