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/2004_AIME_II_Problems/Problem_14 | null | 108 | Consider a string of $n$ $7$ 's, $7777\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression . For example, $7+77+777+7+7=875$ could be obtained from eight $7$ 's in this way. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value $7000$ | [
"Suppose we require $a$ $7$ s, $b$ $77$ s, and $c$ $777$ s to sum up to $7000$ $a,b,c \\ge 0$ ). Then $7a + 77b + 777c = 7000$ , or dividing by $7$ $a + 11b + 111c = 1000$ . Then the question is asking for the number of values of $n = a + 2b + 3c$\nManipulating our equation, we have $a + 2b + 3c = n = 1000 - 9(b + ... |
https://artofproblemsolving.com/wiki/index.php/2015_AIME_I_Problems/Problem_12 | null | 431 | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$ . From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$ | [
"Let $M$ be the desired mean. Then because $\\dbinom{2015}{1000}$ subsets have 1000 elements and $\\dbinom{2015 - i}{999}$ have $i$ as their least element, \\begin{align*} \\binom{2015}{1000} M &= 1 \\cdot \\binom{2014}{999} + 2 \\cdot \\binom{2013}{999} + \\dots + 1016 \\cdot \\binom{999}{999} \\\\ &= \\binom... |
https://artofproblemsolving.com/wiki/index.php/2017_AIME_I_Problems/Problem_11 | null | 360 | Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$ $a_2$ , and $a_3$ be the medians of the numbers in rows $1$ $2$ , and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$ . Let $Q$ be the number of arrangements for which $m = 5$ . ... | [
"Assume that $5 \\in \\{a_1, a_2, a_3\\}$ $m \\neq 5$ , and WLOG, $\\max{(a_1, a_2, a_3)} = 5$ . Then we know that the other two medians in $\\{a_1, a_2, a_3\\}$ and the smallest number of rows 1, 2, and 3 are all less than 5. But there are only 4 numbers less than 5 in $1, 2, 3, \\dots, 9$ , a Contradiction. Thus,... |
https://artofproblemsolving.com/wiki/index.php/1963_AHSME_Problems/Problem_24 | B | 19 | Consider equations of the form $x^2 + bx + c = 0$ . How many such equations have real roots and have coefficients $b$ and $c$ selected
from the set of integers $\{1,2,3, 4, 5,6\}$
$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 19 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 16$ | [
"The discriminant of the quadratic is $b^2 - 4c$ . Since the quadratic has real roots, \\[b^2 - 4c \\ge 0\\] \\[b^2 \\ge 4c\\] If $b = 6$ , then $c$ can be from $1$ to $6$ . If $b = 5$ , then $c$ can also be from $1$ to $6$ . If $b=4$ , then $c$ can be from $1$ to $4$ . If $b=3$ , then $c$ can be $1$ or $2$ . ... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_24 | B | 50 | Consider functions $f$ that satisfy \[|f(x)-f(y)|\leq \frac{1}{2}|x-y|\] for all real numbers $x$ and $y$ . Of all such functions that also satisfy the equation $f(300) = f(900)$ , what is the greatest possible value of \[f(f(800))-f(f(400))?\] $\textbf{(A)}\ 25 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 100 \qquad\tex... | [
"We have \\begin{align*} |f(f(800))-f(f(400))| &\\leq \\frac12|f(800)-f(400)| &&(\\bigstar) \\\\ &\\leq \\frac12\\left|\\frac12|800-400|\\right| \\\\ &= 100, \\end{align*} from which we eliminate answer choices $\\textbf{(D)}$ and $\\textbf{(E)}.$\nNote that \\begin{alignat*}{8} |f(800)-f(300)| &\\leq \\frac12|800-... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12B_Problems/Problem_22 | D | 220 | Consider polynomials $P(x)$ of degree at most $3$ , each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ . How many such polynomials satisfy $P(-1) = -9$
$\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286$ | [
"Suppose that $P(x)=ax^3+bx^2+cx+d.$ This problem is equivalent to counting the ordered quadruples $(a,b,c,d),$ where all of $a,b,c,$ and $d$ are integers from $0$ through $9$ such that \\[P(-1)=-a+b-c+d=-9.\\] Let $a'=9-a$ and $c'=9-c.$ Note that both of $a'$ and $c'$ are integers from $0$ through $9.$ Moreover, t... |
https://artofproblemsolving.com/wiki/index.php/2001_AMC_12_Problems/Problem_25 | null | 4 | Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term $2001$ appear somewhere in the sequence?
$\text{(A) }1 \qquad \text{(B) }2 \qquad \text{(C) }3 \qq... | [
"It never hurts to compute a few terms of the sequence in order to get a feel how it looks like. In our case, the definition is that $\\forall$ (for all) $n>1:~ a_n = a_{n-1}a_{n+1} - 1$ . This can be rewritten as $a_{n+1} = \\frac{a_n +1}{a_{n-1}}$ . We have $a_1=x$ and $a_2=2000$ , and we compute:\n\\begin{align*... |
https://artofproblemsolving.com/wiki/index.php/2008_AIME_I_Problems/Problem_11 | null | 172 | Consider sequences that consist entirely of $A$ 's and $B$ 's and that have the property that every run of consecutive $A$ 's has even length, and every run of consecutive $B$ 's has odd length. Examples of such sequences are $AA$ $B$ , and $AABAA$ , while $BBAB$ is not such a sequence. How many such sequences have len... | [
"Let $a_n$ and $b_n$ denote, respectively, the number of sequences of length $n$ ending in $A$ and $B$ . If a sequence ends in an $A$ , then it must have been formed by appending two $A$ s to the end of a string of length $n-2$ . If a sequence ends in a $B,$ it must have either been formed by appending one $B$ to a... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_18 | B | 338 | Consider systems of three linear equations with unknowns $x$ $y$ , and $z$ \begin{align*} a_1 x + b_1 y + c_1 z & = 0 \\ a_2 x + b_2 y + c_2 z & = 0 \\ a_3 x + b_3 y + c_3 z & = 0 \end{align*} where each of the coefficients is either $0$ or $1$ and the system has a solution other than $x=y=z=0$ .
For example, one such ... | [
"Let $M_1=\\begin{bmatrix}a_1 & b_1 & c_1\\end{bmatrix}, M_2=\\begin{bmatrix}a_2 & b_2 & c_2\\end{bmatrix},$ and $M_3=\\begin{bmatrix}a_3 & b_3 & c_3\\end{bmatrix}.$\nWe wish to count the ordered triples $(M_1,M_2,M_3)$ of row matrices. We perform casework:\nThere are $9+3=12$ ordered triples $(M_1,M_2,M_3).$\nSimi... |
https://artofproblemsolving.com/wiki/index.php/2001_AMC_10_Problems/Problem_11 | C | 800 | Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains $8$ unit squares. The second ring contains $16$ unit squares. If we continue this process, the number of unit squares in the $100^\text{th}$ ring is
[asy] unitsize(3mm); defaultpen(... | [
"We can partition the $n^\\text{th}$ ring into $4$ rectangles: two containing $2n+1$ unit squares and two containing $2n-1$ unit squares.\nThere are $2(2n+1)+2(2n-1)=4n+2+4n-2=8n$ unit squares in the $n^\\text{th}$ ring.\nThus, the $100^\\text{th}$ ring has $8 \\times 100 = \\boxed{800}$ unit squares.",
"We can m... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_8 | B | 42 | Consider the following $100$ sets of $10$ elements each: \begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*} How many of these sets contain exactly two multiples of $7$
$\textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)... | [
"We apply casework to this problem. The only sets that contain two multiples of seven are those for which:\nEach case has $\\left\\lfloor\\frac{100}{7}\\right\\rfloor=14$ sets. Therefore, the answer is $14\\cdot3=\\boxed{42}.$",
"We find a pattern. \\begin{align*} &\\{1,2,3,\\ldots,10\\}, \\\\ &\\{11,12,13,\\ldot... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12B_Problems/Problem_6 | B | 42 | Consider the following $100$ sets of $10$ elements each: \begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*} How many of these sets contain exactly two multiples of $7$
$\textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)... | [
"We apply casework to this problem. The only sets that contain two multiples of seven are those for which:\nEach case has $\\left\\lfloor\\frac{100}{7}\\right\\rfloor=14$ sets. Therefore, the answer is $14\\cdot3=\\boxed{42}.$",
"We find a pattern. \\begin{align*} &\\{1,2,3,\\ldots,10\\}, \\\\ &\\{11,12,13,\\ldot... |
https://artofproblemsolving.com/wiki/index.php/1985_AHSME_Problems/Problem_19 | A | 4 | Consider the graphs of $y = Ax^2$ and $y^2+3 = x^2+4y$ , where $A$ is a positive constant and $x$ and $y$ are real variables. In how many points do the two graphs intersect?
