Because we know that $5^3$ is a factor of $15!$ and $20!$ , the last three digits of both numbers is a $0$ , this means that the difference of the hundreds digits is also $\boxed{\textbf{(A) }0}$ .
3
第 3 题
几何·面积
Ana and Bonita were born on the same date in different years, n years apart. Last year Ana was 5 times as old as Bonita. This year Ana's age is the square of Bonita's age. What is n?
💡 解题思路
Let $A$ be the age of Ana and $B$ be the age of Bonita. Then,
4
第 4 题
综合
A box contains 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 15 balls of a single color will be drawn?
💡 解题思路
We try to find the worst case scenario where we can find the maximum number of balls that can be drawn while getting $<15$ of each color by applying the pigeonhole principle and through this we get a
5
第 5 题
规律与数列
What is the greatest number of consecutive integers whose sum is 45?
💡 解题思路
We might at first think that the answer would be $9$ , because $1+2+3 \dots +n = 45$ when $n = 9$ . But note that the problem says that they can be integers, not necessarily positive. Observe also tha
6
第 6 题
综合
For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?
💡 解题思路
This question is simply asking how many of the listed quadrilaterals are cyclic (since the point equidistant from all four vertices would be the center of the circumscribed circle). A square, a rectan
7
第 7 题
几何·面积
Two lines with slopes \dfrac{1}{2} and 2 intersect at (2,2) . What is the area of the triangle enclosed by these two lines and the line x+y=10 ?
💡 解题思路
Let's first work out the slope-intercept form of all three lines: $(x,y)=(2,2)$ and $y=\frac{x}{2} + b$ implies $2=\frac{2}{2} +b=> 2=1+b$ so $b=1$ , while $y=2x + c$ implies $2= 2 \cdot 2+c=> 2=4+c$
8
第 8 题
几何·面积
The figure below shows line \ell with a regular, infinite, recurring pattern of squares and line segments. [图] 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?
💡 解题思路
Statement $1$ is true. A $180^{\circ}$ rotation about the point half way between an up-facing square and a down-facing square will yield the same figure.
9
第 9 题
规律与数列
What is the greatest three-digit positive integer n for which the sum of the first n positive integers is \underline{not} a divisor of the product of the first n positive integers?
💡 解题思路
The sum of the first $n$ positive integers is $\frac{(n)(n+1)}{2}$ , and we want this to not be a divisor of $n!$ (the product of the first $n$ positive integers). Notice that if and only if $n+1$ wer
10
第 10 题
几何·面积
A rectangular floor that is 10 feet wide and 17 feet long is tiled with 170 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?
💡 解题思路
The number of tiles the bug visits is equal to $1$ plus the number of times it crosses a horizontal or vertical line. As it must cross $16$ horizontal lines and $9$ vertical lines, it must be that the
11
第 11 题
综合
💡 解题思路
Prime factorizing $201^9$ , we get $3^9\cdot67^9$ . A perfect square must have even powers of its prime factors, so our possible choices for our exponents to get a perfect square are $0, 2, 4, 6, 8$ f
12
第 12 题
统计
Melanie computes the mean \mu , the median M , and the modes of the 365 values that are the dates in the months of 2019 . Thus her data consist of 121s , 122s , . . . , 1228s , 1129s , 1130s , and 731s . Let d be the median of the modes. Which of the following statements is true?
💡 解题思路
First of all, $d$ obviously has to be smaller than $M$ , since when calculating $M$ , we must take into account the $29$ s, $30$ s, and $31$ s. So we can eliminate choices $B$ and $C$ . Since there ar
13
第 13 题
几何·面积
Let \triangle ABC be an isosceles triangle with BC = AC and \angle ACB = 40^{\circ} . Construct the circle with diameter \overline{BC} , and let D and E be the other intersection points of the circle with the sides \overline{AC} and \overline{AB} , respectively. Let F be the intersection of the diagonals of the quadrilateral BCDE . What is the degree measure of \angle BFC ?
For a set of four distinct lines in a plane, there are exactly N distinct points that lie on two or more of the lines. What is the sum of all possible values of N ?
