Datasets:

mlfoundations-dev
/

amc_aime_links

source stringclasses 1 value	problem stringlengths 162 2k	solution stringlengths 519 3.16k	messages listlengths 2 2
amc_aime	Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$. [AIME 2002 II Problem 4.gif](https://artofproblemsolving.com/wiki/index.php/File:AIME_2002_II_Problem_4.gif) If ...	1. Identify the number of blocks in the path: For $n > 1$, the path of blocks has $6(n-1)$ blocks in total. This is because there are six sides to the hexagon, and each side (except the first in each direction) contributes an additional block to the length of the previous side. 2. **Calculate the total number of b...	[ { "content": "Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.\n[AIME 2002 II Problem 4.gif](https://artofproblemsolving.com/wiki/index.php/File:AIME_200...
amc_aime	The graph of the line $y=mx+b$ is shown. Which of the following is true? [2004 AMC 12A Problem 5.png](https://artofproblemsolving.com/wiki/index.php/File:2004_AMC_12A_Problem_5.png) $\mathrm {(A)} mb<-1 \qquad \mathrm {(B)} -1<mb<0 \qquad \mathrm {(C)} mb=0 \qquad \mathrm {(D)}$ $0<mb<1 \qquad \mathrm {(E)} mb>1$	1. Identify the slope and y-intercept from the graph: From the graph, we observe that the slope $m$ of the line is negative and appears to be $-\frac{1}{2}$. The y-intercept $b$ is positive and appears to be $\frac{4}{5}$. 2. Calculate the product $mb$: \[ m = -\frac{1}{2}, \quad b = \frac{4}{5} \...	[ { "content": "The graph of the line $y=mx+b$ is shown. Which of the following is true?\n[2004 AMC 12A Problem 5.png](https://artofproblemsolving.com/wiki/index.php/File:2004_AMC_12A_Problem_5.png)\n$\\mathrm {(A)} mb<-1 \\qquad \\mathrm {(B)} -1<mb<0 \\qquad \\mathrm {(C)} mb=0 \\qquad \\mathrm {(D)}$ $0<mb<1 \...
amc_aime	In [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$ with [right angle](https://artofproblemsolving.com/wiki/index.php/Right_angle) $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. [Points](https://artofproblemsolving.com/wiki/index.php/Point) $...	1. Identify the triangle and its properties: Triangle $ABC$ is a right triangle with $CA = 30$, $CB = 16$, and right angle at $C$. Using the Pythagorean theorem, we find the hypotenuse $AB$: \[ AB = \sqrt{CA^2 + CB^2} = \sqrt{30^2 + 16^2} = \sqrt{900 + 256} = \sqrt{1156} = 34. \] 2. **Understanding t...	[ { "content": "In [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$ with [right angle](https://artofproblemsolving.com/wiki/index.php/Right_angle) $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. [Points](https://artofproblemsolving.com/w...
amc_aime	A finite [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digi...	1. Understanding the Sequence Property: The sequence is such that the tens and units digits of each term become the hundreds and tens digits of the next term. This implies that each digit in a position (hundreds, tens, units) of one term will appear in the next position in the next term and eventually cycle back to...	[ { "content": "A finite [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the ...
amc_aime	Two long [cylindrical](https://artofproblemsolving.com/wiki/index.php/Cylinder) tubes of the same length but different [diameters](https://artofproblemsolving.com/wiki/index.php/Diameter) lie [parallel](https://artofproblemsolving.com/wiki/index.php/Parallel) to each other on a [flat surface](https://artofproblemsolvin...	1. Understanding the Problem: Two cylindrical tubes, one with radius 72 and the other with radius 24, are lying parallel on a flat surface. The larger tube rolls over the smaller tube and completes one full revolution, returning to the same point on its circumference, without slipping. 2. Initial Setup: ...	[ { "content": "Two long [cylindrical](https://artofproblemsolving.com/wiki/index.php/Cylinder) tubes of the same length but different [diameters](https://artofproblemsolving.com/wiki/index.php/Diameter) lie [parallel](https://artofproblemsolving.com/wiki/index.php/Parallel) to each other on a [flat surface](http...
amc_aime	In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10$. Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCF...	1. Calculate $BC$ using the Pythagorean Theorem: Since $\triangle ABC$ is a right triangle with $AB = 12$ and $AC = 35$, we apply the Pythagorean Theorem: \[ BC = \sqrt{AB^2 + AC^2} = \sqrt{12^2 + 35^2} = \sqrt{144 + 1225} = \sqrt{1369} = 37. \] 2. **Apply the Angle Bisector Theorem to find $BD$ and $D...	[ { "content": "In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\\overline{BC}$, and $\\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\\overline{AB}$ and $\\overline{AC}$, respectively, so that $AE=3$ and $AF=10$. Given that $EB=9$ and $FC=27$, find the integer closest to the ...
