2014 AMC 10A — Official Competition Problems (February 2014)
📅 2014 A 年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 612 人已练习
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1
第 1 题
综合
What is 10·(\tfrac{1}{2}+\tfrac{1}{5}+\tfrac{1}{10})^{-1}?
💡 解题思路
We have \[10\cdot\left(\frac{1}{2}+\frac{1}{5}+\frac{1}{10}\right)^{-1}\] Making the denominators equal gives \[\implies 10\cdot\left(\frac{5}{10}+\frac{2}{10}+\frac{1}{10}\right)^{-1}\] \[\implies 10
2
第 2 题
整数运算
Roy's cat eats \frac{1}{3} of a can of cat food every morning and \frac{1}{4} of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing 6 cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
💡 解题思路
Each day, the cat eats $\dfrac13+\dfrac14=\dfrac7{12}$ of a can of cat food. Therefore, the cat food will last for $\dfrac{6}{\frac7{12}}=\dfrac{72}7$ days, which is greater than $10$ days but less th
3
第 3 题
应用题
Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for \textdollar 2.50 each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs \textdollar 0.75 for her to make. In dollars, what is her profit for the day?
💡 解题思路
She first sells one-half of her $48$ loaves, or $\frac{48}{2}=24$ loaves. Each loaf sells for $\textdollar 2.50$ , so her total earnings in the morning is equal to \[24\cdot \textdollar 2.50 = \textdo
4
第 4 题
统计
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
💡 解题思路
Let's use casework on the yellow house. The yellow house $(\text{Y})$ is either the $3^\text{rd}$ house or the last house.
5
第 5 题
统计
On an algebra quiz, 10\% of the students scored 70 points, 35\% scored 80 points, 30\% scored 90 points, and the rest scored 100 points. What is the difference between the mean and median score of the students' scores on this quiz?
💡 解题思路
WLOG, let there be $20$ students (the least whole number possible) who took the test. We have $2$ students score $70$ points, $7$ students score $80$ points, $6$ students score $90$ points and $5$ stu
6
第 6 题
行程问题
Suppose that a cows give b gallons of milk in c days. At this rate, how many gallons of milk will d cows give in e days?
💡 解题思路
We need to multiply $b$ by $\frac{d}{a}$ for the new cows and $\frac{e}{c}$ for the new time, so the answer is $b\cdot \frac{d}{a}\cdot \frac{e}{c}=\frac{bde}{ac}$ , or $\boxed{\textbf{(A) } \frac{bde
7
第 7 题
综合
Nonzero real numbers x , y , a , and b satisfy x < a and y < b . How many of the following inequalities must be true? (I)\ x+y < a+b (II)\ x-y < a-b (III)\ xy < ab (IV)\ \frac{x}{y} < \frac{a}{b}
💡 解题思路
Let us denote $a = x + k$ where $k > 0$ and $b = y + l$ where $l > 0$ . We can write that $x + y < x + y + k + l \implies x + y < a + b$ .
8
第 8 题
几何·面积
Which of the following numbers is a perfect square?
💡 解题思路
Note that for all positive $n$ , we have \[\dfrac{n!(n+1)!}{2}\] \[\implies\dfrac{(n!)^2\cdot(n+1)}{2}\] \[\implies (n!)^2\cdot\dfrac{n+1}{2}\]
9
第 9 题
几何·面积
The two legs of a right triangle, which are altitudes, have lengths 2\sqrt3 and 6 . How long is the third altitude of the triangle?
💡 解题思路
We find that the area of the triangle is $\frac{6\times 2\sqrt{3}}{2} =6\sqrt{3}$ . By the Pythagorean Theorem , we have that the length of the hypotenuse is $\sqrt{(2\sqrt{3})^2+6^2}=4\sqrt{3}$ . Dro
10
第 10 题
统计
Five positive consecutive integers starting with a have average b . What is the average of 5 consecutive integers that start with b ?
💡 解题思路
Let $a=1$ . Our list is $\{1,2,3,4,5\}$ with an average of $15\div 5=3$ . Our next set starting with $3$ is $\{3,4,5,6,7\}$ . Our average is $25\div 5=5$ .
11
第 11 题
应用题
A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: 10\% off the listed price if the listed price is at least \textdollar50 Coupon 2: \textdollar 20 off the listed price if the listed price is at least \textdollar100 Coupon 3: 18\% off the amount by which the listed price exceeds \textdollar100 For which of the following listed prices will coupon 1 offer a greater price reduction than either coupon 2 or coupon 3 ?
💡 解题思路
Let the listed price be $x$ . Since all the answer choices are above $\textdollar100$ , we can assume $x > 100$ . Thus the discounts after the coupons are used will be as follows:
12
第 12 题
几何·面积
A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region? [图] (A)\ 27√(3)-9π (B)\ 27√(3)-6π (C)\ 54√(3)-18π (D)\ 54√(3)-12π (E)\ 108√(3)-9π
💡 解题思路
The area of the hexagon is equal to $\dfrac{3(6)^2\sqrt{3}}{2}=54\sqrt{3}$ by the formula for the area of a hexagon.
