Thirty-Ninth Annual Columbus State Invitational Mathematics Tournament

Thirty-Ninth Annual Columbus State Invitational Mathematics Tournament Sponsored by The Columbus State University Department of Mathematics and Philosophy March nd, 0 ************************* Solutions I

. There are 0 people in a room, 0% percent of whom are men. If no men enter or leave the room, how many women must enter the room so that 40% of the total number of people in the room are men? (A) (B) 0 (C) (D) 5 (E) 0 Answer: D There are 8 men 0% of 0. Let x be the number of women to enter the room. Then 8 percentage of men in the room is given by 00 = 40. Solving for x yields 0 + x x = 5.. What is the product of the roots of the equation (x + 4)(x + ) + (x + )(x + ) = 0? (A) (B) 0 (C) 4 (D) 48 (E) 9 Factoring (x + 4)(x + ) + (x + )(x + ) = (x + )(x + 0) = 0, the roots are x = and x = 5, product is 0.. If the domain for the function f(x) = is (, ), which of the following x + x + c best describes all possible values of c? (A) c > (B) c = (C) c < (D) c (E) c > Answer: A Note x + x + c 0, thus the discriminant b 4ac < 0, that is 4 4c < 0 and c >. 4. The quadratic polynomial f(x) satisfies the equation f(x) f(x ) = 4x for all x. If s and t are the coefficients of x and x, respectively, in f(x), what is the value of s + t? (A) (B) (C) (D) 4 (E) 5 Answer B f(x) = sx + tx + c and f(x ) = s(x ) + t(x ) + c. Simplifying and solving sx + tx + c (s(x ) + t(x ) + c) = 4x yields s = and t = therefore s + t =. II

5. Max read on the website of Games and Stuff that a certain 00 dollar computer game was given a % discount; when Max arrived at the store, it was announced at the door that the same game had an additional 7% discount. When Max went to pay for it, he received a third discount of 7%. What single discount would have given the same cost as a series discounts of %, 7%, and 7%? Round your answer to two decimal places. (A) 7.8% (B) 7.00% (C) 58.80% (D) 4.0% (E) 8.40% Answer: C With first discount the cost is 8% of original price, with the second discount the cost is 7% of 8%, which is 7 8 of the original price. With the third discount, the cost 00 00 will be 8% of 7 8 8, which yields 7 8, ie the final cost is 4.0% of original 00 00 00 00 00 cost. Single discount would be 58.80%.. Which of the following is equivalent to 5 + 5 +? (A) + 5 + (B) + 5 + (C) + 5 + (D) 5 + (E) + 5 Answer: E Rationalize the denominator, i.e multiply both the numerator and denominator by the conjugate + 5 which gives 5 + 5 + + 5 + 5 = 5( + 5 ) ( + 5) 8 = + 5. 7. You own a motel and have a pricing structure that encourages rentals of rooms in groups. One room rents for $7, two for $70.50 each, and in general the group rate per room is found by taking $.50 off the base of $7 for each extra room rented. Find a formula for the function R = R(n) that gives the total revenue for renting n rooms to a convention host. (A) R(n) = 70.50 + 7(n + ) (B) R(n) = 70.50 7(n ) (C) R(n) = [7.50(n )]n (D) R(n) = [7.50(n + )]n (E) R(n) = [7.50(n.50)]n III

Answer: C If n rooms are rented then the number of $.50 reductions is (n ). Thus the revenue of each room is [7.50(n )] dollars. Total revenue is R(n) = [7.50(n )]n dollars. 8. If a + b = a + b, then find the value of b a + a b. (A) (B) (C) (D) (E) Answer: A Multiplying both sides of the equation a + b = a + b and + b a + a b + =. Thus b a + a b =. a + b by (a+b) gives a + a + b b 9. If the geometric mean of two positive real numbers a and b (a > b) is equal to 4 and their average is 5, then what is the value of x that satisfies the equation a = b x? (A) (B) (C) (D) 4 (E) 5 = Answer: C We have 0 < b < a, ab = 4 and a + b = 5. So a and b satisfy ab = 4 and a + b = 0. Thus a = 8 and b =, so 8 = x for x =. 0. What is the sum of the real roots of (x x) x = x xx? (A) 8 7 (B) 7 4 (C) 9 4 (D) 4 9 (E) 4 Answer: E Writing in exponent form x x = x x for x > 0 and x. Solving for x in x = x, we get x = 9 4. Note x = is also a root. Sum of roots is 4. The geometric mean of nonnegative numbers a and b is ab. IV

