link stringlengths 75 84 | letter stringclasses 5
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|---|---|---|---|---|
https://artofproblemsolving.com/wiki/index.php/2018_AMC_8_Problems/Problem_14 | D | 18 | Let $N$ be the greatest five-digit number whose digits have a product of $120$ . What is the sum of the digits of $N$
$\textbf{(A) }15\qquad\textbf{(B) }16\qquad\textbf{(C) }17\qquad\textbf{(D) }18\qquad\textbf{(E) }20$ | [
"If we start off with the first digit, we know that it can't be $9$ since $9$ is not a factor of $120$ . We go down to the digit $8$ , which does work since it is a factor of $120$ . Now, we have to know what digits will take up the remaining four spots. To find this result, just divide $\\frac{120}{8}=15$ . The ne... |
https://artofproblemsolving.com/wiki/index.php/2024_AIME_II_Problems/Problem_7 | null | 699 | Let $N$ be the greatest four-digit positive integer with the property that whenever one of its digits is changed to $1$ , the resulting number is divisible by $7$ . Let $Q$ and $R$ be the quotient and remainder, respectively, when $N$ is divided by $1000$ . Find $Q+R$ | [
"We note that by changing a digit to $1$ for the number $\\overline{abcd}$ , we are subtracting the number by either $1000(a-1)$ $100(b-1)$ $10(c-1)$ , or $d-1$ . Thus, $1000a + 100b + 10c + d \\equiv 1000(a-1) \\equiv 100(b-1) \\equiv 10(c-1) \\equiv d-1 \\pmod{7}$ . We can casework on $a$ backwards, finding the m... |
https://artofproblemsolving.com/wiki/index.php/2003_AIME_II_Problems/Problem_2 | null | 120 | Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000? | [
"We want a number with no digits repeating, so we can only use the digits $0-9$ once in constructing our number. To make the greatest number, we want the greatest digit to occupy the leftmost side and the least digit to occupy the rightmost side. Therefore, the last three digits of the greatest number should be an ... |
https://artofproblemsolving.com/wiki/index.php/2001_AIME_II_Problems/Problem_1 | null | 816 | Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$ | [
"The two-digit perfect squares are $16, 25, 36, 49, 64, 81$ . We try making a sequence starting with each one:\nThe largest is $81649$ , so our answer is $\\boxed{816}$"
] |
https://artofproblemsolving.com/wiki/index.php/2015_AIME_II_Problems/Problem_1 | null | 131 | Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$ | [
"If $N$ is $22$ percent less than one integer $k$ , then $N=\\frac{78}{100}k=\\frac{39}{50}k$ . In addition, $N$ is $16$ percent greater than another integer $m$ , so $N=\\frac{116}{100}m=\\frac{29}{25}m$ . Therefore, $k$ is divisible by $50$ and $m$ is divisible by $25$ . Setting these two equal, we have $\\frac{3... |
https://artofproblemsolving.com/wiki/index.php/2018_AIME_I_Problems/Problem_6 | null | 440 | Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$ | [
"Let $a=z^{120}$ . This simplifies the problem constraint to $a^6-a \\in \\mathbb{R}$ . This is true if $\\text{Im}(a^6)=\\text{Im}(a)$ . Let $\\theta$ be the angle $a$ makes with the positive x-axis. Note that there is exactly one $a$ for each angle $0\\le\\theta<2\\pi$ . We are given $\\sin\\theta = \\sin{6\\thet... |
https://artofproblemsolving.com/wiki/index.php/2006_AIME_I_Problems/Problem_4 | null | 124 | Let $N$ be the number of consecutive $0$ 's at the right end of the decimal representation of the product $1!2!3!4!\cdots99!100!.$ Find the remainder when $N$ is divided by $1000$ | [
"A number in decimal notation ends in a zero for each power of ten which divides it. Thus, we need to count both the number of 5s and the number of 2s dividing into our given expression. Since there are clearly more 2s than 5s, it is sufficient to count the number of 5s.\nOne way to do this is as follows: $96$ of... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_II_Problems/Problem_8 | null | 772 | Let $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that have the following properties:
Find $N$ | [
"Let us partition the set $\\{1,2,\\cdots,12\\}$ into $n$ numbers in $A$ and $12-n$ numbers in $B$\nSince $n$ must be in $B$ and $12-n$ must be in $A$ $n\\ne6$ , we cannot partition into two sets of 6 because $6$ needs to end up somewhere, $n\\ne 0$ or $12$ either).\nWe have $\\dbinom{10}{n-1}$ ways of picking the ... |
https://artofproblemsolving.com/wiki/index.php/2013_AIME_I_Problems/Problem_15 | null | 272 | Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions
(a) $0\le A<B<C\le99$ ,
(b) there exist integers $a$ $b$ , and $c$ , and prime $p$ where $0\le b<a<c<p$ ,
(c) $p$ divides $A-a$ $B-b$ , and $C-c$ , and
(d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form ar... | [
"From condition (d), we have $(A,B,C)=(B-D,B,B+D)$ and $(b,a,c)=(a-d,a,a+d)$ . Condition $\\text{(c)}$ states that $p\\mid B-D-a$ $p | B-a+d$ , and $p\\mid B+D-a-d$ . We subtract the first two to get $p\\mid-d-D$ , and we do the same for the last two to get $p\\mid 2d-D$ . We subtract these two to get $p\\mid 3d$ .... |
https://artofproblemsolving.com/wiki/index.php/2003_AIME_I_Problems/Problem_13 | null | 155 | Let $N$ be the number of positive integers that are less than or equal to $2003$ and whose base- $2$ representation has more $1$ 's than $0$ 's. Find the remainder when $N$ is divided by $1000$ | [
"In base- $2$ representation, all positive numbers have a leftmost digit of $1$ . Thus there are ${n \\choose k}$ numbers that have $n+1$ digits in base $2$ notation, with $k+1$ of the digits being $1$ 's.\nIn order for there to be more $1$ 's than $0$ 's, we must have $k+1 > \\frac{n+1}{2} \\implies k > \\frac{n-1... |
https://artofproblemsolving.com/wiki/index.php/2023_AIME_II_Problems/Problem_10 | null | 144 | Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer divisors of $N.$ \[\be... | [
"We replace the numbers which are 0 mod(3) to 0, 1 mod(3) to 1, and 2 mod(3) to 2.\nThen, the problem is equivalent to arranging 4 0's,4 1's, and 4 2's into the grid (and then multiplying by $4!^3$ to account for replacing the remainder numbers with actual numbers) such that no 2 of the same numbers are adjacent.\n... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_I_Problems/Problem_10 | null | 202 | Let $N$ be the number of ways to write $2010$ in the form $2010 = a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0$ , where the $a_i$ 's are integers, and $0 \le a_i \le 99$ . An example of such a representation is $1\cdot 10^3 + 3\cdot 10^2 + 67\cdot 10^1 + 40\cdot 10^0$ . Find $N$ | [
"Note that $a_2\\cdot 10^2 + a_0$ is the base $100$ representation of any number from $0$ to $9999$ , and similarly $10(a_3\\cdot 10^2 + a_1)$ is ten times the base $100$ representation of any number from $0$ to $9999$ . Thus, the number of solutions is just the number of solutions to $2010 = 10a+b$ where $0\\le a,... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10B_Problems/Problem_19 | A | 8 | Let $N$ be the positive integer $7777\ldots777$ , a $313$ -digit number where each digit is a $7$ . Let $f(r)$ be the leading digit of the $r{ }$ th root of $N$ . What is \[f(2) + f(3) + f(4) + f(5)+ f(6)?\] $(\textbf{A})\: 8\qquad(\textbf{B}) \: 9\qquad(\textbf{C}) \: 11\qquad(\textbf{D}) \: 22\qquad(\textbf{E}) \: 29... | [
