Snow College Mathematics Contest

Size: px

Start display at page:

Download "Snow College Mathematics Contest"

Allen Jefferson
5 years ago
Views:

1 Snow College Mathematics Contest key Aril, 08 Senior Division: Grades 0- Form: T Bubble in the single best choice for each question you choose to answer.. If log 0 =0.699 what is log 0 00? log 0 00 = log 0 ( 00) = log 0 + log 0 00 = The number of sots on a articular normal Dalmatian dog is divisible by. When the number of sots is divided by the number of legs, a remainder of results. The sots can also be divided by the total of legs, ears, eyes, and tail to leave a remainder of 6. What is the minimum number of sots? The lcm of and 4 is =. Adding the remainder gives. The lcm of and 9 is =9. Adding the remainder 6 gives as before. The lcm of, 4, and 9 is = 6 so other ossible numbers are + 6 or, in general, + 6n, n =0,,,,.... Three dice are rolled. What is the robability that none of them show a or? Probability of one die not showing or is 4 6 =. Three dice:.

2 4. Define Z = {..., 9, 6,, 0,, 6, 9,...}. If we define rimes in Z as those ositive numbers that cannot be exressed as roducts of smaller ositive elements of the set, what is the sum of the first three ositive comosite (non-rime) numbers in Z? The comosite numbers are, 6, 9,,, 8,... ( ) = 8 = 4 The roerty of the uniqueness (u to ordering) of rime factorization does not hold in this ring; that is, the fundamental theorem of arithmetic does not aly to this system! For examle, and 6 6 are two distinct rime factorizations of 6 in Z.. An equilateral triangle has a side of length a. What is the area of the largest circle which can be drawn within this triangle? 8 a a a a a Draw a right triangle inside with sides r and a and hyotenuse r. a =(r) r ) r = a A = r = a Or use tan 0 =/ =r/(a/). 6. Ashereisinscribedinacube. Whatis the ratio of the volume of the shere to the volume of the cube? Let r be the radius of the shere. The shere volume is 4 r and the cube volume will be (r) =8r. The ratio of the shere volume to the cube volume will be 4 r = 8r Find the measure of an angle that is both the comlement of \A and the sulement of \B if m\a + m\b = Simlify. Let \C be the angle in question. Then m\a + m\c = 90 and m\b + m\c = 80. m\a + m\c + m\b + m\c = 70. m\c = 70 6 (log 64 6) (log 6 64) (log 6 7) (log 6). 4 6 Change of base formula gives attern where all will cancel excet the st numerator and the last denominator leaving (log 6) / (log ) = log 6.

3 9. Acircleandasquarehavethesameerimeter. Then their areas are equal. the area of the circle is the greater. the area of the square is the greater. the area of the circle is times the area of the square. None of these A = P > A = P 4 6 For a given erimeter, a circle encloses the most area of any shae. 0. How many two-digit whole numbers are exactly 7 times the sum of their digits? 0 4 0a + b = 7(a + b) =) a =6b =) a =b {, 4, 6, 84}. Convert the reeating decimal into a fraction. After reducing to lowest terms, find the difference between the denominator and the numerator =) x = = 0 00x = x = x = 60. Tau, who loves eating ieces of i, discovered that when the digits of a three-digit natural number are rearranged to form a second number, the difference between the two numbers is usually divisible by Original number: abc = 00a+0b+c If bac = 00b + 0a + c = 90a 90b If bca = 00b + 0c + a = 99a 90b 9c If cab = 00c + 0a + b = 90a +9b 99c If cba = 00c + 0b + a = 99a 99c If acb = 00a + 0c + b =9b 9c. Given three distinct one-digit numbers (i.e., from the set {,,, 4,, 6, 7, 8, 9}), what is the robability that two of them add u to the third one? There are 9 C = 84 ossible trios. If is the smallest of the three, the numbers, 4,, 6, 7, 8, 9 are all ossible sums, which gives 7 sums. If is the smallest, there are ossible sums, etc. There are = 6 total ossible sums, giving a robability of 6 84 = 4.

get to that sot in the triangle using the rules above.