. Which of the following is the probability of getting exactly three sixes when you roll n fair dice? (A) n!! (C) (E) ( ( ) ( ) n 5 (B) ) ( ) n 5 (D) ( ) n ( ) 5 n!!(n )! n!!(n )! ( ( 5 ) ( ) n 5 ) ( ) n 5 Answer:D From P (X = ) = n!!(n )! (. If y = π, compute the value of (+y). ) ( ) n 5, simplify to get n!!(n )! ( 5 ) ( ) n 5. (A) 8 π (B) 9 π (C) π (D) 9 π (E) 8π From y = π, we have y = 9 π. Thus (+y) = ( y ) = 9 π.. Three straight lines, l, l and l, have slopes, and, respectively. All three lines 4 have the same y-intercept. If the sum of the x-intercepts of the three lines is, what is the y-intercept? (A) (B) (C) 4 (D) 4 (E) 9 Answer: C The equations of the lines l, l and l, are y = x + b, y = x + b and y = 4 x + b respectively. The x-intercepts are x = b, x = b and x = 4b. Summing them yields 9b =, thus b = 4. V

4. Evaluate + + + + +... + 0 + 0. (A) + 0 (B) 0 (C) 0 0 + 0 (D) + 0 (E) 0 Note that = n + n n n n (n ) = n n. Therefore + + + + +... + 0 + 0 = + ( ) + ( ) + + ( 0 0) + ( 0 0). Which telescopes to 0. 5. Three rugs have a combined area of 00 m. By overlapping the rugs to cover a floor area of 40 m, the area which is covered by exactly two layers of rug is 4 m. What area of floor is covered by three layers of rug? (A) m (B) 8 m (C) 4 m (D) m (E) 4 m Let x be the area covered by three layers, then we have one extra layer of 4 m and two extra layers of x m. Thus 40 + 4 + x = 00, giving x = 8. VI

. In figure, the parabola has x- intercepts - and 4, and y-intercept 8. If the parabola passes through the point (, w), what is the value of w? (A) 5 (B) (C) 7 (D) 8 (E) 9 Answer: D Use symmetry f(0) = f(), thus w = 8 (or find the quadratic function). 7. What is the number of triples (x, y, z) such that when any of these numbers is added to the product of the other two, the result is? (A) (B) (C) (D) 4 (E) 5 We have x + yz =, y + xz =, z + xy = From the first two equations we obtain x y + yz xz = 0. Factoring yield ( z)(x y) = 0, thus z =, x = y. From z =, we have (x, y, z) = (,, ). From x = y, we obtain x + xz =, z + x = from the first and third equations. Solving the system for x gives (x ) (x + ) = 0, thus x = and x =. Therefore (x, y, z) = (,, ). 8. Two six sided dice have each of their faces painted either blue or yellow. The first die has five blue faces and one yellow face. When the dice are rolled, the probability that the two top faces show the same color is. How many yellow faces are there on the second die? (A) 5 (B) 4 (C) (D) (E) VII

Answer: C Once rolled the color of the second die must match the first with probability of. This only happens if there are blue and yellow faces on the second die. ( ) 0 9. Write + i in the form a + bi. (A) i (B) + i (C) i (D) (E) i Answer: ( A ) 0 + i = ( + ) 0 i = ( ( ) ) e i π 0 4 = e i(0+) π 4 = e i π 4 = + i. 0. An increasing sequence is formed so that the difference between consecutive terms is a constant. If the first four terms of this sequences are x, y, x + y and x + y +, what is the value of y x? (A) (B) (C) 4 (D) 5 (E) Answer:E Let the common difference be d, so y x = d, x+y y = d and x+y+ (x+y) = d. From the first and third equations we have x = and substituting in third equation we get d =. Thus y x =.. For x < 0, simplify (x x ). (A) x (B) x (C) x- x (D) x (E) x Answer: A For x < 0, (x x ) = x ( x) = x, thus (x x ) < 0 and (x x ) = (x x ) = (x x ) = x. VIII