"We can rewrite $N$ as $\\frac{7}{9}\\cdot 9999\\ldots999 = \\frac{7}{9}\\cdot(10^{313}-1)$ . \nWhen approximating values, as we will shortly do, the minus one will become negligible so we can ignore it.\nWhen we take the power of ten out of the square root, we’ll be multiplying by another power of ten, so the lead... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_12B_Problems/Problem_5 | A | 3 | Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$ . What is the sum of the digits of $N$
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9$ | [
"$N$ must be divisible by every positive integer less than $7$ , or $1, 2, 3, 4, 5,$ and $6$ . Each number that is divisible by each of these is a multiple of their least common multiple. $LCM(1,2,3,4,5,6)=60$ , so each number divisible by these is a multiple of $60$ . The smallest multiple of $60$ is $60$ , so the... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10B_Problems/Problem_23 | C | 9 | Let $N=123456789101112\dots4344$ be the $79$ -digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 44$ | [
"We only need to find the remainders of N when divided by 5 and 9 to determine the answer.\nBy inspection, $N \\equiv 4 \\text{ (mod 5)}$ .\nThe remainder when $N$ is divided by $9$ is $1+2+3+4+ \\cdots +1+0+1+1 +1+2 +\\cdots + 4+3+4+4$ , but since $10 \\equiv 1 \\text{ (mod 9)}$ , we can also write this as $1+2+3 ... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12B_Problems/Problem_19 | C | 9 | Let $N=123456789101112\dots4344$ be the $79$ -digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 44$ | [
"We will consider this number $\\bmod\\ 5$ and $\\bmod\\ 9$ . By looking at the last digit, it is obvious that the number is $\\equiv 4\\bmod\\ 5$ . To calculate the number $\\bmod\\ 9$ , note that\n\\[123456\\cdots 4344 \\equiv 1+2+3+4+5+6+7+8+9+(1+0)+(1+1)+\\cdots+(4+3)+(4+4) \\equiv 1+2+\\cdots+44 \\bmod\\ 9,\\]... |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_12B_Problems/Problem_16 | E | 14 | Let $P$ be a cubic polynomial with $P(0) = k$ $P(1) = 2k$ , and $P(-1) = 3k$ . What is $P(2) + P(-2)$
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ k\qquad\textbf{(C)}\ 6k\qquad\textbf{(D)}\ 7k\qquad\textbf{(E)}\ 14k$ | [
"Let $P(x) = Ax^3+Bx^2 + Cx+D$ . Plugging in $0$ for $x$ , we find $D=k$ , and plugging in $1$ and $-1$ for $x$ , we obtain the following equations: \\[A+B+C+k=2k\\] \\[-A+B-C+k=3k\\] Adding these two equations together, we get \\[2B=3k\\] If we plug in $2$ and $-2$ in for $x$ , we find that \\[P(2)+P(-2) = 8A+4B... |
https://artofproblemsolving.com/wiki/index.php/2020_AIME_II_Problems/Problem_2 | null | 171 | Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$ , and $(0,1)$ . The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than or equal to $\frac12$ can be written as $\frac{m}{n}$ , wher... | [
"The areas bounded by the unit square and alternately bounded by the lines through $\\left(\\frac{5}{8},\\frac{3}{8}\\right)$ that are vertical or have a slope of $1/2$ show where $P$ can be placed to satisfy the condition. One of the areas is a trapezoid with bases $1/16$ and $3/8$ and height $5/8$ . The other are... |
https://artofproblemsolving.com/wiki/index.php/2023_AIME_I_Problems/Problem_5 | null | 106 | Let $P$ be a point on the circle circumscribing square $ABCD$ that satisfies $PA \cdot PC = 56$ and $PB \cdot PD = 90.$ Find the area of $ABCD.$ | [
"Ptolemy's theorem states that for cyclic quadrilateral $WXYZ$ $WX\\cdot YZ + XY\\cdot WZ = WY\\cdot XZ$\nWe may assume that $P$ is between $B$ and $C$ . Let $PA = a$ $PB = b$ $PC = c$ $PD = d$ , and $AB = s$ . We have $a^2 + c^2 = AC^2 = 2s^2$ , because $AC$ is a diameter of the circle. Similarly, $b^2 + d^2 = 2s^... |
https://artofproblemsolving.com/wiki/index.php/1988_AIME_Problems/Problem_12 | null | 441 | Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$ $b$ $c$ , and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$
1988 AIME-12.png | [
"Call the cevians AD, BE, and CF. Using area ratios ( $\\triangle PBC$ and $\\triangle ABC$ have the same base), we have:\n$\\frac {d}{a + d} = \\frac {[PBC]}{[ABC]}$\nSimilarily, $\\frac {d}{b + d} = \\frac {[PCA]}{[ABC]}$ and $\\frac {d}{c + d} = \\frac {[PAB]}{[ABC]}$\nThen, $\\frac {d}{a + d} + \\frac {d}{b + d... |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_12B_Problems/Problem_17 | E | 80 | Let $P$ be the parabola with equation $y=x^2$ and let $Q = (20, 14)$ . There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r$ $m$ $s$ . What is $r + s$
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 52\qquad\textbf{(E)}... | [
"Let $y = m(x - 20) + 14$ . Equating them:\n$x^2 = mx - 20m + 14$\n$x^2 - mx + 20m - 14 = 0$\nFor there to be no solutions, the discriminant must be less than zero:\n$m^2 - 4(20m - 14) < 0$\n$m^2 - 80m + 56 < 0$\nSo $m < 0$ for $r < m < s$ where $r$ and $s$ are the roots of $m^2 - 80m + 56 = 0$ and their sum by Vie... |
https://artofproblemsolving.com/wiki/index.php/2006_AIME_II_Problems/Problem_3 | null | 49 | Let $P$ be the product of the first $100$ positive odd integers . Find the largest integer $k$ such that $P$ is divisible by $3^k .$ | [
"Note that the product of the first $100$ positive odd integers can be written as $1\\cdot 3\\cdot 5\\cdot 7\\cdots 195\\cdot 197\\cdot 199=\\frac{1\\cdot 2\\cdots200}{2\\cdot4\\cdots200} = \\frac{200!}{2^{100}\\cdot 100!}$\nHence, we seek the number of threes in $200!$ decreased by the number of threes in $100!.$\... |
https://artofproblemsolving.com/wiki/index.php/2005_AIME_I_Problems/Problem_6 | null | 45 | Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005.$ Find $\lfloor P\rfloor.$ | [
"The left-hand side of that equation is nearly equal to $(x - 1)^4$ . Thus, we add 1 to each side in order to complete the fourth power and get $(x - 1)^4 = 2006$\nLet $r = \\sqrt[4]{2006}$ be the positive real fourth root of 2006. Then the roots of the above equation are $x = 1 + i^n r$ for $n = 0, 1, 2, 3$ . T... |
https://artofproblemsolving.com/wiki/index.php/2001_AMC_10_Problems/Problem_6 | E | 9 | Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits
of the integer $n$ . For example, $P(23) = 6$ and $S(23) = 5$ . Suppose $N$ is a
two-digit number such that $N = P(N)+S(N)$ . What is the units digit of $N$
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ ... | [
"Denote $a$ and $b$ as the tens and units digit of $N$ , respectively. Then $N = 10a+b$ . It follows that $10a+b=ab+a+b$ , which implies that $9a=ab$ . Since $a\\neq0$ $b=9$ . So the units digit of $N$ is $\\boxed{9}$"
] |
https://artofproblemsolving.com/wiki/index.php/2001_AMC_12_Problems/Problem_2 | E | 9 | Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits
of the integer $n$ . For example, $P(23) = 6$ and $S(23) = 5$ . Suppose $N$ is a
two-digit number such that $N = P(N)+S(N)$ . What is the units digit of $N$
$\text{(A)}\ 2\qquad \text{(B)}\ 3\qquad \text{(C)}\ 6\qquad \text{(D)}\ 8\qquad ... | [
"Denote $a$ and $b$ as the tens and units digit of $N$ , respectively. Then $N = 10a+b$ . It follows that $10a+b=ab+a+b$ , which implies that $9a=ab$ . Since $a\\neq0$ $b=9$ . So the units digit of $N$ is $\\boxed{9}$"
] |
https://artofproblemsolving.com/wiki/index.php/2020_AIME_II_Problems/Problem_11 | null | 71 | Let $P(x) = x^2 - 3x - 7$ , and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$ . David computes each of the three sums $P + Q$ $P + R$ , and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If ... | [
"Let $Q(x) = x^2 + ax + 2$ and $R(x) = x^2 + bx + c$ . We can write the following: \\[P + Q = 2x^2 + (a - 3)x - 5\\] \\[P + R = 2x^2 + (b - 3)x + (c - 7)\\] \\[Q + R = 2x^2 + (a + b)x + (c + 2)\\] Let the common root of $P+Q,P+R$ be $r$ $P+R,Q+R$ be $s$ ; and $P+Q,Q+R$ be $t$ . We then have that the roots of $P+Q... |
https://artofproblemsolving.com/wiki/index.php/2011_AIME_II_Problems/Problem_15 | null | 850 | Let $P(x) = x^2 - 3x - 9$ . A real number $x$ is chosen at random from the interval $5 \le x \le 15$ . The probability that $\lfloor\sqrt{P(x)}\rfloor = \sqrt{P(\lfloor x \rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$ , where $a$ $b$ $c$ $d$ , and $e$ are positive integers. Find $a + b + c + d + ... | [
"Make the substitution $y=2x-3$ , so $P(x)=\\frac{y^2-45}{4}.$ We're looking for solutions to \\[\\bigg\\lfloor{\\sqrt{\\frac{y^2-45}{4}}\\bigg\\rfloor}=\\sqrt{\\frac{\\lfloor{y\\rfloor}^2-45}{4}}\\] with the new bounds $y\\in{[7,27]}$ . Since the left side is an integer, it must be that $\\frac{\\lfloor{y\\rfloor}... |
https://artofproblemsolving.com/wiki/index.php/2016_AIME_I_Problems/Problem_11 | null | 109 | Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$ , and $\left(P(2)\right)^2 = P(3)$ . Then $P(\tfrac72)=\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ | [
"Plug in $x=1$ to get $(1-1)P(1+1) = 0 = (1+2)P(1) \\Rightarrow P(1) = 0$ . Plug in $x=0$ to get $(0-1)P(0+1) = (0+2)P(0)\\Rightarrow P(0) = -\\frac{1}{2}P(1) = 0$ . Plug in $x=-1$ to get $(-1-1)P(-1+1) = (-1+2)P(-1)\\Rightarrow (-2)P(0)=P(-1)\\Rightarrow P(-1) = 0$ . So $P(x) = x(x-1)(x+1)Q(x)$ for some polynomial... |
https://artofproblemsolving.com/wiki/index.php/1999_AHSME_Problems/Problem_17 | C | 118 | Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-19$ , the remainder is $99$ , and when $P(x)$ is divided by $x - 99$ , the remainder is $19$ . What is the remainder when $P(x)$ is divided by $(x-19)(x-99)$
$\mathrm{(A) \ } -x + 80 \qquad \mathrm{(B) \ } x + 80 \qquad \mathrm{(C) \ } -x + 118 \qquad \m... | [
"Since the divisor $(x-19)(x-99)$ is a quadratic, the degree of the remainder is at most linear. We can write $P(x)$ in the form \\[P(x) = Q(x)(x-19)(x-99) + cx+d\\] where $cx+d$ is the remainder.\nBy the Remainder Theorem, plugging in $19$ and $99$ gives us a system of equations. \\[99c+d = 19\\] \\[19c+d = 99\\]\... |
https://artofproblemsolving.com/wiki/index.php/1999_AHSME_Problems/Problem_17 | null | 118 | Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-19$ , the remainder is $99$ , and when $P(x)$ is divided by $x - 99$ , the remainder is $19$ . What is the remainder when $P(x)$ is divided by $(x-19)(x-99)$
$\mathrm{(A) \ } -x + 80 \qquad \mathrm{(B) \ } x + 80 \qquad \mathrm{(C) \ } -x + 118 \qquad \m... | [
"According to the problem statement, there are polynomials $Q(x)$ and $R(x)$ such that $P(x) = Q(x)(x-19) + 99 = R(x)(x-99) + 19$\nFrom the last equality we get $Q(x)(x-19) + 80 = R(x)(x-99)$\nThe value $x=99$ is a root of the polynomial on the right hand side, therefore it must be a root of the one on the left han... |
https://artofproblemsolving.com/wiki/index.php/2005_AIME_II_Problems/Problem_13 | null | 418 | Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17.$ Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2,$ find the product $n_1\cdot n_2.$ | [
"We define $Q(x)=P(x)-x+7$ , noting that it has roots at $17$ and $24$ . Hence $P(x)-x+7=A(x-17)(x-24)$ . In particular, this means that $P(x)-x-3=A(x-17)(x-24)-10$ . Therefore, $x=n_1,n_2$ satisfy $A(x-17)(x-24)=10$ , where $A$ $(x-17)$ , and $(x-24)$ are integers. This cannot occur if $x\\le 17$ or $x\\ge 24$ bec... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_21 | E | 23 | Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$ , the remainder is $x+2$ , and when $P(x)$ is divided by the polynomial $x^2+1$ , the remainder
is $2x+1$ . There is a unique polynomial of least degree with these two properties. What is the sum of
th... | [
"Given that all the answer choices and coefficients are integers, we hope that $P(x)$ has positive integer coefficients.\nThroughout this solution, we will express all polynomials in base $x$ . E.g. $x^2 + x + 1 = 111_{x}$\nWe are given: \\[111a + 12 = 101b + 21 = P(x).\\] We add $111$ and $101$ to each side and ba... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12B_Problems/Problem_20 | E | 23 | Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$ , the remainder is $x+2$ , and when $P(x)$ is divided by the polynomial $x^2+1$ , the remainder
is $2x+1$ . There is a unique polynomial of least degree with these two properties. What is the sum of
th... | [
"Given that all the answer choices and coefficients are integers, we hope that $P(x)$ has positive integer coefficients.\nThroughout this solution, we will express all polynomials in base $x$ . E.g. $x^2 + x + 1 = 111_{x}$\nWe are given: \\[111a + 12 = 101b + 21 = P(x).\\] We add $111$ and $101$ to each side and ba... |
https://artofproblemsolving.com/wiki/index.php/2020_AIME_I_Problems/Problem_14 | null | 85 | Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1.$ Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b.$ Find the sum of all possible values of $(a+b)^2.$ | [
"Either $P(3) = P(4)$ or not. We first see that if $P(3) = P(4)$ it's easy to obtain by Vieta's that $(a+b)^2 = 49$ . Now, take $P(3) \\neq P(4)$ and WLOG $P(3) = P(a), P(4) = P(b)$ . Now, consider the parabola formed by the graph of $P$ . It has vertex $\\frac{3+a}{2}$ . Now, say that $P(x) = x^2 - (3+a)x + c$ . W... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_I_Problems/Problem_6 | null | 406 | Let $P(x)$ be a quadratic polynomial with real coefficients satisfying $x^2 - 2x + 2 \le P(x) \le 2x^2 - 4x + 3$ for all real numbers $x$ , and suppose $P(11) = 181$ . Find $P(16)$ | [
"Let $Q(x) = x^2 - 2x + 2$ $R(x) = 2x^2 - 4x + 3$ Completing the square , we have $Q(x) = (x-1)^2 + 1$ , and $R(x) = 2(x-1)^2 + 1$ , so it follows that $P(x) \\ge Q(x) \\ge 1$ for all $x$ (by the Trivial Inequality ).\nAlso, $1 = Q(1) \\le P(1) \\le R(1) = 1$ , so $P(1) = 1$ , and $P$ obtains its minimum at the poi... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10A_Problems/Problem_21 | D | 47 | Let $P(x)$ be the unique polynomial of minimal degree with the following properties:
The roots of $P(x)$ are integers, with one exception. The root that is not an integer can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime integers. What is $m+n$
$\textbf{(A) }41\qquad\textbf{(B) }43\qquad\textbf{(... | [
"From the problem statement, we know $P(2-2)=0$ $P(9)=0$ and $4P(4)=0$ . Therefore, we know that $0$ $9$ , and $4$ are roots. So, we can factor $P(x)$ as $x(x - 9)(x - 4)(x - a)$ , where $a$ is the unknown root. Since $P(x) - 1 = 0$ , we plug in $x = 1$ which gives $1(-8)(-3)(1 - a) = 1$ , therefore $24(1 - a) = 1 ... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_12A_Problems/Problem_24 | B | 22 | Let $P(x)=(x-1)(x-2)(x-3)$ . For how many polynomials $Q(x)$ does there exist a polynomial $R(x)$ of degree 3 such that $P(Q(x))=P(x) \cdot R(x)$
$\mathrm {(A) } 19 \qquad \mathrm {(B) } 22 \qquad \mathrm {(C) } 24 \qquad \mathrm {(D) } 27 \qquad \mathrm {(E) } 32$ | [
"We can write the problem as\nSince $\\deg P(x) = 3$ and $\\deg R(x) = 3$ $\\deg P(x)\\cdot R(x) = 6$ . Thus, $\\deg P(Q(x)) = 6$ , so $\\deg Q(x) = 2$\nHence, we conclude $Q(1)$ $Q(2)$ , and $Q(3)$ must each be $1$ $2$ , or $3$ . Since a quadratic is uniquely determined by three points, there can be $3*3*3 = 27$ d... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_II_Problems/Problem_7 | null | 136 | Let $P(z)=z^3+az^2+bz+c$ , where a, b, and c are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$ $w+9i$ , and $2w-4$ , where $i^2=-1$ . Find $|a+b+c|$ | [
"Set $w=x+yi$ , so $x_1 = x+(y+3)i$ $x_2 = x+(y+9)i$ $x_3 = 2x-4+2yi$\nSince $a,b,c\\in{R}$ , the imaginary part of $a,b,c$ must be $0$\nStart with a, since it's the easiest one to do: $y+3+y+9+2y=0, y=-3$\nand therefore: $x_1 = x$ $x_2 = x+6i$ $x_3 = 2x-4-6i$\nNow, do the part where the imaginary part of c is 0 si... |
https://artofproblemsolving.com/wiki/index.php/1993_AIME_Problems/Problem_5 | null | 763 | Let $P_0(x) = x^3 + 313x^2 - 77x - 8\,$ . For integers $n \ge 1\,$ , define $P_n(x) = P_{n - 1}(x - n)\,$ . What is the coefficient of $x\,$ in $P_{20}(x)\,$ | [
"Notice that \\begin{align*}P_{20}(x) &= P_{19}(x - 20)\\\\ &= P_{18}((x - 20) - 19)\\\\ &= P_{17}(((x - 20) - 19) - 18)\\\\ &= \\cdots\\\\ &= P_0(x - (20 + 19 + 18 + \\ldots + 2 + 1)).\\end{align*}\nUsing the formula for the sum of the first $n$ numbers, $1 + 2 + \\cdots + 20 = \\frac{20(20+1)}{2} = 210$ . Therefo... |
https://artofproblemsolving.com/wiki/index.php/1990_AIME_Problems/Problem_3 | null | 117 | Let $P_1^{}$ be a regular $r~\mbox{gon}$ and $P_2^{}$ be a regular $s~\mbox{gon}$ $(r\geq s\geq 3)$ such that each interior angle of $P_1^{}$ is $\frac{59}{58}$ as large as each interior angle of $P_2^{}$ . What's the largest possible value of $s_{}^{}$ | [
"The formula for the interior angle of a regular sided polygon is $\\frac{(n-2)180}{n}$\nThus, $\\frac{\\frac{(r-2)180}{r}}{\\frac{(s-2)180}{s}} = \\frac{59}{58}$ . Cross multiplying and simplifying, we get $\\frac{58(r-2)}{r} = \\frac{59(s-2)}{s}$ . Cross multiply and combine like terms again to yield $58rs - 58 \... |
https://artofproblemsolving.com/wiki/index.php/2001_AIME_II_Problems/Problem_4 | null | 67 | Let $R = (8,6)$ . The lines whose equations are $8y = 15x$ and $10y = 3x$ contain points $P$ and $Q$ , respectively, such that $R$ is the midpoint of $\overline{PQ}$ . The length of $PQ$ equals $\frac {m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ | [
"The coordinates of $P$ can be written as $\\left(a, \\frac{15a}8\\right)$ and the coordinates of point $Q$ can be written as $\\left(b,\\frac{3b}{10}\\right)$ . By the midpoint formula, we have $\\frac{a+b}2=8$ and $\\frac{15a}{16}+\\frac{3b}{20}=6$ . Solving for $b$ gives $b= \\frac{80}{7}$ , so the point $Q$ is ... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10A_Problems/Problem_25 | B | 337 | Let $R$ $S$ , and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the $x$ -axis. The left edge of $R$ and the right edge of $S$ are on the $y$ -axis, and $R$ contains $\fra... | [
"Let $r$ be the number of lattice points on the side length of square $R$ $s$ be the number of lattice points on the side length of square $S$ , and $t$ be the number of lattice points on the side length of square $T$ . Note that the actual lengths of the side lengths are the number of lattice points minus $1$ , so... |
https://artofproblemsolving.com/wiki/index.php/1998_AHSME_Problems/Problem_11 | null | 5 | Let $R$ be a rectangle. How many circles in the plane of $R$ have a diameter both of whose endpoints are vertices of $R$
$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }4 \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ }6$ | [
"There are $6$ pairs of vertices of $R$ . However, both diagonals determine the same circle, therefore the answer is $\\boxed{5}$"
] |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_10A_Problems/Problem_25 | C | 2,320 | Let $R$ be a unit square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$ -ray partitional but not $60$ -ray partitional?
$\textbf{(A)}\ 1500 \qquad \textb... | [
"There must be four rays emanating from $X$ that intersect the four corners of the square region. Depending on the location of $X$ , the number of rays distributed among these four triangular sectors will vary. We start by finding the corner-most point that is $100$ -ray partitional (let this point be the bottom-le... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_10A_Problems/Problem_25 | null | 2,320 | Let $R$ be a unit square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$ -ray partitional but not $60$ -ray partitional?
$\textbf{(A)}\ 1500 \qquad \textb... | [
"For the sake of simplicity, let $R$ be a $60 \\times 60$ square and set the bottom-left point as the origin. Then, $R$ has vertices: \\[(0,0), (60,0), (60,60), (0,60).\\]\nNow, let a point in the square have coordinates $(x, y).$\nIn order for the point to be $100-$ ray partitional, we must be able to make $100$ t... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_12A_Problems/Problem_22 | C | 2,320 | Let $R$ be a unit square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$ -ray partitional but not $60$ -ray partitional?
$\textbf{(A)}\ 1500 \qquad \textb... | [
"There must be four rays emanating from $X$ that intersect the four corners of the square region. Depending on the location of $X$ , the number of rays distributed among these four triangular sectors will vary. We start by finding the corner-most point that is $100$ -ray partitional (let this point be the bottom-le... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_12A_Problems/Problem_22 | null | 2,320 | Let $R$ be a unit square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$ -ray partitional but not $60$ -ray partitional?
