tony - 第 142 页 - American Mathematics Competitions

1996年AJHSME 真题及答案

1996年AJHSME 真题：

Problem 1

How many positive factors of 36 are also multiples of 4?

$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$

Problem 2

Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from the number 10, doubles his answer, and then adds 2. Thuy doubles the number 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from the number 10, adds 2 to his number, and then doubles the result. Who gets the largest final answer?

$\text{(A)}\ \text{Jose} \qquad \text{(B)}\ \text{Thuy} \qquad \text{(C)}\ \text{Kareem} \qquad \text{(D)}\ \text{Jose and Thuy} \qquad \text{(E)}\ \text{Thuy and Kareem}$

Problem 3

The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array of 64 squares). The first 8 numbers are written in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the numbers in the four corners will be

$\text{(A)}\ 130 \qquad \text{(B)}\ 131 \qquad \text{(C)}\ 132 \qquad \text{(D)}\ 133 \qquad \text{(E)}\ 134$

Problem 4

$\dfrac{2+4+6+\cdots + 34}{3+6+9+\cdots+51}=$

$\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{2}{3} \qquad \text{(C)}\ \dfrac{3}{2} \qquad \text{(D)}\ \dfrac{17}{3} \qquad \text{(E)}\ \dfrac{34}{3}$

Problem 5

The letters $P$ , $Q$ , $R$ , $S$ , and $T$ represent numbers located on the number line as shown.

$[asy] unitsize(36); draw((-4,0)--(4,0)); draw((-3.9,0.1)--(-4,0)--(-3.9,-0.1)); draw((3.9,0.1)--(4,0)--(3.9,-0.1)); for (int i = -3; i <= 3; ++i) { draw((i,-0.1)--(i,0)); } label("$-3$",(-3,-0.1),S); label("$-2$",(-2,-0.1),S); label("$-1$",(-1,-0.1),S); label("$0$",(0,-0.1),S); label("$1$",(1,-0.1),S); label("$2$",(2,-0.1),S); label("$3$",(3,-0.1),S); draw((-3.7,0.1)--(-3.6,0)--(-3.5,0.1)); draw((-3.6,0)--(-3.6,0.25)); label("$P$",(-3.6,0.25),N); draw((-1.3,0.1)--(-1.2,0)--(-1.1,0.1)); draw((-1.2,0)--(-1.2,0.25)); label("$Q$",(-1.2,0.25),N); draw((0.1,0.1)--(0.2,0)--(0.3,0.1)); draw((0.2,0)--(0.2,0.25)); label("$R$",(0.2,0.25),N); draw((0.8,0.1)--(0.9,0)--(1,0.1)); draw((0.9,0)--(0.9,0.25)); label("$S$",(0.9,0.25),N); draw((1.4,0.1)--(1.5,0)--(1.6,0.1)); draw((1.5,0)--(1.5,0.25)); label("$T$",(1.5,0.25),N); [/asy]$

Which of the following expressions represents a negative number?

$\text{(A)}\ P-Q \qquad \text{(B)}\ P\cdot Q \qquad \text{(C)}\ \dfrac{S}{Q}\cdot P \qquad \text{(D)}\ \dfrac{R}{P\cdot Q} \qquad \text{(E)}\ \dfrac{S+T}{R}$

Problem 6

What is the smallest result that can be obtained from the following process?

Choose three different numbers from the set $\{3,5,7,11,13,17\}$ .

Add two of these numbers.

Multiply their sum by the third number.

$\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 56$

Problem 7

Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, then in how many months from that time will they have the same number of goldfish?

$\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$

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1997年AJHSME 真题及答案

1997年AJHSME 真题：

Problem 1

$\dfrac{1}{10} + \dfrac{9}{100} + \dfrac{9}{1000} + \dfrac{7}{10000} =$

$\text{(A)}\ 0.0026 \qquad \text{(B)}\ 0.0197 \qquad \text{(C)}\ 0.1997 \qquad \text{(D)}\ 0.26 \qquad \text{(E)}\ 1.997$

Problem 2

Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?

$\text{(A)}\ 200 \qquad \text{(B)}\ 202 \qquad \text{(C)}\ 220 \qquad \text{(D)}\ 380 \qquad \text{(E)}\ 398$

Problem 3

Which of the following numbers is the largest?

