ข้อสอบแข่งขัน ลองทำกันเล่นๆ ครับ มีทั้งหมด 25 ข้อ
(1) Fai eats 5 cookies per day while Denton eats 7 cookies every two days. After 18 days, what is the positive difference between the amount of cookies Fai and Denton each ate?
(2) The Samsung Galaxy S5 that is sold at AOPS has a special offer which is $250 each, but there's no sales tax. The Samsung Galaxy S5 that is sold at Samsung is sold for $235 each, with a sales tax of 15%. How much does one save, in dollars when they buy 4 Samsung Galaxy S5 from AOPS rather than at Samsung?
(3) Jonathon has 17 dogs and some number of parrots. He knows that his pets have a total of 20 heads. What is the total count of feet of Jonathon's pets?
(4) Three fair coins are flipped simultaneously. What is the probability that more heads will come up than tails?
(5) We have two sets A and B. It is known that A ∪ B = 15 and A ∩ B = 5. How many elements are in A or B but not in both?
(6) How many primes are there less than 100 that is not divisible by 3?
(7) The maximum number of points that two circles intersect each other at is 2. What is the maximum number of points that two circles out of a total of 89 circles can intersect each other at?
(8) For the AMC 10, a score of 120/150 is required to get to the AIME, or a score in the top 2.5% of all competitors, whichever takes more people. A correct answer is rewarded 6 points, a skipped question is rewarded 1.5 points and an incorrect answer earns no points. Let's suppose that Peter only skips questions or gets questions correct. What is the minimum number of question he needs to answer to ensure a spot in the AIME?
(9) A sphere with radius 12, and a cylinder with the radius of the circular face also 12 is created such that the volume of the sphere and the cylinder are the same. What is the height of the cylinder?
(10) A right-angled triangle is draw with sides 2,1, and an unknown side. What is the sum of all possible areas of this triangle?
(11) Canada and USA are going to face off in a game of hockey in the Winter Olympics 2014, for the semi-finals. Each has a 50% chance of winning the game. In the other semi-final, Russia has a 40% chance of winning against Great Britain. The teams that face of in the final has a 60% chance of winning of the first letter of the name of their country comes first in the alphabet. (C is for Canada, U is for USA, R is for Russia and G is for Great Britain). What is the probability that Team Canada wins the finals?
(12) Rectangle ABCD has a point M on AB such that ∠AMD = ∠CMD. Also, AB/BC = 2. What is the measure of ∠AMD?
(13) John has some number of apples. When he tries to place them into groups of 2, he has 1 left over. When he tries to place them in groups of 3, he has 2 left over. When he places them to groups of 4, there's 3 left. And finally, when he places them into groups of 5, there are 4 left over. What is the least possible number of apples that John has?
(14) There are 26 positive integer solutions for (a,b) where 2a + b = 52. How many positive integer solutions (a,b) are there for the equation 2a + b = 52 when a ≤ 25, b ≥ 25 and b is divisible by 2?
(15) How many integer solutions (a,b) are there for the equation a
3 + 729 = b
3?
(16) A committee composed of Alice, Mark, Ben, Connie and Francisco is about to select three representatives randomly. What is the probability that Connie is excluded from the selection?
(17) There are 28 ways to get from (0,0) to (n,6) on the Cartesian Plane. What is the sum of all possible values of n?
(18) How many trailing zeros, meaning the number of consecutive zeros at the end of a number, does 3890! contain? (Note: 720 has one trailing zero and so does 10)
(19) A farmer owns 6 chicken. Every day, each chicken lays 2 eggs. He then arranges the eggs in any order, after 5 days, into 6 baskets. He puts any number of eggs into each basket, provided that we puts at least 1 egg into each basket. How many ways can we arrange the eggs?
(20) There are three points A, B, C which are connected to form ΔABC where AB = 23, AC = 24 and BC = 25. A line is drawn from A and meets BC at D, where AD = sqrt(9889)/5. What is the length of BD?
(21) In ΔABC, AB = 23, AC = 24, BC = 25 and ∠BAC = 80°. A line is drawn from A and meets BC at D, where CD > BD and BD × CD = 345,000/47
2. What is the measure of ∠BAD?
(22) There is a quadrilateral ABCD inscribed in circle O, with all the vertices of the quadrilateral on the circumference of the circle. Its sides are 14, 13, 12, and 11, in that specific order. What is the area of this cyclic quadrilateral?
(23) James is bored one day and decides to write out the first few terms of the sequence a
n = (n)(n!). His father, a mathematical expert, asks him to find the sum of all the terms he's written. After doing a lot of addition, James finds out that his sum is 362,879. How many terms did James write?
(24) A pyramid has a rectangle base ABCD of integer lengths with an area of 69 and a vertex E. We have that |ΔABE| = 39/2 and |ΔCBE| = 45/2. Also, all the side lengths in the pyramid has integer lengths, as well as the heights of the two triangular faces. What is the volume of the pyramid?
(25) A number is considered
magnificent if the number of the divisors of that number is a perfect square. How many
magnificent are there less than 100?