. Assume a and b are integers between 0 and 9, if a79b is the decimal representation of a number in base 0, such that it is divisible by 7, determine a + b. (A) (B) (C) 4 (D) 5 (E) Answer D Write a79b = 0000 a + 790 + b in mod8. We get + b 0 mod 8, i.e b =. Expressing a79 = 0000 a + 790 + in mod 9, we obtain a + 4 + 0 mod 9, thus a =. Sum a + b = 5.. What is the remainder when 0 0 is divided by 7? (A) (B) 4 (C) (D) (E) Answer:A Write 0 mod 7. Since mod 7, we write 0 0 ( ) 5 mod 7. 4. Three neon lights colored red, blue and green flash at different time intervals. The red light flashes after every 4 seconds, the blue light flashes after every 8 seconds and the green light after every 5 seconds. If all the three lights flash together at 8:00 am, how many times will all three lights flash together by 9:0 am? (A) (B) 5 (C) 8 (D) (E) 4 Consider the least common multiple of 4, 8 and 5 which is 0. The lights flash together every minutes. In 90 minutes, all the three lights will flash together 5 times. 5. If x = a b and y = b log a b b,which expression is equivalent to x y? (A) a b (B) a b b (C) a b b (D) a b (E) a b b b Answer: A x y = ( a b) b log a b b = a log a b = a b. IX

. Find the exact value of the following product sin sin... sin 45 sec 4... sec 89. (A) (B) (C) (D) (E) From complementary angles sin sin... sin 45 sec 4... sec 89 = cos 89 cos 88... cos 4 sin 45 sec 4... sec 89. The right side telescopes to sin 45 = 7. Given x = c + c + c +..., where x is a positive integer, which of the following is a possible values of c? (A) 9 (B) 0 (C) (D) 8 (E) 0 Answer: E Square both sides of the equation x = c + c + c +..., we get x = c+ c + c + c +... i.e x = c + x. From the resulting quadratic equation x x c = 0, we need two numbers whose sum= and product= c. Thus c is a product of two consecutive numbers, namely 4 and 5. 8. In figure, the equation of the line containing AD is y = (x ). BD bisects ADC. If the coordinates of B are (p, q), what is the value of q? (A) (B).5 (C) 0 (D) (E) X

Answer: D Slope of line segmentad is, thus tan ADC = and so ADC = π. We have BDC = π q =. and DC =. Therefore tan BDC = BC DC yields BC =. Thus 9. What is the number of pairs of positive integers (s, t), with s + t 0, that satisfy the equation s + t s + t = 99? (A) 5 (B) 7 (C) 8 (D) 0 (E) 5 Answer: D Simplifying s + t impling t 0 00 s + t = 99, we get s = 99. From s + t 0, we get 99t + t 0 t. Thus t 0 yields 0 such pairs (s, t). 0. The numbers a, a,..., a are positive consecutive integers that sum to 0. Which is the sum a + a +... + a? (A) 858 (B) 859 (C) 8540 (D) 854 (E) 854 From the sum a, a,..., a = 0, we get a = and a =. Since the numbers are consecutive a + a +... + a = a + (a + ) +... + (a + 0). Thus a + a +... + a = ()(4)(7) ( + ) = 859.. The quadratic equations 5x 9x + = 0 and x 7x + = 0 have a common solution. Which is the sum of the other solutions? (A) 5 (B) 4 5 (C) 5 (D) 8 5 (E) 0 5 XI