$\text{(A)}\ 0.97 \qquad \text{(B)}\ 0.979 \qquad \text{(C)}\ 0.9709 \qquad \text{(D)}\ 0.907 \qquad \text{(E)}\ 0.9089$

Problem 4

Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?

$\text{(A)}\ 2250 \qquad \text{(B)}\ 3000 \qquad \text{(C)}\ 4200 \qquad \text{(D)}\ 4350 \qquad \text{(E)}\ 5650$

Problem 5

There are many two-digit multiples of 7, but only two of the multiples have a digit sum of 10. The sum of these two multiples of 7 is

$\text{(A)}\ 119 \qquad \text{(B)}\ 126 \qquad \text{(C)}\ 140 \qquad \text{(D)}\ 175 \qquad \text{(E)}\ 189$

Problem 6

In the number $74982.1035$ the value of the place occupied by the digit 9 is how many times as great as the value of the place occupied by the digit 3?

$\text{(A)}\ 1,000 \qquad \text{(B)}\ 10,000 \qquad \text{(C)}\ 100,000 \qquad \text{(D)}\ 1,000,000 \qquad \text{(E)}\ 10,000,000$

Problem 7

The area of the smallest square that will contain a circle of radius 4 is

$\text{(A)}\ 8 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 128$

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1998年AJHSME 真题及答案

1998年AJHSME真题：

Problem 1

For $x=7$ , which of the following is the smallest?

$\text{(A)}\ \dfrac{6}{x} \qquad \text{(B)}\ \dfrac{6}{x+1} \qquad \text{(C)}\ \dfrac{6}{x-1} \qquad \text{(D)}\ \dfrac{x}{6} \qquad \text{(E)}\ \dfrac{x+1}{6}$

Problem 2

If $\begin{tabular}{r|l}a&b \\ \hline c&d\end{tabular} = \text{a}\cdot \text{d} - \text{b}\cdot \text{c}$ , what is the value of $\begin{tabular}{r|l}3&4 \\ \hline 1&2\end{tabular}$ ?

$\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2$

Problem 3

$\dfrac{\dfrac{3}{8} + \dfrac{7}{8}}{\dfrac{4}{5}} =$

$\text{(A)}\ 1 \qquad \text{(B)} \dfrac{25}{16} \qquad \text{(C)}\ 2 \qquad \text{(D)}\ \dfrac{43}{20} \qquad \text{(E)}\ \dfrac{47}{16}$

Problem 4

How many triangles are in this figure? (Some triangles may overlap other triangles.)

$[asy] draw((0,0)--(42,0)--(14,21)--cycle); draw((14,21)--(18,0)--(30,9)); [/asy]$

$\text{(A)}\ 9 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 5$

Problem 5

Which of the following numbers is largest?

$\text{(A)}\ 9.12344 \qquad \text{(B)}\ 9.123\overline{4} \qquad \text{(C)}\ 9.12\overline{34} \qquad \text{(D)}\ 9.1\overline{234} \qquad \text{(E)}\ 9.\overline{1234}$

Problem 6

Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is

$[asy] for(int a=0; a<4; ++a) { for(int b=0; b<4; ++b) { dot((a,b)); } } draw((0,0)--(0,2)--(1,2)--(2,3)--(2,2)--(3,2)--(3,0)--(2,0)--(2,1)--(1,0)--cycle); [/asy]$

$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$

Problem 7

$100\times 19.98\times 1.998\times 1000=$

$\text{(A)}\ (1.998)^2 \qquad \text{(B)}\ (19.98)^2 \qquad \text{(C)}\ (199.8)^2 \qquad \text{(D)}\ (1998)^2 \qquad \text{(E)}\ (19980)^2$

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1999年AMC8 真题及答案

1999年AMC 8 真题：

Problem 1

$(6?3) + 4 - (2 - 1) = 5$ To make this statement true, the question mark between the 6 and the 3 should be replaced by

$\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \qquad \text{(E)}\ \text{None of these}$

Problem 2

What is the degree measure of the smaller angle formed by the hands of a clock at 10 o clock?

