A box contains a collection of triangular and square tiles. There are 25 tiles in the box, containing 84 edges total. How many square tiles are there in the box?
💡 解题思路
Let $a$ be the amount of triangular tiles and $b$ be the amount of square tiles.
3
第 3 题
综合
Ann made a 3 -step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5 -step staircase? [图]
💡 解题思路
We can see that a $1$ -step staircase requires $4$ toothpicks and a $2$ -step staircase requires $10$ toothpicks. Thus, to go from a $1$ -step to $2$ -step staircase, $6$ additional toothpicks are nee
4
第 4 题
分数与比例
Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?
💡 解题思路
Assign a variable to the number of eggs Mia has, say $m$ . Then, because we are given that Sofia has twice the number of eggs Mia has, Sofia has $2m$ eggs, and Pablo, having three times the number of
5
第 5 题
统计
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80 . After he graded Payton's test, the test average became 81 . What was Payton's score on the test?
💡 解题思路
If the average of the first $14$ peoples' scores was $80$ , then the sum of all of their tests is $14 \cdot 80 = 1120$ . When Payton's score was added, the sum of all of the scores became $15 \cdot 81
6
第 6 题
分数与比例
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller number?
💡 解题思路
Let $a$ be the bigger number and $b$ be the smaller.
7
第 7 题
规律与数列
How many terms are in the arithmetic sequence 13 , 16 , 19 , \dotsc , 70 , 73 ?
💡 解题思路
$73-13 = 60$ , so the amount of terms in the sequence $13$ , $16$ , $19$ , $\dotsc$ , $70$ , $73$ is the same as in the sequence $0$ , $3$ , $6$ , $\dotsc$ , $57$ , $60$ .
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第 8 题
分数与比例
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be 2 : 1 ?
💡 解题思路
This problem can be converted to a system of equations. Let $p$ be Pete's current age and $c$ be Claire's current age.
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第 9 题
立体几何
Two right circular cylinders have the same volume. The radius of the second cylinder is 10\% more than the radius of the first. What is the relationship between the heights of the two cylinders?
💡 解题思路
Let the radius of the first cylinder be $r_1$ and the radius of the second cylinder be $r_2$ . Also, let the height of the first cylinder be $h_1$ and the height of the second cylinder be $h_2$ . We a
10
第 10 题
统计
How many rearrangements of abcd are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either ab or ba .
💡 解题思路
The first thing one would want to do is place a possible letter that works and then stem off of it. For example, if we start with an $a$ , we can only place a $c$ or $d$ next to it. Unfortunately, aft
11
第 11 题
几何·面积
The ratio of the length to the width of a rectangle is 4 : 3 . If the rectangle has diagonal of length d , then the area may be expressed as kd^2 for some constant k . What is k ?
💡 解题思路
Let the rectangle have length $4x$ and width $3x$ . Then by $3-4-5$ triangles (or the Pythagorean Theorem), we have $d = 5x$ , and so $x = \dfrac{d}{5}$ . Hence, the area of the rectangle is $3x \cdot
12
第 12 题
坐标几何
Points ( √(π) , a) and ( √(π) , b) are distinct points on the graph of y^2 + x^4 = 2x^2 y + 1 . What is |a-b| ?
💡 解题思路
Since points on the graph make the equation true, substitute $\sqrt{\pi}$ in to the equation and then solve to find $a$ and $b$ .
13
第 13 题
概率
Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have?
💡 解题思路
Let Claudia have $x$ 5-cent coins and $\left( 12 - x \right)$ 10-cent coins. It is easily observed that any multiple of $5$ between $5$ and $5x + 10(12 - x) = 120 - 5x$ inclusive can be obtained by a
14
第 14 题
时间问题
The diagram below shows the circular face of a clock with radius 20 cm and a circular disk with radius 10 cm externally tangent to the clock face at 12 o' clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction? [图]
💡 解题思路
The below solutions don't do a great job on telling you why the answer is not $\textbf{(D) }\text{6 o' clock}$ . For those who don't understand it, the below video is great on explaining it.