$\textbf{(A)}\ 1500 \qquad \textb... | [
"For the sake of simplicity, let $R$ be a $60 \\times 60$ square and set the bottom-left point as the origin. Then, $R$ has vertices: \\[(0,0), (60,0), (60,60), (0,60).\\]\nNow, let a point in the square have coordinates $(x, y).$\nIn order for the point to be $100-$ ray partitional, we must be able to make $100$ t... |
https://artofproblemsolving.com/wiki/index.php/2011_AIME_I_Problems/Problem_11 | null | 7 | Let $R$ be the set of all possible remainders when a number of the form $2^n$ $n$ a nonnegative integer, is divided by 1000. Let $S$ be the sum of the elements in $R$ . Find the remainder when $S$ is divided by 1000. | [
"Obviously, $2^i$ will have to repeat at some point, and our goal is just to find when it repeats. Suppose $2^a$ is the first time the powers of 2 repeat mod 1000, and that it is the same as $2^b$ where $b < a.$ We have \\[2^a \\equiv 2^b \\mod 1000 \\rightarrow 2^a - 2^b \\equiv 0 \\mod 1000\\] We can factor out a... |
https://artofproblemsolving.com/wiki/index.php/1950_AHSME_Problems/Problem_2 | D | 21 | Let $R=gS-4$ . When $S=8$ $R=16$ . When $S=10$ $R$ is equal to:
$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 21\qquad\textbf{(E)}\ \text{None of these}$ | [
"Our first procedure is to find the value of $g$ . With the given variables' values, we can see that $8g-4=16$ so $g=\\frac{20}{8}=\\frac{5}{2}$\nWith that, we can replace $g$ with $\\frac{5}{2}$ . When $S=10$ , we can see that $10\\times\\frac{5}{2}-4=\\frac{50}{2}-4=25-4=\\boxed{21}$"
] |
https://artofproblemsolving.com/wiki/index.php/2009_AIME_I_Problems/Problem_8 | null | 398 | Let $S = \{2^0,2^1,2^2,\ldots,2^{10}\}$ . Consider all possible positive differences of pairs of elements of $S$ . Let $N$ be the sum of all of these differences. Find the remainder when $N$ is divided by $1000$ | [
"Find the positive differences in all $55$ pairs and you will get $\\boxed{398}$ . \n(This is not recommended unless you can't find any other solutions to this problem)",
"When computing $N$ , the number $2^x$ will be added $x$ times (for terms $2^x-2^0$ $2^x-2^1$ , ..., $2^x - 2^{x-1}$ ), and subtracted $10-x$ t... |
https://artofproblemsolving.com/wiki/index.php/1984_AIME_Problems/Problem_4 | null | 649 | Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$ . However, if $68$ is removed, the average of the remaining numbers drops to $55$ . What is the largest number that can appear in $S$ | [
"Suppose that $S$ has $n$ numbers other than the $68,$ and the sum of these numbers is $s.$\nWe are given that \\begin{align*} \\frac{s+68}{n+1}&=56, \\\\ \\frac{s}{n}&=55. \\end{align*} Clearing denominators, we have \\begin{align*} s+68&=56n+56, \\\\ s&=55n. \\end{align*} Subtracting the equations, we get $68=n+5... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_10A_Problems/Problem_17 | C | 4 | Let $S$ be a set of $6$ integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$ , then $b$ is not a multiple of $a$ . What is the least possible value of an element in $S$
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)... | [
"We start with $2$ because $1$ is not an answer choice. We would have to include every odd number except $1$ to fill out the set, but then $3$ and $9$ would violate the rule, so that won't work.\nExperimentation with $3$ shows it's likewise impossible. You can include $7,11,$ and either $5$ or $10$ (which are alway... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12A_Problems/Problem_12 | C | 4 | Let $S$ be a set of $6$ integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$ , then $b$ is not a multiple of $a$ . What is the least possible value of an element in $S$
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)... | [
"We start with $2$ because $1$ is not an answer choice. We would have to include every odd number except $1$ to fill out the set, but then $3$ and $9$ would violate the rule, so that won't work.\nExperimentation with $3$ shows it's likewise impossible. You can include $7,11,$ and either $5$ or $10$ (which are alway... |
https://artofproblemsolving.com/wiki/index.php/2007_AIME_II_Problems/Problem_10 | null | 710 | Let $S$ be a set with six elements . Let $\mathcal{P}$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$ , not necessarily distinct, are chosen independently and at random from $\mathcal{P}$ . The probability that $B$ is contained in one of $A$ or $S-A$ is $\frac{m}{n^{r}},$ where $m$ $n$ , and $r$ are posit... | [
"Let $|S|$ denote the number of elements in a general set $S$ . We use complementary counting.\nThere is a total of $2^6$ elements in $P$ , so the total number of ways to choose $A$ and $B$ is $(2^6)^2 = 2^{12}$\nNote that the number of $x$ -element subset of $S$ is $\\binom{6}{x}$ . In general, for $0 \\le |A| \\l... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_10A_Problems/Problem_25 | A | 59 | Let $S$ be a square of side length $1$ . Two points are chosen independently at random on the sides of $S$ . The probability that the straight-line distance between the points is at least $\dfrac{1}{2}$ is $\dfrac{a-b\pi}{c}$ , where $a$ $b$ , and $c$ are positive integers with $\gcd(a,b,c)=1$ . What is $a+b+c$
$\te... | [
"Divide the boundary of the square into halves, thereby forming $8$ segments. Without loss of generality, let the first point $A$ be in the bottom-left segment. Then, it is easy to see that any point in the $5$ segments not bordering the bottom-left segment will be distance at least $\\dfrac{1}{2}$ apart from $A$ .... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_12A_Problems/Problem_23 | A | 59 | Let $S$ be a square of side length 1. Two points are chosen independently at random on the sides of $S$ . The probability that the straight-line distance between the points is at least $\frac12$ is $\frac{a-b\pi}{c}$ , where $a,b,$ and $c$ are positive integers and $\text{gcd}(a,b,c) = 1$ . What is $a+b+c$
$\textbf{(A)... | [
"WLOG, let the first point be on the bottom side of the square. The points where the second point could exist are outside a circle of radius 0.5 centered on the first point. The parts of the square that lie in this circle are the distance from the point to the closest side of the square $n$ , the distance from the ... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_12A_Problems/Problem_23 | null | 59 | Let $S$ be a square of side length 1. Two points are chosen independently at random on the sides of $S$ . The probability that the straight-line distance between the points is at least $\frac12$ is $\frac{a-b\pi}{c}$ , where $a,b,$ and $c$ are positive integers and $\text{gcd}(a,b,c) = 1$ . What is $a+b+c$
$\textbf{(A)... | [
"Choose a certain side for one of the points to be on. Let the distance from the point to the vertex on its left be $x$\nWe split this into two cases:\nCase 1: $0\\leq x\\leq \\frac12$\nThe total length of the segments for which the other point can be on such that the straight-line distance between the points is le... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_12A_Problems/Problem_17 | B | 13 | Let $S$ be a subset of $\{1,2,3,\dots,30\}$ with the property that no pair of distinct elements in $S$ has a sum divisible by $5$ . What is the largest possible size of $S$
$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18$ | [
"Of the integers from $1$ to $30$ , there are six each of $0,1,2,3,4\\ (\\text{mod}\\ 5)$ . We can create several rules to follow for the elements in subset $S$ . No element can be $1\\ (\\text{mod}\\ 5)$ if there is an element that is $4\\ (\\text{mod}\\ 5)$ . No element can be $2\\ (\\text{mod}\\ 5)$ if there is ... |
https://artofproblemsolving.com/wiki/index.php/1989_AIME_Problems/Problem_13 | null | 905 | Let $S$ be a subset of $\{1,2,3,\ldots,1989\}$ such that no two members of $S$ differ by $4$ or $7$ . What is the largest number of elements $S$ can have? | [
"We first show that we can choose at most 5 numbers from $\\{1, 2, \\ldots , 11\\}$ such that no two numbers have a difference of $4$ or $7$ . We take the smallest number to be $1$ , which rules out $5,8$ . Now we can take at most one from each of the pairs: $[2,9]$ $[3,7]$ $[4,11]$ $[6,10]$ . Now, $1989 = 180\\cdo... |