$\mathrm{(A) \ }\text{exactly }4 \qquad \mathrm{(B) \ }\text{exactly }2 \qquad$
$\mathrm{(C) \ }\text{at least }1,\text{ but the number varies fo... | [
"Substituting $y = Ax^2$ into the equation $y^2+3 = x^2+4y$ gives \\begin{align*}\\left(Ax^2\\right)+3 = x^2+4\\left(Ax^2\\right) &\\iff A^2x^4+3 = x^2+4Ax^2 \\\\ &\\iff A^2x^4-\\left(4A+1\\right)x^2+3 = 0 \\\\ &\\iff x^2 = \\frac{4A+1 \\pm \\sqrt{4A^2+8A+1}}{2A^2} \\\\ &\\text{(using the quadratic formula)}.\\end{... |
https://artofproblemsolving.com/wiki/index.php/2019_AIME_I_Problems/Problem_1 | null | 342 | Consider the integer \[N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.\] Find the sum of the digits of $N$ | [
"Let's express the number in terms of $10^n$ . We can obtain $(10-1)+(10^2-1)+(10^3-1)+\\cdots+(10^{321}-1)$ . By the commutative and associative property, we can group it into $(10+10^2+10^3+\\cdots+10^{321})-321$ . We know the former will yield $1111....10$ , so we only have to figure out what the last few digits... |
https://artofproblemsolving.com/wiki/index.php/1993_AJHSME_Problems/Problem_9 | D | 4 | Consider the operation $*$ defined by the following table:
\[\begin{tabular}{c|cccc} * & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ 2 & 2 & 4 & 1 & 3 \\ 3 & 3 & 1 & 4 & 2 \\ 4 & 4 & 3 & 2 & 1 \end{tabular}\]
For example, $3*2=1$ . Then $(2*4)*(1*3)=$
$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \t... | [
"Using the chart, $(2*4)=3$ and $(1*3)=3$ . Therefore, $(2*4)*(1*3)=3*3=\\boxed{4}$",
"By the chart, we can see that the \" $*$ \" operation is actually multiplication modulo $5$ . Thus, we can do $(2*4)*(1*3)\\rightarrow(2\\cdot4)\\cdot(1\\cdot3)=8\\cdot3=24\\rightarrow\\boxed{4}$"
] |
https://artofproblemsolving.com/wiki/index.php/1999_AIME_Problems/Problem_15 | null | 408 | Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid? | [
"As shown in the image above, let $D$ $E$ , and $F$ be the midpoints of $\\overline{BC}$ $\\overline{CA}$ , and $\\overline{AB}$ , respectively. Suppose $P$ is the apex of the tetrahedron, and let $O$ be the foot of the altitude from $P$ to $\\triangle ABC$ . The crux of this problem is the following lemma.\nLemm... |
https://artofproblemsolving.com/wiki/index.php/1999_AIME_Problems/Problem_2 | null | 118 | Consider the parallelogram with vertices $(10,45)$ $(10,114)$ $(28,153)$ , and $(28,84)$ . A line through the origin cuts this figure into two congruent polygons . The slope of the line is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers . Find $m+n$ | [
"Let the first point on the line $x=10$ be $(10,45+a)$ where a is the height above $(10,45)$ . Let the second point on the line $x=28$ be $(28, 153-a)$ . For two given points, the line will pass the origin if the coordinates are proportional (such that $\\frac{y_1}{x_1} = \\frac{y_2}{x_2}$ ). Then, we can write th... |
https://artofproblemsolving.com/wiki/index.php/2024_AIME_I_Problems/Problem_6 | null | 294 | Consider the paths of length $16$ that follow the lines from the lower left corner to the upper right corner on an $8\times 8$ grid. Find the number of such paths that change direction exactly four times, as in the examples shown below.
[asy] size(10cm); usepackage("tikz");label("\begin{tikzpicture}[scale=.5]\draw(0,0)... | [
"We divide the path into eight “ $R$ ” movements and eight “ $U$ ” movements. Five sections of alternative $RURUR$ or $URURU$ are necessary in order to make four “turns.” We use the first case and multiply by $2$\nFor $U$ , we have seven ordered pairs of positive integers $(a,b)$ such that $a+b=8$\nFor $R$ , we sub... |
https://artofproblemsolving.com/wiki/index.php/2005_AIME_I_Problems/Problem_14 | null | 936 | Consider the points $A(0,12), B(10,9), C(8,0),$ and $D(-4,7).$ There is a unique square $S$ such that each of the four points is on a different side of $S.$ Let $K$ be the area of $S.$ Find the remainder when $10K$ is divided by $1000$ | [
"Let $(a,b)$ denote a normal vector of the side containing $A$ . Note that $\\overline{AC}, \\overline{BD}$ intersect and hence must be opposite vertices of the square. The lines containing the sides of the square have the form $ax+by=12b$ $ax+by=8a$ $bx-ay=10b-9a$ , and $bx-ay=-4b-7a$ . The lines form a square, so... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_12A_Problems/Problem_20 | B | 6 | Consider the polynomial
\[P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)\]
The coefficient of $x^{2012}$ is equal to $2^a$ . What is $a$
\[\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24\] | [
"The degree of $P(x)$ is $1024+512+256+\\cdots+1=2047$ . We want to find the coefficient of $x^{2012}$ , so we need to omit the powers of $2$ that add up to $2047-2012=35$ . We find that $35=2^0+2^1+2^5$ . From here, we know that the answer is $2^0\\cdot2^1\\cdot2^5=2^6$ . Therefore, the answer is $\\boxed{6.}$"
] |
https://artofproblemsolving.com/wiki/index.php/2003_AIME_II_Problems/Problem_9 | null | 6 | Consider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x) = 0,$ find $P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).$ | [
"When we use long division to divide $P(x)$ by $Q(x)$ , the remainder is $x^2-x+1$\nSo, since $z_1$ is a root, $P(z_1)=(z_1)^2-z_1+1$\nNow this also follows for all roots of $Q(x)$ Now \\[P(z_2)+P(z_1)+P(z_3)+P(z_4)=z_1^2-z_1+1+z_2^2-z_2+1+z_3^2-z_3+1+z_4^2-z_4+1\\]\nNow by Vieta's we know that $-z_4-z_3-z_2-z_1=-1... |
https://artofproblemsolving.com/wiki/index.php/1992_AIME_Problems/Problem_10 | null | 572 | Consider the region $A$ in the complex plane that consists of all points $z$ such that both $\frac{z}{40}$ and $\frac{40}{\overline{z}}$ have real and imaginary parts between $0$ and $1$ , inclusive. What is the integer that is nearest the area of $A$ | [
"Let $z=a+bi \\implies \\frac{z}{40}=\\frac{a}{40}+\\frac{b}{40}i$ . Since $0\\leq \\frac{a}{40},\\frac{b}{40}\\leq 1$ we have the inequality \\[0\\leq a,b \\leq 40\\] which is a square of side length $40$\nAlso, $\\frac{40}{\\overline{z}}=\\frac{40}{a-bi}=\\frac{40a}{a^2+b^2}+\\frac{40b}{a^2+b^2}i$ so we have $0\\... |
https://artofproblemsolving.com/wiki/index.php/1997_AHSME_Problems/Problem_6 | B | 0.5 | Consider the sequence
$1,-2,3,-4,5,-6,\ldots,$
whose $n$ th term is $(-1)^{n+1}\cdot n$ . What is the average of the first $200$ terms of the sequence?