$[asy] draw(circle((0,0),2)); dot((0,0)); for(int i = 0; i < 12; ++i) { dot(2*dir(30*i)); } label("$3$",2*dir(0),W); label("$2$",2*dir(30),WSW); label("$1$",2*dir(60),SSW); label("$12$",2*dir(90),S); label("$11$",2*dir(120),SSE); label("$10$",2*dir(150),ESE); label("$9$",2*dir(180),E); label("$8$",2*dir(210),ENE); label("$7$",2*dir(240),NNE); label("$6$",2*dir(270),N); label("$5$",2*dir(300),NNW); label("$4$",2*dir(330),WNW); [/asy]$

$\text{(A)}\ 30 \qquad \text{(B)}\ 45 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 90$

Problem 3

Which triplet of numbers has a sum NOT equal to 1?

$\text{(A)}\ (1/2,1/3,1/6) \qquad \text{(B)}\ (2,-2,1) \qquad \text{(C)}\ (0.1,0.3,0.6) \qquad \text{(D)}\ (1.1,-2.1,1.0) \qquad \text{(E)}\ (-3/2,-5/2,5)$

Problem 4

The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn?

$[asy] for (int a = 0; a < 6; ++a) { for (int b = 0; b < 6; ++b) { dot((4*a,3*b)); } } draw((0,0)--(20,0)--(20,15)--(0,15)--cycle); draw((0,0)--(16,12)); draw((0,0)--(16,9)); label(rotate(30)*"Bjorn",(12,6.75),SE); label(rotate(37)*"Alberto",(11,8.25),NW); label("$0$",(0,0),S); label("$1$",(4,0),S); label("$2$",(8,0),S); label("$3$",(12,0),S); label("$4$",(16,0),S); label("$5$",(20,0),S); label("$0$",(0,0),W); label("$15$",(0,3),W); label("$30$",(0,6),W); label("$45$",(0,9),W); label("$60$",(0,12),W); label("$75$",(0,15),W); label("H",(6,-2),S); label("O",(8,-2),S); label("U",(10,-2),S); label("R",(12,-2),S); label("S",(14,-2),S); label("M",(-4,11),N); label("I",(-4,9),N); label("L",(-4,7),N); label("E",(-4,5),N); label("S",(-4,3),N); [/asy]$

$\text{(A)}\ 15 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 25 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35$

Problem 5

A rectangular garden 60 feet long and 20 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?

$\text{(A)}\ 100 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 300 \qquad \text{(D)}\ 400 \qquad \text{(E)}\ 500$

Problem 6

Bo, Coe, Flo, Joe, and Moe have different amounts of money. Neither Joe nor Bo has as much money as Flo. Both Bo and Coe have more than Moe. Joe has more than Moe, but less than Bo. Who has the least amount of money?

$\text{(A)}\ \text{Bo} \qquad \text{(B)}\ \text{Coe} \qquad \text{(C)}\ \text{Flo} \qquad \text{(D)}\ \text{Joe} \qquad \text{(E)}\ \text{Moe}$

Problem 7

The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center?

$\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 110 \qquad \text{(D)}\ 120 \qquad \text{(E)}\ 130$

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2023年AMC8 真题及答案

2023年AMC 8 真题：

Problem 1

What is the value of $(8 \times 4 + 2) - (8 + 4 \times 2)$ ?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

Problem 2

A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures? $[asy] //Restored original diagram. Alter it if you would like, but it was made by TheMathGuyd, // Diagram by TheMathGuyd. I even put the lined texture :) // Thank you Kante314 for inspiring thicker arrows. They do look much better size(0,3cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; filldraw(sq,mediumgrey,black); draw((0.75,0)--(1.25,0),currentpen+1,Arrow(size=6)); //folding path sqside = (-0.5,-0.5)--(0.5,-0.5); path rhside = (-0.125,-0.125)--(0.5,-0.5); transform fld = shift((1.75,0))*scale(0.5); draw(fld*sq,black); int i; for(i=0; i<10; i=i+1) { draw(shift(0,0.05*i)*fld*sqside,deepblue); } path rhedge = (-0.125,-0.125)--(-0.125,0.8)--(-0.2,0.85)--cycle; filldraw(fld*rhedge,grey); path sqedge = (-0.5,-0.5)--(-0.5,0.4475)--(-0.575,0.45)--cycle; filldraw(fld*sqedge,grey); filldraw(fld*rh,white,black); int i; for(i=0; i<10; i=i+1) { draw(shift(0,0.05*i)*fld*rhside,deepblue); } draw((2.25,0)--(2.75,0),currentpen+1,Arrow(size=6)); //cutting transform cut = shift((3.25,0))*scale(0.5); draw(shift((-0.01,+0.01))*cut*sq); draw(cut*sq); filldraw(shift((0.01,-0.01))*cut*sq,white,black); int j; for(j=0; j<10; j=j+1) { draw(shift(0,0.05*j)*cut*sqside,deepblue); } draw(shift((0.01,-0.01))*cut*(0,-0.5)--shift((0.01,-0.01))*cut*(0.5,0),dashed); //Answers Below, but already Separated //filldraw(sqA,grey,black); //filldraw(sqB,grey,black); //filldraw(sq,grey,black); //filldraw(sqC,white,black); //filldraw(sq,grey,black); //filldraw(trD,white,black); //filldraw(sq,grey,black); //filldraw(sqE,white,black); [/asy]$