amc_aime	In the pattern below, the cat moves clockwise through the four squares, and the mouse moves counterclockwise through the eight exterior segments of the four squares. [2003amc8prob23a.png](https://artofproblemsolving.com/wiki/index.php/File:2003amc8prob23a.png) If the pattern is continued, where would the cat and m...	To solve this problem, we need to determine the positions of both the cat and the mouse after the 247th move. We will analyze their movements separately and then combine the results to find the correct answer. #### Cat's Movement: 1. Cycle Length: The cat moves through four squares, completing a cycle every 4 move...	[ { "content": "In the pattern below, the cat moves clockwise through the four squares, and the mouse moves counterclockwise through the eight exterior segments of the four squares.\n\n\n[2003amc8prob23a.png](https://artofproblemsolving.com/wiki/index.php/File:2003amc8prob23a.png)\n\n\nIf the pattern is continue...
amc_aime	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](https://artofproblemsolving.com/wiki/in...	1. Understanding the Problem: We are given a triangle $ABC$ with an interior point $P$. Lines are extended from the vertices through $P$ to the opposite sides, creating segments of lengths $a$, $b$, $c$, and $d$. We need to find the product $abc$ given that $a + b + c = 43$ and $d = 3$. 2. Using Ceva's Theorem...	[ { "content": "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 $d = 3$.\n[1988 AIME-12.png](https://artofpro...
amc_aime	In a game of Chomp, two players alternately take bites from a 5-by-7 grid of [unit squares](https://artofproblemsolving.com/wiki/index.php/Unit_square). To take a bite, a player chooses one of the remaining [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)), then removes ("eats") all squares i...	1. Understanding the Game Dynamics: In Chomp, when a player chooses a square, all squares to the right and all squares above that square are also removed. This means that if a square at position $(i, j)$ in a $5 \times 7$ grid is chosen, all squares $(i, k)$ for $k \geq j$ and all squares $(l, j)$ for $l \leq i$ ar...	[ { "content": "In a game of Chomp, two players alternately take bites from a 5-by-7 grid of [unit squares](https://artofproblemsolving.com/wiki/index.php/Unit_square). To take a bite, a player chooses one of the remaining [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)), then removes ...
amc_aime	(Titu Andreescu) Prove that for every [nonnegative](https://artofproblemsolving.com/wiki/index.php/Nonnegative) [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$, the number $7^{7^n}+1$ is the [product](https://artofproblemsolving.com/wiki/index.php/Product) of at least $2n+3$ (not necessarily disti...	We will prove the statement using mathematical induction. #### Base Case: For $n = 0$, we have: \[ 7^{7^0} + 1 = 7^1 + 1 = 8 = 2^3. \] Here, $8$ is the product of $3$ primes (not necessarily distinct, as they are all $2$). This satisfies the condition of having at least $2 \times 0 + 3 = 3$ prime factors. #### Induct...	[ { "content": "(Titu Andreescu) Prove that for every [nonnegative](https://artofproblemsolving.com/wiki/index.php/Nonnegative) [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$, the number $7^{7^n}+1$ is the [product](https://artofproblemsolving.com/wiki/index.php/Product) of at least $2n+3$ ...
amc_aime	Oscar buys $13$ pencils and $3$ erasers for $1.00$. A pencil costs more than an eraser, and both items cost a [whole number](https://artofproblemsolving.com/wiki/index.php/Whole_number) of cents. What is the total cost, in cents, of one pencil and one eraser? $\mathrm{(A)}\ 10\qquad\mathrm{(B)}\ 12\qquad\mathrm{(C)}\ 1...	1. Define Variables: Let $p$ represent the cost of one pencil and $e$ represent the cost of one eraser in cents. We are given the equation: \[ 13p + 3e = 100 \] where $p > e > 0$ and both $p$ and $e$ are whole numbers. 2. Constraints on $p$ and $e$: Since $p$ and $e$ are positive integers, we...	[ { "content": "Oscar buys $13$ pencils and $3$ erasers for $1.00$. A pencil costs more than an eraser, and both items cost a [whole number](https://artofproblemsolving.com/wiki/index.php/Whole_number) of cents. What is the total cost, in cents, of one pencil and one eraser?\n$\\mathrm{(A)}\\ 10\\qquad\\mathrm{(B...
