Problem Solving Problems for Group 3(Due by EOC Feb. 13)

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1 Problem Solving Problems for Group (Due by EOC Feb. ) Eactly How Do You Want Your Million?. Find a positive number that you can add to,000,000 that will give you a larger value than if you multiplied this number by,000,000? Find all such numbers. {Hint: Let the positive number be, and solve,000,000,000,000.} Stand On Your Heads And Get It Together.. a) The sum of two numbers is 50, and their product is 5. Find the sum of their reciprocals. {Hint: y 50, y 5, so divide the first equation by the second equation.} b) The sum of the squares of two numbers is 50, and their product is 5. Find the ratio of the two numbers.. What is the value of if Mind Your Four s And Two s 0,000 0, ? {Hint: Factor 0,000 0, and use the fact that 8.} 4. Solve for if 90 Cover All Your Bases, If It s Within Your Power {Hint: Any number raised to the zero power, ecept zero itself, equals. raised to any power is equal to. - raised to an even power is equal to.} 5. Find the eact value of The Collapse Of Rationalism ,999,000,000. {Hint: Rationalize the denominators. For eample:.}

2 I Cannot Tell A Fib(onacci), My Name Is Lucas. f n f n f n for n,4,5, 6. If f, f, and a) What is the value of f? {Hint: f f f 4, f f f n n Amazingly, f can be represented as f n y for n,,,4,5, b) Find the values of and y. {Hint: f y, f, f y, f 4 4 7, Keep going.}, this should be enough to find values of and y.} c) Alicia always climbs steps,, or 4 at a time. For eample, she climbs 4 steps by ---, --, --,--,-, or 4. In how many ways can she climb 0 steps? {Hint: If 4 f n represents the number of ways to climb to the n th step, then f n f n f n f n.} Highs And Lows In The Classroom. 7. a) The grades on si tests all range from 0 to 00 inclusive. If the average for the si tests is 9, what is the lowest possible grade on any one of the tests? T T T T T T 9 T T T T4 T5 T T 558 T T T4 T5 T6, so T will be as small as possible when T T T T T is as large as possible.} {Hint: b) If the average for the si tests is 6, what is the highest possible grade on any one of the tests? Looky Here Son, This Is A Problem, Not A Chicken. 8. Foghorn C sounds every 4 seconds, and foghorn D sounds every 8 seconds. If they sound together at noon, what time will it be when they net sound together? Foghorn C :00 :00:4 :0:08 :0:4 Foghorn D :00 :00:8 :0:6 sound together {Hint: Every time they sound together after noon will have to be both a multiple of 4 seconds after noon and a multiple of 8 seconds after noon.}

3 The Last One Standing Find the ones digit of 7. {Hint: Look for a pattern: Powers of One s-digit Powers of 7 One s digit } Sorry, I Can t Give You Change For A Dollar. 0. a) What is the largest amount of money in current U.S. coins(pennies, nickels, dimes, quarters, but no half-dollars or dollars) you can have and still not have change for a dollar? {Hint: It s more than 99 cents. For instance: quarters and dimes is $.05, but you can t make change for a dollar.} b) A collection of coins is made up of an equal number of pennies, nickels, dimes, and quarters. What is the largest possible value of the collection which is less than $? c) Trina has two dozen coins, all dimes and nickels, worth between $.7 and $.. What is the least number of dimes she could have?. If ab and a b Solving Without Completely Solving. 8 8, then find a b. {Hint: Start squaring and substituting.} Gerry Benzel s Favorite Problems.. a) A bottle and a cork together cost $.0. If the bottle costs $.00 more than the cork, what does the cork cost? {Hint: Let be the cost of the cork and y the cost of the bottle.} b) A shirt and a tie sell for $9.50. The shirt costs $5.50 more than the tie. What is the cost of the tie?

