link stringlengths 75 84 | letter stringclasses 5
values | answer float64 0 2,935,363,332B | problem stringlengths 14 5.33k | solution listlengths 1 13 |
|---|---|---|---|---|
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12B_Problems/Problem_9 | E | 1,010,000 | What is \[\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ?\]
$\textbf{(A) }100100 \qquad \textbf{(B) }500500\qquad \textbf{(C) }505000 \qquad \textbf{(D) }1001000 \qquad \textbf{(E) }1010000 \qquad$ | [
"Recall that the sum of the first $100$ positive integers is $\\sum^{100}_{k=1} k = \\frac{101\\cdot100}{2}=5050.$ It follows that \\begin{align*} \\sum^{100}_{i=1} \\sum^{100}_{j=1} (i+j) &= \\sum^{100}_{i=1} \\sum^{100}_{j=1}i + \\sum^{100}_{i=1} \\sum^{100}_{j=1}j \\\\ &= \\sum^{100}_{i=1} (100i) + 100 \\sum^{10... |
https://artofproblemsolving.com/wiki/index.php/1988_AJHSME_Problems/Problem_19 | A | 397 | What is the $100\text{th}$ number in the arithmetic sequence $1,5,9,13,17,21,25,...$
$\text{(A)}\ 397 \qquad \text{(B)}\ 399 \qquad \text{(C)}\ 401 \qquad \text{(D)}\ 403 \qquad \text{(E)}\ 405$ | [
"To get from the $1^\\text{st}$ term of an arithmetic sequence to the $100^\\text{th}$ term, we must add the common difference $99$ times. The first term is $1$ and the common difference is $5-1=9-5=13-9=\\cdots = 4$ , so the $100^\\text{th}$ term is \\[1+4(99)=397 \\rightarrow \\boxed{397}\\]",
"Alternatively y... |
https://artofproblemsolving.com/wiki/index.php/1995_AJHSME_Problems/Problem_15 | B | 1 | What is the $100^\text{th}$ digit to the right of the decimal point in the decimal form of $4/37$
$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$ | [
"$\\frac{4}{37}=\\frac{12}{111}=\\frac{108}{999}=0.108108108...$\nSince this repeats every three digits, digit number x = digit number (x mod 3), and the 100th digit = (100 mod 3)th digit = 1st digit = $\\boxed{1}$"
] |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_12B_Problems/Problem_7 | D | 24 | What is the area enclosed by the graph of $|3x|+|4y|=12$
$\mathrm{(A)}\ 6 \qquad \mathrm{(B)}\ 12 \qquad \mathrm{(C)}\ 16 \qquad \mathrm{(D)}\ 24 \qquad \mathrm{(E)}\ 25$ | [
"If we get rid of the absolute values, we are left with the following 4 equations (using the logic that if $|a|=b$ , then $a$ is either $b$ or $-b$ ):\n\\begin{align*} 3x+4y=12 \\\\ -3x+4y=12 \\\\ 3x-4y=12 \\\\ -3x-4y=12 \\end{align*}\nWe can then put these equations in slope-intercept form in order to graph them.\... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12B_Problems/Problem_18 | B | 2 | What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$
$\textbf{(A)}\ \pi+\sqrt{2} \qquad\textbf{(B)}\ \pi+2 \qquad\textbf{(C)}\ \pi+2\sqrt{2} \qquad\textbf{(D)}\ 2\pi+\sqrt{2} \qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$ | [
"Consider the case when $x \\geq 0$ $y \\geq 0$ \\[x^2+y^2=x+y\\] \\[(x - \\frac{1}{2})^2+(y - \\frac{1}{2})^2=\\frac{1}{2}\\] Notice the circle intersect the axes at points $(0, 1)$ and $(1, 0)$ . Find the area of this circle in the first quadrant. The area is made of a semi-circle with radius of $\\frac{\\sqrt{2}... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10B_Problems/Problem_21 | B | 2 | What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$
$\textbf{(A)}\ \pi+\sqrt{2}\qquad\textbf{(B)}\ \pi+2\qquad\textbf{(C)}\ \pi+2\sqrt{2}\qquad\textbf{(D)}\ 2\pi+\sqrt{2}\qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$ | [
"Another way to solve this problem is using cases. \nThough this may seem tedious, we only have to do one case since the area enclosed is symmetrical.\nThe equation for this figure is $x^2+y^2=|x|+|y|$ To make this as easy as possible,\nwe can make both $x$ and $y$ positive. Simplifying the equation for $x$ and $y$... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10B_Problems/Problem_13 | B | 8 | What is the area of the region in the coordinate plane defined by
$| | x | - 1 | + | | y | - 1 | \le 1$
$\text{(A)}\ 2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 12$ | [
"First consider, $|x-1|+|y-1| \\le 1.$ We can see that it is a square with a radius of $1$ (diagonal $\\sqrt{2}$ ). The area of the square is $\\sqrt{2}^2 = 2.$\nNext, we insert an absolute value sign into the equation and get $|x-1|+||y|-1| \\le 1.$ This will double the square reflecting over x-axis.\nSo now we ha... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_9 | B | 8 | What is the area of the region in the coordinate plane defined by
$| | x | - 1 | + | | y | - 1 | \le 1$
$\text{(A)}\ 2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 12$ | [
"First consider, $|x-1|+|y-1| \\le 1.$ We can see that it is a square with a radius of $1$ (diagonal $\\sqrt{2}$ ). The area of the square is $\\sqrt{2}^2 = 2.$\nNext, we insert an absolute value sign into the equation and get $|x-1|+||y|-1| \\le 1.$ This will double the square reflecting over x-axis.\nSo now we ha... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10B_Problems/Problem_2 | B | 6 | What is the area of the shaded figure shown below? [asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8); pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label... | [
"The line of symmetry divides the shaded figure into two congruent triangles, each with base $3$ and height $2.$\nTherefore, the area of the shaded figure is \\[2\\cdot\\left(\\frac12\\cdot3\\cdot2\\right)=2\\cdot3=\\boxed{6}.\\] ~MRENTHUSIASM ~Wilhelm Z",
"To find the area of the shaded figure, we subtract the a... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12B_Problems/Problem_2 | B | 6 | What is the area of the shaded figure shown below? [asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8); pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label... | [
"The line of symmetry divides the shaded figure into two congruent triangles, each with base $3$ and height $2.$\nTherefore, the area of the shaded figure is \\[2\\cdot\\left(\\frac12\\cdot3\\cdot2\\right)=2\\cdot3=\\boxed{6}.\\] ~MRENTHUSIASM ~Wilhelm Z",
"To find the area of the shaded figure, we subtract the a... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_8_Problems/Problem_23 | B | 6 | What is the area of the shaded pinwheel shown in the $5 \times 5$ grid?
[asy] filldraw((2.5,2.5)--(0,1)--(1,1)--(1,0)--(2.5,2.5)--(4,0)--(4,1)--(5,1)--(2.5,2.5)--(5,4)--(4,4)--(4,5)--(2.5,2.5)--(1,5)--(1,4)--(0,4)--cycle, gray, black); int i; for(i=0; i<6; i=i+1) { draw((i,0)--(i,5)); draw((0,i)--(5,i)); } [/asy]
$\tex... | [
"The area of the square around the pinwheel is 25. The area of the pinwheel is equal to $\\text{the square } - \\text{ the white space.}$ Each of the four triangles have a base of 3 units and a height of 2.5 units, and so their combined area is 15 units squared. Then the unshaded space consists of the four triangle... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_8_Problems/Problem_21 | E | 16 | What is the area of the triangle formed by the lines $y=5$ $y=1+x$ , and $y=1-x$
$\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16$ | [
"First, we need to find the coordinates where the graphs intersect.\nWe want the points x and y to be the same. Thus, we set $5=x+1,$ and get $x=4.$ Plugging this into the equation, $y=1-x,$ $y=5$ , and $y=1+x$ intersect at $(4,5)$ , we call this line x.\nDoing the same thing, we get $x=-4.$ Thus, $y=5$ . Also, $y=... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_10B_Problems/Problem_20 | C | 53,332.8 | What is the average (mean) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, and 8 exactly once?