$[asy] // Diagram by TheMathGuyd. size(0,7.5cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; //ANSWERS real sh = 1.5; label("$\textbf{(A)}$",(-0.5,0.5),SW); label("$\textbf{(B)}$",shift((sh,0))*(-0.5,0.5),SW); label("$\textbf{(C)}$",shift((2sh,0))*(-0.5,0.5),SW); label("$\textbf{(D)}$",shift((0,-sh))*(-0.5,0.5),SW); label("$\textbf{(E)}$",shift((sh,-sh))*(-0.5,0.5),SW); filldraw(sqA,mediumgrey,black); filldraw(shift((sh,0))*sqB,mediumgrey,black); filldraw(shift((2*sh,0))*sq,mediumgrey,black); filldraw(shift((2*sh,0))*sqC,white,black); filldraw(shift((0,-sh))*sq,mediumgrey,black); filldraw(shift((0,-sh))*trD,white,black); filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*sqE,white,black); [/asy]$

Problem 3

Wind chill is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation $(\text{wind chill}) = (\text{air temperature}) - 0.7 \times (\text{wind speed}),$ where temperature is measured in degrees Fahrenheit $(^{\circ}\text{F})$ and the wind speed is measured in miles per hour (mph). Suppose the air temperature is $36^{\circ}\text{F}$ and the wind speed is $18$ mph. Which of the following is closest to the approximate wind chill?

$\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 35$

Problem 4

The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7.$ How many of these four numbers are prime? $[asy] /* Made by MRENTHUSIASM */ size(175); void ds(pair p) { filldraw((0.5,0.5)+p--(-0.5,0.5)+p--(-0.5,-0.5)+p--(0.5,-0.5)+p--cycle,mediumgrey); } ds((0.5,4.5)); ds((1.5,3.5)); ds((3.5,1.5)); ds((4.5,0.5)); add(grid(7,7,grey+linewidth(1.25))); int adj = 1; int curUp = 2; int curLeft = 4; int curDown = 6; label("$1$",(3.5,3.5)); for (int len = 3; len<=3; len+=2) { for (int i=1; i<=len-1; ++i) { label("$"+string(curUp)+"$",(3.5+adj,3.5-adj+i)); label("$"+string(curLeft)+"$",(3.5+adj-i,3.5+adj)); label("$"+string(curDown)+"$",(3.5-adj,3.5+adj-i)); ++curDown; ++curLeft; ++curUp; } ++adj; curUp = len^2 + 1; curLeft = len^2 + len + 2; curDown = len^2 + 2*len + 3; } draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(2)); [/asy]$ $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Problem 5

A lake contains $250$ trout, along with a variety of other fish. When a marine biologist catches and releases a sample of $180$ fish from the lake, $30$ are identified as trout. Assume that the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?

$\textbf{(A)}\ 1250 \qquad \textbf{(B)}\ 1500 \qquad \textbf{(C)}\ 1750 \qquad \textbf{(D)}\ 1800 \qquad \textbf{(E)}\ 2000$

Problem 6

The digits $2, 0, 2,$ and $3$ are placed in the expression below, one digit per box. What is the maximum possible value of the expression?

