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
4
第 4 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 $pq=36.$
5
第 5 题
坐标几何
The point (-1, -2) is rotated 270^{\circ} counterclockwise about the point (3, 1) . What are the coordinates of its new position?
💡 解题思路
$(-1,-2)$ is $4$ units west and $3$ units south of $(3,1)$ . Performing a counterclockwise rotation of $270^{\circ}$ , which is equivalent to a clockwise rotation of $90^{\circ}$ , the answer is $3$ u
6
第 6 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,
7
第 7 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 $M+2.$
8
第 8 题
坐标几何
What is the graph of y^4+1=x^4+2y^2 in the coordinate plane?
💡 解题思路
Since the equation has even powers of $x$ and $y$ , let $y'=y^2$ and $x' = x^2$ . Then $y'^2 + 1 = x'^2 + 2y'$ . Rearranging gives $y'^2 - 2y' + 1 = x'^2$ , or $(y'-1)^2=x'^2$ . There are two cases: $
9
第 9 题
规律与数列
The sequence a_0,a_1,a_2,·s is a strictly increasing arithmetic sequence of positive integers such that \[2^{a_7}=2^{27} · a_7.\] What is the minimum possible value of a_2 ?
💡 解题思路
We can rewrite the given equation as $2^{a_7-27}=a_7$ . Hence, $a_7$ must be a power of $2$ and larger than $27$ . The first power of 2 that is larger than $27$ , namely $32$ , does satisfy the equati
10
第 10 题
几何·面积
Regular hexagon ABCDEF has side length 2 . Let G be the midpoint of \overline{AB} , and let H be the midpoint of \overline{DE} . What is the perimeter of GCHF ? [图] ~MRENTHUSIASM
💡 解题思路
Let the center of the hexagon be $O$ . $\triangle AOB$ , $\triangle BOC$ , $\triangle COD$ , $\triangle DOE$ , $\triangle EOF$ , and $\triangle FOA$ are all equilateral triangles with side length $2$
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第 11 题
综合
Let f(n) = ( \frac{-1+i√(3)}{2} )^n + ( \frac{-1-i√(3)}{2} )^n , where i = √(-1) . What is f(2022) ?
💡 解题思路
Converting both summands to exponential form, \begin{align*} -1 + i\sqrt{3} &= 2e^{\frac{2\pi i}{3}}, \\ -1 - i\sqrt{3} &= 2e^{-\frac{2\pi i}{3}} = 2e^{\frac{4\pi i}{3}}. \end{align*} Notice that the
12
第 12 题
概率
Kayla rolls four fair 6 -sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than 4 and at least two of the numbers she rolls are greater than 2 ?
💡 解题思路
We will subtract from one the probability that the first condition is violated and the probability that only the second condition is violated, being careful not to double-count the probability that bo
13
第 13 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.
14
第 14 题
坐标几何
The graph of y=x^2+2x-15 intersects the x -axis at points A and C and the y -axis at point B . What is \tan(\angle ABC) ? [图] ~MRENTHUSIASM
💡 解题思路
First, find $A=(-5,0)$ , $B=(0,-15)$ , and $C=(3,0)$ . Create vectors $\overrightarrow{BA}$ and $\overrightarrow{BC}.$ These can be reduced to $\langle -1, 3 \rangle$ and $\langle 1, 5 \rangle$ , resp
15
第 15 题
数论
One of the following numbers is not divisible by any prime number less than 10. Which is it?
Suppose x and y are positive real numbers such that \[x^y=2^{64} and (\log_2{x})^{\log_2{y}}=2^{7}.\] What is the greatest possible value of \log_2{y} ?
💡 解题思路
Take the base-two logarithm of both equations to get \[y\log_2 x = 64\quad\text{and}\quad (\log_2 y)(\log_2\log_2 x) = 7.\] Now taking the base-two logarithm of the first equation again yields \[\log_
17
第 17 题
规律与数列
How many 4 × 4 arrays whose entries are 0 s and 1 s are there such that the row sums (the sum of the entries in each row) are 1, 2, 3, and 4, in some order, and the column sums (the sum of the entries in each column) are also 1, 2, 3, and 4, in some order? For example, the array \[[ \begin{array}{cccc} 1 & 1 & 1 & 0 ; 0 & 1 & 1 & 0 ; 1 & 1 & 1 & 1 ; 0 & 1 & 0 & 0 ; \end{array} ]\] satisfies the condition.
💡 解题思路
Note that the arrays and the sum configurations have one-to-one correspondence. Furthermore, the row sum configuration and the column sum configuration are independent of each other. Therefore, the an
18
第 18 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:
19
第 19 题
几何·面积
In \triangle{ABC} medians \overline{AD} and \overline{BE} intersect at G and \triangle{AGE} is equilateral. Then \cos(C) can be written as \frac{m\sqrt p}n , where m and n are relatively prime positive integers and p is a positive integer not divisible by the square of any prime. What is m+n+p? [图]
💡 解题思路
Let $AG=AE=EG=2x$ . Since $E$ is the midpoint of $\overline{AC}$ , we must have $EC=2x$ .
20
第 20 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.
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第 21 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
23
第 23 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
24
第 24 题
几何·面积
The figure below depicts a regular 7 -gon inscribed in a unit circle. [图] What is the sum of the 4 th powers of the lengths of all 21 of its edges and diagonals?
💡 解题思路
There are $7$ segments whose lengths are $2 \sin \frac{\pi}{7}$ , $7$ segments whose lengths are $2 \sin \frac{2 \pi}{7}$ , $7$ segments whose lengths are $2 \sin \frac{3\pi}{7}$ .