MOCK AMC 10 (2014)
(1) Fai eats 5 cookies per day while Denton eats 7 cookies every two days. After 18 days, what is the positive difference between the amount of cookies Fai and Denton each ate?
(2) The Samsung Galaxy S5 that is sold at AOPS has a special offer which is $250 each, but there's no sales tax. The Samsung Galaxy S5 that is sold at Samsung is sold for $235 each, with a sales tax of 15%. How much does one save, in dollars when they buy 4 Samsung Galaxy S5 from AOPS rather than at Samsung?
(3) Jonathon has 17 dogs and some number of parrots. He knows that his pets have a total of 20 heads. What is the total count of feet of Jonathon's pets?
(4) Three fair coins are flipped simultaneously. What is the probability that more heads will come up than tails?
(5) We have two sets A and B. It is known that A ∪ B = 15 and A ∩ B = 5. How many elements are in A or B but not in both?
(6) How many primes are there less than 100 that is not divisible by 3?
(7) The maximum number of points that two circles intersect each other at is 2. What is the maximum number of points that two circles out of a total of 89 circles can intersect each other at?
(8) For the AMC 10, a score of 120/150 is required to get to the AIME, or a score in the top 2.5% of all competitors, whichever takes more people. A correct answer is rewarded 6 points, a skipped question is rewarded 1.5 points and an incorrect answer earns no points. Let's suppose that Peter only skips questions or gets questions correct. What is the minimum number of question he needs to answer to ensure a spot in the AIME?
(9) A sphere with radius 12, and a cylinder with the radius of the circular face also 12 is created such that the volume of the sphere and the cylinder are the same. What is the height of the cylinder?
(10) A right-angled triangle is draw with sides 2,1, and an unknown side. What is the sum of all possible areas of this triangle?
(11) Canada and USA are going to face off in a game of hockey in the Winter Olympics 2014, for the semi-finals. Each has a 50% chance of winning the game. In the other semi-final, Russia has a 40% chance of winning against Great Britain. The teams that face of in the final has a 60% chance of winning of the first letter of the name of their country comes first in the alphabet. (C is for Canada, U is for USA, R is for Russia and G is for Great Britain). What is the probability that Team Canada wins the finals?
(12) Rectangle ABCD has a point M on AB such that ∠AMD = ∠CMD. Also, AB/BC = 2. What is the measure of ∠AMD?
(13) John has some number of apples. When he tries to place them into groups of 2, he has 1 left over. When he tries to place them in groups of 3, he has 2 left over. When he places them to groups of 4, there's 3 left. And finally, when he places them into groups of 5, there are 4 left over. What is the least possible number of apples that John has?
(14) There are 26 positive integer solutions for (a,b) where 2a + b = 52. How many positive integer solutions (a,b) are there for the equation 2a + b = 52 when a ≤ 25, b ≥ 25 and b is divisible by 2?
(15) How many integer solutions (a,b) are there for the equation a3 + 729 = b3?
(16) A committee composed of Alice, Mark, Ben, Connie and Francisco is about to select three representatives randomly. What is the probability that Connie is excluded from the selection?
(17) There are 28 ways to get from (0,0) to (n,6) on the Cartesian Plane. What is the sum of all possible values of n?
(18) How many trailing zeros, meaning the number of consecutive zeros at the end of a number, does 3890! contain? (Note: 720 has one trailing zero and so does 10)
(19) A farmer owns 6 chicken. Every day, each chicken lays 2 eggs. He then arranges the eggs in any order, after 5 days, into 6 baskets. He puts any number of eggs into each basket, provided that we puts at least 1 egg into each basket. How many ways can we arrange the eggs?
(20) There are three points A, B, C which are connected to form ΔABC where AB = 23, AC = 24 and BC = 25. A line is drawn from A and meets BC at D, where AD = sqrt(9889)/5. What is the length of BD?
(21) In ΔABC, AB = 23, AC = 24, BC = 25 and ∠BAC = 80°. A line is drawn from A and meets BC at D, where CD > BD and BD × CD = 345,000/472. What is the measure of ∠BAD?
(22) There is a quadrilateral ABCD inscribed in circle O, with all the vertices of the quadrilateral on the circumference of the circle. Its sides are 14, 13, 12, and 11, in that specific order. What is the area of this cyclic quadrilateral?
(23) James is bored one day and decides to write out the first few terms of the sequence an = (n)(n!). His father, a mathematical expert, asks him to find the sum of all the terms he's written. After doing a lot of addition, James finds out that his sum is 362,879. How many terms did James write?
(24) A pyramid has a rectangle base ABCD of integer lengths with an area of 69 and a vertex E. We have that |ΔABE| = 39/2 and |ΔCBE| = 45/2. Also, all the side lengths in the pyramid has integer lengths, as well as the heights of the two triangular faces. What is the volume of the pyramid?
(25) A number is considered magnificent if the number of the divisors of that number is a perfect square. How many magnificent are there less than 100?