4 4. Given these rules, how many diﬀerent aths can you take to sell SNOWMATH? Begin at the to Move only down For each move, go to one of the letters directly below S N N O O O 0 W W W W M M M M M 0 A A A A T T T 70 H H Each entry in Pascal s triangle is the number of ways to get to that sot in the triangle using the rules above. So there are ways to get to each H: + ( + ) + ( + + ) + ( ) + ( ) + ( + + ) + ( + ) = 40 mh 4 mh 4 mh 48 mh 0 mh The answer is the same for any size triangle, but let s assume a secific case for ease of illustration: s = 0 mi, so the total tri is 60 mi. The first leg takes (0 mi)/(0 mi/h) = 4 h; likewise, the second leg takes h, and the last leg takes h, for a total of 9 h. The average seed is 60 mi/9 h = 40 mi/h. Marilyn vos Savant in Parade, Dec. 7, Towns A, B, and C are at the corners of a triangle with equal sides. A car travels at constant seed from A to B at 0 mh, from B to C at 40 mh, and from C back to A at 60 mh. What is the average seed for the round tri? Two semicircles are in a quarter-circle. What is the area of the shaded region in ft? Divide the overla of the semicircles in half with a diagonal. Then rotate each half around the oint of intersection creating a shaded region with a quarter-circle outer edge and a straight line below. The white triangle has an area of 8 ft and the full quarter-circle has an area of 4 ft. 7. Matrix X is an inverse to matrix A if AX = XA = I. Which of the following cannot have an inverse matrix? a b, then det = ad bc. c d A is invertible iﬀ det 6= 0. If A =

5 8. How many different ositive integers less than 08 are multiles of 0 or 8 (inclusive)? The number of multiles of 0 less than 08 is 08 0 = 00. For 8, there are 08 8 =. However, these sets overla at the multiles of 80, of which there are =. ) there are 00 + = 0 different numbers that are multiles of 0 or What is 08 mod 6? = Driving on a certain street, Prof. B can hit stolights green 90% of the time. What is the robability that he will hit exactly two of the next three lights green, assuming indeendent events? Probability of hitting the first two lights green and the last not green is There are =different (mutually exclusive) ways to do this.. At a certain business, ackages are delivered through a square delivery chute, with each side of the chute measuring ft.suosea rectangular box of dimensions 6ft 4ft x is to be delivered through the chute. What is the maximum value of x? 4 4 If the long dimension of the inner rectangle is 4, then each of the gray lines must have a length of, as we are bisecting a isosceles right triangle with hyotenuse 4. The diagonal of the chute is 8 or, so the width of the box is x = ().. If f(x) =x, findf(f(f())) f() = 7 f(7) = 9 f(9) =. Suose f(ln x) = x.findf (x). ln x (ln x) e x e x ln x If f(ln x) = x, then y = f(x) = e x. To find f (x), switch x and y and solve for y. To solve x = e y,we have x = e y which gives y =lnx.

6 X 4. Simlify the sum i n if i =. 0 i i n= i P 4 n= in = i i + = 0 and since P = 4(8) +, n= in = i i.. An aquarium on a level table has a square base 0 in wide and is 8in tall. When tilted, the water in it just covers one of the 0 in 8inends but only three-fourths of the square bottom. What is the deth of the water when the bottom is again level? in in. in 4in 6in When tilted, the water forms a triangular rism with volume (7.in)(0 in)(8in) = 00 in. With the square base (0 in)(0 in) =00 in, the height will be in. 6. Suose the Earth is a erfect shere and that there is a steel belt fitting snugly around it at the equator. If the length of the belt were increased by 0 feet, how far above the Earth would the belt be raised if it remained circular and centered around the Earth? Less than inch Between inch and inches Between inches and foot Between foot and feet More than feet The change in the circumference of the belt is C = 0 and C = r so r = C/. 7. In regular octagon ABCDEF GH, what is the fraction of the total octagon area found in rectangle ADEH? 7 G H F s A The area of a regular olygon is a where a is the aothem length (erendicular distance from center to one side) and is the erimeter. If s is the length of a side, then =8s and the area of the octagon is 4as. The rectangle width will be s and length a. 8. Shannon takes her favorite number, adds to it, multilies the answer by 0, subtracts 0 from the result and then dros the final 0. If Shannon s (correct) answer is 9, what is her favorite number? = E a B D 0(x+) 0 0 = x + =) x =6 9. If the standard order of oerations is reversed (addition and subtraction are done first and exonentiation is done last), what is the value of ^+? ( )^( + ) = 6 = 7776 C