$\textbf{(A) }48000\qquad\textbf{(B) }49999.5\qquad\textbf{(C) }53332.8\qquad\textbf{(D) }55555\qquad\textbf{(E) }56432.8$ | [
"We first look at how many times each number will appear in each slot. If we fix a number in a slot, then there are $4! = 24$ ways to arrange the other numbers, so each number appears in each spot $24$ times. Therefore, the sum of all such numbers is $24 \\times (1+3+5+7+8) \\times (11111) = 24 \\times 24 \\times 1... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_11 | C | 45 | What is the degree measure of the acute angle formed by lines with slopes $2$ and $\frac{1}{3}$
$\textbf{(A)} ~30\qquad\textbf{(B)} ~37.5\qquad\textbf{(C)} ~45\qquad\textbf{(D)} ~52.5\qquad\textbf{(E)} ~60$ | [
"Remind that $\\text{slope}=\\dfrac{\\Delta y}{\\Delta x}=\\tan \\theta$ where $\\theta$ is the angle between the slope and $x$ -axis. $k_1=2=\\tan \\alpha$ $k_2=\\dfrac{1}{3}=\\tan \\beta$ . The angle formed by the two lines is $\\alpha-\\beta$ $\\tan(\\alpha-\\beta)=\\dfrac{\\tan\\alpha-\\tan\\beta}{1+\\tan\\alph... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_11 | null | 45 | What is the degree measure of the acute angle formed by lines with slopes $2$ and $\frac{1}{3}$
$\textbf{(A)} ~30\qquad\textbf{(B)} ~37.5\qquad\textbf{(C)} ~45\qquad\textbf{(D)} ~52.5\qquad\textbf{(E)} ~60$ | [
"We can take any two lines of this form, since the angle between them will always be the same. Let's take $y=2x$ for the line with slope of 2 and $y=\\frac{1}{3}x$ for the line with slope of 1/3. Let's take 3 lattice points and create a triangle. Let's use $(0,0)$ $(1,2)$ , and $(3,1)$ . The distance between the or... |
https://artofproblemsolving.com/wiki/index.php/1999_AMC_8_Problems/Problem_2 | C | 60 | 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),SS... | [
"At $10:00$ , the hour hand will be on the $10$ while the minute hand on the $12$\nThis makes them $\\frac{1}{6}$ th of a circle apart, and $\\frac{1}{6}\\cdot360^{\\circ}=\\boxed{60}$",
"We know that the full clock is a circle, and therefore has 360 degrees. Considering that there are $12$ numbers, the distance ... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10A_Problems/Problem_1 | D | 2,003 | What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers?
$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \ } 2003\qquad \mathrm{(E) \ } 4006$ | [
"The first $2003$ even counting numbers are $2,4,6,...,4006$\nThe first $2003$ odd counting numbers are $1,3,5,...,4005$\nThus, the problem is asking for the value of $(2+4+6+...+4006)-(1+3+5+...+4005)$\n$(2+4+6+...+4006)-(1+3+5+...+4005) = (2-1)+(4-3)+(6-5)+...+(4006-4005)$\n$= 1+1+1+...+1 = \\boxed{2003}$",
"Us... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_12A_Problems/Problem_1 | D | 2,003 | What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers?
$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \ } 2003\qquad \mathrm{(E) \ } 4006$ | [
"The first $2003$ even counting numbers are $2,4,6,...,4006$\nThe first $2003$ odd counting numbers are $1,3,5,...,4005$\nThus, the problem is asking for the value of $(2+4+6+...+4006)-(1+3+5+...+4005)$\n$(2+4+6+...+4006)-(1+3+5+...+4005) = (2-1)+(4-3)+(6-5)+...+(4006-4005)$\n$= 1+1+1+...+1 = \\boxed{2003}$",
"Us... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_10A_Problems/Problem_14 | A | 80 | What is the greatest integer less than or equal to \[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\]
$\textbf{(A) }80\qquad \textbf{(B) }81 \qquad \textbf{(C) }96 \qquad \textbf{(D) }97 \qquad \textbf{(E) }625\qquad$ | [
"We write \\[\\frac{3^{100}+2^{100}}{3^{96}+2^{96}}=\\frac{3^{96}}{3^{96}+2^{96}}\\cdot\\frac{3^{100}}{3^{96}}+\\frac{2^{96}}{3^{96}+2^{96}}\\cdot\\frac{2^{100}}{2^{96}}=\\frac{3^{96}}{3^{96}+2^{96}}\\cdot 81+\\frac{2^{96}}{3^{96}+2^{96}}\\cdot 16.\\] Hence we see that our number is a weighted average of 81 and 16,... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_10A_Problems/Problem_5 | D | 90 | What is the greatest number of consecutive integers whose sum is $45?$
$\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$ | [
"We might at first think that the answer would be $9$ , because $1+2+3 \\dots +n = 45$ when $n = 9$ . But note that the problem says that they can be integers, not necessarily positive. Observe also that every term in the sequence $-44, -43, \\cdots, 44, 45$ cancels out except $45$ . Thus, the answer is, intuitive... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_12A_Problems/Problem_4 | D | 90 | What is the greatest number of consecutive integers whose sum is $45?$
$\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$ | [
"We might at first think that the answer would be $9$ , because $1+2+3 \\dots +n = 45$ when $n = 9$ . But note that the problem says that they can be integers, not necessarily positive. Observe also that every term in the sequence $-44, -43, \\cdots, 44, 45$ cancels out except $45$ . Thus, the answer is, intuitive... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_10B_Problems/Problem_12 | C | 22 | What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$
$\textbf{(A) } 11 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 22 \qquad\textbf{(D) } 23 \qquad\textbf{(E) } 27$ | [
"Observe that $2019_{10} = 5613_7$ . To maximize the sum of the digits, we want as many $6$ s as possible (since $6$ is the highest value in base $7$ ), and this will occur with either of the numbers $4666_7$ or $5566_7$ . Thus, the answer is $4+6+6+6 = 5+5+6+6 = \\boxed{22}$",
"Note that all base $7$ numbers wit... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_10A_Problems/Problem_9 | B | 996 | What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?
