We find that $AP + DP = 5 = AD$ . Because $ABCD$ is a rhombus, we get that AD = AB = 5. Notice that using Pythagorean Theorem, we have that $AB^2 = AP^2 + BP^2$ , which simplifies to $25 = AP^2 + 9 ==
3
第 3 题
数字运算
How many three-digit positive integers have an odd number of even digits?
💡 解题思路
We use simple case work to solve this problem.
4
第 4 题
行程问题
A donkey suffers an attack of hiccups and the first hiccup happens at 4:00 one afternoon. Suppose that the donkey hiccups regularly every 5 seconds. At what time does the donkey’s 700 th hiccup occur? (A) 15 seconds after 4:58(B) 20 seconds after 4:58(C) 25 seconds after 4:58(D) 30 seconds after 4:58(E) 35 seconds after 4:58
💡 解题思路
Since the donkey hiccupped the 1st hiccup at $4:00$ , it hiccupped for $5 \cdot (700-1) = 3495$ seconds, which is $58$ minutes and $15$ seconds, so the answer is $\boxed{\textbf{(A) }15 \text{ seconds
5
第 5 题
综合
💡 解题思路
We apply the difference of squares to the denominator, and then regroup factors: \begin{align*} \frac{\left(1+\frac13\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\ri
6
第 6 题
数论
How many of the first ten numbers of the sequence 121, 11211, 1112111, \ldots are prime numbers?
💡 解题思路
The $n$ th term of this sequence is \[\sum_{k=n}^{2n}10^k + \sum_{k=0}^{n}10^k = 10^n\sum_{k=0}^{n}10^k + \sum_{k=0}^{n}10^k = \left(10^n+1\right)\sum_{k=0}^{n}10^k.\] It follows that the terms are \b
7
第 7 题
整数运算
For how many values of the constant k will the polynomial x^{2}+kx+36 have two distinct integer roots?
💡 解题思路
Let $p$ and $q$ be the roots of $x^{2}+kx+36.$ By Vieta's Formulas , we have $p+q=-k$ and $pq=36.$
8
第 8 题
数论
Consider the following 100 sets of 10 elements each: &\{1,2,3,\ldots,10\}, ; &\{11,12,13,\ldots,20\}, ; &\{21,22,23,\ldots,30\}, ; &\vdots ; &\{991,992,993,\ldots,1000\}. How many of these sets contain exactly two multiples of 7 ?
💡 解题思路
There are \(\text{floor}\left(\frac{1000}{7}\right) = 142\) numbers divisible by 7. We split these into 100 sets containing 10 numbers each, giving us 1.42 multiples of 7 per set. After the first set,
9
第 9 题
规律与数列
The sum \[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+·s+\frac{2021}{2022!}\] can be expressed as a-\frac{1}{b!} , where a and b are positive integers. What is a+b ?
💡 解题思路
For all positive integers $n,$ we have \[\frac{n}{(n+1)!}=\frac{(n+1)-1}{(n+1)!}=\frac{1}{n!}-\frac{1}{(n+1)!}.\] Note that the original sum is a telescoping series: \begin{align*} \frac{1}{2!}+\frac{
10
第 10 题
统计
Camila writes down five positive integers. The unique mode of these integers is 2 greater than their median, and the median is 2 greater than their arithmetic mean. What is the least possible value for the mode?
💡 解题思路
Let $M$ be the median. It follows that the two largest integers are both $M+2.$
11
第 11 题
逻辑推理
All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"? (A) All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS. (B) No school that sold more T-shirts than Euclid HS is bigger than Euclid HS. (C) All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS. (D) All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS. (E) All schools smaller than Euclid HS sold more T-shirts than Euclid HS.
💡 解题思路
Let $B$ denote a school that is bigger than Euclid HS, and $M$ denote a school that sold more T-shirts than Euclid HS.
12
第 12 题
概率
A pair of fair 6 -sided dice is rolled n times. What is the least value of n such that the probability that the sum of the numbers face up on a roll equals 7 at least once is greater than \frac{1}{2} ?
💡 解题思路
Rolling a pair of fair $6$ -sided dice, the probability of getting a sum of $7$ is $\frac16:$ Regardless what the first die shows, the second die has exactly one outcome to make the sum $7.$ We consid
13
第 13 题
数论
The positive difference between a pair of primes is equal to 2 , and the positive difference between the cubes of the two primes is 31106 . What is the sum of the digits of the least prime that is greater than those two primes?
💡 解题思路
Denote the two primes as \( a \) and \( b \). Then,
14
第 14 题
规律与数列
Suppose that S is a subset of \{ 1, 2, 3, \ldots , 25 \} such that the sum of any two (not necessarily distinct) elements of S is never an element of S. What is the maximum number of elements S may contain?