amc_aime	The number of [digits](https://artofproblemsolving.com/wiki/index.php/Digit) in $4^{16}5^{25}$ (when written in the usual [base](https://artofproblemsolving.com/wiki/index.php/Base_number) $10$ form) is $\mathrm{(A) \ }31 \qquad \mathrm{(B) \ }30 \qquad \mathrm{(C) \ } 29 \qquad \mathrm{(D) \ }28 \qquad \mathrm{(E) \ }...	1. Rewrite the expression using properties of exponents: The given expression is $4^{16}5^{25}$. We can express $4$ as $2^2$, so: \[ 4^{16} = (2^2)^{16} = 2^{32} \] Therefore, the expression becomes: \[ 4^{16}5^{25} = 2^{32}5^{25} \] 2. Combine powers of 2 and 5 to form powers of 10: ...	[ { "content": "The number of [digits](https://artofproblemsolving.com/wiki/index.php/Digit) in $4^{16}5^{25}$ (when written in the usual [base](https://artofproblemsolving.com/wiki/index.php/Base_number) $10$ form) is\n$\\mathrm{(A) \\ }31 \\qquad \\mathrm{(B) \\ }30 \\qquad \\mathrm{(C) \\ } 29 \\qquad \\mathrm...
amc_aime	The [set](https://artofproblemsolving.com/wiki/index.php/Set) $\{3,6,9,10\}$ is augmented by a fifth element $n$, not equal to any of the other four. The [median](https://artofproblemsolving.com/wiki/index.php/Median) of the resulting set is equal to its [mean](https://artofproblemsolving.com/wiki/index.php/Mean). What...	We are given the set $\{3,6,9,10\}$ and an additional element $n$, which is distinct from the other elements. We need to find the sum of all possible values of $n$ such that the median and the mean of the augmented set are equal. #### Case 1: Median is $6$ For $6$ to be the median, $n$ must be less than or equal to $6...	[ { "content": "The [set](https://artofproblemsolving.com/wiki/index.php/Set) $\\{3,6,9,10\\}$ is augmented by a fifth element $n$, not equal to any of the other four. The [median](https://artofproblemsolving.com/wiki/index.php/Median) of the resulting set is equal to its [mean](https://artofproblemsolving.com/wi...
amc_aime	Odell and Kershaw run for $30$ minutes on a [circular](https://artofproblemsolving.com/wiki/index.php/Circle) track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on th...	1. Determine the Circumference of Each Track: - Odell's track radius = $50$ meters, so the circumference is $C_O = 2\pi \times 50 = 100\pi$ meters. - Kershaw's track radius = $60$ meters, so the circumference is $C_K = 2\pi \times 60 = 120\pi$ meters. 2. Calculate the Speed in Terms of Radians per Minute...	[ { "content": "Odell and Kershaw run for $30$ minutes on a [circular](https://artofproblemsolving.com/wiki/index.php/Circle) track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ ...
amc_aime	Many calculators have a [reciprocal](https://artofproblemsolving.com/wiki/index.php/Reciprocal) key $\boxed{\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\boxed{00004}$ and the $\boxed{\frac{1}{x}}$ key is depressed, then the display becomes $\boxed{000.2...	1. Define the function: Let $f(x) = \frac{1}{x}$ represent the operation of the reciprocal key on the calculator. 2. Apply the function: We need to determine how many times we must apply $f(x)$ to return to the original number. Start by applying $f(x)$ to the number 32: \[ f(32) = \frac{1}{32} \] 3. ...	[ { "content": "Many calculators have a [reciprocal](https://artofproblemsolving.com/wiki/index.php/Reciprocal) key $\\boxed{\\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\\boxed{00004}$ and the $\\boxed{\\frac{1}{x}}$ key is depressed, then the di...
amc_aime	In a magic triangle, each of the six [whole numbers](https://artofproblemsolving.com/wiki/index.php/Whole_number) $10-15$ is placed in one of the [circles](https://artofproblemsolving.com/wiki/index.php/Circle) so that the sum, $S$, of the three numbers on each side of the [triangle](https://artofproblemsolving.com/wik...	1. Assign Variables to Circles: Let the numbers in the circles be $a$, $b$, $c$, $d$, $e$, and $f$ starting from the top circle and moving clockwise. 2. Set Up Equations for Each Side of the Triangle: - The sum of the numbers on the first side is $S = a + b + c$. - The sum of the numbers on the second si...	[ { "content": "In a magic triangle, each of the six [whole numbers](https://artofproblemsolving.com/wiki/index.php/Whole_number) $10-15$ is placed in one of the [circles](https://artofproblemsolving.com/wiki/index.php/Circle) so that the sum, $S$, of the three numbers on each side of the [triangle](https://artof...