$\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$ | [
"The sum of the first $n$ positive integers is $\\frac{(n)(n+1)}{2}$ , and we want this not to be a divisor of $n!$ (the product of the first $n$ positive integers). Notice that if and only if $n+1$ were composite, all of its factors would be less than or equal to $n$ , which means they would be able to cancel with... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_10B_Problems/Problem_23 | D | 6 | What is the hundreds digit of $2011^{2011}?$
$\textbf{(A) } 1 \qquad \textbf{(B) } 4 \qquad \textbf{(C) }5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 9$ | [
"Since $2011 \\equiv 11 \\pmod{1000},$ we know that $2011^{2011} \\equiv 11^{2011} \\pmod{1000}.$\nTo compute this, we use a clever application of the binomial theorem\n$\\begin{aligned} 11^{2011} &= (1+10)^{2011} \\\\ &= 1 + \\dbinom{2011}{1} \\cdot 10 + \\dbinom{2011}{2} \\cdot 10^2 + \\cdots \\end{aligned}$\nI... |
https://artofproblemsolving.com/wiki/index.php/1983_AIME_Problems/Problem_8 | null | 61 | What is the largest $2$ -digit prime factor of the integer $n = {200\choose 100}$ | [
"Expanding the binomial coefficient , we get ${200 \\choose 100}=\\frac{200!}{100!100!}$ . Let the required prime be $p$ ; then $10 \\le p < 100$ . If $p > 50$ , then the factor of $p$ appears twice in the denominator. Thus, we need $p$ to appear as a factor at least three times in the numerator, so $3p<200$ . The ... |
https://artofproblemsolving.com/wiki/index.php/1992_AJHSME_Problems/Problem_3 | D | 28 | What is the largest difference that can be formed by subtracting two numbers chosen from the set $\{ -16,-4,0,2,4,12 \}$
$\text{(A)}\ 10 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 28 \qquad \text{(E)}\ 48$ | [
"To maximize anything of the form $a-b$ , we maximize $a$ and minimize $b$ . The maximal element of the set is $12$ and the minimal element is $-16$ , so the maximal difference is \\[12-(-16)=28\\rightarrow \\boxed{28}.\\]"
] |
https://artofproblemsolving.com/wiki/index.php/1984_AIME_Problems/Problem_14 | null | 38 | What is the largest even integer that cannot be written as the sum of two odd composite numbers? | [
"Take an even positive integer $x$ $x$ is either $0 \\bmod{6}$ $2 \\bmod{6}$ , or $4 \\bmod{6}$ . Notice that the numbers $9$ $15$ $21$ , ... , and in general $9 + 6n$ for nonnegative $n$ are odd composites. We now have 3 cases:\nIf $x \\ge 18$ and is $0 \\bmod{6}$ $x$ can be expressed as $9 + (9+6n)$ for some nonn... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10B_Problems/Problem_18 | D | 15 | What is the largest integer that is a divisor of
\[(n+1)(n+3)(n+5)(n+7)(n+9)\]
for all positive even integers $n$
$\text {(A) } 3 \qquad \text {(B) } 5 \qquad \text {(C) } 11 \qquad \text {(D) } 15 \qquad \text {(E) } 165$ | [
"For all consecutive odd integers, one of every five is a multiple of 5 and one of every three is a multiple of 3. The answer is $3 \\cdot 5 = 15$ , so ${\\boxed{15}$ is the correct answer.",
"We'll just test all the answer choices.\nNote that for any 3 consecutive odd integers, there will be exactly one multiple... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_12B_Problems/Problem_12 | D | 15 | What is the largest integer that is a divisor of
\[(n+1)(n+3)(n+5)(n+7)(n+9)\]
for all positive even integers $n$
$\text {(A) } 3 \qquad \text {(B) } 5 \qquad \text {(C) } 11 \qquad \text {(D) } 15 \qquad \text {(E) } 165$ | [
"For all consecutive odd integers, one of every five is a multiple of 5 and one of every three is a multiple of 3. The answer is $3 \\cdot 5 = 15$ , so ${\\boxed{15}$ is the correct answer.",
"We'll just test all the answer choices.\nNote that for any 3 consecutive odd integers, there will be exactly one multiple... |
https://artofproblemsolving.com/wiki/index.php/1999_AHSME_Problems/Problem_7 | B | 3 | What is the largest number of acute angles that a convex hexagon can have?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$ | [
"The sum of the interior angles of a hexagon is $720$ degrees. In a convex polygon, each angle must be strictly less than $180$ degrees.\nSix acute angles can only sum to less than $90\\cdot 6 = 540$ degrees, so six acute angles could not form a hexagon.\nFive acute angles and one obtuse angle can only sum to less... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10B_Problems/Problem_6 | B | 4 | What is the largest number of solid $2\text{-in} \times 2\text{-in} \times 1\text{-in}$ blocks that can fit in a $3\text{-in} \times 2\text{-in}\times3\text{-in}$ box?
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$ | [
"We find that the volume of the larger block is $18$ , and the volume of the smaller block is $4$ . Dividing the two, we see that only a maximum of four $2$ by $2$ by $1$ blocks can fit inside the $3$ by $3$ by $2$ block. Drawing it out, we see that such a configuration is indeed possible. Therefore, the answer is ... |
https://artofproblemsolving.com/wiki/index.php/1987_AIME_Problems/Problem_8 | null | 112 | What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$ | [
"Flip the fractions and subtract one from all sides to yield \\[\\frac{7}{8}>\\frac{k}{n}>\\frac{6}{7}.\\] Multiply both sides by $56n$ to get \\[49n>56k>48n.\\] This is equivalent to find the largest value of $n$ such that there is only one multiple of 56 within the open interval between $48n$ and $49n$ . If $n=1... |
https://artofproblemsolving.com/wiki/index.php/1995_AIME_Problems/Problem_10 | null | 215 | What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer? | [
"Let our answer be $n$ . Write $n = 42a + b$ , where $a, b$ are positive integers and $0 \\leq b < 42$ . Then note that $b, b + 42, ... , b + 42(a-1)$ are all primes.\nIf $b$ is $0\\mod{5}$ , then $b = 5$ because $5$ is the only prime divisible by $5$ . We get $n = 215$ as our largest possibility in this case.\nIf ... |
https://artofproblemsolving.com/wiki/index.php/1987_AIME_Problems/Problem_2 | null | 137 | What is the largest possible distance between two points , one on the sphere of radius 19 with center $(-2,-10,5)$ and the other on the sphere of radius 87 with center $(12,8,-16)$ | [
"The distance between the two centers of the spheres can be determined via the distance formula in three dimensions: $\\sqrt{(12 - (-2))^2 + (8 - (-10))^2 + (-16 - 5)^2} = \\sqrt{14^2 + 18^2 + 21^2} = 31$ . The largest possible distance would be the sum of the two radii and the distance between the two centers, mak... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_8_Problems/Problem_15 | C | 32 | What is the largest power of $2$ that is a divisor of $13^4 - 11^4$
$\textbf{(A)}\mbox{ }8\qquad \textbf{(B)}\mbox{ }16\qquad \textbf{(C)}\mbox{ }32\qquad \textbf{(D)}\mbox{ }64\qquad \textbf{(E)}\mbox{ }128$ | [
"First, we use difference of squares on $13^4 - 11^4 = (13^2)^2 - (11^2)^2$ to get $13^4 - 11^4 = (13^2 + 11^2)(13^2 - 11^2)$ . Using difference of squares again and simplifying, we get $(169 + 121)(13+11)(13-11) = 290 \\cdot 24 \\cdot 2 = (2\\cdot 8 \\cdot 2) \\cdot (3 \\cdot 145)$ . Realizing that we don't need t... |
https://artofproblemsolving.com/wiki/index.php/1991_AJHSME_Problems/Problem_8 | D | 12 | What is the largest quotient that can be formed using two numbers chosen from the set $\{ -24, -3, -2, 1, 2, 8 \}$
$\text{(A)}\ -24 \qquad \text{(B)}\ -3 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24$ | [
"Let the two chosen numbers be $a$ and $b$ . To maximize the quotient, we first have either $a,b>0$ or $a,b<0$ , and from there we maximize $|a|$ and minimize $|b|$\nFor the case $a,b<0$ , we have $a=-24$ and $b=-2$ , which gives us $(-24)/(-2)=12$ . For the case $a,b>0$ , we have $a=8$ and $b=1$ , which gives us... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_10A_Problems/Problem_9 | D | 1 | What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$
$\textbf{(A)} ~0\qquad\textbf{(B)} ~\frac{1}{4}\qquad\textbf{(C)} ~\frac{1}{2} \qquad\textbf{(D)} ~1 \qquad\textbf{(E)} ~2$ | [
"Expanding, we get that the expression is $x^2+2xy+y^2+x^2y^2-2xy+1$ or $x^2+y^2+x^2y^2+1$ . By the Trivial Inequality (all squares are nonnegative) the minimum value for this is $\\boxed{1}$ , which can be achieved at $x=y=0$",
"We expand the original expression, then factor the result by grouping: \\begin{align... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_12A_Problems/Problem_7 | D | 1 | What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$
$\textbf{(A)} ~0\qquad\textbf{(B)} ~\frac{1}{4}\qquad\textbf{(C)} ~\frac{1}{2} \qquad\textbf{(D)} ~1 \qquad\textbf{(E)} ~2$ | [
"Expanding, we get that the expression is $x^2+2xy+y^2+x^2y^2-2xy+1$ or $x^2+y^2+x^2y^2+1$ . By the Trivial Inequality (all squares are nonnegative) the minimum value for this is $\\boxed{1}$ , which can be achieved at $x=y=0$",
"We expand the original expression, then factor the result by grouping: \\begin{align... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_10A_Problems/Problem_19 | B | 2,018 | What is the least possible value of \[(x+1)(x+2)(x+3)(x+4)+2019\] where $x$ is a real number?