$\textbf{(A)}-\!1\qquad\textbf{(B)}-\!0.5\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 0.5\qquad\textbf{(E)}\ 1$ | [
"The average of a list is the sum of all numbers divided by the size of the list.\nThe sum of the list can be found by adding the numbers in pairs: $(1 + -2) + (3 + -4) + ... + (199 + -200)$\nThe sum of each pair is $-1$ , and there are $100$ pairs, so the total sum is $-100$\nThere are $200$ numbers on the list, s... |
https://artofproblemsolving.com/wiki/index.php/2021_AIME_I_Problems/Problem_10 | null | 59 | Consider the sequence $(a_k)_{k\ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k\ge 1$ , if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , then
\[a_{k+1} = \frac{m + 18}{n+19}.\] Determine the sum of all positive integers $j$ such that the rational number $a_... | [
"We know that $a_{1}=\\tfrac{t}{t+1}$ when $t=2020$ so $1$ is a possible value of $j$ . Note also that $a_{2}=\\tfrac{2038}{2040}=\\tfrac{1019}{1020}=\\tfrac{t}{t+1}$ for $t=1019$ . Then $a_{2+q}=\\tfrac{1019+18q}{1020+19q}$ unless $1019+18q$ and $1020+19q$ are not relatively prime which happens when $q+1$ divides ... |
https://artofproblemsolving.com/wiki/index.php/1989_AHSME_Problems/Problem_10 | C | 16 | Consider the sequence defined recursively by $u_1=a$ (any positive number), and $u_{n+1}=-1/(u_n+1)$ $n=1,2,3,...$ For which of the following values of $n$ must $u_n=a$
$\mathrm{(A) \ 14 } \qquad \mathrm{(B) \ 15 } \qquad \mathrm{(C) \ 16 } \qquad \mathrm{(D) \ 17 } \qquad \mathrm{(E) \ 18 }$ | [
"Repeatedly applying the function, and simplifying, we get \\[a,\\quad-\\frac1{a+1},\\quad-\\frac{a+1}a,\\] and then $a$ again. So $a$ must appear at every third term after $u_1$ . The only option given of the form $1+3k$ is $\\boxed{16}$"
] |
https://artofproblemsolving.com/wiki/index.php/1987_AHSME_Problems/Problem_29 | A | 15 | Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even
and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$ , the sum of the digits of $n$ is
$\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 17 \qquad \textbf{(C)}\ 19 \qquad \textbf{(D)}\ 2... | [
"If $n$ is even, then $t_{(n/2)}$ would be negative, which is not possible. Therefore, $n$ is odd. With this function, backwards thinking is the key. If $t_x < 1$ , then $x$ is odd, and $t_{(x-1)} = \\frac{1}{t_{x}}$ . Otherwise, you keep on subtracting 1 and halving x until $t_\\frac{x}{2^{n}} < 1$ .\nWe can use t... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_12A_Problems/Problem_21 | B | 1,999 | Consider the sequence of numbers: $4,7,1,8,9,7,6,\dots$ For $n>2$ , the $n$ -th term of the sequence is the units digit of the sum of the two previous terms. Let $S_n$ denote the sum of the first $n$ terms of this sequence. The smallest value of $n$ for which $S_n>10,000$ is:
$\text{(A) }1992 \qquad \text{(B) }1999 \qq... | [
"The sequence is infinite. As there are only $100$ pairs of digits, sooner or later a pair of consecutive digits will occur for the second time. As each next digit only depends on the previous two, from this point on the sequence will be periodic.\n(Additionally, as every two consecutive digits uniquely determine t... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_12B_Problems/Problem_21 | C | 810 | Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$ . No three of these parabolas have a common point. How many points in the plane are on ... | [
"Being on two parabolas means having the same distance from the common focus and both directrices. In particular, you have to be on an angle bisector of the directrices, and clearly on the same \"side\" of the directrices as the focus. So it's easy to see there are at most two solutions per pair of parabolae. Conve... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_12B_Problems/Problem_21 | null | 810 | Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$ . No three of these parabolas have a common point. How many points in the plane are on ... | [
"Through similar reasoning as above in Solution I, determine that two parabolas that have a common focus intersect zero times if there directrixes are parallel and the focus lies on the same side of both directrixes, and intersect twice otherwise. Thereby, as each parabola will intersect $30-3 = 27$ other parabolas... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_10A_Problems/Problem_15 | B | 1 | Consider the set of all fractions $\frac{x}{y}$ , where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$ , the value of the fraction is increased by $10\%$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2... | [
"You can create the equation $\\frac{x+1}{y+1}=\\frac{11x}{10y}$\nCross multiplying and combining like terms gives $xy + 11x - 10y = 0$\nThis can be factored into $(x - 10)(y + 11) = -110$\n$x$ and $y$ must be positive, so $x > 0$ and $y > 0$ , so $x - 10> -10$ and $y + 11 > 11$\nUsing the factors of 110, we can ge... |
https://artofproblemsolving.com/wiki/index.php/2009_AIME_I_Problems/Problem_11 | null | 600 | Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$ . Find the number of such distinct triangles whose area is a positive integer. | [
"Let the two points $P$ and $Q$ be defined with coordinates; $P=(x_1,y_1)$ and $Q=(x_2,y_2)$\nWe can calculate the area of the parallelogram with the determinant of the matrix of the coordinates of the two points(shoelace theorem).\n$\\det \\left(\\begin{array}{c} P \\\\ Q\\end{array}\\right)=\\det \\left(\\begin{a... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_16 | B | 21 | Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$ . The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$
$\textbf{(A)}~20\qquad\textbf{(B)}~21\qquad\textbf{(C)}~22\qquad\textbf{(D)}~23\q... | [
"First, substitute in $z=a+bi$\n\\[|1+(a+bi)+(a+bi)^2|=4\\] \\[|(1+a+a^2-b^2)+ (b+2ab)i|=4\\] \\[(1+a+a^2-b^2)^2+ (b+2ab)^2=16\\] \\[(1+a+a^2-b^2)^2+ b^2(1+4a+4a^2)=16\\]\nLet $p=b^2$ and $q=1+a+a^2$\n\\[(q-p)^2+ p(4q-3)=16\\] \\[p^2-2pq+q^2 + 4pq -3p=16\\]\nWe are trying to maximize $b=\\sqrt p$ , so we'll turn ... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_10B_Problems/Problem_10 | B | 9 | Consider the set of numbers $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}$ . The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)} 101$ | [
"The requested ratio is \\[\\dfrac{10^{10}}{10^9 + 10^8 + \\ldots + 10 + 1}.\\] Using the formula for a geometric series, we have \\[10^9 + 10^8 + \\ldots + 10 + 1 = \\dfrac{10^{10} - 1}{10 - 1} = \\dfrac{10^{10} - 1}{9},\\] which is very close to $\\dfrac{10^{10}}{9},$ so the ratio is very close to $\\boxed{9}.$",... |
https://artofproblemsolving.com/wiki/index.php/2003_AIME_I_Problems/Problem_5 | null | 505 | Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $3$ by $4$ by $5$ units. Given that the volume of this set is $\frac{m + n\pi}{p},$ where $m, n,$ and $p$ are positive integers , and $n$ and $p$ are relatively prime , find $m + n + p.$ | [
"The set can be broken into several parts: the big $3\\times 4 \\times 5$ parallelepiped, $6$ external parallelepipeds that each share a face with the large parallelepiped and have a height of $1$ , the $1/8$ spheres (one centered at each vertex of the large parallelepiped), and the $1/4$ cylinders connecting each ... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_10B_Problems/Problem_2 | E | 27 | Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$ | [
"Since a counterexample must be a value of $n$ which is not prime, $n$ must be composite, so we eliminate $\\text{A}$ and $\\text{C}$ . Now we subtract $2$ from the remaining answer choices, and we see that the only time $n-2$ is not prime is when $n = \\boxed{27}$"
] |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_12B_Problems/Problem_2 | E | 27 | Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$ | [
"Since a counterexample must be a value of $n$ which is not prime, $n$ must be composite, so we eliminate $\\text{A}$ and $\\text{C}$ . Now we subtract $2$ from the remaining answer choices, and we see that the only time $n-2$ is not prime is when $n = \\boxed{27}$"
] |
https://artofproblemsolving.com/wiki/index.php/1963_AHSME_Problems/Problem_26 | E | 4 | Consider the statements:
$\textbf{(1)}\ p\text{ }\wedge\sim q\wedge r\qquad\textbf{(2)}\ \sim p\text{ }\wedge\sim q\wedge r\qquad\textbf{(3)}\ p\text{ }\wedge\sim q\text{ }\wedge\sim r\qquad\textbf{(4)}\ \sim p\text{ }\wedge q\text{ }\wedge r$
where $p,q$ , and $r$ are propositions. How many of these imply the truth of... | [
"Statement $1$ states that $p$ is true and $q$ is false. Therefore, $p \\rightarrow q$ is false, because a premise being true and a conclusion being false is, itself, false. This means that $(p \\rightarrow q) \\rightarrow X$ , where $X$ is any logical statement (or series of logical statements) must be true - if... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_8_Problems/Problem_2 | D | 100 | Consider these two operations: \begin{align*} a \, \blacklozenge \, b &= a^2 - b^2\\ a \, \bigstar \, b &= (a - b)^2 \end{align*} What is the output of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6?$
$\textbf{(A) } {-}20 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 100 \qquad \textbf{(E) } 220$ | [
"We can substitute $5$ $3$ , and $6$ into the functions' definitions: \\begin{align*} (5 \\, \\blacklozenge \\, 3) \\, \\bigstar \\, 6 &= \\left(5^2-3^2\\right) \\, \\bigstar \\, 6 \\\\ (5 \\, \\blacklozenge \\, 3) \\, \\bigstar \\, 6 &= \\left(25-9\\right) \\, \\bigstar \\, 6 \\\\ &= 16 \\, \\bigstar \\, 6 \\\\ &=... |
https://artofproblemsolving.com/wiki/index.php/1993_AJHSME_Problems/Problem_11 | C | 70 | Consider this histogram of the scores for $81$ students taking a test:
[asy] unitsize(12); draw((0,0)--(26,0)); draw((1,1)--(25,1)); draw((3,2)--(25,2)); draw((5,3)--(23,3)); draw((5,4)--(21,4)); draw((7,5)--(21,5)); draw((9,6)--(21,6)); draw((11,7)--(19,7)); draw((11,8)--(19,8)); draw((11,9)--(19,9)); draw((11,10)--(1... | [
"Since $81$ students took the test, the median is the score of the $41^{st}$ student. The five rightmost intervals include $2+3+6+12+16=39$ students, so the $41^{st}$ one must lie in the next interval, which is $\\boxed{70}$"
] |
https://artofproblemsolving.com/wiki/index.php/1997_AHSME_Problems/Problem_27 | D | 24 | Consider those functions $f$ that satisfy $f(x+4)+f(x-4) = f(x)$ for all real $x$ . Any such function is periodic, and there is a least common positive period $p$ for all of them. Find $p$
$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 32$ | [
"Recall that $p$ is the fundamental period of function $f$ iff $p$ is the smallest positive $p$ such that $f(x) = f(x + p)$ for all $x$\nIn this case, we know that $f(x+ 4) + f(x - 4) = f(x)$ . Plugging in $x+4$ in for $x$ to get the next equation in the recursion, we also get $f(x + 8) + f(x) = f(x + 4)$ . Addin... |
https://artofproblemsolving.com/wiki/index.php/1996_AHSME_Problems/Problem_27 | D | 13 | Consider two solid spherical balls, one centered at $\left(0, 0,\frac{21}{2}\right)$ with radius $6$ , and the other centered at $(0, 0, 1)$ with radius $\frac{9}{2}$ . How many points with only integer coordinates (lattice points) are there in the intersection of the balls?
$\text{(A)}\ 7\qquad\text{(B)}\ 9\qquad\text... | [
"The two equations of the balls are\n\\[x^2 + y^2 + \\left(z - \\frac{21}{2}\\right)^2 \\le 36\\]\n\\[x^2 + y^2 + (z - 1)^2 \\le \\frac{81}{4}\\]\nNote that along the $z$ axis, the first ball goes from $10.5 \\pm 6$ , and the second ball goes from $1 \\pm 4.5$ . The only integer value that $z$ can be is $z=5$\nPlu... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_8_Problems/Problem_9 | B | 23 | Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resultin... | [
"Of the six shapes used to create the polygon, the triangle and octagon are adjacent to the others on one side, and the others are adjacent on two sides. In the triangle and octagon $3+8-2(1)=9$ sides are on the outside of the final polygon. In the other shapes $4+5+6+7-4(2) = 14$ sides are on the outside. The resu... |
https://artofproblemsolving.com/wiki/index.php/2020_AIME_II_Problems/Problem_13 | null | 60 | Convex pentagon $ABCDE$ has side lengths $AB=5$ $BC=CD=DE=6$ , and $EA=7$ . Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$ | [
"Assume the incircle touches $AB$ $BC$ $CD$ $DE$ $EA$ at $P,Q,R,S,T$ respectively. Then let $PB=x=BQ=RD=SD$ $ET=y=ES=CR=CQ$ $AP=AT=z$ . So we have $x+y=6$ $x+z=5$ and $y+z$ =7, solve it we have $x=2$ $z=3$ $y=4$ . Let the center of the incircle be $I$ , by SAS we can proof triangle $BIQ$ is congruent to triangle $D... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12A_Problems/Problem_24 | E | 84 | Convex quadrilateral $ABCD$ has $AB = 18, \angle{A} = 60^\circ,$ and $\overline{AB} \parallel \overline{CD}.$ In some order, the lengths of the four sides form an arithmetic progression, and side $\overline{AB}$ is a side of maximum length. The length of another side is $a.$ What is the sum of all possible values of $a... | [
"Let $E$ be a point on $\\overline{AB}$ such that $BCDE$ is a parallelogram. Suppose that $BC=ED=b, CD=BE=c,$ and $DA=d,$ so $AE=18-c,$ as shown below. We apply the Law of Cosines to $\\triangle ADE:$ \\begin{align*} AD^2 + AE^2 - 2\\cdot AD\\cdot AE\\cdot\\cos 60^\\circ &= DE^2 \\\\ d^2 + (18-c)^2 - d(18-c) &= b^... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_10A_Problems/Problem_23 | A | 6 | Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$ . Diagonals $AC$ and $BD$ intersect at $E$ $AC = 14$ , and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$
$\textbf{(A)}\ \frac {9}{2}\qquad \textbf{(B)}\ \frac {50}{11}\qquad \textbf{(C)}\ \frac {21}{4}\qquad \textbf{(D)}\ \frac {17}{3}\qquad ... | [
"The easiest way for the areas of the triangles to be equal would be if they were congruent [1] . A way for that to work would be if $ABCD$ were simply an isosceles trapezoid! Since $AC = 14$ and $AE:EC = 3:4$ (look at the side lengths and you'll know why!), $\\boxed{6}$"
] |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_10A_Problems/Problem_23 | null | 6 | Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$ . Diagonals $AC$ and $BD$ intersect at $E$ $AC = 14$ , and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$
$\textbf{(A)}\ \frac {9}{2}\qquad \textbf{(B)}\ \frac {50}{11}\qquad \textbf{(C)}\ \frac {21}{4}\qquad \textbf{(D)}\ \frac {17}{3}\qquad ... | [
"Using the fact that $[AED] = [BEC]$ and the fact that $\\triangle AEB \\sim \\triangle EDC$ (which should be trivial given the two equal triangles) we have that\n\\[\\frac{AE}{DC} = \\frac{BE}{EC} = \\frac{9}{12}\\]\nWe know that $DC=EC,$ so we have\n\\[\\frac{AE}{EC} = \\frac{BE}{EC} = \\frac{3}{4}\\]\nThus\n\\[\... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12A_Problems/Problem_20 | A | 6 | Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$ . Diagonals $AC$ and $BD$ intersect at $E$ $AC = 14$ , and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$
$\textbf{(A)}\ \frac {9}{2}\qquad \textbf{(B)}\ \frac {50}{11}\qquad \textbf{(C)}\ \frac {21}{4}\qquad \textbf{(D)}\ \frac {17}{3}\qquad ... | [
"The easiest way for the areas of the triangles to be equal would be if they were congruent [1] . A way for that to work would be if $ABCD$ were simply an isosceles trapezoid! Since $AC = 14$ and $AE:EC = 3:4$ (look at the side lengths and you'll know why!), $\\boxed{6}$"
] |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12A_Problems/Problem_20 | null | 6 | Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$ . Diagonals $AC$ and $BD$ intersect at $E$ $AC = 14$ , and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$
$\textbf{(A)}\ \frac {9}{2}\qquad \textbf{(B)}\ \frac {50}{11}\qquad \textbf{(C)}\ \frac {21}{4}\qquad \textbf{(D)}\ \frac {17}{3}\qquad ... | [
"Using the fact that $[AED] = [BEC]$ and the fact that $\\triangle AEB \\sim \\triangle EDC$ (which should be trivial given the two equal triangles) we have that\n\\[\\frac{AE}{DC} = \\frac{BE}{EC} = \\frac{9}{12}\\]\nWe know that $DC=EC,$ so we have\n\\[\\frac{AE}{EC} = \\frac{BE}{EC} = \\frac{3}{4}\\]\nThus\n\\[\... |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_12B_Problems/Problem_9 | B | 36 | Convex quadrilateral $ABCD$ has $AB=3$ $BC=4$ $CD=13$ $AD=12$ , and $\angle ABC=90^{\circ}$ , as shown. What is the area of the quadrilateral?