amc_aime	The [product](https://artofproblemsolving.com/wiki/index.php/Product) $8\times .25\times 2\times .125 =$ $\text{(A)}\ \frac18 \qquad \text{(B)}\ \frac14 \qquad \text{(C)}\ \frac12 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2$	1. Convert Decimals to Fractions: The given decimals are converted to fractions for easier multiplication. - $0.25$ as a fraction is $\frac{1}{4}$. - $0.125$ as a fraction is $\frac{1}{8}$. 2. Set Up the Product: Substitute the decimals with their fractional equivalents in the product. \[ 8 \times ...	[ { "content": "The [product](https://artofproblemsolving.com/wiki/index.php/Product) $8\\times .25\\times 2\\times .125 =$\n$\\text{(A)}\\ \\frac18 \\qquad \\text{(B)}\\ \\frac14 \\qquad \\text{(C)}\\ \\frac12 \\qquad \\text{(D)}\\ 1 \\qquad \\text{(E)}\\ 2$", "role": "user" }, { "content": "1. **Con...
amc_aime	([Sam Vandervelde](https://artofproblemsolving.com/wiki/index.php/Sam_Vandervelde)) At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an...	#### Step-by-step Proof: 1. Graph Interpretation: - Consider each mathematician as a vertex in a graph $G$. - An edge exists between two vertices if the corresponding mathematicians are friends. - The problem is then to color each vertex (mathematician) with one of two colors (representing two dining roo...	[ { "content": "([Sam Vandervelde](https://artofproblemsolving.com/wiki/index.php/Sam_Vandervelde)) At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a ...
amc_aime	Six ants simultaneously stand on the six [vertices](https://artofproblemsolving.com/wiki/index.php/Vertex) of a regular [octahedron](https://artofproblemsolving.com/wiki/index.php/Octahedron), with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjace...	We approach this problem by counting the number of ways ants can do their desired migration, and then multiply this number by the probability that each case occurs. Let the octahedron be labeled as $ABCDEF$, with points $B, C, D, E$ being coplanar. Then the ant from $A$ and the ant from $F$ must move to plane $BCDE$....	[ { "content": "Six ants simultaneously stand on the six [vertices](https://artofproblemsolving.com/wiki/index.php/Vertex) of a regular [octahedron](https://artofproblemsolving.com/wiki/index.php/Octahedron), with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to ...
amc_aime	Let $A_1A_2A_3$ be a [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) and let $\omega_1$ be a [circle](https://artofproblemsolving.com/wiki/index.php/Circle) in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots,...	#### Step-by-step Proof: 1. Setup and Notation: Let $\triangle A_1A_2A_3$ be a triangle and $\omega_1$ be a circle passing through $A_1$ and $A_2$. Define $\omega_k$ for $k = 2, 3, \dots, 7$ such that each $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1}$, with $A_{n+3} ...	[ { "content": "Let $A_1A_2A_3$ be a [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) and let $\\omega_1$ be a [circle](https://artofproblemsolving.com/wiki/index.php/Circle) in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\\omega_2, \\omega_3, \\dots, \\omega_7$ such...
amc_aime	A number $x$ is $2$ more than the product of its [reciprocal](https://artofproblemsolving.com/wiki/index.php/Reciprocal) and its additive [inverse](https://artofproblemsolving.com/wiki/index.php/Inverse). In which [interval](https://artofproblemsolving.com/wiki/index.php/Interval) does the number lie? $\textbf{(A) }\ -...	Let's analyze the given problem step by step. 1. Understanding the terms: - The reciprocal of a number $x$ is $\frac{1}{x}$. - The additive inverse of a number $x$ is $-x$. 2. Setting up the equation: The problem states that $x$ is $2$ more than the product of its reciprocal and its additive ...	[ { "content": "A number $x$ is $2$ more than the product of its [reciprocal](https://artofproblemsolving.com/wiki/index.php/Reciprocal) and its additive [inverse](https://artofproblemsolving.com/wiki/index.php/Inverse). In which [interval](https://artofproblemsolving.com/wiki/index.php/Interval) does the number ...