$\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$ | [
"Grouping the first and last terms and two middle terms gives $(x^2+5x+4)(x^2+5x+6)+2019$ , which can be simplified to $(x^2+5x+5)^2-1+2019$ . Noting that squares are nonnegative, and verifying that $x^2+5x+5=0$ for some real $x$ , the answer is $\\boxed{2018}$",
"Let $a=x+\\tfrac{5}{2}$ . Then the expression $(x... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10A_Problems/Problem_3 | D | 6 | What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$ | [
"The volume of the cube is $V_{\\text{cube}}=6^3=216,$ and the volume of a clay ball is $V_{\\text{ball}}=\\frac43\\cdot\\pi\\cdot2^3=\\frac{32}{3}\\pi.$\nSince the balls can be reshaped but not compressed, the maximum number of balls that can completely fit inside a cube is \\[\\left\\lfloor\\frac{V_{\\text{cube}}... |
https://artofproblemsolving.com/wiki/index.php/1999_AHSME_Problems/Problem_12 | C | 3 | What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $y=p(x)$ and $y=q(x)$ , each with leading coefficient 1?
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$ | [
"The intersections of the two polynomials, $p(x)$ and $q(x)$ , are precisely the roots of the equation $p(x)=q(x) \\rightarrow p(x) - q(x) = 0$ . Since the leading coefficients of both polynomials are $1$ , the degree of $p(x) - q(x) = 0$ is at most three, and the maximum point of intersection is three, because a t... |
https://artofproblemsolving.com/wiki/index.php/2001_AMC_10_Problems/Problem_4 | E | 6 | What is the maximum number of possible points of intersection of a circle and a triangle?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$ | [
"Circle-triangle problem.PNG\nWe can draw a circle and a triangle, such that each side is tangent to the circle. This means that each side would intersect the circle at one point.\nYou would then have $3$ points, but what if the circle was bigger? Then, each side would intersect the circle at 2 points.\nTherefore, ... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_12P_Problems/Problem_13 | D | 17 | What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, ... k_n$ for which
\[k^2_1 + k^2_2 + ... + k^2_n = 2002?\]
$\text{(A) }14 \qquad \text{(B) }15 \qquad \text{(C) }16 \qquad \text{(D) }17 \qquad \text{(E) }18$ | [
"Note that $k^2_1 + k^2_2 + ... + k^2_n = 2002 \\leq \\frac{n(n+1)(2n+1)}{6}$\nWhen $n = 17$ $\\frac{n(n+1)(2n+1)}{6} = \\frac{(17)(18)(35)}{6} = 1785 < 2002$\nWhen $n = 18$ $\\frac{n(n+1)(2n+1)}{6} = 1785 + 18^2 = 2109 > 2002$\nTherefore, we know $n \\leq 17$\nNow we must show that $n = 17$ works. We replace one o... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_8_Problems/Problem_20 | D | 10 | What is the measure of the acute angle formed by the hands of the clock at 4:20 PM?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$ | [
"Imagine the clock as a circle. The minute hand will be at the 4 at 20 minutes past the hour. The central angle formed between $4$ and $5$ is $30$ degrees (since it is 1/12 of a full circle, 360). By $4:20$ , the hour hand would have moved $\\frac{1}{3}$ way from 4 to 5 since $\\frac{20}{60}$ is reducible to $\\fr... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A_Problems/Problem_11 | C | 1,976.5 | What is the median of the following list of $4040$ numbers $?$ \[1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2\] $\textbf{(A)}\ 1974.5\qquad\textbf{(B)}\ 1975.5\qquad\textbf{(C)}\ 1976.5\qquad\textbf{(D)}\ 1977.5\qquad\textbf{(E)}\ 1978.5$ | [
"We can see that $44^2=1936$ which is less than 2020. Therefore, there are $2020-44=1976$ of the $4040$ numbers greater than $2020$ . Also, there are $2020+44=2064$ numbers that are less than or equal to $2020$\nSince there are $44$ duplicates/extras, it will shift up our median's placement down $44$ . Had the list... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12A_Problems/Problem_8 | C | 1,976.5 | What is the median of the following list of $4040$ numbers $?$ \[1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2\] $\textbf{(A)}\ 1974.5\qquad\textbf{(B)}\ 1975.5\qquad\textbf{(C)}\ 1976.5\qquad\textbf{(D)}\ 1977.5\qquad\textbf{(E)}\ 1978.5$ | [
"We can see that $44^2=1936$ which is less than 2020. Therefore, there are $2020-44=1976$ of the $4040$ numbers greater than $2020$ . Also, there are $2020+44=2064$ numbers that are less than or equal to $2020$\nSince there are $44$ duplicates/extras, it will shift up our median's placement down $44$ . Had the list... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_12A_Problems/Problem_15 | C | 26 | What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{123456789}{2^{26}\cdot 5^4}$ as a decimal?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104$ | [
"We can rewrite the fraction as $\\frac{123456789}{2^{22} \\cdot 10^4} = \\frac{12345.6789}{2^{22}}$ . Since the last digit of the numerator is odd, a $5$ is added to the right if the numerator is divided by $2$ , and this will continuously happen because $5$ , itself, is odd. Indeed, this happens twenty-two times ... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_8_Problems/Problem_3 | D | 4 | What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal $\overline{BD}$ of square $ABCD$
[asy] defaultpen(linewidth(1)); for ( int x = 0; x < 5; ++x ) { draw((0,x)--(4,x)); draw((x,0)--(x,4)); } fill((1,0)--(2,0)--(2,1)--(1,1)--cycle); fill((0,3... | [
"Rotating square $ABCD$ counterclockwise $45^\\circ$ so that the line of symmetry $BD$ is a vertical line makes it easier to see that $\\boxed{4}$ squares need to be colored to match its corresponding square."
] |
https://artofproblemsolving.com/wiki/index.php/2000_AMC_8_Problems/Problem_7 | B | 280 | What is the minimum possible product of three different numbers of the set $\{-8,-6,-4,0,3,5,7\}$
$\text{(A)}\ -336 \qquad \text{(B)}\ -280 \qquad \text{(C)}\ -210 \qquad \text{(D)}\ -192 \qquad \text{(E)}\ 0$ | [
"The only way to get a negative product using three numbers is to multiply one negative number and two positives or three negatives. Only two reasonable choices\nexist: $(-8)\\times(-6)\\times(-4) = (-8)\\times(24) = -192$ and $(-8)\\times5\\times7 = (-8)\\times35 = -280$ .\nThe latter is smaller, so $\\boxed{280}$... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_12A_Problems/Problem_22 | A | 49 | What is the minimum value of $f(x)=\left|x-1\right| + \left|2x-1\right| + \left|3x-1\right| + \cdots + \left|119x - 1 \right|$
$\textbf{(A)}\ 49 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 53$ | [
"If we graph each term separately, we will notice that all of the zeros occur at $\\frac{1}{m}$ , where $m$ is any integer from $1$ to $119$ , inclusive: $|mx-1|=0\\implies mx=1\\implies x=\\frac{1}{m}$\nThe minimum value of $f(x)$ occurs where the absolute value of the sum of the slopes is at a minimum $\\ge 0$ , ... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_12A_Problems/Problem_22 | null | 49 | What is the minimum value of $f(x)=\left|x-1\right| + \left|2x-1\right| + \left|3x-1\right| + \cdots + \left|119x - 1 \right|$
$\textbf{(A)}\ 49 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 53$ | [
"By the triangle inequality, $|x-1|+|2x-1|+|3x-1|+\\cdots + |119x-1| \\geq |(x-1)+(2x-1)+\\cdots+(119x)-1|.$ However, we may change signs of some of these terms to cancel out the $x$ 's.\nSince the minimum exists, we want all the $x$ s to cancel out. Thus, we want to find some $n$ such that \\[1+2+3+...+n=(n+1)+(n+... |
https://artofproblemsolving.com/wiki/index.php/1989_AJHSME_Problems/Problem_10 | D | 150 | What is the number of degrees in the smaller angle between the hour hand and the minute hand on a clock that reads seven o'clock?