[asy] pair A=(0,0), B=(-3,0), C=(-3,-4), D=(48/5,-36/5); draw(A--B--C--D--A); label("$A$",A,N); label("$B$",B,NW); label("$C$",C,SW); label("$D$",D,E); draw(rightanglemark(A,B,... | [
"Note that by the pythagorean theorem, $AC=5$ . Also note that $\\angle CAD$ is a right angle because $\\triangle CAD$ is a right triangle. The area of the quadrilateral is the sum of the areas of $\\triangle ABC$ and $\\triangle CAD$ which is equal to \\[\\frac{3\\times4}{2} + \\frac{5\\times12}{2} = 6 + 30 = \\... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_21 | D | 13 | Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if neces... | [
"Let $n$ be the number of steps. We have\n\\[\\left\\lceil \\frac{n}{2} \\right\\rceil - 19 = \\left\\lceil \\frac{n}{5} \\right\\rceil\\]\nWe will proceed to solve this equation via casework.\nCase $1$ $\\left\\lceil \\frac{n}{2} \\right\\rceil = \\frac{n}{2}$\nOur equation becomes $\\frac{n}{2} - 19 = \\frac{n}{5... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_21 | null | 13 | Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if neces... | [
"We're looking for natural numbers $x$ such that $\\left \\lceil{\\frac{x}{5}}\\right \\rceil + 19 = \\left \\lceil{\\frac{x}{2}}\\right \\rceil$\nLet's call $x = 10a + b$ . We now have $2a + \\left \\lceil{\\frac{b}{5}}\\right \\rceil + 19 = 5a + \\left \\lceil{\\frac{b}{2}}\\right \\rceil$ , or\n$19 - 3a = \\le... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_12B_Problems/Problem_21 | D | 13 | Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessar... | [
"We can translate this wordy problem into this simple equation:\n\\[\\left\\lceil \\frac{s}{2} \\right\\rceil - 19 = \\left\\lceil \\frac{s}{5} \\right\\rceil\\]\nWe will proceed to solve this equation via casework.\nCase 1: $\\left\\lceil \\frac{s}{2} \\right\\rceil = \\frac{s}{2}$\nOur equation becomes $\\frac{s}... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10A_Problems/Problem_7 | C | 3 | Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run?
$\ma... | [
"Crystal runs north one mile, then her next two moves can be broken up into four individual moves: for her northeast section, it forms a $45-45-90$ triangle whose legs are each $\\frac{\\sqrt{2}}{2}$ . For her southeast section, it is also a $45-45-90$ triangle whose legs are each $\\frac{\\sqrt{2}}{2}$ . Notice th... |
https://artofproblemsolving.com/wiki/index.php/2012_AIME_I_Problems/Problem_8 | null | 89 | Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\tfrac{p}{q}... | [
"Define a coordinate system with $D$ at the origin and $C,$ $A,$ and $H$ on the $x$ $y$ , and $z$ axes respectively. Then $D=(0,0,0),$ $M=(.5,1,0),$ and $N=(1,0,.5).$ It follows that the plane going through $D,$ $M,$ and $N$ has equation $2x-y-4z=0.$ Let $Q = (1,1,.25)$ be the intersection of this plane and edge $B... |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_10B_Problems/Problem_14 | D | 37 | Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a $3$ -digit number with $a\ge1$ and $a+b+c\le7$ . At the end of the trip, the odometer showed $cba$ miles. What is $a^2+b^2+c^2$
$\textb... | [
"We can set up an algebraic equation for this problem.\nFrom what's given, we have that $100c+10b+a=55x+100a+10b+c$\nThis simplifies to be $0=55x+99a-99c\\implies -55x=99a-99c$\nFactoring, we get that $-55x=99(a-c)\\implies x=-\\frac{9(a-c)}{5}$\nHence, notice that we want $a-c=-5$ so that $x=9$\nThe only pair that... |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_12B_Problems/Problem_10 | D | 37 | Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a \geq{1}$ and $a+b+c \leq{7}$ . At the end of the trip, the odometer showed $cba$ miles. What is $a^2+b^2+c^2?$
$\t... | [
"We know that the number of miles she drove is divisible by $5$ , so $a$ and $c$ must either be the equal or differ by $5$ . We can quickly conclude that the former is impossible, so $a$ and $c$ must be $5$ apart. Because we know that $c > a$ and $a + c \\le 7$ and $a \\ge 1$ , we find that the only possible value... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_8_Problems/Problem_1 | A | 1 | Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the greatest number of additional cars she must buy in order to be able to arrange all her cars this way?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 4 \qquad \textb... | [
"The least multiple of 6 greater than 23 is 24. So she will need to add $24-23=\\boxed{1}$ more model car.\n~avamarora",
"6 x 4 = 24, which is 1 more than 23. So, the answer is $\\boxed{1}$"
] |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10A_Problems/Problem_10 | E | 18 | Daniel finds a rectangular index card and measures its diagonal to be $8$ centimeters.
Daniel then cuts out equal squares of side $1$ cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area o... | [
" Label the bottom left corner of the larger rectangle (without the square cut out) as $A$ and the top right as $D$ $w$ is the width of the rectangle and $\\ell$ is the length. Now we have vertices $E, F, G, H$ as vertices of the irregular octagon created by cutting out the squares. Let $I, J$ be the two closest ve... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10A_Problems/Problem_17 | B | 54 | Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$ -day period will exactly two friends visit her?
$\textbf{... | [
"The $365$ -day time period can be split up into $6$ $60$ -day time periods, because after $60$ days, all three of them visit again (Least common multiple of $3$ $4$ , and $5$ ).\nYou can find how many times each pair of visitors can meet by finding the LCM of their visiting days and dividing that number by 60.\nRe... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_II_Problems/Problem_4 | null | 52 | Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $... | [
"There are $12 \\cdot 11 = 132$ possible situations ( $12$ choices for the initially assigned gate, and $11$ choices for which gate Dave's flight was changed to). We are to count the situations in which the two gates are at most $400$ feet apart.\nIf we number the gates $1$ through $12$ , then gates $1$ and $12$ ha... |
https://artofproblemsolving.com/wiki/index.php/2009_AIME_II_Problems/Problem_8 | null | 41 | Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $m$ and $n$ be relatively prime positive integers such that $\dfrac mn$ is the probability that the number of times Dave rolls his die is equal to or within... | [
"There are many almost equivalent approaches that lead to summing a geometric series. For example, we can compute the probability of the opposite event. Let $p$ be the probability that Dave will make at least two more throws than Linda. Obviously, $p$ is then also the probability that Linda will make at least two m... |
https://artofproblemsolving.com/wiki/index.php/2018_AIME_I_Problems/Problem_15 | null | 59 | David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\text{ }B,\text{ }C$ , which can each be inscribed in a circle with radius $1$ . Let $\varphi_A$ denote the measure of the acute angle made by the diagonals of quadrilateral $A$ , and define $\varp... | [
"Suppose our four sides lengths cut out arc lengths of $2a$ $2b$ $2c$ , and $2d$ , where $a+b+c+d=180^\\circ$ . Then, we only have to consider which arc is opposite $2a$ . These are our three cases, so \\[\\varphi_A=a+c\\] \\[\\varphi_B=a+b\\] \\[\\varphi_C=a+d\\] Our first case involves quadrilateral $ABCD$ with $... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10A_Problems/Problem_19 | C | 5 | Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that \[\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{17}=\frac{h}{L_{17}}\] What is the remainder when $h$ is divided by $17$
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \... | [
"Notice that $L_{17}$ contains the highest power of every prime below $17$ since higher primes cannot divide $L_{17}$ . Thus, $L_{17}=16\\cdot 9 \\cdot 5 \\cdot 7 \\cdot 11 \\cdot 13 \\cdot 17$\nWhen writing the sum under a common fraction, we multiply the denominators by $L_{17}$ divided by each denominator. Howev... |
https://artofproblemsolving.com/wiki/index.php/2024_AIME_I_Problems/Problem_12 | null | 385 | Define $f(x)=|| x|-\tfrac{1}{2}|$ and $g(x)=|| x|-\tfrac{1}{4}|$ . Find the number of intersections of the graphs of \[y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).\] | [
"If we graph $4g(f(x))$ , we see it forms a sawtooth graph that oscillates between $0$ and $1$ (for values of $x$ between $-1$ and $1$ , which is true because the arguments are between $-1$ and $1$ ). Thus by precariously drawing the graph of the two functions in the square bounded by $(0,0)$ $(0,1)$ $(1,1)$ , and ... |
https://artofproblemsolving.com/wiki/index.php/2009_AIME_II_Problems/Problem_7 | null | 401 | Define $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. When $\sum_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$ | [
"First, note that $(2n)!! = 2^n \\cdot n!$ , and that $(2n)!! \\cdot (2n-1)!! = (2n)!$\nWe can now take the fraction $\\dfrac{(2i-1)!!}{(2i)!!}$ and multiply both the numerator and the denominator by $(2i)!!$ . We get that this fraction is equal to $\\dfrac{(2i)!}{(2i)!!^2} = \\dfrac{(2i)!}{2^{2i}(i!)^2}$\nNow we c... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10A_Problems/Problem_6 | C | 0 | Define $x \heartsuit y$ to be $|x-y|$ for all real numbers $x$ and $y$ . Which of the following statements is not true?