amc_aime	The [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) enclosed by the solid lines in the figure consists of 4 [congruent](https://artofproblemsolving.com/wiki/index.php/Congruent) [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) joined [edge](https://artofproblemsolving.com/w...	To solve this problem, we need to determine how many of the nine positions for the additional square allow the resulting figure to be folded into a cube with one face missing. We start by understanding the structure of the given figure and the implications of adding a square at each position. #### Step 1: Understand ...	[ { "content": "The [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) enclosed by the solid lines in the figure consists of 4 [congruent](https://artofproblemsolving.com/wiki/index.php/Congruent) [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) joined [edge](https://art...
amc_aime	Let $a$ and $c$ be fixed [positive numbers](https://artofproblemsolving.com/wiki/index.php/Natural_numbers). For each [real number](https://artofproblemsolving.com/wiki/index.php/Real_number) $t$ let $(x_t, y_t)$ be the [vertex](https://artofproblemsolving.com/wiki/index.php/Vertex) of the [parabola](https://artofprobl...	1. Identify the vertex of the parabola: The vertex of a parabola given by the equation $y = ax^2 + bx + c$ can be found using the formula for the x-coordinate of the vertex, $x_v = -\frac{b}{2a}$. 2. Substitute $x_v$ into the parabola equation: Plugging $x_v = -\frac{b}{2a}$ into the equation of the parabola,...	[ { "content": "Let $a$ and $c$ be fixed [positive numbers](https://artofproblemsolving.com/wiki/index.php/Natural_numbers). For each [real number](https://artofproblemsolving.com/wiki/index.php/Real_number) $t$ let $(x_t, y_t)$ be the [vertex](https://artofproblemsolving.com/wiki/index.php/Vertex) of the [parabo...
amc_aime	A box contains $11$ balls, numbered $1, 2, 3, ... 11$. If $6$ balls are drawn simultaneously at random, what is the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that the sum of the numbers on the balls drawn is odd? $\mathrm{(A) \ }\frac{100}{231} \qquad \mathrm{(B) \ }\frac{115}{231} \qqua...	To solve this problem, we need to calculate the probability that the sum of the numbers on the six balls drawn is odd. We start by noting that there are 6 odd-numbered balls (1, 3, 5, 7, 9, 11) and 5 even-numbered balls (2, 4, 6, 8, 10) in the box. The sum of the numbers on the balls drawn is odd if and only if the nu...	[ { "content": "A box contains $11$ balls, numbered $1, 2, 3, ... 11$. If $6$ balls are drawn simultaneously at random, what is the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that the sum of the numbers on the balls drawn is odd?\n$\\mathrm{(A) \\ }\\frac{100}{231} \\qquad \\mathrm{...
amc_aime	At Euclid High School, the number of students taking the [AMC 10](https://artofproblemsolving.com/wiki/index.php/AMC_10) was $60$ in 2002, $66$ in 2003, $70$ in 2004, $76$ in 2005, $78$ and 2006, and is $85$ in 2007. Between what two consecutive years was there the largest percentage increase? $\text{(A)}\ 2002\ \text{...	To find the largest percentage increase in the number of students taking the AMC 10 between consecutive years, we calculate the percentage increase for each interval: 1. From 2002 to 2003: \[ \frac{66 - 60}{60} \times 100\% = \frac{6}{60} \times 100\% = 10\% \] 2. From 2003 to 2004: \[ \frac{70...	[ { "content": "At Euclid High School, the number of students taking the [AMC 10](https://artofproblemsolving.com/wiki/index.php/AMC_10) was $60$ in 2002, $66$ in 2003, $70$ in 2004, $76$ in 2005, $78$ and 2006, and is $85$ in 2007. Between what two consecutive years was there the largest percentage increase?\n$\...
amc_aime	A transformation of the first [quadrant](https://artofproblemsolving.com/wiki/index.php/Quadrant) of the [coordinate plane](https://artofproblemsolving.com/wiki/index.php/Coordinate_plane) maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The [vertices](https://artofproblemsolving.com/wiki/index.php/Vertex) ...	1. Transform the vertices of quadrilateral $ABCD$: - For $A = (900, 300)$, the transformed point $A'$ is given by: \[ A' = (\sqrt{900}, \sqrt{300}) = (30, \sqrt{300}) \] - For $B = (1800, 600)$, the transformed point $B'$ is given by: \[ B' = (\sqrt{1800}, \sqrt{600}) = (30\sqrt{2}, 3...	[ { "content": "A transformation of the first [quadrant](https://artofproblemsolving.com/wiki/index.php/Quadrant) of the [coordinate plane](https://artofproblemsolving.com/wiki/index.php/Coordinate_plane) maps each point $(x,y)$ to the point $(\\sqrt{x},\\sqrt{y}).$ The [vertices](https://artofproblemsolving.com...