4 Two Smokin Good Problems In One.. a) Mrs. Puffem, a heavy smoker for many years finally decided to stop smoking altogether. I ll finish the 7 cigarettes I have left, she said to herself, and never smoke another one. It was Mrs. Puffem s practice to smoke eactly two-thirds of each complete cigarette(the cigarettes are filterless). It did not take her long to discover that with the aid of some tape, she could stick three butts together to make a new complete cigarette. With 7 cigarettes on hand, how many complete cigarettes can she smoke before she gives up smoking forever, and what portion of a cigarette will remain? {Hint: With 7 complete cigarettes, she can smoke 7 complete ones and assemble 9 new complete ones, keep going.} b) Hobo Hal makes his cigars by connecting 5 cigar butts. Hal smokes 4 of a cigar and 5 then stops, leaving a cigar butt. If Hal finds 65 cigar butts, how many cigars will he be able to make and smoke? Grazin In The Grass Is A Gas, Baby, Can You Dig It? 4. a) A horse is tethered by a rope to a corner on the outside of a square corral that is 0 feet on each side. The horse can graze at a distance of 8 feet from the corner of the corral where the rope is tied. What is the total grazing area for the horse? {Hint: } b) Do the same calculation if the horse can graze at a distance of feet from the corner of the corral where the rope is tied.

5 Round And Round With Donald And Hillary 5. Donald and Hillary are racing cars around a track. Donald can make a complete circuit in 7 seconds, and Hillary completes a circuit in 68 seconds. a) If they start together at the starting line, how many seconds will it take for Hillary to pass Donald at the starting line for the first time. {Hint: Every time Donald reaches the starting line must be a multiple of 7 seconds, and every time Hillary reaches the starting line must be a multiple of 68 seconds. So they will both be at the starting line at common multiples of 7 and 68.} b) If they start together, how many laps will Donald have completed when Hillary has completed one more lap than Donald? {Hint: Let n be the number of laps completed by Donald, then 7n 68n.} 6. If a It Squares; It Cubes; It Does It All! b 0 and ab 5, then what is the value of a ab b? {Hint: a b a ab b a b.} It s Just Gotta Be a) Show that for any integer n, 6 divides n n. n n n n n n n } {Hint: b) Show that for any integer n, 6 divides 5 n n. Real-valued Function, What s Your Function? 04 f f for all 0. Find 8. a) Suppose that f is a real-valued function with f. {Hint: Plug in and 007, and solve for f.} b) Suppose that f is a real-valued function with f f f. for all 0. Find

6 On The Mark, Off The Mark, Or Bull s Eye? 9. In the following figure, the curves are concentric circles with the indicated radii. Which shaded region has the larger area, the inner circle or the outer ring? 4 5 Calculate the area of each region and check your visual estimation ability. Don t Put All Of Your Eggs In One Basket! 0. If eggs are taken from a basket two at a time, then one egg remains in the basket. If eggs are taken three at a time from the same basket, then two eggs remain in the basket. If eggs are taken four at a time from the same basket, then three eggs remain in the basket. If eggs are taken five at a time from the same basket, then four eggs remain in the basket. If eggs are taken si at a time from the same basket, then five eggs remain in the basket. If eggs are taken seven at a time from the same basket, then no eggs remain in the basket. What is the fewest possible number of eggs in the basket? {Hint: If N is the number of eggs in the basket, then N must be a multiple of, N must be a multiple of, N must be a multiple of 4, N must be a multiple of 5, and N must be a multiple of 6. So N must be a multiple of the LCM of,, 4, 5, and 6. Also, N must be a multiple of 7.}

7 . Observe that Sometimes Reduction Can Get In The Way Of Induction Use inductive reasoning to make a conjecture about the value of 4 n. Use your conjecture to find the value of 4,000,000. Middleaged At 40?. The counting numbers are written in the pattern below. Find the middle number of the 40 th row Shifty Four.. The leftmost digit of a 6 digit number is 4. If it s shifted to be the rightmost digit, the new number is one-fourth of the original number. What s the original number? {Hint: If the original number is 4abcde, then 0 4 4abcde4, and if you subtract the right number from 0 4, you ll have the new number.} Zero Is My Hero. 4. If a and b are two unequal numbers, and a b, then find the eact numerical value of a b.

