Math Stars Regional Competition Sample Team Relays Round Problem Set A

Transcription

1 Math Stars 2016 Regional Competition Sample Team Relays Round Problem Set A School/Team Code Grade(s) Team Members Team Captain DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Number of Problems: 5 in this problem set Time Allotted: 45 minutes for all five problem sets combined Scientific calculators are permitted, but books or other aids are not permitted Answer in exact form (i.e. integer, common fraction...etc.) and round only when asked to do so. No units need to be provided after your answers. Please record only final answers in the blanks in the left-hand column of the competition paper. If your team completes the problems before time is called, use the remaining time to check your answers. Form Code A F 0 5 B G 1 6 C H 2 7 D I 3 8 E J 4 9 Total Correct Scorer s Initials CSSMA Major Sponsors University of Toronto UBC Math Club Canadian Mathematical Society Expii.inc. Various PAC committees

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3 1. 1. Determine the value of the following expression: Two standard 6-faced dice are rolled. How many ways are there to roll a total of a, where a is the answer to the previous problem? ABC has side lengths BC = b+1, AC = b+3, and AB = b+5, where b is the answer to the previous problem. What is the area of ABC? Determine the value of c, where c is the answer to the previous problem Given a circle with center O and chord AB of the circle. The acute AOB = (400 d), where d is the answer to the previous problem. Extend AO and BO beyond O so that each of these line segments intersects the circle at different points C and D respectively. What is the measure, in degrees, of CDO?

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5 Math Stars 2016 Regional Competition Sample Team Relays Round Problem Set B School/Team Code Grade(s) Team Members Team Captain DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Number of Problems: 5 in this problem set Time Allotted: 45 minutes for all five problem sets combined Scientific calculators are permitted, but books or other aids are not permitted Answer in exact form (i.e. integer, common fraction...etc.) and round only when asked to do so. No units need to be provided after your answers. Please record only final answers in the blanks in the left-hand column of the competition paper. If your team completes the problems before time is called, use the remaining time to check your answers. Form Code A F 0 5 B G 1 6 C H 2 7 D I 3 8 E J 4 9 Total Correct Scorer s Initials CSSMA Major Sponsors University of Toronto UBC Math Club Canadian Mathematical Society Expii.inc. Various PAC committees

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7 1. 1. If = x + x y, find the value of 7@ What is the minimum number of coins needed to make a sum of $20.15, if no more than a 2 coins of any single kind is allowed to be used and no bills are to be used? Note: a is the answer to the previous problem, and we live in Canada, where there are no pennies anymore The square of the answer to the previous problem can be written in the form p q r s 3 + t 1 for some whole numbers p, q, r, s, t. If 0 p, q, r, s, t 2, then what s the five digit integer pqrst? In a garden there are n roses, n 7 tulips, n 10 daffodils, and n 30 dandelions, where n is the remainder when the answer to the previous problem is divided by 82. How many flowers must one pick to make sure he or she has picked 2 pairs of flowers with the same color? The numbers 36, 27, d 4, d 14, d 18, d 15, 23, and 17 are grouped in pairs so that the sum of each pair is the same. Which number is paired with d 14? Note: d is the answer to the previous problem.

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9 Math Stars 2016 Regional Competition Sample Team Relays Round Problem Set C School/Team Code Grade(s) Team Members Team Captain DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Number of Problems: 5 in this problem set Time Allotted: 45 minutes for all five problem sets combined Scientific calculators are permitted, but books or other aids are not permitted Answer in exact form (i.e. integer, common fraction...etc.) and round only when asked to do so. No units need to be provided after your answers. Please record only final answers in the blanks in the left-hand column of the competition paper. If your team completes the problems before time is called, use the remaining time to check your answers. Form Code A F 0 5 B G 1 6 C H 2 7 D I 3 8 E J 4 9 Total Correct Scorer s Initials CSSMA Major Sponsors University of Toronto UBC Math Club Canadian Mathematical Society Expii.inc. Various PAC committees