$\text{(A)}\ 50^\circ \qquad \text{(B)}\ 120^\circ \qquad \text{(C)}\ 135^\circ \qquad \text{(D)}\ 150^\circ \qquad \text{(E)}\ 165^\circ$ | [
"The smaller angle makes up $5/12$ of the circle which is the clock. A circle is $360^\\circ$ , so the measure of the smaller angle is \\[\\frac{5}{12}\\cdot 360^\\circ = 150^\\circ \\rightarrow \\boxed{150}\\]"
] |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12A_Problems/Problem_12 | C | 167 | What is the number of terms with rational coefficients among the $1001$ terms in the expansion of $\left(x\sqrt[3]{2}+y\sqrt{3}\right)^{1000}?$
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 166 \qquad\textbf{(C)}\ 167 \qquad\textbf{(D)}\ 500 \qquad\textbf{(E)}\ 501$ | [
"By the Binomial Theorem, each term in the expansion is of the form \\[\\binom{1000}{k}\\left(x\\sqrt[3]{2}\\right)^k\\left(y\\sqrt{3}\\right)^{1000-k}=\\binom{1000}{k}2^{\\frac k3}3^{\\frac{1000-k}{2}}x^k y^{1000-k},\\] where $k\\in\\{0,1,2,\\ldots,1000\\}.$\nThis problem is equivalent to counting the values of $k... |
https://artofproblemsolving.com/wiki/index.php/2024_AMC_8_Problems/Problem_1 | B | 2 | What is the ones digit of \[222,222-22,222-2,222-222-22-2?\] $\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$ | [
"We can rewrite the expression as \\[222,222-(22,222+2,222+222+22+2).\\]\nWe note that the units digit of the addition is $0$ because all the units digits of the five numbers are $2$ and $5*2=10$ , which has a units digit of $0$\nNow, we have something with a units digit of $0$ subtracted from $222,222$ . The units... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10B_Problems/Problem_24 | E | 16 | 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 $0\le w\le1$
$\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16$ | [
" Notice that we are given a parametric form of the region, and $w$ is used in both $x$ and $y$ . We first fix $u$ and $v$ to $0$ , and graph $(-3w,4w)$ from $0\\le w\\le1$ . When $w$ is $0$ , we have the point $(0,0)$ , and when $w$ is $1$ , we have the point $(-3,4)$ . We see that since this is a directly proport... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_8_Problems/Problem_19 | A | 180 | What is the perimeter of trapezoid $ABCD$
[asy]size(3inch, 1.5inch); pair a=(0,0), b=(18,24), c=(68,24), d=(75,0), f=(68,0), e=(18,0); draw(a--b--c--d--cycle); draw(b--e); draw(shift(0,2)*e--shift(2,2)*e--shift(2,0)*e); label("30", (9,12), W); label("50", (43,24), N); label("25", (71.5, 12), E); label("24", (18, 12), E... | [
"Draw altitudes from $B$ and $C$ to base $AD$ to create a rectangle and two right triangles. The side opposite $BC$ is equal to $50$ . The bases of the right triangles can be found using Pythagorean or special triangles to be $18$ and $7$ . Add it together to get $AD=18+50+7=75$ . The perimeter is $75+30+50+25=\\bo... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_8_Problems/Problem_9 | C | 1,003 | What is the product of $\frac{3}{2}\times\frac{4}{3}\times\frac{5}{4}\times\cdots\times\frac{2006}{2005}$
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 1002\qquad\textbf{(C)}\ 1003\qquad\textbf{(D)}\ 2005\qquad\textbf{(E)}\ 2006$ | [
"The numerator in each fraction cancels out with the denominator of the next fraction. There are only two numbers that didn't cancel: $\\frac{2006}{2}=\\boxed{1003}$"
] |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_11 | E | 81 | What is the product of all real numbers $x$ such that the distance on the number line between $\log_6x$ and $\log_69$ is twice the distance on the number line between $\log_610$ and $1$
$\textbf{(A) } 10 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 25 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 81$ | [
"First, notice that there must be two such numbers: one greater than $\\log_69$ and one less than it. Furthermore, they both have to be the same distance away, namely $2(\\log_610 - 1)$ . Let these two numbers be $\\log_6a$ and $\\log_6b$ . Because they are equidistant from $\\log_69$ , we have $\\frac{\\log_6a + \... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_11 | null | 81 | What is the product of all real numbers $x$ such that the distance on the number line between $\log_6x$ and $\log_69$ is twice the distance on the number line between $\log_610$ and $1$
$\textbf{(A) } 10 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 25 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 81$ | [
"Let $a = 2 \\cdot |\\log_6 10 - 1| = |\\log_6 9 - \\log_6 x| = |\\log_6 \\frac{9}{x}|$\n$\\pm a = \\log_6 \\frac{9}{x} \\implies 6^{\\pm a} = b^{\\pm 1} = \\frac{9}{x} \\implies x = 9 \\cdot b^{\\pm 1}$\n$9b^1 \\cdot 9b^{-1} = \\boxed{81}$"
] |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_19 | C | 1 | What is the product of all solutions to the equation \[\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023\]
$\textbf{(A)} ~(\log_{2023}7\cdot \log_{2023}289)^2\qquad\textbf{(B)} ~\log_{2023}7\cdot \log_{2023}289\qquad\textbf{(C)} ~1$ $\textbf{(D)} ~\log_{7}2023\cdot \log_{289}2023\qquad\textbf{(E)} ~(\log_7 2023\cdot\... | [
"For $\\log_{7x}2023\\cdot \\log_{289x}2023=\\log_{2023x}2023$ , transform it into $\\dfrac{\\ln 289+\\ln 7}{\\ln 7 + \\ln x}\\cdot \\dfrac{\\ln 289+\\ln 7}{\\ln 289 + \\ln x}=\\dfrac{\\ln 289+\\ln 7}{\\ln 289+\\ln 7+\\ln x}$ . Replace $\\ln x$ with $y$ . Because we want to find the product of all solutions of $x$ ... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_10B_Problems/Problem_19 | A | 64 | What is the product of all the roots of the equation \[\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}.\]
$\textbf{(A)}\ -64 \qquad\textbf{(B)}\ -24 \qquad\textbf{(C)}\ -9 \qquad\textbf{(D)}\ 24 \qquad\textbf{(E)}\ 576$ | [
"First, square both sides, and isolate the absolute value. \\begin{align*} 5|x|+8&=x^2-16\\\\ 5|x|&=x^2-24\\\\ |x|&=\\frac{x^2-24}{5}. \\\\ \\end{align*} Solve for the absolute value and factor.\nCase 1: $x=\\frac{x^2-24}{5}$\nMultiplying both sides by $5$ gives us \\[5x=x^2-24.\\] Rearranging and factoring, we hav... |
https://artofproblemsolving.com/wiki/index.php/1983_AIME_Problems/Problem_3 | null | 20 | What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$ | [
"If we were to expand by squaring, we would get a quartic polynomial , which isn't always the easiest thing to deal with.\nInstead, we substitute $y$ for $x^2+18x+30$ , so that the equation becomes $y=2\\sqrt{y+15}$\nNow we can square; solving for $y$ , we get $y=10$ or $y=-6$ . The second root is extraneous since ... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_8_Problems/Problem_10 | C | 330 | What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?