https://artofproblemsolving.com/wiki/index.php/2012_AIME_II_Problems/Problem_7 | null | 32 | Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$ . Find the remainder when $N$ is divided by $1000$ | [
"Okay, an exercise in counting (lots of binomials to calculate!). In base 2, the first number is $11111111$ , which is the only way to choose 8 1's out of 8 spaces, or $\\binom{8}{8}$ . What about 9 spaces? Well, all told, there are $\\binom{9}{8}=9$ , which includes the first 1. Similarly, for 10 spaces, there... |
https://artofproblemsolving.com/wiki/index.php/2018_AIME_I_Problems/Problem_1 | null | 600 | Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$ | [
"Let the linear factors be $(x+c)(x+d)$\nThen, $a=c+d$ and $b=cd$\nWe know that $1\\le a\\le 100$ and $b\\ge 0$ , so $c$ and $d$ both have to be non-negative\nHowever, $a$ cannot be $0$ , so at least one of $c$ and $d$ must be greater than $0$ , ie positive.\nAlso, $a$ cannot be greater than $100$ , so $c+d$ must b... |
https://artofproblemsolving.com/wiki/index.php/2015_AIME_I_Problems/Problem_9 | null | 494 | Let $S$ be the set of all ordered triple of integers $(a_1,a_2,a_3)$ with $1 \le a_1,a_2,a_3 \le 10$ . Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |$ for all $n\ge 4$ . Find the number of such sequences for which $a_n=0$ for some $n$ | [
"Let $a_1=x, a_2=y, a_3=z$ . First note that if any absolute value equals 0, then $a_n=0$ .\nAlso note that if at any position, $a_n=a_{n-1}$ , then $a_{n+2}=0$ .\nThen, if any absolute value equals 1, then $a_n=0$ .\nTherefore, if either $|y-x|$ or $|z-y|$ is less than or equal to 1, then that ordered triple meets... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_12A_Problems/Problem_25 | C | 80 | Let $S$ be the set of all points with coordinates $(x,y,z)$ , where $x$ $y$ , and $z$ are each chosen from the set $\{0,1,2\}$ . How many equilateral triangles all have their vertices in $S$
$(\mathrm {A}) \ 72\qquad (\mathrm {B}) \ 76 \qquad (\mathrm {C})\ 80 \qquad (\mathrm {D}) \ 84 \qquad (\mathrm {E})\ 88$ | [
"For this solution, we will just find as many solutions as possible, until it becomes intuitive that there are no more size of triangles left.\nFirst, try to make three of its vertices form an equilateral triangle. This we find is possible by taking any vertex , and connecting the three adjacent vertices into a tri... |
https://artofproblemsolving.com/wiki/index.php/2013_AIME_II_Problems/Problem_12 | null | 540 | Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$ , where $a$ $b$ , and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$ | [
"Every cubic with real coefficients has to have either three real roots or one real and two nonreal roots which are conjugates. This follows from Vieta's formulas\nThe real root $r$ must be one of $-20$ $20$ $-13$ , or $13$ . By Viète's formulas, $a=-(r+\\omega+\\omega^*)$ $b=|\\omega|^2+r(\\omega+\\omega^*)$ , and... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_10B_Problems/Problem_19 | C | 117 | Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$
$\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$ | [
"The prime factorization of $100,000$ is $2^5 \\cdot 5^5$ . Thus, we choose two numbers $2^a5^b$ and $2^c5^d$ where $0 \\le a,b,c,d \\le 5$ and $(a,b) \\neq (c,d)$ , whose product is $2^{a+c}5^{b+d}$ , where $0 \\le a+c \\le 10$ and $0 \\le b+d \\le 10$\nNotice that this is similar to choosing a divisor of $100,000... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_12B_Problems/Problem_14 | C | 117 | Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$
$\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$ | [
"The prime factorization of $100,000$ is $2^5 \\cdot 5^5$ . Thus, we choose two numbers $2^a5^b$ and $2^c5^d$ where $0 \\le a,b,c,d \\le 5$ and $(a,b) \\neq (c,d)$ , whose product is $2^{a+c}5^{b+d}$ , where $0 \\le a+c \\le 10$ and $0 \\le b+d \\le 10$\nNotice that this is similar to choosing a divisor of $100,000... |
https://artofproblemsolving.com/wiki/index.php/2023_AIME_II_Problems/Problem_5 | null | 719 | Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be... | [
"Denote $r = \\frac{a}{b}$ , where $\\left( a, b \\right) = 1$ .\nWe have $55 r = \\frac{55a}{b}$ .\nSuppose $\\left( 55, b \\right) = 1$ , then the sum of the numerator and the denominator of $55r$ is $55a + b$ .\nThis cannot be equal to the sum of the numerator and the denominator of $r$ $a + b$ .\nTherefore, $\\... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_22 | E | 136 | Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$ $x^{2}+y^{2}=64$ , and $(x-5)^{2}+y^{2}=3$ . What is the sum of the areas of all circles in $S$
$\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\... | [
"We denote by $C_1$ the circle that has the equation $x^2 + y^2 = 4$ .\nWe denote by $C_2$ the circle that has the equation $x^2 + y^2 = 64$ .\nWe denote by $C_3$ the circle that has the equation $(x-5)^2 + y^2 = 3$\nWe denote by $C_0$ a circle that is tangent to $C_1$ $C_2$ and $C_3$ .\nWe denote by $\\left( u, v ... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12B_Problems/Problem_21 | E | 136 | Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$ $x^{2}+y^{2}=64$ , and $(x-5)^{2}+y^{2}=3$ . What is the sum of the areas of all circles in $S$
$\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\... | [
"We denote by $C_1$ the circle that has the equation $x^2 + y^2 = 4$ .\nWe denote by $C_2$ the circle that has the equation $x^2 + y^2 = 64$ .\nWe denote by $C_3$ the circle that has the equation $(x-5)^2 + y^2 = 3$\nWe denote by $C_0$ a circle that is tangent to $C_1$ $C_2$ and $C_3$ .\nWe denote by $\\left( u, v ... |
https://artofproblemsolving.com/wiki/index.php/2004_AIME_II_Problems/Problem_10 | null | 913 | Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$ 's. If a number is chosen at random from $S,$ the probability that it is divisible by $9$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$ | [
"Any number from 1 to $2^{40}$ can be represented in binary with 40 digits (because $2^{40}$ has 41) with leading zeroes if necessary. Therefore the number of sets where there are exactly two 1’s in this binary representation is just $\\binom {40}{2}$ because we’re choosing 2 1s to go in 40 digit slots. This is equ... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_10B_Problems/Problem_25 | E | 85 | Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | [
"I know that I want about $\\frac{2}{3}$ of the box of integer coordinates above my line. There are a total of 30 integer coordinates in the desired range for each axis which gives a total of 900 lattice points. I estimate that the slope, m, is $\\frac{2}{3}$ . Now, although there is probably an easier solution, I ... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_12B_Problems/Problem_25 | E | 85 | Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | [
"I know that I want about $\\frac{2}{3}$ of the box of integer coordinates above my line. There are a total of 30 integer coordinates in the desired range for each axis which gives a total of 900 lattice points. I estimate that the slope, m, is $\\frac{2}{3}$ . Now, although there is probably an easier solution, I ... |
https://artofproblemsolving.com/wiki/index.php/2004_AIME_I_Problems/Problem_12 | null | 14 | Let $S$ be the set of ordered pairs $(x, y)$ such that $0 < x \le 1, 0<y\le 1,$ and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ The not... | [
"$\\left\\lfloor\\log_2\\left(\\frac{1}{x}\\right)\\right\\rfloor$ is even when\n\\[x \\in \\left(\\frac{1}{2},1\\right) \\cup \\left(\\frac{1}{8},\\frac{1}{4}\\right) \\cup \\left(\\frac{1}{32},\\frac{1}{16}\\right) \\cup \\cdots\\]\nLikewise: $\\left\\lfloor\\log_5\\left(\\frac{1}{y}\\right)\\right\\rfloor$ is ev... |
https://artofproblemsolving.com/wiki/index.php/1997_AIME_Problems/Problem_13 | null | 66 | Let $S$ be the set of points in the Cartesian plane that satisfy
If a model of $S$ were built from wire of negligible thickness, then the total length of wire required would be $a\sqrt{b}$ , where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime number. Find $a+b$ | [
