2023AMC10B-AMC12B卷真题答案公布,快来对答案! – SM国际教育-专注AMC/欧几里得国际数学竞赛,AP/IB/ALEVL/IGCSE标化考试!原偶爱数学!

2023年AMC10/12 B卷于昨日落幕！ 整理版AMC10/12B卷真题与解析

机构教研老师快马加鞭整理本次B卷解析，帮助同学们更清楚地分析这次考试的重难点，评估成绩，同时为同学们之后的学习和备考一些建议。

AMC10/12B卷解析讲座

B卷试题评析— Comments on Version B —01AMC10B

10B的难度如我们所预期，还是偏简单的，在AB卷同时出题的情况下，A简单，B就难不了太多，不过B卷的简单并不意味着好做，对于想要冲击AIME晋级的同学，可能做题的时候会遇到一些卡顿，非常考验大家的应试技巧，在该跳的时候跳，该磨的时候磨。

简单题部分，第5题和第10题是比较容易让同学们陷入困顿的，题目的描述比较长，而解决路径也并不是非常直白，一个需要引入两个未知数，一个需要画图操作去进行推理，都不是容易一眼看出结果的题目。

而中档题部分的11-15题也对大家的训练量提出了要求，如果不怎么练题，或者练得不多，那么无论是的11题不定方程求解集数还是12题数形结合，15题的连续阶乘的质因数分解都会让学生觉得难受，若是强行举例子死算都会比较费时间。13题则是极少情况的复杂的绝对值函数图像求面积问题出现在这么靠前的位置，以往的安排可能放在19题左右更合适，14题求整数解则要求学生灵活应对，若是想要正解，则变成了一道中考风味的题目了。

更难部分的题目反而给人没有超预期的挑战了，16和17，22,25题都是陈题，想必对于我们AMC10Level3的学员非常轻松，19题要在概率计算的时候考虑几何概型或者使用期望加以解决，有些新颖。

这份试卷对于AIME晋级目标的学生来说会比较合适，对于冲击1%的同学属于偏简单的，其中的整数解、绝对值函数、几何概型也是新瓶装旧酒，新手们做起来会吃力，高手们还是可以游刃有余的。

02AMC12B

2023秋季的AMC12B总体来说回归到了AMC的“正常”水平，不管是知识点分布，题意理解的强度还是技巧的丰富程度都比较传统。

在1-10的简单题当中，保持了阅读量和一些复杂的计算，并不容易轻易完成，并且有一个第9题消耗学生时间，对于绝对值图像不熟练的同学很难快速得到答案。

在11-17题当中，有经典的counting和可选递推的问题，也有一些逻辑判断，并且几何的考察回归到了2010年以前的风格，使用Loc且没有图；同时Frobenius硬币的题目也消耗学生一些时间，很少有人学过三枚硬币的公式，所以必须在考场上Bash，这对于心态是个不小的考验。

最后的一些难题部分，20,21,23反而是最困难的，并且今年的所谓冷门考点是22题的函数方程，这个问题学生容易意识到cos，但是取非常巧妙的构造去发现f(1)不容易想到。

总的来说，今年AMC12B知识技巧分布均匀，难度适中，最后几题偏简单，题目内容陈旧且有迹可循，更加接近国内的考试风格

分数线预测— Cut-off Points Prediction —

10B 预测分数线

AIME:99分

HR5%:105分

DHR1%:120分

12B 预测分数线

AIME:88.5分

HR5%:106.5分

DHR1%:130.5分

10A 预测分数线

AIME:103.5分

HR5%:109.5分

DHR1%:133.5分

12A 预测分数线

AIME:90分

HR5%:108分

DHR1%:133.5分

查询考试成绩

考试后2-4周

查询分数线及证书

考试后6-8周

查询方式

中国区组委会

届时大家会收到邮件通知，我们也会出推送提醒。可以登录邮件中提供的官方网站，或直接进入官网https://math-table.com/，进入成绩查询系统查询。

阿思丹组委会

届时阿思丹会在微信弹出通知，各位考生可通过微信登录“阿斯丹国际理科测评”小程序，点击“我的报名”-点击“AMC10/12”-点击“分数查询”，进行成绩查询。

B卷考点— Exam Points for Version A —AMC10B考点整理

AMC12B考点整理

AMC10B Problem 1

Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only $\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?

$\textbf{(A) }\frac{1}{12}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{1}{6}\qquad\textbf{(D) }\frac{1}{8}\qquad\textbf{(E) }\frac{2}{9}$

Problem 2

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\%$ sales tax on the discounted price. He had $43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?