amc_aime	Placing no more than one $\text{X}$ in each small [square](https://artofproblemsolving.com/wiki/index.php/Square), what is the greatest number of $\text{X}$'s that can be put on the grid shown without getting three $\text{X}$'s in a row vertically, horizontally, or diagonally? $\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad...	1. Understanding the Grid: The grid is a $4 \times 4$ square grid, which means there are 16 small squares in total. 2. Objective: We need to place the maximum number of $\text{X}$'s such that no three $\text{X}$'s are aligned vertically, horizontally, or diagonally. 3. Using the Pigeonhole Principle: If w...	[ { "content": "Placing no more than one $\\text{X}$ in each small [square](https://artofproblemsolving.com/wiki/index.php/Square), what is the greatest number of $\\text{X}$'s that can be put on the grid shown without getting three $\\text{X}$'s in a row vertically, horizontally, or diagonally?\n$\\text{(A)}\\ 2...
amc_aime	Find the number of ordered triples $(a,b,c)$ where $a$, $b$, and $c$ are positive [integers](https://artofproblemsolving.com/wiki/index.php/Integer), $a$ is a [factor](https://artofproblemsolving.com/wiki/index.php/Factor) of $b$, $a$ is a factor of $c$, and $a+b+c=100$.	1. Express $b$ and $c$ in terms of $a$: Let $b = ax$ and $c = ay$ where $x$ and $y$ are positive integers. This is because $a$ is a factor of both $b$ and $c$. 2. Rewrite the equation $a+b+c=100$: Substituting $b = ax$ and $c = ay$ into the equation, we get: \[ a + ax + ay = 100 \implies a(1 + ...	[ { "content": "Find the number of ordered triples $(a,b,c)$ where $a$, $b$, and $c$ are positive [integers](https://artofproblemsolving.com/wiki/index.php/Integer), $a$ is a [factor](https://artofproblemsolving.com/wiki/index.php/Factor) of $b$, $a$ is a factor of $c$, and $a+b+c=100$.", "role": "user" }, ...
amc_aime	[Square](https://artofproblemsolving.com/wiki/index.php/Square) $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt {50}$ and $BE = 1$. What is the area of the inner square $EFGH$? $(\mathrm {A}) \ 25 \qquad (\mathrm {B}...	1. Understanding the Problem Setup: - We have two squares, $ABCD$ and $EFGH$, where $EFGH$ is inside $ABCD$. - Each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. - The side length of $ABCD$ is given as $\sqrt{50}$. - The distance from vertex $B$ of square $ABCD$ to the nearest point...	[ { "content": "[Square](https://artofproblemsolving.com/wiki/index.php/Square) $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\\sqrt {50}$ and $BE = 1$. What is the area of the inner square $EFGH$?\n$(\\mathrm {A})...
amc_aime	If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$? $\textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30}$ Video Solution by MegaMath (LaTeX) [https://www.youtube.com/watch?v=L54A79X1ezc&t=2s](https://artofproblemsolving.comhttps://ww...	We are given the equations: 1. \( y + 4 = (x-2)^2 \) 2. \( x + 4 = (y-2)^2 \) We are also given that \( x \neq y \) and need to find \( x^2 + y^2 \). #### Step 1: Manipulate the equations First, we rewrite the equations: - From equation 1: \( y = (x-2)^2 - 4 \) - From equation 2: \( x = (y-2)^2 - 4 \) #### Step 2: ...	[ { "content": "If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \\neq y$, what is the value of $x^2+y^2$?\n$\\textbf{(A) }10\\qquad\\textbf{(B) }15\\qquad\\textbf{(C) }20\\qquad\\textbf{(D) }25\\qquad\\textbf{(E) }\\text{30}$\n\nVideo Solution by MegaMath (LaTeX)\n[https://www.youtube.com/watch?v=L54A79X1ezc&t=2s](http...
amc_aime	Let the [set](https://artofproblemsolving.com/wiki/index.php/Set) $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}.$ Susan makes a list as follows: for each two-element subset of $\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.	1. Order the Elements: First, we order the elements of the set $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}$ in descending order for easier management: \[ \mathcal{S} = \{34, 21, 13, 8, 5, 3, 2, 1\} \] 2. Count the Subsets: Each element in the set $\mathcal{S}$ will be paired with every other element ...	[ { "content": "Let the [set](https://artofproblemsolving.com/wiki/index.php/Set) $\\mathcal{S} = \\{8, 5, 1, 13, 34, 3, 21, 2\\}.$ Susan makes a list as follows: for each two-element subset of $\\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.",...