"Let $f(x) = \\Big|\\big||x|-2\\big|-1\\Big|$ $f(x) \\ge 0$ . Then $f(x) + f(y) = 1 \\Longrightarrow f(x), f(y) \\le 1 \\Longrightarrow x, y \\le 4$ . We only have a $4\\times 4$ area, so guessing points and graphing won't be too bad of an idea. Since $f(x) = f(-x)$ , there's a symmetry about all four quadrants , s... |
https://artofproblemsolving.com/wiki/index.php/2001_AIME_I_Problems/Problem_10 | null | 200 | Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime ... | [
"The distance between the $x$ $y$ , and $z$ coordinates must be even so that the midpoint can have integer coordinates. Therefore,\nHowever, we have $3\\cdot 4\\cdot 5 = 60$ cases where we have simply taken the same point twice, so we subtract those. Therefore, our answer is $\\frac {5\\cdot 8\\cdot 13 - 60}{60\\cd... |
https://artofproblemsolving.com/wiki/index.php/2020_AIME_I_Problems/Problem_9 | null | 77 | Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\tfrac{m}{n},$ where $m$ and $n$ are ... | [
"\nFirst, prime factorize $20^9$ as $2^{18} \\cdot 5^9$ . Denote $a_1$ as $2^{b_1} \\cdot 5^{c_1}$ $a_2$ as $2^{b_2} \\cdot 5^{c_2}$ , and $a_3$ as $2^{b_3} \\cdot 5^{c_3}$\nIn order for $a_1$ to divide $a_2$ , and for $a_2$ to divide $a_3$ $b_1\\le b_2\\le b_3$ , and $c_1\\le c_2\\le c_3$ . We will consider each c... |
https://artofproblemsolving.com/wiki/index.php/2020_AIME_I_Problems/Problem_4 | null | 93 | Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020,$ and when the last four digits are removed, the result is a divisor of $N.$ For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020.$ Find the sum of all the digits of all the numbers in $S.$ For exampl... | [
"We note that any number in $S$ can be expressed as $a(10,000) + 2,020$ for some integer $a$ . The problem requires that $a$ divides this number, and since we know $a$ divides $a(10,000)$ , we need that $a$ divides 2020. Each number contributes the sum of the digits of $a$ , as well as $2 + 0 + 2 +0 = 4$ . Since $2... |
https://artofproblemsolving.com/wiki/index.php/2021_AIME_I_Problems/Problem_15 | null | 285 | Let $S$ be the set of positive integers $k$ such that the two parabolas \[y=x^2-k~~\text{and}~~x=2(y-20)^2-k\] intersect in four distinct points, and these four points lie on a circle with radius at most $21$ . Find the sum of the least element of $S$ and the greatest element of $S$ | [
"Note that $y=x^2-k$ is an upward-opening parabola with the vertex at $(0,-k),$ and $x=2(y-20)^2-k$ is a rightward-opening parabola with the vertex at $(-k,20).$ We consider each condition separately:\nTaking the intersection of Conditions 1 and 2 produces $5\\leq k\\leq280.$ Therefore, the answer is $5+280=\\boxed... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_12A_Problems/Problem_10 | D | 143 | Let $S$ be the set of positive integers $n$ for which $\tfrac{1}{n}$ has the repeating decimal representation $0.\overline{ab} = 0.ababab\cdots,$ with $a$ and $b$ different digits. What is the sum of the elements of $S$
$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 44\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 143\qquad\textbf{... | [
"Let us begin by working with the condition $0.\\overline{ab} = 0.ababab\\cdots,$ . Let $x = 0.ababab\\cdots$ . So, $100x-x = ab \\Rightarrow x = \\frac{ab}{99}$ . In order for this fraction $x$ to be in the form $\\frac{1}{n}$ $99$ must be a multiple of $ab$ . Hence the possibilities of $ab$ are $1,3,9,11,33,99$ .... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10B_Problems/Problem_12 | B | 49 | Let $S$ be the set of sides and diagonals of a regular pentagon. A pair of elements of $S$ are selected at random without replacement. What is the probability that the two chosen segments have the same length?
$\textbf{(A) }\frac{2}5\qquad\textbf{(B) }\frac{4}9\qquad\textbf{(C) }\frac{1}2\qquad\textbf{(D) }\frac{5}9\qq... | [
"The problem is simply asking how many ways are there to choose two sides or two diagonals. Hence, the probability is \\[\\dfrac{\\binom{5}{2} + \\binom{5}{2}}{\\binom{10}{2}} = \\dfrac{10+10}{45} = \\dfrac{20}{45}=\\boxed{49}\\]"
] |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_10A_Problems/Problem_22 | D | 668 | Let $S$ be the set of the $2005$ smallest positive multiples of $4$ , and let $T$ be the set of the $2005$ smallest positive multiples of $6$ . How many elements are common to $S$ and $T$
$\textbf{(A) } 166\qquad \textbf{(B) } 333\qquad \textbf{(C) } 500\qquad \textbf{(D) } 668\qquad \textbf{(E) } 1001$ | [
"Since the least common multiple $\\mathrm{lcm}(4,6)=12$ , the elements that are common to $S$ and $T$ must be multiples of $12$\nSince $4\\cdot2005=8020$ and $6\\cdot2005=12030$ , several multiples of $12$ that are in $T$ won't be in $S$ , but all multiples of $12$ that are in $S$ will be in $T$ . So we just need ... |
https://artofproblemsolving.com/wiki/index.php/2000_AIME_I_Problems/Problem_11 | null | 248 | Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10$ | [
"Since all divisors of $1000 = 2^35^3$ can be written in the form of $2^{m}5^{n}$ , it follows that $\\frac{a}{b}$ can also be expressed in the form of $2^{x}5^{y}$ , where $-3 \\le x,y \\le 3$ . Thus every number in the form of $a/b$ will be expressed one time in the product\n\\[(2^{-3} + 2^{-2} + 2^{-1} + 2^{0} +... |
https://artofproblemsolving.com/wiki/index.php/1986_AIME_Problems/Problem_8 | null | 141 | Let $S$ be the sum of the base $10$ logarithms of all the proper divisors (all divisors of a number excluding itself) of $1000000$ . What is the integer nearest to $S$ | [
"The prime factorization of $1000000 = 2^65^6$ , so there are $(6 + 1)(6 + 1) = 49$ divisors, of which $48$ are proper. The sum of multiple logarithms of the same base is equal to the logarithm of the products of the numbers.\nWriting out the first few terms, we see that the answer is equal to \\[\\log 1 + \\log 2 ... |
https://artofproblemsolving.com/wiki/index.php/1960_AHSME_Problems/Problem_27 | E | 2,700 | Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\frac{1}{2}$ times the
exterior angle at the same vertex. Then
$\textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular}\qquad \\ \textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \... | [
"Let $a_n$ be the interior angle of the nth vertex, and let $b_n$ be the exterior angle of the nth vertex. From the conditions in the problem, \\[a_n = 7.5b_n\\] That means \\[a_1 + a_2 \\cdots a_n = 7.5(b_1 + b_2 \\cdots b_n)\\] Since the sum of the exterior angles of a polygon is $360^{\\circ}$ , the equation ca... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10A_Problems/Problem_20 | D | 1,239 | Let $S(n)$ equal the sum of the digits of positive integer $n$ . For example, $S(1507) = 13$ . For a particular positive integer $n$ $S(n) = 1274$ . Which of the following could be the value of $S(n+1)$
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$ | [
"Note that $n \\equiv S(n) \\pmod{9}$ . This can be seen from the fact that $\\sum_{k=0}^{n}10^{k}a_k \\equiv \\sum_{k=0}^{n}a_k \\pmod{9}$ . Thus, if $S(n) = 1274$ , then $n \\equiv 5 \\pmod{9}$ , and thus $n+1 \\equiv S(n+1) \\equiv 6 \\pmod{9}$ . The only answer choice that satisfies $n+1 \\equiv 6 \\pmod{9}$ is... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12A_Problems/Problem_18 | D | 1,239 | Let $S(n)$ equal the sum of the digits of positive integer $n$ . For example, $S(1507) = 13$ . For a particular positive integer $n$ $S(n) = 1274$ . Which of the following could be the value of $S(n+1)$
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$ | [
"Note that $n\\equiv S(n)\\bmod 9$ , so $S(n+1)-S(n)\\equiv n+1-n = 1\\bmod 9$ . So, since $S(n)=1274\\equiv 5\\bmod 9$ , we have that $S(n+1)\\equiv 6\\bmod 9$ . The only one of the answer choices $\\equiv 6\\bmod 9$ is $\\boxed{1239}$",
"One possible value of $S(n)$ would be $1275$ , but this is not any of the ... |
https://artofproblemsolving.com/wiki/index.php/2018_AIME_I_Problems/Problem_14 | null | 351 | Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$ . From any vertex of the heptagon except $E$ , the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$ , the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that ... | [
