Datasets:

mlfoundations-dev
/

aops_forum_plot

source stringclasses 1 value	problem stringlengths 226 4.47k	solution stringlengths 1.28k 5.19k	messages listlengths 2 2
aops_forum	Let $I$ be the incenter of triangle $ABC$ ; $H_B, H_C$ the orthocenters of triangles $ACI$ and $ABI$ respectively; $K$ the touching point of the incircle with the side $BC$ . Prove that $H_B, H_C$ and K are collinear. Proposed by M.Plotnikov	1. Define Points and Setup: Let \( I \) be the incenter of triangle \( \triangle ABC \). Let \( H_B \) and \( H_C \) be the orthocenters of triangles \( \triangle ACI \) and \( \triangle ABI \) respectively. Let \( K \) be the point where the incircle of \( \triangle ABC \) touches the side \( BC \). 2. **Inter...	[ { "content": " Let $I$ be the incenter of triangle $ABC$ ; $H_B, H_C$ the orthocenters of triangles $ACI$ and $ABI$ respectively; $K$ the touching point of the incircle with the side $BC$ . Prove that $H_B, H_C$ and K are collinear.\nProposed by M.Plotnikov", "role": "user" }, { "con...
aops_forum	p1. Hitori is trying to guess a three-digit integer with three different digits in five guesses to win a new guitar. She guesses $819$ , and is told that exactly one of the digits in her guess is in the answer, but it is in the wrong place. Next, she guesses $217$ , and is told that exactly one of the digits is i...	To solve the problem, we need to compute the product of the function \( f(x) = \log_x(2x) \) for specific values of \( x \). Let's break down the steps: 1. Express \( f(x) \) in a simpler form: \[ f(x) = \log_x(2x) \] Using the change of base formula for logarithms, we get: \[ \log_x(2x) = \frac{...	[ { "content": "p1. Hitori is trying to guess a three-digit integer with three different digits in five guesses to win a new guitar. She guesses $819$ , and is told that exactly one of the digits in her guess is in the answer, but it is in the wrong place. Next, she guesses $217$ , and is told that exactly ...
aops_forum	Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.	1. Understanding the Problem: We need to show that no matter how 15 points are placed within a circle of radius 2 (including the border), there will always be a smaller circle of radius 1 (including the border) that contains at least three of these points. 2. Using the Pigeonhole Principle: The Pigeonhol...	[ { "content": "Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.", "role": "user" }, { "content": "1. **Understanding the Pr...
aops_forum	The graphs of four functions of the form $y = x^2 + ax + b$ , where a and b are real coefficients, are plotted on the coordinate plane. These graphs have exactly four points of intersection, and at each one of them, exactly two graphs intersect. Prove that the sum of the largest and the smallest $x$ -coordinates of t...	1. Define the functions and their intersections: Let the four functions be \( f_1, f_2, f_3, f_4 \) where \( f_i(x) = x^2 + a_i x + b_i \) for \( i = 1, 2, 3, 4 \). The graphs of these functions intersect exactly at four points, and at each intersection point, exactly two graphs intersect. 2. **Intersection con...	[ { "content": "The graphs of four functions of the form $y = x^2 + ax + b$ , where a and b are real coefficients, are plotted on the coordinate plane. These graphs have exactly four points of intersection, and at each one of them, exactly two graphs intersect. Prove that the sum of the largest and the smallest ...
aops_forum	Let $ABC$ be an acute triangle. Squares $AA_1A_2A_3$ , $BB_1B_2B_3$ and $CC_1C_2C_3$ are located such that the lines $A_1A_2$ , $B_1B_2$ , $C_1C_2$ pass through the points $B$ , $C$ and $A$ respectively and the lines $A_2A_3$ , $B_2B_3$ , $C_2C_3$ pass through the points $C$ , $A$ and $B$ respe...	### Part (a): Prove that the lines \(AA_2\), \(B_1B_2\), and \(C_1C_3\) intersect at one point. 1. Construct the squares: - Construct squares \(AA_1A_2A_3\), \(BB_1B_2B_3\), and \(CC_1C_2C_3\) on the sides of the triangle \(ABC\) such that the lines \(A_1A_2\), \(B_1B_2\), and \(C_1C_2\) pass through the points...	[ { "content": "Let $ABC$ be an acute triangle. Squares $AA_1A_2A_3$ , $BB_1B_2B_3$ and $CC_1C_2C_3$ are located such that the lines $A_1A_2$ , $B_1B_2$ , $C_1C_2$ pass through the points $B$ , $C$ and $A$ respectively and the lines $A_2A_3$ , $B_2B_3$ , $C_2C_3$ pass through the points $C$ ,...
aops_forum	In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find $\alpha.$ [asy] unitsize(40); import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffttww = rgb(1,0.2...	1. Identify the given information and the goal: - We are given a circle with center \( O \). - Three angles \(\alpha\), \(\beta\), and \(\gamma\) are equal. - We need to find the value of \(\alpha\). 2. Understand the geometric configuration: - Since \(O\) is the center of the circle, \(OA\), \(OB\...	[ { "content": "In the following diagram, $O$ is the center of the circle. If three angles $\\alpha, \\beta$ and $\\gamma$ be equal, find $\\alpha.$ [asy]\nunitsize(40);\nimport graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0....