$[asy] // Diagram by TheMathGuyd. I can compress this later size(5cm); real w=2.2; pair O,I,J; O=(0,0);I=(1,0);J=(0,1); path bsqb = O--I; path bsqr = I--I+J; path bsqt = I+J--J; path bsql = J--O; path lsqb = shift((1.2,0.75))*scale(0.5)*bsqb; path lsqr = shift((1.2,0.75))*scale(0.5)*bsqr; path lsqt = shift((1.2,0.75))*scale(0.5)*bsqt; path lsql = shift((1.2,0.75))*scale(0.5)*bsql; draw(bsqb,dashed); draw(bsqr,dashed); draw(bsqt,dashed); draw(bsql,dashed); draw(lsqb,dashed); draw(lsqr,dashed); draw(lsqt,dashed); draw(lsql,dashed); label(scale(3)*"$\times$",(w,1/3)); draw(shift(1.3w,0)*bsqb,dashed); draw(shift(1.3w,0)*bsqr,dashed); draw(shift(1.3w,0)*bsqt,dashed); draw(shift(1.3w,0)*bsql,dashed); draw(shift(1.3w,0)*lsqb,dashed); draw(shift(1.3w,0)*lsqr,dashed); draw(shift(1.3w,0)*lsqt,dashed); draw(shift(1.3w,0)*lsql,dashed); [/asy]$

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18$

Problem 7

A rectangle, with sides parallel to the $x$ -axis and $y$ -axis, has opposite vertices located at $(15, 3)$ and $(16, 5)$ . A line drawn through points $A(0, 0)$ and $B(3, 1)$ . Another line is drawn through points $C(0, 10)$ and $D(2, 9)$ . How many points on the rectangle lie on at least one of the two lines? $[asy] usepackage("mathptmx"); size(9cm); draw((0,-.5)--(0,11),EndArrow(size=.15cm)); draw((1,0)--(1,11),mediumgray); draw((2,0)--(2,11),mediumgray); draw((3,0)--(3,11),mediumgray); draw((4,0)--(4,11),mediumgray); draw((5,0)--(5,11),mediumgray); draw((6,0)--(6,11),mediumgray); draw((7,0)--(7,11),mediumgray); draw((8,0)--(8,11),mediumgray); draw((9,0)--(9,11),mediumgray); draw((10,0)--(10,11),mediumgray); draw((11,0)--(11,11),mediumgray); draw((12,0)--(12,11),mediumgray); draw((13,0)--(13,11),mediumgray); draw((14,0)--(14,11),mediumgray); draw((15,0)--(15,11),mediumgray); draw((16,0)--(16,11),mediumgray); draw((-.5,0)--(17,0),EndArrow(size=.15cm)); draw((0,1)--(17,1),mediumgray); draw((0,2)--(17,2),mediumgray); draw((0,3)--(17,3),mediumgray); draw((0,4)--(17,4),mediumgray); draw((0,5)--(17,5),mediumgray); draw((0,6)--(17,6),mediumgray); draw((0,7)--(17,7),mediumgray); draw((0,8)--(17,8),mediumgray); draw((0,9)--(17,9),mediumgray); draw((0,10)--(17,10),mediumgray); draw((-.13,1)--(.13,1)); draw((-.13,2)--(.13,2)); draw((-.13,3)--(.13,3)); draw((-.13,4)--(.13,4)); draw((-.13,5)--(.13,5)); draw((-.13,6)--(.13,6)); draw((-.13,7)--(.13,7)); draw((-.13,8)--(.13,8)); draw((-.13,9)--(.13,9)); draw((-.13,10)--(.13,10)); draw((1,-.13)--(1,.13)); draw((2,-.13)--(2,.13)); draw((3,-.13)--(3,.13)); draw((4,-.13)--(4,.13)); draw((5,-.13)--(5,.13)); draw((6,-.13)--(6,.13)); draw((7,-.13)--(7,.13)); draw((8,-.13)--(8,.13)); draw((9,-.13)--(9,.13)); draw((10,-.13)--(10,.13)); draw((11,-.13)--(11,.13)); draw((12,-.13)--(12,.13)); draw((13,-.13)--(13,.13)); draw((14,-.13)--(14,.13)); draw((15,-.13)--(15,.13)); draw((16,-.13)--(16,.13)); label(scale(.7)*"$1$", (1,-.13), S); label(scale(.7)*"$2$", (2,-.13), S); label(scale(.7)*"$3$", (3,-.13), S); label(scale(.7)*"$4$", (4,-.13), S); label(scale(.7)*"$5$", (5,-.13), S); label(scale(.7)*"$6$", (6,-.13), S); label(scale(.7)*"$7$", (7,-.13), S); label(scale(.7)*"$8$", (8,-.13), S); label(scale(.7)*"$9$", (9,-.13), S); label(scale(.7)*"$10$", (10,-.13), S); label(scale(.7)*"$11$", (11,-.13), S); label(scale(.7)*"$12$", (12,-.13), S); label(scale(.7)*"$13$", (13,-.13), S); label(scale(.7)*"$14$", (14,-.13), S); label(scale(.7)*"$15$", (15,-.13), S); label(scale(.7)*"$16$", (16,-.13), S); label(scale(.7)*"$1$", (-.13,1), W); label(scale(.7)*"$2$", (-.13,2), W); label(scale(.7)*"$3$", (-.13,3), W); label(scale(.7)*"$4$", (-.13,4), W); label(scale(.7)*"$5$", (-.13,5), W); label(scale(.7)*"$6$", (-.13,6), W); label(scale(.7)*"$7$", (-.13,7), W); label(scale(.7)*"$8$", (-.13,8), W); label(scale(.7)*"$9$", (-.13,9), W); label(scale(.7)*"$10$", (-.13,10), W); dot((0,0),linewidth(4)); label(scale(.75)*"$A$", (0,0), NE); dot((3,1),linewidth(4)); label(scale(.75)*"$B$", (3,1), NE); dot((0,10),linewidth(4)); label(scale(.75)*"$C$", (0,10), NE); dot((2,9),linewidth(4)); label(scale(.75)*"$D$", (2,9), NE); draw((15,3)--(16,3)--(16,5)--(15,5)--cycle,linewidth(1.125)); dot((15,3),linewidth(4)); dot((16,3),linewidth(4)); dot((16,5),linewidth(4)); dot((15,5),linewidth(4)); [/asy]$ $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