$\textbf{(A) }$46\qquad\textbf{(B) }$50\qquad\textbf{(C) }$48\qquad\textbf{(D) }$47\qquad\textbf{(E) }$49$

Problem 3

A $3-4-5$ right triangle is inscribed in circle $A$ , and a $5-12-13$ right triangle is inscribed in circle $B$ . What is the ratio of the area of circle $A$ to the area of circle $B$ ?

$\textbf{(A) }\frac{9}{25}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{1 }{5}\qquad\textbf{(D) }\frac{25}{169}\qquad\textbf{(E) }\frac{4}{25}$

Problem 4

Jackson’s paintbrush makes a narrow strip with a width of 6.5 millimeters. Jackson has enough paint to make a strip 25 meters long. How many square centimeters of paper could Jackson cover with paint?

$\textbf{(A) }162,500\qquad\textbf{(B) }162.5\qquad\textbf{(C) }1,625\qquad\textbf{(D) }1,625,000\qquad\textbf{(E) }16,250$

Problem 5

Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$ . Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$ . How many numbers are written on the blackboard?

$\textbf{(A) }10\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Problem 6

Let $L_1 = 1$ , $L_2 = 3$ , and $L_{n+2} = L_{n+1}+L_n$ for $n \geq 1$ . How many terms in the sequence $L_1, L_2, L_3, \cdots, L_{2023}$ are even?

$\textbf{(A) }673\qquad\textbf{(B) }1011\qquad\textbf{(C) }675\qquad\textbf{(D) }1010\qquad\textbf{(E) }674$

Problem 7

Square ABCD is rotated 20 degrees clockwise about its center to obtain square EFGH, as shown below. What is the degree measure of $\angle{EAB}$ ?

$\textbf{(A) }24\qquad\textbf{(B) }35\qquad\textbf{(C) }30\qquad\textbf{(D) }32\qquad\textbf{(E) }20$

Problem 8

What is the units digit of $2022^{2023} + 2023^{2022}$ ?

$\text{(A)}\ 7 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 3$

Problem 9

The numbers 16 and 25 are a pair of consecutive postive squares whose difference is 9. How many pairs of consecutive positive perfect squares have a difference of less than or equal to 2023?

$\text{(A)}\ 674 \qquad \text{(B)}\ 1011 \qquad \text{(C)}\ 1010 \qquad \text{(D)}\ 2019 \qquad \text{(E)}\ 2017$

Problem 10

You are playing a game. A $2$ $\times$ $1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3$ $\times$ $3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A “turn” consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?

$\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 6$

Problem 11

Suzanne went to the bank and withdrew 800 dollars. The teller gave her this amount using 20 dollar bills, 50 dollar bills, and 100 dollar bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?

$\textbf{(A) }45\qquad\textbf{(B) }21\qquad\textbf{(C) }36\qquad\textbf{(D) }28\qquad\textbf{(E) }32$

Problem 12

When the roots of the polynomial

$P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}$

are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is $P(x)$ positive?

$\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$

Problem 13

What is the area of the region in the coordinate plane defined by

$| | x | - 1 | + | | y | - 1 | \le 1$ ?

$\text{(A)}\ 2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 12$

Problem 14

How many ordered pairs of integers $(m,n)$ satisfy the equation $m^2+mn+n^2 = m^2n^2$ ?

$\textbf{(A) }7\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }5$

Problem 15

What is the least positive integer $m$ such that $m \cdot 2! \cdot 3!\cdot 4!\cdot 5! \dots 16!$ is a perfect square?

$\textbf{(A) }30\qquad\textbf{(B) }30030\qquad\textbf{(C) }70\qquad\textbf{(D) }1430\qquad\textbf{(E) }1001$

Problem 16

Define an $upno$ to be a positive integer of 2 or more digits where the digits are strictly increasing moving left to right. Similarly, define a $downno$ to be a positive integer of 2 or more digits where the digits are strictly decreasing moving left to right. For instance, the number 258 is an upno and 8620 is a downno. Let ? equal the total number of $upnos$ and let ? equal the total number of $downnos$ . What is |? − ?|?

$\textbf{(A)}~512\qquad\textbf{(B)}~10\qquad\textbf{(C)}~0\qquad\textbf{(D)}~9\qquad\textbf{(E)}~511$

Problem 17

A rectangular box ? has distinct edge lengths ?, ?, and ?. The sum of the lengths of all 12 edges of ? is 13, the sum of the areas of all 6 faces of ? is $\dfrac{11}{2}$ , and the volume of ? is $\dfrac{1}{2}$ . What is the length of the longest interior diagonal connecting two vertices of ? ?

