首发文字版，2019amc10a晋级2019 AIME cutoff 分数线待公布
- a square
- a rectangle that is not a square
- a rhombus that is not a square
- a parallelogram that is not a rectangle or a rhombus
- an isosceles trapezoid that is not a parallelogram
8.The figure below shows line with a regular, infinite, recurring pattern of squares and line segments.
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
- some rotation around a point of line
- some translation in the direction parallel to line
- the reflection across line
- some reflection across a line perpendicular to line
9.What is the greatest three-digit positive integer for which the sum of the first positive integers is a divisor of the product of the first positive integers?
10.A rectangular floor that is feet wide and feet long is tiled with one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?
11.How many positive integer divisors of are perfect squares or perfect cubes (or both)?
12.Melanie computes the mean , the median , and the modes of the values that are the dates in the months of . Thus her data consist of , , . . . , , , , and . Let be the median of the modes. Which of the following statements is true?
13.Let be an isosceles triangle with and . Contruct the circle with diameter , and let and be the other intersection points of the circle with the sides and , respectively. Let be the intersection of the diagonals of the quadrilateral . What is the degree measure of
14.For a set of four distinct lines in a plane, there are exactly distinct points that lie on two or more of the lines. What is the sum of all possible values of ?
for all Then can be written as , where and are relatively prime positive inegers. What is
16.The figure below shows circles of radius within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius
17.A child builds towers using identically shaped cubes of different color. How many different towers with a height cubes can the child build with red cubes, blue cubes, and green cubes? (One cube will be left out.)
where is a real number?
20.The numbers are randomly placed into the squares of a grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
21.A sphere with center has radius . A triangle with sides of length and is situated in space so that each of its sides is tangent to the sphere. What is the distance between and the plane determined by the triangle?
22.Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval . Two random numbers and are chosen independently in this manner. What is the probability that ?
23.Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number , then Todd must say the next two numbers ( and ), then Tucker must say the next three numbers (, , ), then Tadd must say the next four numbers (, , , ), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number is reached. What is the th number said by Tadd?
24.Let , , and be the distinct roots of the polynomial . It is given that there exist real numbers , , and such thatfor all . What is ?
an integer? (Recall that .)