💡 解题思路
It is possible to obtain $0$ , $1$ , $3$ , $4$ , $5$ , and $6$ points of intersection, as demonstrated in the following figures:
15
第 15 题
数论
A sequence of numbers is defined recursively by a_1 = 1 , a_2 = \frac{3}{7} , and \[a_n=\frac{a_{n-2} · a_{n-1}}{2a_{n-2} - a_{n-1}}\] for all n ≥ 3 Then a_{2019} can be written as \frac{p}{q} , where p and q are relatively prime positive integers. What is p+q ?
💡 解题思路
Using the recursive formula, we find $a_3=\frac{3}{11}$ , $a_4=\frac{3}{15}$ , and so on. It appears that $a_n=\frac{3}{4n-1}$ , for all $n$ . Setting $n=2019$ , we find $a_{2019}=\frac{3}{8075}$ , so
16
第 16 题
几何·面积
The figure below shows 13 circles of radius 1 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 1 ? [图]
A child builds towers using identically shaped cubes of different colors. How many different towers with a height 8 cubes can the child build with 2 red cubes, 3 blue cubes, and 4 green cubes? (One cube will be left out.)
💡 解题思路
Arranging eight cubes is the same as arranging the nine cubes first, and then removing the last cube. In other words, there is a one-to-one correspondence between every arrangement of nine cubes, and
18
第 18 题
分数与比例
For some positive integer k , the repeating base- k representation of the (base-ten) fraction \frac{7}{51} is 0.\overline{23}_k = 0.232323..._k . What is k ?
💡 解题思路
We can expand the fraction $0.\overline{23}_k$ as follows: $0.\overline{23}_k = 2\cdot k^{-1} + 3 \cdot k^{-2} + 2 \cdot k^{-3} + 3 \cdot k^{-4} + \cdots$
19
第 19 题
综合
What is the least possible value of \[(x+1)(x+2)(x+3)(x+4)+2019\] where x is a real number?
💡 解题思路
Grouping the first and last terms and two middle terms gives $(x^2+5x+4)(x^2+5x+6)+2019$ , which can be simplified to $(x^2+5x+5)^2-1+2019$ . Noting that squares are nonnegative, and verifying that $x
20
第 20 题
几何·面积
The numbers 1,2,\dots,9 are randomly placed into the 9 squares of a 3 × 3 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?
💡 解题思路
Note that odd sums can only be formed by $(e,e,o)$ or $(o,o,o),$ so we focus on placing the evens: we need to have each even be with another even in each row/column. Because there are only $5$ odd num
21
第 21 题
几何·面积
A sphere with center O has radius 6 . A triangle with sides of length 15, 15, and 24 is situated in space so that each of its sides is tangent to the sphere. What is the distance between O and the plane determined by the triangle? 3D: [图] Plane through triangle: [图]
💡 解题思路
The triangle is placed on the sphere so that its three sides are tangent to the sphere. The cross-section of the sphere created by the plane of the triangle is also the incircle of the triangle. To fi
22
第 22 题
概率
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 [0,1] . Two random numbers x and y are chosen independently in this manner. What is the probability that |x-y| > \tfrac{1}{2} ?
💡 解题思路
There are several cases depending on what the first coin flip is when determining $x$ and what the first coin flip is when determining $y$ .
23
第 23 题
计数
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 1 , then Todd must say the next two numbers ( 2 and 3 ), then Tucker must say the next three numbers ( 4 , 5 , 6 ), then Tadd must say the next four numbers ( 7 , 8 , 9 , 10 ), 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 10,000 is reached. What is the 2019 th number said by Tadd?
💡 解题思路
Define a round as one complete rotation through each of the three children, and define a turn as the portion when one child says his numbers (similar to how a game is played).
24
第 24 题
综合
Let p , q , and r be the distinct roots of the polynomial x^3 - 22x^2 + 80x - 67 . It is given that there exist real numbers A , B , and C such that \[\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\] for all s\not\in\{p,q,r\} . What is \tfrac1A+\tfrac1B+\tfrac1C ?
💡 解题思路
Multiplying both sides by $(s-p)(s-q)(s-r)$ yields \[1 = A(s-q)(s-r) + B(s-p)(s-r) + C(s-p)(s-q)\] As this is a polynomial identity, and it is true for infinitely many $s$ , it must be true for all $s
25
第 25 题
整数运算
For how many integers n between 1 and 50 , inclusive, is \[\frac{(n^2-1)!}{(n!)^n}\] an integer? (Recall that 0! = 1 .)