$\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$

Problem 18

Suppose ?, ?, and ? are positive integers such that $\dfrac{a}{14}+\dfrac{b}{15}=\dfrac{c}{210}$ .

Which of the following statements are necessarily true?

I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$ .

II. If $\gcd(c,210)=1$ , then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.

III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$ .

$\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$

Problem 19

Sonya the frog chooses a point uniformly at random lying within the square $[0, 6]$ $\times$ $[0, 6]$ in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from $[0, 1]$ and a direction uniformly at random from {north, south, east, west}. All he choices are independent. She now hops the distance in the chosen direction. What is the probability that she lands outside the square?

$\textbf{(A) } \frac{1}{6} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{4} \qquad \textbf{(D) } \frac{1}{10} \qquad \textbf{(E) } \frac{1}{9}$

Problem 20

Four congruent semicircles are drawn on the surface of a sphere with radius 2, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$ . What is ??

$\textbf{(A) } 32 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 48 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 27$

Problem 21

Each of 2023 balls is randomly placed into one of 3 bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?

$\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$

Problem 22

How many distinct values of ? satisfy $\lfloor{x}\rfloor^2-3x+2=0.$ where $\lfloor{x}\rfloor$ denotes the largest integer less than or equal to ??

$\textbf{(A) } \text{an infinite number} \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 0$

Problem 23

An arithmetic sequence of positive integers has $n\ge3$ terms, initial term $a$ , and common difference $$d>1$” width=”41″ height=”13″>. Carl wrote down all the terms in this sequence correctly except for one term, which was off by <img loading=$ . The sum of the terms he wrote down was $222$ . What is $a+d+n$ ?

$\textbf{(A) } 24 \qquad \textbf{(B) } 20 \qquad \textbf{(C) } 22 \qquad \textbf{(D) } 28 \qquad \textbf{(E) } 26$

Problem 24

What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0\le u\le1$ , $0\le v\le1,$ and $0\le w\le1$ ?

$\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16$

Problem 25

A regular pentagon with area $1+\sqrt5$ is printed on paper and cut out. All five vertices are folded to the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?

$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$

AMC10/12B卷 1-4题相同

AMC12 Problem 1

Problem 2

Problem 3

A $3-4-5$ right triangle is inscribed in circle $A$ , and a $5-12-13$ right triangle is inscribed in circle $B$ . What is the ratio of the area of circle $A$ to the area of circle $B$ ?

Problem 4

Problem 5

You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A “turn” consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensue that at least one of your guessed squares is covered by the rectangle?

$\textbf{(A)}~3\qquad\textbf{(B)}~5\qquad\textbf{(C)}~4\qquad\textbf{(D)}~8\qquad\textbf{(E)}~6$

Problem 6

When the roots of the polynomial.

are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is $P(x)$ positive?

Problem 7

For how many integers $n$ does the expression $\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}}$ represent a real number, where log denotes the base $10$ logarithm?

$\textbf{(A) }900 \qquad \textbf{(B) }2\qquad \textbf{(C) }902 \qquad \textbf{(D) } 2 \qquad \textbf{(E) }901$

Problem 8

How many nonempty subsets $B$ of $\{0, 1, 2, 3, \dots, 12\}$ have the property that the number of elements in $B$ is equal to the least element of $B$ ? For example, $B = \{4, 6, 8, 11\}$ satisfies the condition.

$\textbf{(A)}\ 256 \qquad\textbf{(B)}\ 136 \qquad\textbf{(C)}\ 108 \qquad\textbf{(D)}\ 144 \qquad\textbf{(E)}\ 156$

Problem 9

What is the area of the region in the coordinate plane defined by

$\left||x|-1\right|+\left||y|-1\right|\leq 1?$

$\textbf{(A)}~4\qquad\textbf{(B)}~8\qquad\textbf{(C)}~10\qquad\textbf{(D)}~12\qquad\textbf{(E)}~15$

Problem 10

In the $xy$ -plane, a circle of radius $4$ with center on the positive $x$ -axis is tangent to the $y$ -axis at the origin, and a circle with radius $10$ with center on the positive $y$ -axis is tangent to the $x$ -axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?

$\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}} \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}} \qquad\textbf{(E)}\ \dfrac{2}{5}$

Problem 11

What is the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other?

