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2019AIME II 完整真题及答案

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2019年aime II bo

测试时间:北美时间2019年3月21日,中国时间2019年3月22日

Question 1

Two different points,??and?, lie on the same side of line??so that??and??are congruent with?, and?. The intersection of these two triangular regions has area?, where??and??are relatively prime positive integers. Find?.

Question 2

Lily pads??lie in a row on a pond. A frog makes a sequence of jumps starting on pad?. From any pad??the frog jumps to either pad??or pad??chosen randomly with probability??and independently of other jumps. The probability that the frog visits pad??is?, where?and??are relatively prime positive integers. Find?.

Question 3

Find the number of?-tuples of positive integers??that satisfy the following system of equations:

Question 4

A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is?, where??and??are relatively prime positive integers. Find?.

Question 5

Four ambassadors and one advisor for each of them are to be seated at a round table with??chairs numbered in order??to?. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are??ways for the?people to be seated at the table under these conditions. Find the remainder when??is divided by?.

Question 6

In a Martian civilization, all logarithms whose bases are not specified as assumed to be base?, for some fixed?. A Martian student writes downand finds that this system of equations has a single real number solution?. Find?.

Question 7

Triangle??has side lengths?, and?. Lines?, and??are drawn parallel to?, and?, respectively, such that the intersections of?, and??with the interior of??are segments of lengths?, and?, respectively. Find the perimeter of the triangle whose sides lie on lines?, and?.

The eighth problem

The polynomial??has real coefficients not exceeding??and?. Find the remainder when??is divided by?.

Question 9

Call a positive integer??–pretty?if??has exactly??positive divisors and??is divisible by?. For example,??is?-pretty. Let??be the sum of the positive integers less than??that are?-pretty. Find?.

Question 10

There is a unique angle??between??and??such that for nonnegative integers??the value of??is positive when??is a multiple of?, and negative otherwise. The degree measure of??is?, where??and??are relatively prime positive integers. Find?.

Question 11

Triangle??has side lengths??and??Circle??passes through??and is tangent to line??at??Circle?passes through??and is tangent to line??at??Let??be the intersection of circles??and??not equal to??Then??where??and??are relatively prime positive integers. Find

Question 12

For??call a finite sequence??of positive integers?progressive?if??and??divides??for?. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to

13 questions

Regular octagon??is inscribed in a circle of area??Point??lies inside the circle so that the region bounded by??and the minor arc??of the circle has area??while the region bounded by??and the minor arc??of the circle has area??There is a positive integer??such that the area of the region bounded by??and the minor arc??of the circle is equal to??Find

Question 14

Find the sum of all positive integers??such that, given an unlimited supply of stamps of denominations??and??cents,??cents is the greatest postage that cannot be formed.

15 questions

In acute triangle??points??and??are the feet of the perpendiculars from??to??and from??to?, respectively. Line??intersects the circumcircle of??in two distinct points,??and?. Suppose?,?, and?. The value of??can be written in the form??where??and??are positive relatively prime integers. Find?.

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