$\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 165 \qquad \textbf{(C)}\ 330 \qquad \textbf{(D)}\ 625 \qquad \textbf{(E)}\ 660$ | [
"To find either the LCM or the GCF of two numbers, always prime factorize first.\nThe prime factorization of $180 = 3^2 \\times 5 \\times 2^2$\nThe prime factorization of $594 = 3^3 \\times 11 \\times 2$\nThen, to find the LCM, we have to find the greatest power of all the numbers there are; if one number is one ... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10B_Problems/Problem_22 | D | 201 | What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$
$\textbf{(A) } 100 \qquad\textbf{(B) } 101 \qquad\textbf{(C) } 200 \qquad\textbf{(D) } 201 \qquad\textbf{(E) } 202$ | [
"Completing the square, then difference of squares:\n\\begin{align*} 2^{202} + 202 &= (2^{101})^2 + 2\\cdot 2^{101} + 1 - 2\\cdot 2^{101} + 201\\\\ &= (2^{101} + 1)^2 - 2^{102} + 201\\\\ &= (2^{101} - 2^{51} + 1)(2^{101} + 2^{51} + 1) + 201. \\end{align*}\nThus, we see that the remainder is surely $\\boxed{201}$",
... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_10B_Problems/Problem_21 | D | 4 | What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by 8?
$\mathrm{(A)}\ 0\qquad \mathrm{(B)}\ 1\qquad \mathrm{(C)}\ 2\qquad \mathrm{(D)}\ 4\qquad \mathrm{(E)}\ 6$ | [
"The sum of any four consecutive powers of 3 is divisible by $3^0 + 3^1 + 3^2 +3^3 = 40$ and hence is divisible by 8. Therefore\nis divisible by 8. So the required remainder is $3^0 + 3^1 = \\boxed{4}$"
] |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_10B_Problems/Problem_21 | null | 4 | What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by 8?
$\mathrm{(A)}\ 0\qquad \mathrm{(B)}\ 1\qquad \mathrm{(C)}\ 2\qquad \mathrm{(D)}\ 4\qquad \mathrm{(E)}\ 6$ | [
"We have $3^2 = 9 \\equiv 1 \\pmod 8$ . Hence for any $k$ we have $3^{2k}\\equiv 1^k = 1 \\pmod 8$ , and then $3^{2k+1} = 3\\cdot 3^{2k} \\equiv 3\\cdot 1 = 3 \\pmod 8$\nTherefore our sum gives the same remainder modulo $8$ as $1 + 3 + 1 + 3 + 1 + \\cdots + 1 + 3$ . There are $2010$ terms in the sum, hence there a... |
https://artofproblemsolving.com/wiki/index.php/1974_AHSME_Problems/Problem_4 | D | 50 | What is the remainder when $x^{51}+51$ is divided by $x+1$
$\mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 49 \qquad \mathrm{(D) \ } 50 \qquad \mathrm{(E) \ }51$ | [
"From the Remainder Theorem , the remainder when $x^{51}+51$ is divided by $x+1$ is $(-1)^{51}+51=-1+51=50, \\boxed{50}$"
] |
https://artofproblemsolving.com/wiki/index.php/1986_USAMO_Problems/Problem_3 | null | 337 | What is the smallest integer $n$ , greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be \[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\] | [
"Let's first obtain an algebraic expression for the root mean square of the first $n$ integers, which we denote $I_n$ . By repeatedly using the identity $(x+1)^3 = x^3 + 3x^2 + 3x + 1$ , we can write \\[1^3 + 3\\cdot 1^2 + 3 \\cdot 1 + 1 = 2^3,\\] \\[1^3 + 3 \\cdot(1^2 + 2^2) + 3 \\cdot (1 + 2) + 1 + 1 = 3^3,\\] a... |
https://artofproblemsolving.com/wiki/index.php/1975_AHSME_Problems/Problem_29 | C | 970 | What is the smallest integer larger than $(\sqrt{3}+\sqrt{2})^6$
$\textbf{(A)}\ 972 \qquad \textbf{(B)}\ 971 \qquad \textbf{(C)}\ 970 \qquad \textbf{(D)}\ 969 \qquad \textbf{(E)}\ 968$ | [
"$(\\sqrt{3}+\\sqrt{2})^6=(5+2\\sqrt{6})^3=(5+2\\sqrt{6})(49+20\\sqrt{6})=(485+198\\sqrt{6})$ Then, find that $\\sqrt{6}$ is about $2.449$ . Finally, multiply and add to find that the smallest integer higher is $\\boxed{970}$",
"Let's evaluate $(\\sqrt{3}+\\sqrt{2})^6 + (\\sqrt{3}-\\sqrt{2})^6$ . We see that all ... |
https://artofproblemsolving.com/wiki/index.php/1974_AHSME_Problems/Problem_10 | B | 2 | What is the smallest integral value of $k$ such that \[2x(kx-4)-x^2+6=0\] has no real roots?
$\mathrm{(A)\ } -1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 4 \qquad \mathrm{(E) \ }5$ | [
"Expanding, we have $2kx^2-8x-x^2+6=0$ , or $(2k-1)x^2-8x+6=0$ . For this quadratic not to have real roots, it must have a negative discriminant. Therefore, $(-8)^2-4(2k-1)(6)<0\\implies 64-48k+24<0\\implies k>\\frac{11}{6}$ . From here, we can easily see that the smallest integral value of $k$ is $2, \\boxed{2}$"
... |
https://artofproblemsolving.com/wiki/index.php/1978_AHSME_Problems/Problem_18 | null | 2,501 | What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<.01$
$\textbf{(A) }2499\qquad \textbf{(B) }2500\qquad \textbf{(C) }2501\qquad \textbf{(D) }10,000\qquad \textbf{(E) }\text{There is no such integer}$ | [
"Adding $\\sqrt{n - 1}$ to both sides, we get \\[\\sqrt{n} < \\sqrt{n - 1} + 0.01.\\] Squaring both sides, we get \\[n < n - 1 + 0.02 \\sqrt{n - 1} + 0.0001,\\] which simplifies to \\[0.9999 < 0.02 \\sqrt{n - 1},\\] or \\[\\sqrt{n - 1} > 49.995.\\] Squaring both sides again, we get \\[n - 1 > 2499.500025,\\] so $n ... |
https://artofproblemsolving.com/wiki/index.php/1993_AIME_Problems/Problem_6 | null | 495 | What is the smallest positive integer that can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers? | [
"Let the desired integer be $n$ . From the information given, it can be determined that, for positive integers $a, \\ b, \\ c$\n$n = 9a + 36 = 10b + 45 = 11c + 55$\nThis can be rewritten as the following congruences:\n$n \\equiv 0 \\pmod{9}$\n$n \\equiv 5 \\pmod{10}$\n$n \\equiv 0 \\pmod{11}$\nSince 9 and 11 are re... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_8_Problems/Problem_18 | A | 3,127 | What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?
$\textbf{(A)}\hspace{.05in}3127\qquad\textbf{(B)}\hspace{.05in}3133\qquad\textbf{(C)}\hspace{.05in}3137\qquad\textbf{(D)}\hspace{.05in}3139\qquad\textbf{(E)}\hspace{.05in}3149$ | [
"The problem states that the answer cannot be a perfect square or have prime factors less than $50$ . Therefore, the answer will be the product of at least two different primes greater than $50$ . The two smallest primes greater than $50$ are $53$ and $59$ . Multiplying these two primes, we obtain the number $3127$... |
https://artofproblemsolving.com/wiki/index.php/2000_AIME_II_Problems/Problem_4 | null | 180 | What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors? | [
"We use the fact that the number of divisors of a number $n = p_1^{e_1}p_2^{e_2} \\cdots p_k^{e_k}$ is $(e_1 + 1)(e_2 + 1) \\cdots (e_k + 1)$ . If a number has $18 = 2 \\cdot 3 \\cdot 3$ factors, then it can have at most $3$ distinct primes in its factorization.\nDividing the greatest power of $2$ from $n$ , we hav... |
https://artofproblemsolving.com/wiki/index.php/1976_AHSME_Problems/Problem_21 | B | 9 | What is the smallest positive odd integer $n$ such that the product $2^{1/7}2^{3/7}\cdots2^{(2n+1)/7}$ is greater than $1000$ ?