Answer: A Equating 5x 9x+ = x 7x+ and solving for x yields the common solution x =. Since (x ) is a factor, solve for a in (x )(5x a) = 5x 9x + which yields a = 9 which implies x = is the other solution of 5 5x 9x + = 0. Similarly solve for b in (x )(x b) = x 7x + to get x = as the other solution of 7 x 7x + = 0. The sum of other solutions is. 5. Let a n equal the integer closest to n. For example a = a since = and.4 and a = since a.7. What is the sum + + + + + + + equal to? a a a a 4 a 08 a 09 a 0 (A) 8 (B) 9 (C) 0 (D) (E) Answer: C Trying a few terms in the sequence; a n = for values of n, a n = for 4 values of n, a n = for values of n,...a n = k for k values of n. Since 0 0 and we have includded all a n 0, implying 0 values of n. Thus + + + + + + + a a a a 4 a 08 a 09 = () + 4 ( ) + ( ) +... + 0 ( 0. In figure, AB = BC = CD = DA = BD = and AE = CE = 4. What is the length of line segment corresponding to DE? ) = 0. (A) 4 0 (B) (C) 7 (D) 0 (E) a 0 XII

Answer: D Let F be the intercestion of AC and BD. Note ABD = 0 and AF =. Thus F E = 4 ( ) =. Therefore DE = F E F D = 0. 4. Two points are chosen at random on a circle. What is the probability that the distance between them is more than the radius of the circle? (A) 4 (B) (C) (D) 4 (E) Measure one radius from a point A on the circle, swing an arc from A any point on arc BAC is less than one radius away from A. This is of the circle. Any point on the other part which is of the circumfrence is greater than one radius away from point A. Thus Probability is. 5. The equation 0 = x + 4y has only two solution pairs (x, y) of positive integers. Considering these pairs, which is the largest value of x + y? (A) 4 (B) 4 (C) 44 (D) 45 (E) 4 Answer: D Consider x < y, then x and y 7 yield the pair (, 7). For x > y, we have y and x 7 yielding the pair (4, ). Thus the largest value of x + y = 45.. Find the sum of all possible values of x satisfying the logarithmic equation log 5x+9 ( x + x + 9 ) + log x+ ( 5x + 4x + 7 ) = 4. (A) 5 (B) (C) 0 (D) (E) Answer:A Change of base yields log x+(x + ) log x+ 5x + 9 + log x+ ((5x + 9)(x + )) = 4. Simplifying log x+ 5x + 9 +log x+ ((5x + 9))+ = 4. Let u = log x+ (5x+9) and solve the resulting quadratic equation. The solutions are x = 0, x =, and x =. Therefore sum of all possible values of x is 5. XIII

7. In figure 4, ABCDEF GH is a cube of side-lengths equal to meters. On the side AD there is an ant at the midpoint M of it. The ant travels on the faces of this cube to the point N located meters away from G on F G. What is the shortest distance (in meters) that this ant can travel to arrive at N? (A) 549 (B) 548 (C) 547 (D) 54 (E) 585 Answer:A Openning up the cube into nets, the shortest distance from M to N is through face CDHG. 8. A line l is parallel to the line y = 5 4 x + 95 and it intersects the x-axis and y-axis at 4 points A and B, respectively. If l also passes through (, 5), how many points with integer coordinates are there on the line segment AB (including the endpoints of AB)? (A) 4 (B) 5 (C) (D) 7 (E) 8 The equation of line l is y = 5(x 9). To obtain integer coordinates x 9 must be an 4 4 integer. Therefore x =, x = 7, x =, x = 5 and x = 9. XIV

9. Assume that two real numbers a and b satisfy a + ab + b = and t = ab a b. What is the range for the values of t? (A) 0 t (B) t (C) t (D) 4 t (E) t 4 Answer: C We have t + = ab and (a + b) = t+ 0. So a and b are real roots of t + x ± x + t + = 0. So t+ (t + t) 0. Ths t 40. In figure 5 we have three circles of radii, and tangent to each other at points P, Q and R. The triangle formed by their centers area equal to 0 a. What is the value of a? (A) 7 (B) 7 (C) 7 (D) 74 (E) 75 Answer: E Using Hero s Formula A = p(p α)(p β)(p γ), where p = α + β + γ 75()()() = 75(0). Thus a = 75. we have XV