$\mathrm{(A) \ } x \heartsuit y = y \heartsuit x$ for all $x$ and $y$
$\mathrm{(B) \ } 2(x \heartsuit y) = (2x) \heartsuit (2y)$ for all $x$ and $y$
$\mathrm{(C) \ } x \heartsuit 0 = x$ for all $x$
$\... | [
"We start by looking at the answers. Examining statement C, we notice:\n$x \\heartsuit 0 = |x-0| = |x|$\n$|x| \\neq x$ when $x<0$ , but statement C says that it does for all $x$\nTherefore the statement that is not true is $\\boxed{0}$"
] |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_12A_Problems/Problem_6 | C | 0 | Define $x \heartsuit y$ to be $|x-y|$ for all real numbers $x$ and $y$ . Which of the following statements is not true?
$\mathrm{(A) \ } x \heartsuit y = y \heartsuit x$ for all $x$ and $y$
$\mathrm{(B) \ } 2(x \heartsuit y) = (2x) \heartsuit (2y)$ for all $x$ and $y$
$\mathrm{(C) \ } x \heartsuit 0 = x$ for all $x$
$\... | [
"We start by looking at the answers. Examining statement C, we notice:\n$x \\heartsuit 0 = |x-0| = |x|$\n$|x| \\neq x$ when $x<0$ , but statement C says that it does for all $x$\nTherefore the statement that is not true is $\\boxed{0}$"
] |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_1 | A | 2 | Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\]
$\textbf{(A)}\ {-}2 \qquad \textbf{(B)}\ {-}1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$ | [
"We have \\begin{align*} (1\\diamond(2\\diamond3))-((1\\diamond2)\\diamond3) &= |1-|2-3|| - ||1-2|-3| \\\\ &= |1-1| - |1-3| \\\\ &= 0-2 \\\\ &= \\boxed{2} ~MRENTHUSIASM",
"Observe that the $\\diamond$ function is simply the positive difference between two numbers. Thus, we evaluate: the difference between $2$ and... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12B_Problems/Problem_1 | A | 2 | Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\]
$\textbf{(A)}\ {-}2 \qquad \textbf{(B)}\ {-}1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$ | [
"We have \\begin{align*} (1\\diamond(2\\diamond3))-((1\\diamond2)\\diamond3) &= |1-|2-3|| - ||1-2|-3| \\\\ &= |1-1| - |1-3| \\\\ &= 0-2 \\\\ &= \\boxed{2} ~MRENTHUSIASM",
"Observe that the $\\diamond$ function is simply the positive difference between two numbers. Thus, we evaluate: the difference between $2$ and... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A_Problems/Problem_17 | E | 5,100 | Define \[P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).\] How many integers $n$ are there such that $P(n)\leq 0$
$\textbf{(A) } 4900 \qquad \textbf{(B) } 4950\qquad \textbf{(C) } 5000\qquad \textbf{(D) } 5050 \qquad \textbf{(E) } 5100$ | [
"We perform casework on $P(n)\\leq0:$\nTogether, the answer is $100+5000=\\boxed{5100}.$",
"Notice that $P(x)$ is nonpositive when $x$ is between $100^2$ and $99^2, 98^2$ and $97^2, \\cdots$ $2^2$ and $1^2$ (inclusive), because there are an odd number of negatives, which means that the number of values equals \\[... |
https://artofproblemsolving.com/wiki/index.php/2003_AIME_II_Problems/Problem_3 | null | 192 | Define a $\text{good~word}$ as a sequence of letters that consists only of the letters $A$ $B$ , and $C$ - some of these letters may not appear in the sequence - and in which $A$ is never immediately followed by $B$ $B$ is never immediately followed by $C$ , and $C$ is never immediately followed by $A$ . How many seven... | [
"There are three letters to make the first letter in the sequence. However, after the first letter (whatever it is), only two letters can follow it, since one of the letters is restricted. Therefore, the number of seven-letter good words is $3*2^6=192$\nTherefore, there are $\\boxed{192}$ seven-letter good words.",... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_II_Problems/Problem_11 | null | 68 | Define a T-grid to be a $3\times3$ matrix which satisfies the following two properties:
Find the number of distinct T-grids | [
"The T-grid can be considered as a tic-tac-toe board: five $1$ 's (or X's) and four $0$ 's (or O's).\nThere are only $\\dbinom{9}{5} = 126$ ways to fill the board with five $1$ 's and four $0$ 's. Now we just need to subtract the number of bad grids. Bad grids are ones with more than one person winning, or where so... |
https://artofproblemsolving.com/wiki/index.php/1998_AIME_Problems/Problem_15 | null | 761 | Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $... | [
"We can draw a comparison between the domino a set of 40 points (labeled 1 through 40) in which every point is connected with every other point. The connections represent the dominoes.\nYou need to have all even number of segments coming from each point except 0 or 2 which have an odd number of segments coming from... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12A_Problems/Problem_7 | B | 2,018 | Define a function on the positive integers recursively by $f(1) = 2$ $f(n) = f(n-1) + 1$ if $n$ is even, and $f(n) = f(n-2) + 2$ if $n$ is odd and greater than $1$ . What is $f(2017)$
$\textbf{(A)}\ 2017 \qquad\textbf{(B)}\ 2018 \qquad\textbf{(C)}\ 4034 \qquad\textbf{(D)}\ 4035 \qquad\textbf{(E)}\ 4036$ | [
"This is a recursive function, which means the function refers back to itself to calculate subsequent terms. To solve this, we must identify the base case, $f(1)=2$ . We also know that when $n$ is odd, $f(n)=f(n-2)+2$ . Thus we know that $f(2017)=f(2015)+2$ . Thus we know that n will always be odd in the recursion ... |
https://artofproblemsolving.com/wiki/index.php/1992_AIME_Problems/Problem_15 | null | 396 | Define a positive integer $n^{}_{}$ to be a factorial tail if there is some positive integer $m^{}_{}$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails? | [
"Let the number of zeros at the end of $m!$ be $f(m)$ . We have $f(m) = \\left\\lfloor \\frac{m}{5} \\right\\rfloor + \\left\\lfloor \\frac{m}{25} \\right\\rfloor + \\left\\lfloor \\frac{m}{125} \\right\\rfloor + \\left\\lfloor \\frac{m}{625} \\right\\rfloor + \\left\\lfloor \\frac{m}{3125} \\right\\rfloor + \\cdot... |
https://artofproblemsolving.com/wiki/index.php/2004_AIME_I_Problems/Problem_8 | null | 199 | Define a regular $n$ -pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that
There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars... | [
"We use the Principle of Inclusion-Exclusion (PIE).\nIf we join the adjacent vertices of the regular $n$ -star, we get a regular $n$ -gon. We number the vertices of this $n$ -gon in a counterclockwise direction: $0, 1, 2, 3, \\ldots, n-1.$\nA regular $n$ -star will be formed if we choose a vertex number $m$ , wher... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10A_Problems/Problem_13 | D | 9 | Define a sequence recursively by $F_{0}=0,~F_{1}=1,$ and $F_{n}=$ the remainder when $F_{n-1}+F_{n-2}$ is divided by $3,$ for all $n\geq 2.$ Thus the sequence starts $0,1,1,2,0,2,\ldots$ What is $F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}?$
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf... | [