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2022年USAMO 真题及答案

2022年USAMO 真题：

Day 1

Problem 1

Let $a$ and $b$ be positive integers. The cells of an $(a+b+1)\times (a+b+1)$ grid are colored amber and bronze such that there are at least $a^2+ab-b$ amber cells and at least $b^2+ab-a$ bronze cells. Prove that it is possible to choose $a$ amber cells and $b$ bronze cells such that no two of the $a+b$ chosen cells lie in the same row or column.

Problem 2

Let $b\geq2$ and $w\geq2$ be fixed integers, and $n=b+w$ . Given are $2b$ identical black rods and $2w$ identical white rods, each of side length 1.

We assemble a regular $2n-$ gon using these rods so that parallel sides are the same color. Then, a convex $2b$ -gon $B$ is formed by translating the black rods, and a convex $2w$ -gon $W$ is formed by translating the white rods. An example of one way of doing the assembly when $b=3$ and $w=2$ is shown below, as well as the resulting polygons $B$ and $W$ .

Prove that the difference of the areas of $B$ and $W$ depends only on the numbers $b$ and $w$ , and not on how the $2n$ -gon was assembled.

Problem 3

Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all functions $f:\mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that for all $x,y\in \mathbb{R}_{>0}$ we have $f(x) = f(f(f(x)) + y) + f(xf(y)) f(x+y).$

Day 2

Problem 4

Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

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2021年USAMO 真题及答案

2021年USAMO 真题：

Day 1

$\textbf{Note:}$ For any geometry problem whose statement begins with an asterisk $(*)$ , the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.

Problem 1

$(*)$ Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that $\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.$ Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.

Problem 2

The Planar National Park is a subset of the Euclidean plane consisting of several trails which meet at junctions. Every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions.

A visitor walks through the park as follows: she begins at a junction and starts walking along a trail. At the end of that first trail, she enters a junction and turns left. On the next junction she turns right, and so on, alternating left and right turns at each junction. She does this until she gets back to the junction where she started. What is the largest possible number of times she could have entered any junction during her walk, over all possible layouts of the park?

Problem 3

Let $n \geq 2$ be an integer. An $n \times n$ board is initially empty. Each minute, you may perform one of three moves: If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells. If all cells in a column have a stone, you may remove all stones from that column. If all cells in a row have a stone, you may remove all stones from that row. $[asy] unitsize(20); draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)); fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--cycle, grey); draw((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--(0.2, 3.8), linewidth(2)); draw((0,2)--(4,2)); draw((2,4)--(2,0)); [/asy]$ For which $n$ is it possible that, after some non-zero number of moves, the board has no stones?

Day 2

Problem 4

A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$ , there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$ . (The elements $s$ and $t$ could be equal.)