"This is easily solved by recursion/dynamic programming. To simplify the problem somewhat, let us imagine the seven vertices on a line $E \\leftrightarrow P_4 \\leftrightarrow P_5 \\leftrightarrow S \\leftrightarrow P_1 \\leftrightarrow P_2 \\leftrightarrow P_3 \\leftrightarrow E$ . We can count the number of left/... |
https://artofproblemsolving.com/wiki/index.php/1993_AIME_Problems/Problem_8 | null | 365 | Let $S\,$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S\,$ so that the union of the two subsets is $S\,$ ? The order of selection does not matter; for example, the pair of subsets $\{a, c\},\{b, c, d, e, f\}$ represents the same selection as the pair $... | [
"Call the two subsets $m$ and $n.$ For each of the elements in $S,$ we can assign it to either $m,n,$ or both. This gives us $3^6$ possible methods of selection. However, because the order of the subsets does not matter, each possible selection is double counted, except the case where both $m$ and $n$ contain all $... |
https://artofproblemsolving.com/wiki/index.php/1992_AIME_Problems/Problem_5 | null | 660 | Let $S^{}_{}$ be the set of all rational numbers $r^{}_{}$ $0^{}_{}<r<1$ , that have a repeating decimal expansion in the form $0.abcabcabc\ldots=0.\overline{abc}$ , where the digits $a^{}_{}$ $b^{}_{}$ , and $c^{}_{}$ are not necessarily distinct. To write the elements of $S^{}_{}$ as fractions in lowest terms, how ma... | [
"We consider the method in which repeating decimals are normally converted to fractions with an example:\n$x=0.\\overline{176}$\n$\\Rightarrow 1000x=176.\\overline{176}$\n$\\Rightarrow 999x=1000x-x=176$\n$\\Rightarrow x=\\frac{176}{999}$\nThus, let $x=0.\\overline{abc}$\n$\\Rightarrow 1000x=abc.\\overline{abc}$\n$\... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_12A_Problems/Problem_21 | E | 102 | Let $S_1=\{(x,y)|\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}$ and $S_2=\{(x,y)|\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}$
What is the ratio of the area of $S_2$ to the area of $S_1$
$\mathrm{(A) \ } 98\qquad \mathrm{(B) \ } 99\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D) \ } 101\qquad \mathrm{(E) \ } 102$ | [
"Looking at the constraints of $S_1$\n$x+y > 0$\n$\\log_{10}(1+x^2+y^2)\\le 1+\\log_{10}(x+y)$\n$\\log_{10}(1+x^2+y^2)\\le \\log_{10} 10 +\\log_{10}(x+y)$\n$\\log_{10}(1+x^2+y^2)\\le \\log_{10}(10x+10y)$\n$1+x^2+y^2 \\le 10x+10y$\n$x^2-10x+y^2-10y \\le -1$\n$x^2-10x+25+y^2-10y+25 \\le 49$\n$(x-5)^2 + (y-5)^2 \\le (... |
https://artofproblemsolving.com/wiki/index.php/2008_AIME_I_Problems/Problem_7 | null | 708 | Let $S_i$ be the set of all integers $n$ such that $100i\leq n < 100(i + 1)$ . For example, $S_4$ is the set ${400,401,402,\ldots,499}$ . How many of the sets $S_0, S_1, S_2, \ldots, S_{999}$ do not contain a perfect square? | [
"The difference between consecutive squares is $(x + 1)^2 - x^2 = 2x + 1$ , which means that all squares above $50^2 = 2500$ are more than $100$ apart.\nThen the first $26$ sets ( $S_0,\\cdots S_{25}$ ) each have at least one perfect square because the differences between consecutive squares in them are all less th... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_15 | D | 400 | Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence that has a common difference of $2$ . The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$ . What is $S_{20}$
$\textbf{(A) } 340 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 380 \qquad \textbf{(D) } 400 \qquad \textbf{(E) } 420$ | [
"Suppose that the first number of the arithmetic sequence is $a$ . We will try to compute the value of $S_{n}$ . First, note that the sum of an arithmetic sequence is equal to the number of terms multiplied by the median of the sequence. The median of this sequence is equal to $a + n - 1$ . Thus, the value of $S_{n... |
https://artofproblemsolving.com/wiki/index.php/2006_AIME_II_Problems/Problem_14 | null | 63 | Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer. | [
"Let $K = \\sum_{i=1}^{9}{\\frac{1}{i}}$ . Examining the terms in $S_1$ , we see that $S_1 = K + 1$ since each digit $n$ appears once and 1 appears an extra time. Now consider writing out $S_2$ . Each term of $K$ will appear 10 times in the units place and 10 times in the tens place (plus one extra 1 will appear... |
https://artofproblemsolving.com/wiki/index.php/1982_USAMO_Problems/Problem_2 | null | 2 | Let $S_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$
$(*)$ $\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}$
for $(m,n)=(2,3),(3,2),(2,5)$ , or $(5,2)$ . Determine all other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=0$ | [
"Claim Both $m,n$ can not be even.\nProof $x+y+z=0$ $\\implies x=-(y+z)$\nSince $\\frac{S_{m+n}}{m+n} = \\frac{S_m S_n}{mn}$\nby equating cofficient of $y^{m+n}$ on LHS and RHS ,get\n$\\frac{2}{m+n}=\\frac{4}{mn}$\n$\\implies \\frac{m}{2} + \\frac {n}{2} = \\frac{m\\cdot n}{2\\cdot2}$\nSo we have, $\\frac{m}{2} \\... |
https://artofproblemsolving.com/wiki/index.php/1990_AIME_Problems/Problem_13 | null | 184 | Let $T = \{9^k : k ~ \mbox{is an integer}, 0 \le k \le 4000\}$ . Given that $9^{4000}_{}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T_{}^{}$ have 9 as their leftmost digit? | [
"Lemma : For all positive integers n, there's exactly one n-digit power of 9 that does not have a left-most digit 9\n(Not-so-rigorous) Proof: One can prove by contradiction that there must be at least either one or two n-digit power of 9 for all n. If there is exactly 1 n-digit power of 9, then such a number $m$ ca... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A_Problems/Problem_23 | A | 12 | Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$ -axis, and reflection across the... | [
"\nFirst, any combination of motions we can make must reflect $T$ an even number of times. This is because every time we reflect $T$ , it changes orientation. Once $T$ has been flipped once, no combination of rotations will put it back in place because it is the mirror image; however, flipping it again changes it b... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12A_Problems/Problem_20 | A | 12 | Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$ -axis, and reflection across the... | [
"\nFirst, any combination of motions we can make must reflect $T$ an even number of times. This is because every time we reflect $T$ , it changes orientation. Once $T$ has been flipped once, no combination of rotations will put it back in place because it is the mirror image; however, flipping it again changes it b... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10A_Problems/Problem_18 | A | 359 | Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$ -axis. What is the least positive
integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point... | [
"Let $P=(r,\\theta)$ be a point in polar coordinates, where $\\theta$ is in degrees.\nRotating $P$ by $k^{\\circ}$ counterclockwise around the origin gives the transformation $(r,\\theta)\\rightarrow(r,\\theta+k^{\\circ}).$ Reflecting $P$ across the $y$ -axis gives the transformation $(r,\\theta)\\rightarrow(r,180^... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_18 | A | 359 | Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$ -axis. What is the least positive
integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point... | [
"Let $P=(r,\\theta)$ be a point in polar coordinates, where $\\theta$ is in degrees.\nRotating $P$ by $k^{\\circ}$ counterclockwise around the origin gives the transformation $(r,\\theta)\\rightarrow(r,\\theta+k^{\\circ}).$ Reflecting $P$ across the $y$ -axis gives the transformation $(r,\\theta)\\rightarrow(r,180^... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_8_Problems/Problem_7 | A | 11 | Let $Z$ be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of $Z$
$\textbf{(A) }11\qquad\textbf{(B) }19\qquad\textbf{(C) }101\qquad\textbf{(D) }111\qquad\textbf{(E) }1111$ | [
"To check, if a number is divisible by 19, take its unit digit and multiply it by 2, then add the result to the rest of the number, and repeat this step until the number is reduced to two digits. If the result is divisible by 19, then the original number is also divisible by 19. Or we could just try to divide the e... |
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