(In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$ .)
$\textbf{(A) }7\qquad \textbf{(B) }9\qquad \textbf{(... | [
"Combine the terms in the product to get $2^{\\frac{1+3+5+ \\dots +(2n-1)+(2n+1)}{7}}$\nThe exponent can be simplified to \\[\\frac{1+3+5+ \\dots +(2n-1)+(2n+1)}{7} \\Rightarrow \\frac{\\frac{n(1+(2n+1))}{2}}{7} \\Rightarrow \\frac{n^2}{7}.\\]\nWe want this inequality to be true with the smallest positive odd integ... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_8_Problems/Problem_3 | C | 5 | What is the smallest possible average of four distinct positive even integers?
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$ | [
"In order to get the smallest possible average, we want the 4 even numbers to be as small as possible. The first 4 positive even numbers are 2, 4, 6, and 8. Their average is $\\frac{2+4+6+8}{4}=\\boxed{5}$"
] |
https://artofproblemsolving.com/wiki/index.php/1996_AJHSME_Problems/Problem_6 | C | 36 | What is the smallest result that can be obtained from the following process?
$\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 56$ | [
"Since we want the smallest possible result, and we are only adding and multiplying positive numbers over $1$ , we can \"prune\" the set to the three smallest numbers $\\{3,5,7\\}$ . Using bigger numbers will create bigger sums and bigger products.\nFrom there, compute the $3$ ways you can do the two operations:\n... |
https://artofproblemsolving.com/wiki/index.php/1990_AJHSME_Problems/Problem_1 | C | 1,047 | What is the smallest sum of two $3$ -digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one of the six boxes in this addition problem?
[asy] unitsize(12); draw((0,0)--(10,0)); draw((-1.5,1.5)--(-1.5,2.5)); draw((-1,2)--(-2,2)); draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); draw((1,4)--(3,4... | [
"Let the two three-digit numbers be $\\overline{abc}$ and $\\overline{def}$ . Their sum is equal to $100(a+d)+10(b+e)+(c+f)$\nTo minimize this, we need to minimize the contribution of the $100$ factor, so we let $a=4$ and $d=5$ . Similarly, we let $b=6$ $e=7$ , and then $c=8$ and $f=9$ . The sum is \\[100(9)+10(... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_8_Problems/Problem_8 | D | 48 | What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$
$\textbf{(A) }24\qquad\textbf{(B) }29\qquad\textbf{(C) }43\qquad\textbf{(D) }48\qquad \textbf{(E) }57$ | [
"We know from the Triangle Inequality that the last side, $s$ , fulfills $s<5+19=24$ . Adding $5+19$ to both sides of the inequality, we get $s+5+19<48$ , and because $s+5+19$ is the perimeter of our triangle, $\\boxed{48}$ is our answer."
] |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_10B_Problems/Problem_8 | B | 12 | What is the sum of all integer solutions to $1<(x-2)^2<25$
$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 25$ | [
"$(x-2)^2$ = perfect square.\n1 < perfect square < 25\nPerfect square can equal: 4, 9, or 16\nSolve for $x$\n$(x-2)^2=4$\n$x=4,0$\nand\n$(x-2)^2=9$\n$x=5,-1$\nand\n$(x-2)^2=16$\n$x=6,-2$\nThe sum of all integer solutions is\n$4+5+6+0+(-1)+(-2)=\\boxed{12}$"
] |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12A_Problems/Problem_5 | E | 10 | What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common?
$\textbf{(A) }3 \qquad\textbf{(B) }4 \qquad\textbf{(C) }5 \qquad\textbf{(D) }6 \qquad\textbf{(E) }10$ | [
"We factor $x^2-3x+2$ into $(x-1)(x-2)$ . Thus, either $1$ or $2$ is a root of $x^2-5x+k$ . If $1$ is a root, then $1^2-5\\cdot1+k=0$ , so $k=4$ . If $2$ is a root, then $2^2-5\\cdot2+k=0$ , so $k=6$ . The sum of all possible values of $k$ is $\\boxed{10}$"
] |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12B_Problems/Problem_10 | E | 380 | What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 40^\circ,\sin 40^\circ), (\cos 60^\circ,\sin 60^\circ), \text{ and } (\cos t^\circ,\sin t^\circ)\] is isosceles?
$\textbf{(A)} \: 100 \qquad\textbf{(B)} \: 150 \qquad\textbf{(C)}... | [
"Let $A = (\\cos 40^{\\circ}, \\sin 40^{\\circ}), B = (\\cos 60^{\\circ}, \\sin 60^{\\circ}),$ and $C = (\\cos t^{\\circ}, \\sin t^{\\circ}).$ We apply casework to the legs of isosceles $\\triangle ABC:$\nTogether, the sum of all such possible values of $t$ is $20+80+50+230=\\boxed{380}.$"
] |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A_Problems/Problem_5 | C | 18 | What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$
$\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 25$ | [
"Split the equation into two cases, where the value inside the absolute value is positive and nonpositive.\nCase 1:\nThe equation yields $x^2-12x+34=2$ , which is equal to $(x-4)(x-8)=0$ . Therefore, the two values for the positive case is $4$ and $8$\nCase 2:\nSimilarly, taking the nonpositive case for the value i... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_10B_Problems/Problem_13 | A | 5 | What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?
$\textbf{(A) } -5 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } \frac{15}{4} \qquad\textbf{(E) } \frac{35}{4}$ | [
"The mean is $\\frac{4+6+8+17+x}{5}=\\frac{35+x}{5}$\nThere are three possibilities for the median: it is either $6$ $8$ , or $x$\nLet's start with $6$\n$\\frac{35+x}{5}=6$ has solution $x=-5$ , and the sequence is $-5, 4, 6, 8, 17$ , which does have median $6$ , so this is a valid solution.\nNow let the median be ... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_12B_Problems/Problem_7 | A | 5 | What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?