7 0. Simlify. tan t sin t cos t tan t sin t sin t cos t cos t Use tan t = sin t/ cos t and simlify the fraction. cos t =sin t.. Different shades of ink, red, and white can be made by mixing whole numbers of quarts of red and white aint. Shades are different if the ratio of red to white is different. Find the number of different ossible shades that can be made from at most 4 quarts of red and quarts of white aint = 7 red white 0 any 0,,,, 4,,,,, 4, 4,,. Eric the Shee is at the end of a line of 0 shee waiting to be shorn. But being imatient, Eric sneaks u the line two laces every time the shearer takes a shee from the front to be shorn. So, for examle, while the first shee is being shorn, Eric moves ahead so that there are two shee behind him in line. If at some oint it is only ossible for Eric to move one lace, he does so. How many shee get shorn before Eric? When the st shee is taken, Eric moves to number 48. When the 6th shee is taken Eric has moved u laces, making him #8. There is one shee (#7) ahead of him. When the shearer takes number 7, Eric is next.. How many strings of length 7 of a s, b s, and c s are there such that all the a s, if any, show u at the beginning? For examle: aaaabcb is accetable but abbbbab is not The sum of a finite geometric sequence = 8 =. 4. Find the values of c such that the trinomial x +cx 4 can be factored over integers. {, 9, 9, } { 0, 4,, 9} { 4,,, 4} {,,, } {,, 9,,, 9,, } The ossible combinations of factors of 4 are: and 4, and 4, and 7, and and 7. The sums of the airs are the ossible values for c.

The ratio between any two successive elements must be the same. x x 0 = =) x = 0 x 0 40 The elevation of 0 makes a 0-6090 triangle and the deression of 4 makes a 4-4-90 triangle.

8 . From my balcony, I measure the angle of elevation to the to of the neighboring building to be 0, and the angle of deression to the base to be 4. If my (erendicular) distance from the building is 0 m, how tall is the building in meters? If 0, x, and x are the first three terms of a geometric sequence, what is x? The ratio between any two successive elements must be the same. x x 0 = =) x = 0 x 0 40 The elevation of 0 makes a triangle and the deression of 4 makes a triangle. The oosite sides, x and y in the figure, make u the height of the building. Since the legs of the triangle are the same length, y = 0 m. In a triangle the sides are x, x, and x, 0 0 so x = 0 m, so x = =. 8. Car A is mi ahead of car B, which is going in the same direction. 8 min later car A is only mi ahead of car B. On average, how much faster is car B traveling? 6. Simlify comletely:r q 0 x40 x40 x0 40 x0 x0 x 0 x x x (((x40 )/ )/ )/ = x40/0 = mh 7. mh 0 mh mh not enough info Let di and vi be the distance and seed cari travels in the 8 min. Then da = 8vA and db = 8vB = da +. Substitute and subtract to get 8(vB va ) = mi ) (vb va ) = 8 min = 60 8 mh. Shorter: Car B travels mi more in mi 8 min: 8 min = 60 8 mh = 7. mh x4

9 9. Arollofaertowelscontains00aer towels. The tube around which the towels are rolled is cm in diameter. Including the tube, the whole roll is 4 cm in diameter. What is the diameter of the roll (and tube) when there only 0 aer towels left? 6cm 7cm 8cm 0 cm cm The annular cross-sectional area of the aer towels on the full roll is (7 ) cm = 48 cm. If the new radius of the roll is R, then (R ) cm = 4 cm =) R =cm 40. During shooting ractice, a basketball layer takes one ste closer if she misses a shot, and one ste farther away if she makes a shot. After a while, she notices she is two stes farther away than when she began. What is the most we can say about her shooting ercentage P (i.e., shots made shots taken)? % <P ale 0% P>0% P>67% P>7% not enough info If she s farther away, then P>0%. Without knowing how many shots she took, however, there s no way to know how much better than 0%. If she took just two shots P = 00%; ifshetook thousands, P is slightly above 0%.

7. The olynomial P (x) =x, 9x 4 +9x, 4x +4x, 4 can be written in the form P (x) =(x, ) n Q(x), where n is a ositive integer and Q(x) is not divisible

7. The olynomial P (x) =x, 9x 4 +9x, 4x +4x, 4 can be written in the form P (x) =(x, ) n Q(x), where n is a ositive integer and Q(x) is not divisible . Which of the following intervals contains all of the real zeros of the function g(t) =t, t + t? (a) (,; ) (b) ( ; ) (c) (,; ) (d) (; ) (e) (,; ) 4 4. The exression ((a), + b), is equivalent to which