Log1 Contest Round 3 Theta Individual. 4 points each

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1 0 04 Log Contest Round Theta Individual Name: 4 points each 0 degrees Celsius is equivalent to degrees Fahrenheit, and 0 degrees Celsius is equivalent to degrees Fahrenheit. The number of degrees Fahrenheit is a firstdegree polynomial in terms of the number of degrees Celsius. How many degrees are in the Fahrenheit temperature that is exactly twice its equivalent Celsius temperature? 4 Write in simplest radical form: 9 If a and d are real numbers, find the value of a kd Find an expression for the inverse function f. k0 x if x e f x. 5 One-half of one-third of one-fourth of one-fifth of one sixth of a number equals. Find the number. 7 Solve for matrix X: 5 points each 5 7 X Find the imaginary solution to the equation z i z i 0, where i. 8 Mikey was buying some Life TM cereal since he likes it. In the first store he spent half of his money plus $ on Life TM cereal. In the second store he spent one-third of his remaining money plus $ on Life TM cereal. In the third store he spent one-fourth of his remaining money on Life TM cereal. If Mikey came home with $, how much did he have before he started buying Life TM cereal, in dollars? 9 The three terms in the sequence log 5, log x, log 480 form an arithmetic progression. What is the value of x? ABC has a right angle at B. The altitude of the triangle from B to AC intersects AC at point D. If AD 5 and CD 7, find the area enclosed by ABC.

2 points each Thom Yorke has identical Valentine s Day chocolates that that he is going to distribute to 5 kids, one of whom is Kid A. Kid A cannot receive more than Valentine s Day chocolates, or else he will get a toothache. In how many ways can Thom Yorke distribute the Valentine s Day chocolates? Assume the chocolates are indistinguishable from each other. A positive integer k is added to each number in the sequence 4, 5, 04 to yield a new sequence, and the new sequence consists of the squares of three consecutive terms of an arithmetic sequence. Find the value of k. A box contains one chip labeled A, two chips labeled B, three chips labeled C, and so on up to twenty-six chips labeled Z. If chips are removed from this box one at a time without noting the letter labeled on the chips, what is the minimum number of chips that must be removed from the box in order to guarantee that there are at least ten chips removed with the same label? 4 The region bounded by the graphs of the y -axis. Find the volume of the resulting solid. y x and the -axis x is revolved about 5 Find the positive difference between the lengths of the radii of the circumscribed and inscribed circles of the triangle whose side lengths are 7, 8, and.

3 0 04 Log Contest Round Alpha Individual Name: 4 points each 0 degrees Celsius is equivalent to degrees Fahrenheit, and 0 degrees Celsius is equivalent to degrees Fahrenheit. The number of degrees Fahrenheit is a firstdegree polynomial in terms of the number of degrees Celsius. How many degrees are in the Fahrenheit temperature that is exactly twice its equivalent Celsius temperature? If a and d are real numbers, find the value of a kd Find an expression for the inverse function f. k0 x if x e f x. 4 One-half of one-third of one-fourth of one-fifth of one sixth of a number equals. Find the number. 5 Solve for matrix X: 5 7 X points each Find the imaginary solution to the equation z i z i 0, where i. 7 Mikey was buying some Life TM cereal since he likes it. In the first store he spent half of his money plus $ on Life TM cereal. In the second store he spent one-third of his remaining money plus $ on Life TM cereal. In the third store he spent one-fourth of his remaining money on Life TM cereal. If Mikey came home with $, how much did he have before he started buying Life TM cereal, in dollars? 8 How many five-digit positive integers are such that all five digits are of the same parity (odd or even)? 9 The three terms in the sequence log 5, log x, log 480 form an arithmetic progression. What is the value of x? Thom Yorke has identical Valentine s Day chocolates that that he is going to distribute to 5 kids, one of whom is Kid A. Kid A cannot receive more than Valentine s Day chocolates, or else he will get a toothache. In how many ways can Thom Yorke distribute the Valentine s Day chocolates? Assume the chocolates are indistinguishable from each other.