amc_aime	Two [squares](https://artofproblemsolving.com/wiki/index.php/Square) of a $7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a [rotation](https://artofproblemsolving.com/wiki/index.php/Rotation) in the plane boa...	1. Total Number of Ways to Paint Two Squares Yellow: The total number of squares on a $7 \times 7$ checkerboard is $49$. We need to choose $2$ squares out of these $49$ to paint them yellow. The number of ways to do this is given by the combination formula $\binom{n}{k}$, which represents the number of ways to...	[ { "content": "Two [squares](https://artofproblemsolving.com/wiki/index.php/Square) of a $7\\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a [rotation](https://artofproblemsolving.com/wiki/index.php/Rot...
amc_aime	The [average](https://artofproblemsolving.com/wiki/index.php/Average) weight of $6$ boys is $150$ pounds and the average weight of $4$ girls is $120$ pounds. The average weight of the $10$ children is $\text{(A)}\ 135\text{ pounds} \qquad \text{(B)}\ 137\text{ pounds} \qquad \text{(C)}\ 138\text{ pounds} \qquad \text...	1. Calculate the total weight of the boys: Given that the average weight of 6 boys is 150 pounds, we use the formula for average: \[ \text{Average} = \frac{\text{Total Sum}}{\text{Number of Items}} \] Rearranging to find the total sum: \[ S_B = 6 \times 150 = 900 \text{ pounds} \] 2. **Ca...	[ { "content": "The [average](https://artofproblemsolving.com/wiki/index.php/Average) weight of $6$ boys is $150$ pounds and the average weight of $4$ girls is $120$ pounds. The average weight of the $10$ children is \n$\\text{(A)}\\ 135\\text{ pounds} \\qquad \\text{(B)}\\ 137\\text{ pounds} \\qquad \\text{(C)}...
amc_aime	What is the $100\text{th}$ number in the [arithmetic sequence](https://artofproblemsolving.com/wiki/index.php/Arithmetic_sequence): $1,5,9,13,17,21,25,...$? $\text{(A)}\ 397 \qquad \text{(B)}\ 399 \qquad \text{(C)}\ 401 \qquad \text{(D)}\ 403 \qquad \text{(E)}\ 405$	1. Identify the first term and common difference: The given arithmetic sequence is $1, 5, 9, 13, 17, 21, 25, \ldots$. The first term ($a_1$) is $1$. The common difference ($d$) can be calculated as the difference between any two consecutive terms: \[ d = 5 - 1 = 4 \] 2. **Use the formula for the $n$-t...	[ { "content": "What is the $100\\text{th}$ number in the [arithmetic sequence](https://artofproblemsolving.com/wiki/index.php/Arithmetic_sequence): $1,5,9,13,17,21,25,...$?\n$\\text{(A)}\\ 397 \\qquad \\text{(B)}\\ 399 \\qquad \\text{(C)}\\ 401 \\qquad \\text{(D)}\\ 403 \\qquad \\text{(E)}\\ 405$", "role": "...
amc_aime	How many [ordered pairs](https://artofproblemsolving.com/wiki/index.php/Ordered_pair) $(m,n)$ of positive [integers](https://artofproblemsolving.com/wiki/index.php/Integer), with $m \ge n$, have the property that their squares differ by $96$? $\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ ...	To find the number of ordered pairs $(m,n)$ of positive integers such that $m \ge n$ and $m^2 - n^2 = 96$, we start by factoring the expression for the difference of squares: \[ m^2 - n^2 = (m+n)(m-n) = 96. \] We need to find all pairs of factors of $96$ that can be expressed as $(m+n, m-n)$. Since $m$ and $n$ are in...	[ { "content": "How many [ordered pairs](https://artofproblemsolving.com/wiki/index.php/Ordered_pair) $(m,n)$ of positive [integers](https://artofproblemsolving.com/wiki/index.php/Integer), with $m \\ge n$, have the property that their squares differ by $96$? \n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 4 \\qquad \\t...