Given this information, find all possible values for the number of elements of $S$ .

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2020年USAMO 真题及答案

2020年USAMO 真题：

Day 1

Problem 1

Let $ABC$ be a fixed acute triangle inscribed in a circle $\omega$ with center $O$ . A variable point $X$ is chosen on minor arc $AB$ of $\omega$ , and segments $CX$ and $AB$ meet at $D$ . Denote by $O_1$ and $O_2$ the circumcenters of triangles $ADX$ and $BDX$ , respectively. Determine all points $X$ for which the area of triangle $OO_1O_2$ is minimized.

Problem 2

An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:

$\bullet$ The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot 2020^2$ possible positions for a beam.)

$\bullet$ No two beams have intersecting interiors.

$\bullet$ The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.

What is the smallest positive number of beams that can be placed to satisfy these conditions?

Problem 3

Let $p$ be an odd prime. An integer $x$ is called a quadratic non-residue if $p$ does not divide $x - t^2$ for any integer $t$ .

Denote by $A$ the set of all integers $a$ such that $1 \le a < p$ , and both $a$ and $4 - a$ are quadratic non-residues. Calculate the remainder when the product of the elements of $A$ is divided by $p$ .

Day 2

Problem 4

Suppose that $(a_1, b_1), (a_2, b_2), \ldots , (a_{100}, b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i, j)$ satisfying $1 \le i < j \le 100$ and $|a_ib_j - a_j b_i|=1$ . Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.

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2018年USAMO 真题及答案

2018年USAMO 真题：

Day 1

Note: For any geometry problem whose statement begins with an asterisk ( $*$ ), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.

Problem 1

Let $a,b,c$ be positive real numbers such that $a+b+c=4\sqrt[3]{abc}$ . Prove that $2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.$

Problem 2

Find all functions $f:(0,\infty) \rightarrow (0,\infty)$ such that

$f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1$ for all $x,y,z >0$ with $xyz =1.$

Problem 3

For a given integer $n\ge 2,$ let $\{a_1,a_2,…,a_m\}$ be the set of positive integers less than $n$ that are relatively prime to $n.$ Prove that if every prime that divides $m$ also divides $n,$ then $a_1^k+a_2^k + \dots + a_m^k$ is divisible by $m$ for every positive integer $k.$

Day 2

Problem 4

Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers $a_1 + k, a_2 + 2k, \dots, a_p + pk$ produce at least $\tfrac{1}{2} p$ distinct remainders upon division by $p$ .

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2017年USAMO 真题及答案

2017年USAMO 真题：

Day 1

Problem 1

Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a > 1$ and $b > 1$ such that $a^b + b^a$ is divisible by $a + b.$

Problem 2

Let $m_1, m_2, \ldots, m_n$ be a collection of $n$ positive integers, not necessarily distinct. For any sequence of integers $A = (a_1, \ldots, a_n)$ and any permutation $w = w_1, \ldots, w_n$ of $m_1, \ldots, m_n$ , define an $A$ -inversion of $w$ to be a pair of entries $w_i, w_j$ with $i < j$ for which one of the following conditions holds: $a_i \ge w_i > w_j,$ $w_j > a_i \ge w_i,$ or $w_i > w_j > a_i.$ Show that, for any two sequences of integers $A = (a_1, \ldots, a_n)$ and $B = (b_1, \ldots, b_n)$ , and for any positive integer $k$ , the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $A$ -inversions is equal to the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $B$ -inversions.

Problem 3

( $*$ ) Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I$ . Ray $AI$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$ ; the circle with diameter $\overline{DM}$ cuts $\Omega$ again at $K$ . Lines $MK$ and $BC$ meet at $S$ , and $N$ is the midpoint of $\overline{IS}$ . The circumcircles of $\triangle KID$ and $\triangle MAN$ intersect at points $L_1$ and $L_2$ . Prove that $\Omega$ passes through the midpoint of either $\overline{IL_1}$ or $\overline{IL_2}$ .

Day 2

Problem 4

Let $P_1$ , $P_2$ , $\dots$ , $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$ , other than $(1,0)$ . Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$ , $R_2$ , $\dots$ , $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$ . Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$ , and so on, until we have labeled all of the blue points $B_1, \dots, B_n$ . Show that the number of counterclockwise arcs of the form $R_i \to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \dots, R_n$ of the red points.

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