4 points each A positive integer k is added to each number in the sequence 4, 5, 04 to yield a new sequence, and the new sequence consists of the squares of three consecutive terms of an arithmetic sequence. Find the value of k. Given that 0a b and, what is the numerical value of a b a b? a b 8ab A box contains one chip labeled A, two chips labeled B, three chips labeled C, and so on up to twenty-six chips labeled Z. If chips are removed from this box one at a time without noting the letter labeled on the chips, what is the minimum number of chips that must be removed from the box in order to guarantee that there are at least ten chips removed with the same label? 4 5 The region bounded by the graphs of y x and the x -axis is revolved about the y -axis. Find the volume of the resulting solid. 4 Aand B are acute angles satisfying AB 45. If tan A, find the value of tanb. 7

5 0 04 Log Contest Round Mu Individual Name: 4 points each 0 degrees Celsius is equivalent to degrees Fahrenheit, and 0 degrees Celsius is equivalent to degrees Fahrenheit. The number of degrees Fahrenheit is a firstdegree polynomial in terms of the number of degrees Celsius. How many degrees are in the Fahrenheit temperature that is exactly twice its equivalent Celsius temperature? If a and d are real numbers, find the value of a kd. k0 dy Find the solution y f x to the differential equation when x. x x dx, where y 0 4 One-half of one-third of one-fourth of one-fifth of one sixth of a number equals. Find the number. 5 Find the tens digit of the quantity points each Find the imaginary solution to the equation z i z i 0, where i. 7 How many five-digit positive integers are such that all five digits are of the same parity (odd or even)? 8 Find the area enclosed by the curves with equations y x x and y x x. 9 The three terms in the sequence log 5, log x, log 480 form an arithmetic progression. What is the value of x? Thom Yorke has identical Valentine s Day chocolates that that he is going to distribute to 5 kids, one of whom is Kid A. Kid A cannot receive more than Valentine s Day chocolates, or else he will get a toothache. In how many ways can Thom Yorke distribute the Valentine s Day chocolates? Assume the chocolates are indistinguishable from each other.

6 Given that 0a b and If a b 8ab points each, what is the numerical value of a b a b? 9 x 9 x, where x 0, find the value of x. A box contains one chip labeled A, two chips labeled B, three chips labeled C, and so on up to twenty-six chips labeled Z. If chips are removed from this box one at a time without noting the letter labeled on the chips, what is the minimum number of chips that must be removed from the box in order to guarantee that there are at least ten chips removed with the same label? 4 Evaluate as a single fraction: x 4x 0 5 If f x ax bx, where a and b are real numbers, has a point of inflection at find the numerical value of a b. dx,,

7 0 04 Log Contest Round Theta Individual Name: 4 points each 0 degrees Celsius is equivalent to degrees Fahrenheit, and 0 degrees Celsius is equivalent to degrees Fahrenheit. The number of degrees Fahrenheit is a firstdegree polynomial in terms of the number of degrees Celsius. How many degrees are in the Fahrenheit temperature that is exactly twice its equivalent Celsius temperature? 4 Write in simplest radical form: 9 If a and d are real numbers, find the value of a kd Find an expression for the inverse function f. k0 x if x e f x. 5 One-half of one-third of one-fourth of one-fifth of one sixth of a number equals. Find the number. 0 9 a 55d ln x 0 7 Solve for matrix X: 5 points each 5 7 X Find the imaginary solution to the equation z i z i 0, where i. i 8 Mikey was buying some Life TM cereal since he likes it. In the first store he spent half of his money plus $ on Life TM cereal. In the second store he spent one-third of his remaining money plus $ on Life TM cereal. In the third store he spent one-fourth of his remaining money on Life TM cereal. If Mikey came home with $, how much did he have before he started buying Life TM cereal, in dollars? 9 The three terms in the sequence log 5, log x, log 480 form an arithmetic progression. What is the value of x? ABC has a right angle at B. The altitude of the triangle from B to AC intersects AC at point D. If AD 5 and CD 7, find the area enclosed by ABC

8 points each Thom Yorke has identical Valentine s Day chocolates that that he is going to distribute to 5 kids, one of whom is Kid A. Kid A cannot receive more than Valentine s Day chocolates, or else he will get a toothache. In how many ways can Thom Yorke distribute the Valentine s Day chocolates? Assume the chocolates are indistinguishable from each other. A positive integer k is added to each number in the sequence 4, 5, 04 to yield a new sequence, and the new sequence consists of the squares of three consecutive terms of an arithmetic sequence. Find the value of k. A box contains one chip labeled A, two chips labeled B, three chips labeled C, and so on up to twenty-six chips labeled Z. If chips are removed from this box one at a time without noting the letter labeled on the chips, what is the minimum number of chips that must be removed from the box in order to guarantee that there are at least ten chips removed with the same label? 4 The region bounded by the graphs of the y -axis. Find the volume of the resulting solid. y x and the -axis x is revolved about 5 Find the positive difference between the lengths of the radii of the circumscribed and inscribed circles of the triangle whose side lengths are 7, 8, and