"A pattern starts to emerge as the function is continued. The repeating pattern is $0,1,1,2,0,2,2,1\\ldots$ The problem asks for the sum of eight consecutive terms in the sequence. Because there are eight numbers in the repeating pattern, we just need to find the sum of the numbers in the sequence, which is $\\boxe... |
https://artofproblemsolving.com/wiki/index.php/2020_AIME_II_Problems/Problem_8 | null | 101 | Define a sequence recursively by $f_1(x)=|x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n>1$ . Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500,000$ | [
"First it will be shown by induction that the zeros of $f_n$ are the integers $a, {a+2,} {a+4,} \\dots, {a + n(n-1)}$ , where $a = n - \\frac{n(n-1)}2.$\nThis is certainly true for $n=1$ . Suppose that it is true for $n = m-1 \\ge 1$ , and note that the zeros of $f_m$ are the solutions of $|x - m| = k$ , where $k$ ... |
https://artofproblemsolving.com/wiki/index.php/2020_AIME_II_Problems/Problem_6 | null | 626 | Define a sequence recursively by $t_1 = 20$ $t_2 = 21$ , and \[t_n = \frac{5t_{n-1}+1}{25t_{n-2}}\] for all $n \ge 3$ . Then $t_{2020}$ can be expressed as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ | [
"Let $t_n=\\frac{s_n}{5}$ . Then, we have $s_n=\\frac{s_{n-1}+1}{s_{n-2}}$ where $s_1 = 100$ and $s_2 = 105$ . By substitution, we find $s_3 = \\frac{53}{50}$ $s_4=\\frac{103}{105\\cdot50}$ $s_5=\\frac{101}{105}$ $s_6=100$ , and $s_7=105$ . So $s_n$ has a period of $5$ . Thus $s_{2020}=s_5=\\frac{101}{105}$ . So, $... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_10B_Problems/Problem_24 | C | 81,242 | Define a sequence recursively by $x_0=5$ and \[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\] for all nonnegative integers $n.$ Let $m$ be the least positive integer such that
\[x_m\leq 4+\frac{1}{2^{20}}.\]
In which of the following intervals does $m$ lie?
$\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [8... | [
"We first prove that $x_n > 4$ for all $n \\ge 0$ , by induction. Observe that \\[x_{n+1} - 4 = \\frac{x_n^2 + 5x_n + 4 - 4(x_n+6)}{x_n+6} = \\frac{(x_n - 4)(x_n+5)}{x_n+6}.\\] so (since $x_n$ is clearly positive for all $n$ , from the initial definition), $x_{n+1} > 4$ if and only if $x_{n} > 4$\nWe similarly prov... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_12B_Problems/Problem_22 | C | 81,242 | Define a sequence recursively by $x_0=5$ and \[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\] for all nonnegative integers $n.$ Let $m$ be the least positive integer such that
\[x_m\leq 4+\frac{1}{2^{20}}.\]
In which of the following intervals does $m$ lie?
$\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [8... | [
"We first prove that $x_n > 4$ for all $n \\ge 0$ , by induction. Observe that \\[x_{n+1} - 4 = \\frac{x_n^2 + 5x_n + 4 - 4(x_n+6)}{x_n+6} = \\frac{(x_n - 4)(x_n+5)}{x_n+6}.\\] so (since $x_n$ is clearly positive for all $n$ , from the initial definition), $x_{n+1} > 4$ if and only if $x_{n} > 4$\nWe similarly prov... |
https://artofproblemsolving.com/wiki/index.php/2011_AIME_II_Problems/Problem_6 | null | 80 | Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a<b<c<d \le 10$ , and $a+d>b+c$ . How many interesting ordered quadruples are there? | [
"We first start out when the value of $a=1$\nDoing casework, we discover that $d=5,6,7,8,9,10$ . We quickly find a pattern.\nNow, doing this for the rest of the values of $a$ and $d$ , we see that the answer is simply:\n$(1)+(2)+(1+3)+(2+4)+(1+3+5)+(2+4+6)+(1)+(2)+(1+3)+(2+4)$ $+(1+3+5)+(1)+(2)+(1+3)+(2+4)+(1)+(2)+... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_12A_Problems/Problem_23 | D | 11 | Define binary operations $\diamondsuit$ and $\heartsuit$ by \[a \, \diamondsuit \, b = a^{\log_{7}(b)} \qquad \text{and} \qquad a \, \heartsuit \, b = a^{\frac{1}{\log_{7}(b)}}\] for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3\, \heartsu... | [
"By definition, the recursion becomes $a_n = \\left(n^{\\frac1{\\log_7(n-1)}}\\right)^{\\log_7(a_{n-1})}=n^{\\frac{\\log_7(a_{n-1})}{\\log_7(n-1)}}$ . By the change of base formula, this reduces to $a_n = n^{\\log_{n-1}(a_{n-1})}$ . Thus, we have $\\log_n(a_n) = \\log_{n-1}(a_{n-1})$ . Thus, for each positive in... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_10B_Problems/Problem_2 | E | 16 | Define the operation $\star$ by $a \star b = (a+b)b.$ What is $(3 \star 5) - (5 \star 3)?$
$\textbf{(A) } -16 \qquad\textbf{(B) } -8 \qquad\textbf{(C) } 0 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 16$ | [
"Substitute and simplify. \\[(3+5)5 - (5+3)3 = (3+5)2 = 8\\cdot2 = \\boxed{16}\\]",
"Note that $(a \\star b) - (b \\star a) = (a+b)b - (b+a)a= (a+b)(b-a)= b^2 - a^2$ . We can substitute $a=3$ and $b=5$ to get $5^2 - 3^2 = \\boxed{16}$"
] |
https://artofproblemsolving.com/wiki/index.php/2015_AIME_II_Problems/Problem_13 | null | 628 | Define the sequence $a_1, a_2, a_3, \ldots$ by $a_n = \sum\limits_{k=1}^n \sin{k}$ , where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$ | [
"If $n = 1$ $a_n = \\sin(1) > 0$ . Then if $n$ satisfies $a_n < 0$ $n \\ge 2$ , and \\[a_n = \\sum_{k=1}^n \\sin(k) = \\cfrac{1}{\\sin{1}} \\sum_{k=1}^n\\sin(1)\\sin(k) = \\cfrac{1}{2\\sin{1}} \\sum_{k=1}^n\\cos(k - 1) - \\cos(k + 1) = \\cfrac{1}{2\\sin(1)} [\\cos(0) + \\cos(1) - \\cos(n) - \\cos(n + 1)].\\] Since... |
https://artofproblemsolving.com/wiki/index.php/1986_AIME_Problems/Problem_4 | null | 181 | Determine $3x_4+2x_5$ if $x_1$ $x_2$ $x_3$ $x_4$ , and $x_5$ satisfy the system of equations below. | [
"Adding all five equations gives us $6(x_1 + x_2 + x_3 + x_4 + x_5) = 6(1 + 2 + 4 + 8 + 16)$ so $x_1 + x_2 + x_3 + x_4 + x_5 = 31$ . Subtracting this from the fourth given equation gives $x_4 = 17$ and subtracting it from the fifth given equation gives $x_5 = 65$ , so our answer is $3\\cdot17 + 2\\cdot65 = \\boxed... |
https://artofproblemsolving.com/wiki/index.php/1953_AHSME_Problems/Problem_36 | C | 36 | Determine $m$ so that $4x^2-6x+m$ is divisible by $x-3$ . The obtained value, $m$ , is an exact divisor of:
$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 64$ | [
"Since the given expression is a quadratic, the factored form would be $(x-3)(4x+y)$ , where $y$ is a value such that $-12x+yx=-6x$ and $-3(y)=m$ . The only number that fits the first equation is $y=6$ , so $m=-18$ . The only choice that is a multiple of 18 is $\\boxed{36}$"
] |
https://artofproblemsolving.com/wiki/index.php/1984_AIME_Problems/Problem_15 | null | 36 | Determine $x^2+y^2+z^2+w^2$ if | [
"Rewrite the system of equations as \\[\\frac{x^{2}}{t-1}+\\frac{y^{2}}{t-3^{2}}+\\frac{z^{2}}{t-5^{2}}+\\frac{w^{2}}{t-7^{2}}=1.\\] This equation is satisfied when $t \\in \\{4, 16, 36, 64\\}$ . After clearing fractions, for each of the values $t=4,16,36,64$ , we have the equation \\[x^2P_1(t)+y^2P_3(t)+z^2P_5(t)+... |