4. Find the limit lim x 0 5x + 8. x (A) 5 (B) 7 (C) (D) 4 (E) Answer:A Using L Hopital s rule lim x 0 5x + 8 x = lim x 0 5 (5x + 8) = 5. 4. Two circles with radii a and b are tangent to each other as shown in figure. The ray OA contains the diameter of each circle, and the ray OB is tangent to each circle. Which of the following is equivalent to cos θ? (A) a + b b a (D) a b a + b (B) ab a + b (E) a + b ab (C) b a ab Note sin θ = a h = b a(a + b). Solving for h = h + a + b b a and cos θ = ab a + b. 4. The equation 9x = a + ln x has a unique positive solution in x if a is a certain positive real number. What is the value of 0e a? (A) 4 (B) (C) (D) (E) 0 XVI

Answer:E The function f(x) = 9x ln x has a derivative equal to f(x) = 7x which shows x that f has a global minimum at x = of f() = ln( ) = + ln. Clearly a must be this minimum, and so 0e a / = 0e ln = 0. 44. The square ABCD in figure 7 has side lengths 4 meters. Point E is on AC with AC = 4EC. A circle centered at E is tangent to two sides of the square. AG is tangent to the circle at F. What is the length AF? (A) (B) 7 (C) (D) 4 (E) 4 Diagonal AC has length 4, AE = and F C =. The radius EF =, hence AF = 7. 45. If every solution of the equation (cos x) (cos x) = 0 is a solution of the equation a(cos x) + b(cos x) = 0, what is the value of a + b? (A) (B) (C) (D) 4 (E) 5 Answer: C Use quadratic formula to obtain cos x = 5. Plugging in cos x = 5 into equation a(cos x) +b(cos x) = 0, yields a = and b = 4. Hence sum a+b =. XVII

4. In figure 8, the circle with center P has radius and is tangent to both the positive x-axis and the positive y-axis, as shown. Also, the circle with center Q has radius and is tangent to both the positive x-axis and the circle with center P. The line L is tangent to both circles. What is the y-intercept of L? (A) 8 (B) + (C) 0+ (D) 9+ (E) 0+ Answer: D Let A and B indicate the y and x-intercepts of line L respectively. Let QB = x and using similarity x = 4 + x, we get x =. We have F B = and OBA = 0. By similarity we find EF =. From tan OBA = OA, and we obtain OA = 9 +. OB 47. The graph of the function g is the reflection of the graph of f(x) = x + e x defined for all real x about the line y = x. What is g dg () (i.e. dx ())? (A) (B) (C) 4 (D) 5 (E) Answer: A From the given information g is the inverse of f. So, g(f(x)) = x implies g (f(x))f (x) =. Then for x = 0 we get g (f(0))f (0) = or g () =. 48. Knowing that there is exactly one pair (x, y) of two positive integers x and y satisfying the equation x + y = 0, what is x + y? (A) 0 (B) (C) (D) (E) 5 XVIII

Answer: D Decompose 0 = ()() and observe that = 5 + = 5 +. Hence, we can take x = (5) and y = (), which turns out to be the only solution of the proposed equation. Then x + y = (). 49. How many ordered pairs (m, n) of positive integers are solutions of the equation 0 m + n =? (A) (B) (C) 4 (D) (E) 8 Answer: D m = n = gives pair (m, n). Consider m < n and find values m such that 0 < m < for which n = m is an positive integer. This yields 9 pairs (m, n). m 0 Consider n < m and find values of n such that < n < for which m = 0n n is a positive integer. This yields pairs (m, n). Total number of ordered pairs (m, n) is. 50. In figure 9, P QR is a right triangle with A and B on P Q. Also, C is on QR, and BC is parallel to P R. If AB =, P A =, P R = 4, and the area of ACR is 5, what is BQ? (A) (B).5 (C) (D) (E) Answer: A Let BQ = x and BC = h. Area of P QR = 0 + x, area of P RA =, RAC = 5, ACB = h and CBQ = xh. Equating the areas we obtain the equation xh + h 4x + = 0, by similarity 5+x = x we obtain the equation xh 4x = 5h. 4 h XIX

Solving the two equations we obtain h = and x =. XX