$\textbf{(A) } -5 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } \frac{15}{4} \qquad\textbf{(E) } \frac{35}{4}$ | [
"The mean is $\\frac{4+6+8+17+x}{5}=\\frac{35+x}{5}$\nThere are three possibilities for the median: it is either $6$ $8$ , or $x$\nLet's start with $6$\n$\\frac{35+x}{5}=6$ has solution $x=-5$ , and the sequence is $-5, 4, 6, 8, 17$ , which does have median $6$ , so this is a valid solution.\nNow let the median be ... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10B_Problems/Problem_13 | C | 92 | What is the sum of all the solutions of $x = \left|2x-|60-2x|\right|$
$\textbf{(A)}\ 32 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 92 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 124$ | [
"We evaluate this in cases:\nCase 1 $x<30$\nWhen $x<30$ we are going to have $60-2x>0$ . When $x>0$ we are going to have $|x|>0\\implies x>0$ and when $-x>0$ we are going to have $|x|>0\\implies -x>0$ . Therefore we have $x=|2x-(60-2x)|$ $x=|2x-60+2x|\\implies x=|4x-60|$\nSubcase 1 $30>x>15$\nWhen $30>x>15$ we ar... |
https://artofproblemsolving.com/wiki/index.php/1999_AHSME_Problems/Problem_6 | D | 7 | What is the sum of the digits of the decimal form of the product $2^{1999}\cdot 5^{2001}$
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 10$ | [
"$2^{1999}\\cdot5^{2001}=2^{1999}\\cdot5^{1999}\\cdot5^{2}=25\\cdot10^{1999}$ , a number with the digits \"25\" followed by 1999 zeros. The sum of the digits in the decimal form would be $2+5=7$ , thus making the answer $\\boxed{7}$"
] |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_10A_Problems/Problem_5 | E | 81 | What is the sum of the digits of the square of $\text 111111111$
$\mathrm{(A)}\ 18\qquad\mathrm{(B)}\ 27\qquad\mathrm{(C)}\ 45\qquad\mathrm{(D)}\ 63\qquad\mathrm{(E)}\ 81$ | [
"Using the standard multiplication algorithm, $111,111,111^2=12,345,678,987,654,321,$ whose digit sum is $\\boxed{81.}$",
"We note that\n$1^2 = 1$\n$11^2 = 121$\n$111^2 = 12321$\nand $1,111^2 = 1234321$\nWe can clearly see the pattern: If $X$ is $111\\cdots111$ , with $n$ ones (and for the sake of simplicity, ass... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_8_Problems/Problem_9 | B | 12 | What is the sum of the distinct prime integer divisors of $2016$
$\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }16\qquad\textbf{(D) }49\qquad \textbf{(E) }63$ | [
"The prime factorization is $2016=2^5\\times3^2\\times7$ . Since the problem is only asking us for the distinct prime factors, we have $2,3,7$ . Their desired sum is then $\\boxed{12}$",
"We notice that $9 \\mid 2016$ , since $2+0+1+6 = 9$ , and $9 \\mid 9$ . We can divide $2016$ by $9$ to get $224$ . This is d... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_12B_Problems/Problem_9 | C | 8 | What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$
$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$ | [
"Looking at the prime numbers under $12$ , we see that there are $\\left\\lfloor\\frac{12}{2}\\right\\rfloor+\\left\\lfloor\\frac{12}{2^2}\\right\\rfloor+\\left\\lfloor\\frac{12}{2^3}\\right\\rfloor=6+3+1=10$ factors of $2$ $\\left\\lfloor\\frac{12}{3}\\right\\rfloor+\\left\\lfloor\\frac{12}{3^2}\\right\\rfloor=4+1... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_8_Problems/Problem_4 | C | 7.5 | What is the sum of the mean, median, and mode of the numbers $2,3,0,3,1,4,0,3$
$\textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 7.5\qquad\textbf{(D)}\ 8.5\qquad\textbf{(E)}\ 9$ | [
"Putting the numbers in numerical order we get the list $0,0,1,2,3,3,3,4.$ The mode is $3.$ The median is $\\frac{2+3}{2}=2.5.$ The average is $\\frac{0+0+1+2+3+3+3+4}{8}=\\frac{16}{8}=2.$ The sum of all three is $3+2.5+2=\\boxed{7.5}$"
] |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_8_Problems/Problem_14 | C | 77 | What is the sum of the prime factors of $2010$
$\textbf{(A)}\ 67\qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 77\qquad\textbf{(D)}\ 201\qquad\textbf{(E)}\ 210$ | [
"First, we must find the prime factorization of $2010$ $2010=2\\cdot 3 \\cdot 5 \\cdot 67$ . We add the factors up to get $\\boxed{77}$"
] |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10A_Problems/Problem_18 | B | 1 | What is the sum of the reciprocals of the roots of the equation $\frac{2003}{2004}x+1+\frac{1}{x}=0$
$\mathrm{(A) \ } -\frac{2004}{2003}\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } \frac{2003}{2004}\qquad \mathrm{(D) \ } 1\qquad \mathrm{(E) \ } \frac{2004}{2003}$ | [
"Multiplying both sides by $x$\n$\\frac{2003}{2004}x^{2}+1x+1=0$\nLet the roots be $a$ and $b$\nThe problem is asking for $\\frac{1}{a}+\\frac{1}{b}= \\frac{a+b}{ab}$\nBy Vieta's formulas\n$a+b=(-1)^{1}\\frac{1}{\\frac{2003}{2004}}=-\\frac{2004}{2003}$\n$ab=(-1)^{2}\\frac{1}{\\frac{2003}{2004}}=\\frac{2004}{2003}$\... |
https://artofproblemsolving.com/wiki/index.php/1986_AIME_Problems/Problem_1 | null | 337 | What is the sum of the solutions to the equation $\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}$ | [
"Let $y = \\sqrt[4]{x}$ . Then we have $y(7 - y) = 12$ , or, by simplifying, \\[y^2 - 7y + 12 = (y - 3)(y - 4) = 0.\\]\nThis means that $\\sqrt[4]{x} = y = 3$ or $4$\nThus the sum of the possible solutions for $x$ is $4^4 + 3^4 = \\boxed{337}$"
] |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_10B_Problems/Problem_11 | C | 4 | What is the tens digit in the sum $7!+8!+9!+...+2006!$
$\textbf{(A) } 1\qquad \textbf{(B) } 3\qquad \textbf{(C) } 4\qquad \textbf{(D) } 6\qquad \textbf{(E) } 9$ | [
"Since $10!$ is divisible by $100$ , any factorial greater than $10!$ is also divisible by $100$ . The last two digits of all factorials greater than $10!$ are $00$ , so the last two digits of $10!+11!+...+2006!$ are $00$ . \n(*)\nSo all that is needed is the tens digit of the sum $7!+8!+9!$\n$7!+8!+9!=5040+40320+3... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10B_Problems/Problem_8 | A | 0 | What is the tens digit of $2015^{2016}-2017?$
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 8$ | [
"Since we only need the tens digits, we only need to care about the multiplication of tens and ones. (If you want to use mathematical terms then we only need to look at the exponents in $mod 100$ .) We will use the \" $\\equiv$ \" sign to denote congruence in modulus, basically taking the last two digits and ignori... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10B_Problems/Problem_8 | null | 0 | What is the tens digit of $2015^{2016}-2017?$
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 8$ | [
"Notice that, for $n\\ge 2$ $2015^n\\equiv 15^n$ is congruent to $25\\pmod{100}$ when $n$ is even and $75\\pmod{100}$ when $n$ is odd. (Check for yourself). Since $2016$ is even, $2015^{2016} \\equiv 25\\pmod{100}$ and $2015^{2016}-2017 \\equiv 25 - 17 \\equiv \\underline{0}8\\pmod{100}$\nSo the answer is $\\textb... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_8_Problems/Problem_22 | D | 4 | What is the tens digit of $7^{2011}$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }7$ | [
"Since we want the tens digit, we can find the last two digits of $7^{2011}$ . We can do this by using modular arithmetic. \\[7^1\\equiv 07 \\pmod{100}.\\] \\[7^2\\equiv 49 \\pmod{100}.\\] \\[7^3\\equiv 43 \\pmod{100}.\\] \\[7^4\\equiv 01 \\pmod{100}.\\] We can write $7^{2011}$ as $(7^4)^{502}\\times 7^3$ . Using... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10A_Problems/Problem_16 | C | 7 | What is the units digit of $13^{2003}$
$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 9$ | [
"$13^{2003}\\equiv 3^{2003}\\pmod{10}$\nSince $3^4=81\\equiv1\\pmod{10}$\n$3^{2003}=(3^{4})^{500}\\cdot3^{3}\\equiv1^{500}\\cdot27\\equiv7\\pmod{10}$\nTherefore, the units digit is $7 \\Rightarrow\\boxed{7}$",
"Since we are looking for the units digit of $13^{2003}$ , we only have to focus on the units digit of t... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10B_Problems/Problem_8 | A | 7 | What is the units digit of $2022^{2023} + 2023^{2022}$
$\text{(A)}\ 7 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 3$ | [
"$2022^{2023} + 2023^{2022} \\equiv 2^3 + 3^2 \\equiv 17 \\equiv 7$ (mod 10). $\\boxed{7}$ ~andliu766",
"When looking at the units digit patterns of the powers of $2$ , we see that\n$2^1=$ , units digit $2$\n$2^2=$ , units digit $4$\n$2^3=$ , units digit $8$\n$2^4=$ , units digit $6$\n$2^5=$ , units digit $2... |
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