https://artofproblemsolving.com/wiki/index.php/1973_USAMO_Problems/Problem_4 | null | 111 | Determine all the roots real or complex , of the system of simultaneous equations | [
"Let $P(t)=t^3-at^2+bt-c$ have roots x, y, and z. Then \\[0=P(x)+P(y)+P(z)=3-3a+3b-3c\\] using our system of equations, so $P(1)=0$ . Thus, at least one of x, y, and z is equal to 1; without loss of generality, let $x=1$ . Then we can use the system of equations to find that $y=z=1$ as well, and so $\\boxed{1,1,1}$... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_8_Problems/Problem_11 | B | 7 | Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132.$
$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad \textbf{(E) }12$ | [
"We can write the two digit number in the form of $10a+b$ ; reverse of $10a+b$ is $10b+a$ . The sum of those numbers is: \\[(10a+b)+(10b+a)=132\\] \\[11a+11b=132\\] \\[a+b=12\\] We can use brute force to find order pairs $(a,b)$ such that $a+b=12$ . Since $a$ and $b$ are both digits, both $a$ and $b$ have to be int... |
https://artofproblemsolving.com/wiki/index.php/1977_AHSME_Problems/Problem_25 | null | 250 | Determine the largest positive integer $n$ such that $1005!$ is divisible by $10^n$
$\textbf{(A) }102\qquad \textbf{(B) }112\qquad \textbf{(C) }249\qquad \textbf{(D) }502\qquad \textbf{(E) }\text{none of the above}\qquad$ | [
"We first observe that since there will be more 2s than 5s in $1005!$ , we are looking for the largest $n$ such that $5^n$ divides $1005!$ . We will use the fact that:\n\\[n = \\left \\lfloor {\\frac{1005}{5^1}}\\right \\rfloor + \\left \\lfloor {\\frac{1005}{5^2}}\\right \\rfloor + \\left \\lfloor {\\frac{1005}{5^... |
https://artofproblemsolving.com/wiki/index.php/2005_AIME_II_Problems/Problem_5 | null | 54 | Determine the number of ordered pairs $(a,b)$ of integers such that $\log_a b + 6\log_b a=5, 2 \leq a \leq 2005,$ and $2 \leq b \leq 2005.$ | [
"The equation can be rewritten as $\\frac{\\log b}{\\log a} + 6 \\frac{\\log a}{\\log b} = \\frac{(\\log b)^2+6(\\log a)^2}{\\log a \\log b}=5$ Multiplying through by $\\log a \\log b$ and factoring yields $(\\log b - 3\\log a)(\\log b - 2\\log a)=0$ . Therefore, $\\log b=3\\log a$ or $\\log b=2\\log a$ , so eithe... |
https://artofproblemsolving.com/wiki/index.php/1984_AIME_Problems/Problem_5 | null | 512 | Determine the value of $ab$ if $\log_8a+\log_4b^2=5$ and $\log_8b+\log_4a^2=7$ | [
"Use the change of base formula to see that $\\frac{\\log a}{\\log 8} + \\frac{2 \\log b}{\\log 4} = 5$ ; combine denominators to find that $\\frac{\\log ab^3}{3\\log 2} = 5$ . Doing the same thing with the second equation yields that $\\frac{\\log a^3b}{3\\log 2} = 7$ . This means that $\\log ab^3 = 15\\log 2 \\Lo... |
https://artofproblemsolving.com/wiki/index.php/1985_AHSME_Problems/Problem_17 | B | 4.2 | Diagonal $DB$ of rectangle $ABCD$ is divided into three segments of length $1$ by parallel lines $L$ and $L'$ that pass through $A$ and $C$ and are perpendicular to $DB$ . The area of $ABCD$ , rounded to the one decimal place, is
[asy] defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=or... | [
"Let $E$ be the point of intersection of $L$ and $\\overline{BD}$ . Then, because $AE$ is the altitude to the hypotenuse of right triangle $ABD$ , triangles $ADE$ and $BAE$ are similar, giving \\[\\frac{AE}{BE} = \\frac{ED}{EA},\\] and so \\begin{align*}AE &= \\sqrt{BE \\cdot ED} \\\\ &= \\sqrt{(1+1)(1)} \\\\ &= \\... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_10B_Problems/Problem_12 | A | 3 | Distinct points $A$ $B$ $C$ , and $D$ lie on a line, with $AB=BC=CD=1$ . Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$ . A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?
$\text{(A) } 3 \qquad \text{(... | [
"Consider the classical formula for triangle area: $\\frac 12 \\cdot b \\cdot h$ . \nEach of the triangles that we can make has exactly one side lying on one of the two parallel lines. If we pick this side to be the base, the height will always be the same - it will be the distance between the two lines.\nHence eac... |
https://artofproblemsolving.com/wiki/index.php/1983_AHSME_Problems/Problem_30 | C | 20 | Distinct points $A$ and $B$ are on a semicircle with diameter $MN$ and center $C$ .
The point $P$ is on $CN$ and $\angle CAP = \angle CBP = 10^{\circ}$ . If $\stackrel{\frown}{MA} = 40^{\circ}$ , then $\stackrel{\frown}{BN}$ equals
[asy] import geometry; import graph; unitsize(2 cm); pair A, B, C, M, N, P; M = (-1,... | [
"Since $\\angle CAP = \\angle CBP = 10^\\circ$ , quadrilateral $ABPC$ is cyclic (as shown below) by the converse of the theorem \"angles inscribed in the same arc are equal\".\n\nSince $\\angle ACM = 40^\\circ$ $\\angle ACP = 140^\\circ$ , so, using the fact that opposite angles in a cyclic quadrilateral sum to $18... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10A_Problems/Problem_17 | D | 7 | Distinct points $P$ $Q$ $R$ $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS}$
$\textbf{(A) } 3 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 3\sqrt{5} \qquad \textbf{(D) } 7 \qquad \text... | [
"Because $P$ $Q$ $R$ , and $S$ are lattice points, there are only a few coordinates that actually satisfy the equation. The coordinates are $(\\pm 3,\\pm 4), (\\pm 4, \\pm 3), (0,\\pm 5),$ and $(\\pm 5,0).$ We want to maximize $PQ$ and minimize $RS.$ They also have to be non perfect squares, because they are both i... |
https://artofproblemsolving.com/wiki/index.php/1998_AJHSME_Problems/Problem_6 | B | 6 | Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is
[asy] for(int a=0; a<4; ++a) { for(int b=0; b<4; ++b) { dot((a,b)); } } draw((0,0)--(0,2)--(1,2)--(2,3)--(2,2)--(3,2)--(3,0)--(2,0)--(2,1)--(1,0)--cycle); [/asy]
$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad... | [
"We could count the area contributed by each square on the $3 \\times 3$ grid:\nTop-left: $0$\nTop: Triangle with area $\\frac{1}{2}$\nTop-right: $0$\nLeft: Square with area $1$\nCenter: Square with area $1$\nRight: Square with area $1$\nBottom-left: Square with area $1$\nBottom: Triangle with area $\\frac{1}{2}$\n... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_10A_Problems/Problem_5 | D | 4 | Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. ... | [
"Dave and Doug paid $8+2=10$ dollars in total. Doug paid for three slices of plain pizza, which cost $\\frac{3}{8}\\cdot 8=3$ . Dave paid $10-3=7$ dollars. Dave paid $7-3=\\boxed{4}$ more dollars than Doug."
] |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_12A_Problems/Problem_5 | D | 4 | Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. ... | [
"Dave and Doug paid $8+2=10$ dollars in total. Doug paid for three slices of plain pizza, which cost $\\frac{3}{8}\\cdot 8=3$ . Dave paid $10-3=7$ dollars. Dave paid $7-3=\\boxed{4}$ more dollars than Doug."
] |
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