💡 解题思路
Let $M$ be the largest number in $S$ . We categorize numbers $\left\{ 1, 2, \ldots , M-1 \right\}$ (except $\frac{M}{2}$ if $M$ is even) into $\left\lfloor \frac{M-1}{2} \right\rfloor$ groups, such th
15
第 15 题
规律与数列
Let S_n be the sum of the first n terms of an arithmetic sequence that has a common difference of 2 . The quotient \frac{S_{3n}}{S_n} does not depend on n . What is S_{20} ?
💡 解题思路
Let's say that our sequence is \[a, a+2, a+4, a+6, a+8, a+10, \ldots.\] Then, since the value of n doesn't matter in the quotient $\frac{S_{3n}}{S_n}$ , we can say that \[\frac{S_{3}}{S_1} = \frac{S_{
16
第 16 题
几何·面积
The diagram below shows a rectangle with side lengths 4 and 8 and a square with side length 5 . Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle? [图]
💡 解题思路
Let us label the points on the diagram.
17
第 17 题
数论
One of the following numbers is not divisible by any prime number less than 10. Which is it?
Consider systems of three linear equations with unknowns x , y , and z , a_1 x + b_1 y + c_1 z & = 0 ; a_2 x + b_2 y + c_2 z & = 0 ; a_3 x + b_3 y + c_3 z & = 0 where each of the coefficients is either 0 or 1 and the system has a solution other than x=y=z=0 . For example, one such system is \[\{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \}\] with a nonzero solution of \{x,y,z\} = \{1, -1, 1\} . How many such systems of equations are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)
💡 解题思路
Let $M_1=\begin{bmatrix}a_1 & b_1 & c_1\end{bmatrix}, M_2=\begin{bmatrix}a_2 & b_2 & c_2\end{bmatrix},$ and $M_3=\begin{bmatrix}a_3 & b_3 & c_3\end{bmatrix}.$
19
第 19 题
几何·面积
Each square in a 5 × 5 grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:
💡 解题思路
There are two cases for the initial configuration:
20
第 20 题
几何·角度
Let ABCD be a rhombus with \angle ADC = 46^\circ . Let E be the midpoint of \overline{CD} , and let F be the point on \overline{BE} such that \overline{AF} is perpendicular to \overline{BE} . What is the degree measure of \angle BFC ? [图] ~MRENTHUSIASM
💡 解题思路
Extend segments $\overline{AD}$ and $\overline{BE}$ until they meet at point $G$ .
21
第 21 题
几何·面积
Let P(x) be a polynomial with rational coefficients such that when P(x) is divided by the polynomial x^2 + x + 1 , the remainder is x+2 , and when P(x) is divided by the polynomial x^2+1 , the remainder is 2x+1 . There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
💡 解题思路
Given that all the answer choices and coefficients are integers, we hope that $P(x)$ has positive integer coefficients.
22
第 22 题
几何·面积
Let S be the set of circles in the coordinate plane that are tangent to each of the three circles with equations x^{2}+y^{2}=4 , x^{2}+y^{2}=64 , and (x-5)^{2}+y^{2}=3 . What is the sum of the areas of all circles in S ?
Ant Amelia starts on the number line at 0 and crawls in the following manner. For n=1,2,3, Amelia chooses a time duration t_n and an increment x_n independently and uniformly at random from the interval (0,1). During the n th step of the process, Amelia moves x_n units in the positive direction, using up t_n minutes. If the total elapsed time has exceeded 1 minute during the n th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most 3 steps in all. What is the probability that Amelia’s position when she stops will be greater than 1 ?
💡 解题思路
Let $x$ and $y$ be random variables that are independently and uniformly distributed in the interval $(0,1).$ Note that \[P(x+y\leq 1)=\frac{\frac12\cdot1^2}{1^2}=\frac12,\] as shown below: [asy] /* M
24
第 24 题
综合
💡 解题思路
We have \begin{align*} |f(f(800))-f(f(400))| &\leq \frac12|f(800)-f(400)| &&(\bigstar) \\ &\leq \frac12\left|\frac12|800-400|\right| \\ &= 100, \end{align*} from which we eliminate answer choices $\te
25
第 25 题
综合
💡 解题思路
In binary numbers, we have \[S_n = (x_{n-1} x_{n-2} x_{n-3} x_{n-4} \ldots x_{2} x_{1} x_{0})_2.\] It follows that \[8S_n = (x_{n-1} x_{n-2} x_{n-3} x_{n-4} \ldots x_{2} x_{1} x_{0}000)_2.\] We obtain