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11 1. 1. What s the diameter of a circle with circumference 50π? 2. % 2. If x is a% larger than z, and y is (a 25)% larger than z, then x is what percentage larger than y? Note: a is the answer to the previous problem. 3. m 3. A coconut falls from a b 2 meter tall tree with a constant velocity at b b 10m/s, but a wind blows the coconut left at 20m/s while it s falling. You re running rightwards towards the tree from 15m away at 3m/s when the coconut just began falling. How far will the coconut be away from your feet six seconds later (in meters)? Note: b is the integer answer to the previous problem How many positive integer divisors does 2015n have, if n is the answer to the previous problem? Hint: prime factorize 2015n Let!! be the function such that: n (n 2) if n > 0 and odd n!! = n (n 2) if n > 0 and even 1 if n = 1, 0 Determine the number of zeroes at the end of the decimal representation of k!!, if k is the answer to the previous problem.

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13 Math Stars 2016 Regional Competition Sample Team Relays Round Problem Set D School/Team Code Grade(s) Team Members Team Captain DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Number of Problems: 5 in this problem set Time Allotted: 45 minutes for all five problem sets combined Scientific calculators are permitted, but books or other aids are not permitted Answer in exact form (i.e. integer, common fraction...etc.) and round only when asked to do so. No units need to be provided after your answers. Please record only final answers in the blanks in the left-hand column of the competition paper. If your team completes the problems before time is called, use the remaining time to check your answers. Form Code A F 0 5 B G 1 6 C H 2 7 D I 3 8 E J 4 9 Total Correct Scorer s Initials CSSMA Major Sponsors University of Toronto UBC Math Club Canadian Mathematical Society Expii.inc. Various PAC committees

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15 1. 1. What s the remainder when 1000 is divided by 200? What s the smallest integer greater than a that s divisible by 9, if a is the answer to the previous problem? W XY Z is a rectangle where Y Z is equal to b + 3. If the area of XY Z is equal to 3b + 3, what is the perimeter of the rectangle? Note: b is the sum of the digits of the answer to the previous problem What is the area of BAD with vertices B(c, 3), A(c, 5) and D(7, 8), where c is the remainder when the answer to the previous problem is divided by 11? Square EF GH has one vertex on each side of square ABCD. Point E is on AB with AE : EB = d : 1, where d + 1 is the largest single-digit positive integer divisor of the answer to the previous problem. The ratio of the area of EF GH to the area of ABCD can be written as a common fraction in lowest terms as m n, where m, n are integers. Determine the value of m + n.

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17 Math Stars 2016 Regional Competition Sample Team Relays Round Problem Set E School/Team Code Grade(s) Team Members Team Captain DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Number of Problems: 5 in this problem set Time Allotted: 45 minutes for all five problem sets combined Scientific calculators are permitted, but books or other aids are not permitted Answer in exact form (i.e. integer, common fraction...etc.) and round only when asked to do so. No units need to be provided after your answers. Please record only final answers in the blanks in the left-hand column of the competition paper. If your team completes the problems before time is called, use the remaining time to check your answers. Form Code A F 0 5 B G 1 6 C H 2 7 D I 3 8 E J 4 9 Total Correct Scorer s Initials CSSMA Major Sponsors University of Toronto UBC Math Club Canadian Mathematical Society Expii.inc. Various PAC committees

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19 1. 1. Two standard six-sided dice are tossed. The probability of rolling two 1 s can be written as a common fraction in lowest terms as m n where m, n are integers. Determine the value of m n The circumference of a circle is a 9π, where a is the answer to the previous problem. What s the area of the largest square that can fit completely inside this circle? A triangle has side lengths of 6, b, and x. What s the minimum possible integer value for x, if b is the answer to the previous problem? The dwarf planet Pluto was officially named on May 1, in a year between 1900 and c, where c is the answer to the previous problem. The sum of the digits of the year is a prime, and the number of years that has passed since then (as of 2015) is a multiple of 17. In what year was Pluto officially named? A d 1900 page booklet is flipped from its first page to its last page, where d is the answer to the previous problem. However, every page whose page number is divisible by another page number (excluding the first page) already flipped past is ripped out. What is the sum of all of the page numbers that is not ripped out?

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