9 0 04 Log Contest Round Alpha Individual Name: 4 points each 0 degrees Celsius is equivalent to degrees Fahrenheit, and 0 degrees Celsius is equivalent to degrees Fahrenheit. The number of degrees Fahrenheit is a firstdegree polynomial in terms of the number of degrees Celsius. How many degrees are in the Fahrenheit temperature that is exactly twice its equivalent Celsius temperature? If a and d are real numbers, find the value of a kd Find an expression for the inverse function f. k0 x if x e f x. 4 One-half of one-third of one-fourth of one-fifth of one sixth of a number equals. Find the number. 5 Solve for matrix X: 5 7 X a 55d ln x points each Find the imaginary solution to the equation z i z i 0, where i. i 7 Mikey was buying some Life TM cereal since he likes it. In the first store he spent half of his money plus $ on Life TM cereal. In the second store he spent one-third of his remaining money plus $ on Life TM cereal. In the third store he spent one-fourth of his remaining money on Life TM cereal. If Mikey came home with $, how much did he have before he started buying Life TM cereal, in dollars? 8 How many five-digit positive integers are such that all five digits are of the same parity (odd or even)? 9 The three terms in the sequence log 5, log x, log 480 form an arithmetic progression. What is the value of x? Thom Yorke has identical Valentine s Day chocolates that that he is going to distribute to 5 kids, one of whom is Kid A. Kid A cannot receive more than Valentine s Day chocolates, or else he will get a toothache. In how many ways can Thom Yorke distribute the Valentine s Day chocolates? Assume the chocolates are indistinguishable from each other

10 points each A positive integer k is added to each number in the sequence 4, 5, 04 to yield a new sequence, and the new sequence consists of the squares of three consecutive terms of an arithmetic sequence. Find the value of k. Given that 0a b and a b 8ab, what is the numerical value of a b a b? A box contains one chip labeled A, two chips labeled B, three chips labeled C, and so on up to twenty-six chips labeled Z. If chips are removed from this box one at a time without noting the letter labeled on the chips, what is the minimum number of chips that must be removed from the box in order to guarantee that there are at least ten chips removed with the same label? 4 5 The region bounded by the graphs of y x and the x -axis is revolved about the y -axis. Find the volume of the resulting solid. 4 Aand B are acute angles satisfying AB 45. If tan A, find the value of tanb

11 0 04 Log Contest Round Mu Individual Name: 4 points each 0 degrees Celsius is equivalent to degrees Fahrenheit, and 0 degrees Celsius is equivalent to degrees Fahrenheit. The number of degrees Fahrenheit is a firstdegree polynomial in terms of the number of degrees Celsius. How many degrees are in the Fahrenheit temperature that is exactly twice its equivalent Celsius temperature? If a and d are real numbers, find the value of a kd. k0 dy Find the solution y f x to the differential equation when x. x x dx, where y 0 4 One-half of one-third of one-fourth of one-fifth of one sixth of a number equals. Find the number. 5 Find the tens digit of the quantity 0 a 55d y x x x points each Find the imaginary solution to the equation z i z i 0, where i. i 7 How many five-digit positive integers are such that all five digits are of the same parity (odd or even)? 8 Find the area enclosed by the curves with equations y x x and y x x 9 The three terms in the sequence log 5, log x, log 480 form an arithmetic progression. What is the value of x? Thom Yorke has identical Valentine s Day chocolates that that he is going to distribute to 5 kids, one of whom is Kid A. Kid A cannot receive more than Valentine s Day chocolates, or else he will get a toothache. In how many ways can Thom Yorke distribute the Valentine s Day chocolates? Assume the chocolates are indistinguishable from each other

12 Given that 0a b and If a b 8ab points each, what is the numerical value of a b a b? 9 x 9 x, where x 0, find the value of x. 4 5 A box contains one chip labeled A, two chips labeled B, three chips labeled C, and so on up to twenty-six chips labeled Z. If chips are removed from this box one at a time without noting the letter labeled on the chips, what is the minimum number of chips that must be removed from the box in order to guarantee that there are at least ten chips removed with the same label? 4 Evaluate as a single fraction: x 4x 0 5 If f x ax bx, where a and b are real numbers, has a point of inflection at find the numerical value of a b. dx,,