amc_aime	(Gregory Galparin) Let $\mathcal{P}$ be a [convex polygon](https://artofproblemsolving.com/wiki/index.php/Convex_polygon) with $n$ sides, $n\ge3$. Any set of $n - 3$ diagonals of $\mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $\mathcal{P}$ into $n - 2$ triangles. If $\ma...	1. Label the Vertices and Consider a Diagonal: Label the vertices of the convex polygon $\mathcal{P}$ as $P_0, P_1, P_2, \ldots, P_{n-1}$. Consider a diagonal $d = \overline{P_aP_{a+k}}$, where $k \le n/2$. 2. Partitioning by the Diagonal: The diagonal $d$ partitions $\mathcal{P}$ into two regions, $\mathcal{Q...	[ { "content": "(Gregory Galparin) Let $\\mathcal{P}$ be a [convex polygon](https://artofproblemsolving.com/wiki/index.php/Convex_polygon) with $n$ sides, $n\\ge3$. Any set of $n - 3$ diagonals of $\\mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $\\mathcal{P}$ into ...
amc_aime	The [arithmetic mean](https://artofproblemsolving.com/wiki/index.php/Arithmetic_mean) of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ doesn't contain the digit $\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 2 \qquad\mathrm{(C)}\ ...	1. Identify the pattern in the set: The set given is $\{9, 99, 999, 9999, \ldots, 999999999\}$. Each number in the set is composed of repeated digits of $9$, increasing by one $9$ at each step. 2. Express each number as a sum: Each number in the set can be expressed as a sum of $9$s multiplied by powers of $10...	[ { "content": "The [arithmetic mean](https://artofproblemsolving.com/wiki/index.php/Arithmetic_mean) of the nine numbers in the set $\\{9, 99, 999, 9999, \\ldots, 999999999\\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ doesn't contain the digit\n$\\mathrm{(A)}\\ 0 \\qquad\\mathr...
amc_aime	[Pdfresizer.com-pdf-convert.png](https://artofproblemsolving.com/wiki/index.php/File:Pdfresizer.com-pdf-convert.png) In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, and sides $BC$ and $ED$. Each side has length $1$. Also, $\angle FAB = \angle BCD = 60^\circ$. The area of t...	1. Identify the Shape and Symmetry: The given figure can be visualized as composed of several triangles. Notably, the problem states that sides $AF$ and $CD$ are parallel, as are $AB$ and $EF$, and $BC$ and $ED$. Each side has a length of $1$, and the angles $\angle FAB$ and $\angle BCD$ are each $60^\circ$. 2. **...	[ { "content": "[Pdfresizer.com-pdf-convert.png](https://artofproblemsolving.com/wiki/index.php/File:Pdfresizer.com-pdf-convert.png)\nIn the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, \nand sides $BC$ and $ED$. Each side has length $1$. Also, $\\angle FAB = \\angle BCD =...
amc_aime	$\diamondsuit$ and $\Delta$ are [whole numbers](https://artofproblemsolving.com/wiki/index.php/Whole_number) and $\diamondsuit \times \Delta =36$. The largest possible value of $\diamondsuit + \Delta$ is $\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 20\ \qquad \text{(E)}\ 37$	To find the largest possible value of $\diamondsuit + \Delta$ given that $\diamondsuit \times \Delta = 36$, we need to consider all pairs of whole numbers $(\diamondsuit, \Delta)$ that multiply to 36. We will then calculate the sum $\diamondsuit + \Delta$ for each pair and determine the maximum. 1. Factorize 36: W...	[ { "content": "$\\diamondsuit$ and $\\Delta$ are [whole numbers](https://artofproblemsolving.com/wiki/index.php/Whole_number) and $\\diamondsuit \\times \\Delta =36$. The largest possible value of $\\diamondsuit + \\Delta$ is \n$\\text{(A)}\\ 12 \\qquad \\text{(B)}\\ 13 \\qquad \\text{(C)}\\ 15 \\qquad \\text{(...
amc_aime	[Rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The [inscribed circle](https://artofproblemsolving.com/wiki/index.php/Inscribed_circle) of [triangle](https://artofproblemsolving.c...	#### Step 1: Calculate $BE$ and $DF$ Given that $AB = 63$ and $AE = CF = 84$, we can find $BE$ and $DF$ using the Pythagorean theorem in triangles $ABE$ and $CDF$ respectively: \[ BE = DF = \sqrt{AB^2 + AE^2} = \sqrt{63^2 + 84^2} = \sqrt{3969 + 7056} = \sqrt{11025} = 105. \] #### Step 2: Calculate $EF$ Since $BC = 44...	[ { "content": "[Rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The [inscribed circle](https://artofproblemsolving.com/wiki/index.php/Inscribed_circle) of [triangle](https:/...