$\textbf{(A) }\frac 54 \qquad \textbf{(B) } \frac 87 \qquad \textbf{(C)} \frac{5\sqrt2}4 \qquad \textbf{(D) } \frac 32 \qquad \textbf{(E) } \frac{3\sqrt3}4$

Problem 12

For complex numbers $u=a+bi$ and $v=c+di$ , define the binary operation $\otimes$ by $u\otimes v=ac+bdi.$ Suppose $z$ is a complex number such that $z\otimes z=z^{2}+40$ . What is $|z|$ ?

$\textbf{(A)}~\sqrt{10}\qquad\textbf{(B)}~3\sqrt{2}\qquad\textbf{(C)}~2\sqrt{6}\qquad\textbf{(D)}~6\qquad\textbf{(E)}~5\sqrt{2}$

Problem 13

A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$ . The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$ , the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$ , and the volume of $\mathcal{P}$ is $\frac{1}{2}$ . What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$ ?

Problem 14

For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?

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

Problem 15

Suppose $a$ , $b$ , and $c$ are positive integers such that $\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.$ Which of the following statements are necessarily true?

I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$ .

II. If $\gcd(c,210)=1$ , then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.

III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$ .

Problem 16

In Coinland, there are three types of coins, each worth $6, 10,$ and $15.$ What is the sum of the digits of the maximum amount of money that is impossible to have?

$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }7\qquad\textbf{(D) }11\qquad\textbf{(E) }9$

Problem 17

Triangle ABC has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$ ?

$\textbf{(A) }12\sqrt{3}\qquad\textbf{(B) }8\sqrt{6}\qquad\textbf{(C) }14\sqrt{2}\qquad\textbf{(D) }20\sqrt{2}\qquad\textbf{(E) }15\sqrt{3}$

Problem 18

Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda’s average on all the quizzes she took during the first semester was 3 points higher than Zelda’s average on all the quizzes she took during the first semester. Yolanda’s average on all the quizzes she took during the second semester was 18 points higher than her average for the first semester and was again 3 points higher than Zelda’s average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?

$\textbf{(A)}$ Yolanda’s quiz average for the academic year was 22 points higher than Zelda’s.

$\textbf{(B)}$ Zelda’s quiz average for the academic year was higher than Yolanda’s.

$\textbf{(C)}$ Yolanda’s quiz average for the academic year was 3 points higher than Zelda’s.

$\textbf{(D)}$ Zelda’s quiz average for the academic year equaled Yolanda’s.

$\textbf{(E)}$ If Zelda had scored 3 points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.

Problem 19

Each of $2023$ balls is placed in on of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?

Problem 20

Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position?

$\textbf{(A)}~\frac{1}{6}\qquad\textbf{(B)}~\frac{1}{5}\qquad\textbf{(C)}~\frac{\sqrt{3}}{8}\qquad\textbf{(D)}~\frac{\arctan \frac{1}{2}}{\pi}\qquad\textbf{(E)}~\frac{2\arcsin \frac{1}{4}}{\pi}$

Problem 21

A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is $3\sqrt3$ inches, its top diameter is $6$ inches, and its bottom diameter is $12$ inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?

$\textbf{(A) } 6 + 3\pi\qquad \textbf{(B) }6 + 6\pi\qquad \textbf{(C) } 6\sqrt3 \qquad \textbf{(D) } 6\sqrt5 \qquad \textbf{(E) } 6\sqrt3 + \pi$

Problem 22

A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ $f(a + b) + f(a - b) = 2f(a) f(b).$ Which one of the following cannot be the value of $f(1)?$

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

Problem 23

When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$ ?

$\textbf{(A)}~6\qquad\textbf{(B)}~8\qquad\textbf{(C)}~9\qquad\textbf{(D)}~10\qquad\textbf{(E)}~11$

Problem 24

Suppose that $a$ , $b$ , $c$ and $d$ are positive integers satisfying all of the following relations.

\[\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3\]

\[\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3\]

\[\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3\]

\[\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2\]

\[\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2\]

\[\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2\]

What is $\text{gcd}(a,b,c,d)$ ?

$\textbf{(A)}~30\qquad\textbf{(B)}~45\qquad\textbf{(C)}~3\qquad\textbf{(D)}~15\qquad\textbf{(E)}~6$

Problem 25

A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?

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AMC10B Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

AMC12 Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

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Problem 19

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Problem 22

Problem 23

Problem 24

Problem 25

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