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第 15 题
数论
Consider the set of all fractions \frac{x}{y} , where x and y are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by 1 , the value of the fraction is increased by 10\% ?
💡 解题思路
You can create the equation $\frac{x+1}{y+1}=\frac{11x}{10y}$
16
第 16 题
综合
If y+4 = (x-2)^2, x+4 = (y-2)^2 , and x ≠ y , what is the value of x^2+y^2 ?
💡 解题思路
Note that we can add the two equations to yield the equation
17
第 17 题
几何·面积
A line that passes through the origin intersects both the line x = 1 and the line y=1+ \frac{√(3)}{3} x . The three lines create an equilateral triangle. What is the perimeter of the triangle?
💡 解题思路
Since the triangle is equilateral and one of the sides is a vertical line, the triangle must have a horizontal line of symmetry, and therefore the other two sides will have opposite slopes. The slope
18
第 18 题
分数与比例
Hexadecimal (base-16) numbers are written using numeric digits 0 through 9 as well as the letters A through F to represent 10 through 15 . Among the first 1000 positive integers, there are n whose hexadecimal representation contains only numeric digits. What is the sum of the digits of n ?
💡 解题思路
Notice that $1000$ is $3E8$ when converted to hexadecimal ( $3 \cdot 16^2 + 14 \cdot 16^1 + 8 \cdot 16^0$ ). We will proceed by constructing numbers that consist of only numeric digits in hexadecimal.
19
第 19 题
几何·面积
The isosceles right triangle ABC has right angle at C and area 12.5 . The rays trisecting \angle ACB intersect AB at D and E . What is the area of \bigtriangleup CDE ?
💡 解题思路
$\triangle ADC$ can be split into a $45-45-90$ right triangle and a $30-60-90$ right triangle by dropping a perpendicular from $D$ to side $AC$ . Let $F$ be where that perpendicular intersects $AC$ .
20
第 20 题
几何·面积
A rectangle with positive integer side lengths in cm has area Acm^2 and perimeter Pcm . Which of the following numbers cannot equal A+P ?
💡 解题思路
Let the rectangle's length be $a$ and its width be $b$ . Its area is $ab$ and the perimeter is $2a+2b$ .
21
第 21 题
立体几何
Tetrahedron ABCD has AB=5 , AC=3 , BC=4 , BD=4 , AD=3 , and CD=\tfrac{12}5\sqrt2 . What is the volume of the tetrahedron?
💡 解题思路
Drop altitudes of triangle $ABC$ and triangle $ABD$ down from $C$ and $D$ , respectively. Both will hit the same point; let this point be $T$ . Because both triangle $ABC$ and triangle $ABD$ are 3-4-5
22
第 22 题
概率
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
💡 解题思路
We will count how many valid standing arrangements there are (counting rotations as distinct), and divide by $2^8 = 256$ at the end. We casework on how many people are standing.
23
第 23 题
规律与数列
The zeroes of the function f(x)=x^2-ax+2a are integers. What is the sum of the possible values of a?
💡 解题思路
By Vieta's Formula, $a$ is the sum of the integral zeros of the function, and so $a$ is integral.
24
第 24 题
几何·面积
For some positive integers p , there is a quadrilateral ABCD with positive integer side lengths, perimeter p , right angles at B and C , AB=2 , and CD=AD . How many different values of p<2015 are possible?
💡 解题思路
Let $BC = x$ and $CD = AD = y$ be positive integers. Drop a perpendicular from $A$ to $CD$ to show that, using the Pythagorean Theorem, that \[x^2 + (y - 2)^2 = y^2.\] Simplifying yields $x^2 - 4y + 4
25
第 25 题
几何·面积
Let S be a square of side length 1 . Two points are chosen independently at random on the sides of S . The probability that the straight-line distance between the points is at least \dfrac{1}{2} is \dfrac{a-bπ}{c} , where a , b , and c are positive integers with \gcd(a,b,c)=1 . What is a+b+c ?
💡 解题思路
Divide the boundary of the square into halves, thereby forming $8$ segments. Without loss of generality, let the first point $A$ be in the bottom-left segment. Then, it is easy to see that any point i