13
第 13 题
几何·面积
Equilateral \triangle ABC has side length 1 , and squares ABDE , BCHI , CAFG lie outside the triangle. What is the area of hexagon DEFGHI ? [图]
💡 解题思路
The area of the equilateral triangle is $\dfrac{\sqrt{3}}{4}$ . The area of the three squares is $3\times 1=3$ .
14
第 14 题
几何·面积
The y -intercepts, P and Q , of two perpendicular lines intersecting at the point A(6,8) have a sum of zero. What is the area of \triangle APQ ?
💡 解题思路
[asy]//Needs refining (hmm I think it's fine --bestwillcui1) size(12cm); fill((0,10)--(6,8)--(0,-10)--cycle,rgb(.7,.7,.7)); for(int i=-2;i<=8;i+=1) draw((i,-12)--(i,12),grey); for(int j=-12;j<=12;j+=1
15
第 15 题
行程问题
David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour, but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by 15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many miles is the airport from his home?
💡 解题思路
Note that he drives at $50$ miles per hour after the first hour and continues doing so until he arrives.
16
第 16 题
几何·面积
In rectangle ABCD , AB=1 , BC=2 , and points E , F , and G are midpoints of \overline{BC} , \overline{CD} , and \overline{AD} , respectively. Point H is the midpoint of \overline{GE} . What is the area of the shaded region? [图]
💡 解题思路
Denote $D=(0,0)$ . Then $A= (0,2), F = \left(\frac12,0\right), H = \left(\frac12,1\right)$ . Let the intersection of $AF$ and $DH$ be $X$ , and the intersection of $BF$ and $CH$ be $Y$ . Then we want
17
第 17 题
概率
Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?
💡 解题思路
First, we note that there are $1, 2, 3, 4,$ and $5$ ways to get sums of $2, 3, 4, 5, 6$ respectively--this is not too hard to see. With any specific sum, there is exactly one way to attain it on the o
18
第 18 题
几何·面积
A square in the coordinate plane has vertices whose y -coordinates are 0 , 1 , 4 , and 5 . What is the area of the square?
💡 解题思路
Let the points be $A=(x_1,0)$ , $B=(x_2,1)$ , $C=(x_3,5)$ , and $D=(x_4,4)$
19
第 19 题
立体几何
Four cubes with edge lengths 1 , 2 , 3 , and 4 are stacked as shown. What is the length of the portion of \overline{XY} contained in the cube with edge length 3 ? [图]
💡 解题思路
By Pythagorean Theorem in three dimensions, the distance $XY$ is $\sqrt{4^2+4^2+10^2}=2\sqrt{33}$ .
20
第 20 题
数论
The product (8)(888\dots8) , where the second factor has k digits, is an integer whose digits have a sum of 1000 . What is k ?
💡 解题思路
We can list the first few numbers in the form $8 \cdot (8....8)$
21
第 21 题
坐标几何
Positive integers a and b are such that the graphs of y=ax+5 and y=3x+b intersect the x -axis at the same point. What is the sum of all possible x -coordinates of these points of intersection?
💡 解题思路
Note that when $y=0$ , the $x$ values of the equations should be equal by the problem statement. We have that \[0 = ax + 5 \implies x = -\dfrac{5}{a}\] \[0 = 3x+b \implies x= -\dfrac{b}{3}\] Which mea
22
第 22 题
几何·面积
In rectangle ABCD , \overline{AB}=20 and \overline{BC}=10 . Let E be a point on \overline{CD} such that \angle CBE=15^\circ . What is \overline{AE} ?
💡 解题思路
Note that $\tan 15^\circ=2-\sqrt{3}=\frac{EC}{10} \Rightarrow EC=20-10 \sqrt 3$ . (It is important to memorize the sin, cos, and tan values of $15^\circ$ and $75^\circ$ .) Therefore, we have $DE=10\sq
23
第 23 题
几何·面积
A rectangular piece of paper whose length is \sqrt3 times the width has area A . The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area B . What is the ratio \frac{B}{A} ? [图]
💡 解题思路
Find the midpoint of the dotted line. Draw a line perpendicular to it. From the point this line intersects the top of the paper, draw lines to each endpoint of the dotted line. These two lines plus th
24
第 24 题
分数与比例
A sequence of natural numbers is constructed by listing the first 4 , then skipping one, listing the next 5 , skipping 2 , listing 6 , skipping 3 , and on the n th iteration, listing n+3 and skipping n . The sequence begins 1,2,3,4,6,7,8,9,10,13 . What is the 500,\!000 th number in the sequence?
💡 解题思路
If we list the rows by iterations, then we get
25
第 25 题
整数运算
The number 5^{867} is between 2^{2013} and 2^{2014} . How many pairs of integers (m,n) are there such that 1≤ m≤ 2012 and \[5^n<2^m<2^{m+2}<5^{n+1}?\]
💡 解题思路
Between any two consecutive powers of $5$ there are either $2$ or $3$ powers of $2$ (because $2^2<5^1<2^3$ ). Consider the intervals $(5^0,5^1),(5^1,5^2),\dots (5^{866},5^{867})$ . We want the number