8 Slide Your Way Into An Answer. 5. What fractional part of the figure is shaded(assuming that line segments that appear parallel actually are, all angles are right angles, and the vertical line segments are equally spaced.)? See How Everything Lines Up. 6. Given the following incomplete distance chart for 4 points in a plane, find the distance from A to B. A B C D A 0? 9 B? C 5 0 D Sister Act. 7. Three sisters leave home on the same day. One returns every 5 days, another returns every 4 days, and the third returns every days. How many days until all three sisters meet at home again for the first time? The People Under The Stairs. 8. Three rectangles are connected as in the figure. The first rectangle is by ; the second rectangle is 4 by ; the third rectangle is 8 by 4. A line is drawn from a verte of the smallest rectangle to a verte of the largest rectangle. Find the area of the shaded region

9 What s The Difference? 9. Consider the two overlapping rectangles below: 8 feet 0 feet y 7 feet 5 feet What is the difference between the areas of the non-overlapping regions of the rectangles? {Hint: The area of the first non-overlapping region is 80 y. Find the area of the second non-overlapping region, and subtract it from the first.} Mission: Impossible/Possible Perimeters. 0. A rectangle has the following sides. Find all possible numerical values of the perimeter of the rectangle. 4y 8 y 0 6

10 Mary, Mary, Not So Contrary.. Lottie and Lucy Hill are both 90 years old. Mary Jones, on the other hand, is half again as old as she was when she was half again as old as she was when she lacked 5 years of being half as old as she is now. How old is Mary? Hint: M 5 is half again as old as she was when she was half again as old as she was when she lacked 5 years of being half as old as she is now, and M is her age now. Which One Are You Rooting For?. Which number is larger, the 0,000,000 th root of 0 or the,000,000 th root of? Hint: Raise both numbers to the 0,000,000 th power to decide. Sometimes You Got To Kiss A Lot Of Frogs.. There are some princes and frogs in a fairy-tale. As a group, they have 5 heads and 94 feet. How many princes, and how many frogs are there? I Guess I Have To Spell It Out For You. 4. How can 9 horses be used to fill the following row of 0 horse stalls? {Hint: See the title of the problem.} Irrationally Yours. 5. Show that there is an irrational number so that is a rational number. {Hint: Give a try for.} Eight Is Not Enough. 6. What is the fewest number of cards that can be selected without looking from a standard 5- card deck to guarantee that at least cards from each of the 4 suits will be selected? {Hint: What if you get hearts followed by diamonds followed by?}

11 Don t Bank On The Old Switcheroo. 7. An absentminded bank teller switches the dollars and cents when he cashed a check for Mr. Spencer, giving him dollars instead of cents, and cents instead of dollars. After buying a five cent newspaper, Mr. Spencer discovered he had left eactly twice as much as his original check. What was the original amount of the check? {Hint: If D = original number of dollars and C = original number of cents, then the original amount of the check in cents is X 00D C and the incorrect amount of the check in cents is Y 00C D. This leads to the equation 00C D 5 00D C, which can be rearranged into 98C99 D 5 or 99D 5 C. We need to find D so that C is a whole number of at most -digits.} 98 D 99 D C Pyramid Power. 8. In the following pyramid, the number in each stone is found by adding together the numbers in the two stones below it. Complete the pyramid

12 You May Be A King Of Comedy, But Do You Know How A Clock Works? 9. On a recent episode of Who Wants to Be a Millionaire with Cedric the Entertainer, the following question appeared. For which of the following times will the minute and hour hands of a clock form a right angle? a) 4:05 b) 5:0 c) :5 d) :50 The contestant chose answer a) and he was told that he was correct. He wasn t correct, in fact, none of the options are correct. For t measured in minutes after midnight, M t 6t represents the cumulative angle of the minute hand in degrees, and H t t represents the cumulative angle of the hour hand in degrees. In order for the two hands to form a right angle, the difference between the cumulative angle of the minute hand and the cumulative angle of the hour hand must be an odd multiple of 90. So we get that M t H t n 90; n,, t n n n 6t t n 90; n,, 90;,, 80 t ; n,, a) Use the previous formula to find the number of times from one midnight to the net that the minute and hour hands form a right angle. {Hint: 80 n # of minutes in a 4 hour period.} b) Use the same reasoning to find a formula for the times(in minutes after midnight) from one midnight to the net(inclusive) that the minute and hour hands point in eactly the same direction, and the number of times that it occurs. c) Use the same reasoning to find a formula for the times(in minutes after midnight) from one midnight to the net that the minute and hour hands point in eactly opposite directions, and the number of times that it occurs. Triangulate Your Answer. 40. What number is missing from the third triangle? Why?