13 0 04 Log Contest Round Individual Solutions Mu Al Th Solution Using the given information, the relationship is F.8C. Therefore, F.8 0.5F 0.9F 0.F F Since this is an arithmetic series with terms, a kd a a d a d a 55 d. x f e f x x e x f x ln x y x x dx x x x C k0. Plugging in C. Therefore, y x x x x x,0 yields 0 C X X X X Using Euler s theorem, 7 7 mod0. Therefore, mod0, so we only need to see what the tens digit of 0, , so the tens digit is. i 8i 4 7 is. In modulus i i 4 i i i z or i, so the imaginary solution is i. Working backwards, if Mikey came home with $ after spending one-fourth of his money in the third store, he went into the third store with $8. In the second store he spent one-third of his money plus $, so adding the $ yields $4, and if he spent onethird of his money, he went into the second store with $. Finally, he spent half of his money plus $ in the first store, so adding the $ yields $7, and if he spent half of his money, he went into the first store with $54. If all five digits are odd, then there are 5 choices for each digit, making a total of 5 5 numbers with all five digits odd. If all five digits are even, then there are 4 choices for the first digit (0 can t be the first digit) and 5 choices for the other four 4 digits, making a total of numbers with all five digits even. This makes a total of such numbers.

14 To find the intersection values, set the equations equal to each other: x x x x 0 5x 5x 5x x x 0 or x. Both curves are parabolas, so the one that opens downward will be the upper curve, and the area is x x x xdx 5x 5xdx x x Since log 5, log x, log 480 forms an arithmetic progression, 5, x, 480 forms a geometric progression. Therefore, x Because the triangle is right, BD The hypotenuse has length, so the enclosed area is 5 5. Examining the distribution casewise, if Kid A gets 0 chocolates, then Thom Yorke 4 distributes chocolates to 4 kids, which can be done in 8 ways. If Kid A gets chocolate, then Thom Yorke distributes 9 chocolates to 4 kids, 49 which can be done in 0 ways. If Kid A gets chocolates, then Thom Yorke distributes 8 chocolates to 4 kids, which can be done in ways. This yields a total of total distributions. From the given information, 4 k ad a d ad, 5k a, and 04 k ad a d ad. Adding the first and third equations yields 8 k a d 4 k a d. Plugging in the second equation yields 4 k 5 k d d 9d (since d 0). Now, taking the third equation and subtracting the first equation yields 80 4ad a a 5. Therefore, 5 k 5 5 k. a b First, since 0a b, 0. Now, if a b 8ab ab 40ab x. 8ab ab ab 9 a b x, then x a b 0 a b a b ab a b a b ab Cubing both sides of the equation, 9 x 9 x 8 x 9 x 9 x x 8 x x 80 x 4 5. At worst case, all of the chips labeled with letters from A to I can be drawn since there are not ten chips labeled with any one of the letters; this is chips. Of the remaining 7 letters, again at worst case, nine of each lettered chip could be drawn, which is 79 5 chips. Therefore, if one more chip was drawn, there must be chips with one specific letter removed. This makes the minimum number of chips removed

15 The graph is a semicircle symmetric to the y -axis, so the revolution yields a hemisphere with radius of length 4. Therefore, the volume is x x First, split the integral: dx dx dx. For the first 4x 4x 4x integral, use the substitution u 4 x, du xdx ; for the second integral, use the substitution v x, dv dx. This becomes du dv u 4v v x u arcsin 4x arcsin. Therefore, 4x arcsin 0 0 x 4x 0 x. 9 Using Hero s formula, since the semiperimeter of the triangle is 4, the area enclosed by the triangle is Therefore, the inscribed circle has radius of length 4 and the circumscribed circle has radius of length The positive difference in these lengths is tan B tan A tan B 4 4 tan 45 tan A B 7 tan B tan B tan Atan B tan B 7 tan B tan B 7 7, is on the graph of f, f 7a 9b 9a b. Additionally,. Since f has a point of inflection at Since f ' x ax bx f '' x ax b dx,, 0 f ' 8a b. Solving this system of equations yields a and b, so 9 8 ab. 9

Gauss School and Gauss Math Circle 2017 Gauss Math Tournament Grade 7-8 (Sprint Round 50 minutes)

Gauss School and Gauss Math Circle 2017 Gauss Math Tournament Grade 7-8 (Sprint Round 50 minutes) 1. Compute. 2. Solve for x: 3. What is the sum of the negative integers that satisfy the inequality 2x