aops_forum	A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field...	1. Define the side length of the squares: Let the side length of the squares be \( a \). The field is to be partitioned into congruent square plots, so the number of squares along the length (52 meters) and the width (24 meters) must be integers. Let \( n \) be the number of squares along the length and \( m \) ...	[ { "content": "A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to ...
aops_forum	Let $l$ be the tangent line at the point $P(s,\ t)$ on a circle $C:x^2+y^2=1$ . Denote by $m$ the line passing through the point $(1,\ 0)$ , parallel to $l$ . Let the line $m$ intersects the circle $C$ at $P'$ other than the point $(1,\ 0)$ . Note : if $m$ is the line $x=1$ , then $P'$ is consider...	Given the circle \( C: x^2 + y^2 = 1 \) and a point \( P(s, t) \) on the circle, we need to find the tangent line at \( P \) and the line \( m \) passing through \( (1, 0) \) parallel to the tangent line. We then find the intersection of \( m \) with the circle at a point \( P' \) other than \( (1, 0) \). 1. **Express...	[ { "content": "Let $l$ be the tangent line at the point $P(s,\\ t)$ on a circle $C:x^2+y^2=1$ . Denote by $m$ the line passing through the point $(1,\\ 0)$ , parallel to $l$ . Let the line $m$ intersects the circle $C$ at $P'$ other than the point $(1,\\ 0)$ .\n\nNote : if $m$ is the line $x=1...
aops_forum	Let $n,k$ be positive integers such that $n>k$ . There is a square-shaped plot of land, which is divided into $n\times n$ grid so that each cell has the same size. The land needs to be plowed by $k$ tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side ...	To determine the minimum possible number of unplowed cells, we need to analyze the movement of the tractors on the \( n \times n \) grid. Each tractor starts at the lower-left corner and can only move up or right until it reaches the upper-right corner. 1. Invariant Analysis: Each tractor must make exactly \( ...	[ { "content": "Let $n,k$ be positive integers such that $n>k$ . There is a square-shaped plot of land, which is divided into $n\\times n$ grid so that each cell has the same size. The land needs to be plowed by $k$ tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell...
aops_forum	There is a rectangular plot of size $1 \times n$ . This has to be covered by three types of tiles - red, blue and black. The red tiles are of size $1 \times 1$ , the blue tiles are of size $1 \times 1$ and the black tiles are of size $1 \times 2$ . Let $t_n$ denote the number of ways this can be done. For exampl...	### Part (a): Prove that \( t_{2n+1} = t_n(t_{n-1} + t_{n+1}) \) for all \( n > 1 \). 1. Consider the term \( t_n t_{n-1} \): - We first place the first \( n \) tiles in the first \( n \) blocks in \( t_n \) ways. - Then, we place a black tile covering the \((n+1)\)-th and \((n+2)\)-th blocks. - Finally, ...	[ { "content": "There is a rectangular plot of size $1 \\times n$ . This has to be covered by three types of tiles - red, blue and black. The red tiles are of size $1 \\times 1$ , the blue tiles are of size $1 \\times 1$ and the black tiles are of size $1 \\times 2$ . Let $t_n$ denote the number of ways th...
aops_forum	Let $n\geq 2$ be an integer. $3n$ distinct points are plotted on the circle, where A and B perform the following operation : Firstly, $A$ picks exactly 2 points which haven't been connected yet and connects them by a segment. Secondly, B picks exactly 1 point with no piece and place a piece. Prove that, after con...	1. Define the Problem and Notations: - Let \( n \geq 2 \) be an integer. - There are \( 3n \) distinct points plotted on a circle. - Player A picks exactly 2 points which haven't been connected yet and connects them by a segment. - Player B picks exactly 1 point with no piece and places a piece. - We...	[ { "content": "Let $n\\geq 2$ be an integer. $3n$ distinct points are plotted on the circle, where A and B perform the following operation : Firstly, $A$ picks exactly 2 points which haven't been connected yet and connects them by a segment. Secondly, B picks exactly 1 point with no piece and place a piece...
aops_forum	Let $ABC$ be a triangle such that $AB < BC$ . Plot the height $BH$ with $H$ in $AC$ . Let I be the incenter of triangle $ABC$ and $M$ the midpoint of $AC$ . If line $MI$ intersects $BH$ at point $N$ , prove that $BN < IM$ .	1. Define the problem and setup the geometry: - Let \(ABC\) be a triangle with \(AB < BC\). - Let \(BH\) be the height from \(B\) to \(AC\), with \(H\) on \(AC\). - Let \(I\) be the incenter of \(\triangle ABC\). - Let \(M\) be the midpoint of \(AC\). - Let line \(MI\) intersect \(BH\) at point \(N\)...	[ { "content": "Let $ABC$ be a triangle such that $AB < BC$ . Plot the height $BH$ with $H$ in $AC$ . Let I be the incenter of triangle $ABC$ and $M$ the midpoint of $AC$ . If line $MI$ intersects $BH$ at point $N$ , prove that $BN < IM$ .", "role": "user" }, { "content": "1. **Defi...
aops_forum	In the acute-angled triangle $ABC$ , the altitudes $BP$ and $CQ$ were drawn, and the point $T$ is the intersection point of the altitudes of $\Delta PAQ$ . It turned out that $\angle CTB = 90 {} ^ \circ$ . Find the measure of $\angle BAC$ . (Mikhail Plotnikov)	1. Define the problem and given conditions: - We have an acute-angled triangle \(ABC\). - Altitudes \(BP\) and \(CQ\) are drawn. - Point \(T\) is the intersection point of the altitudes of \(\Delta PAQ\). - It is given that \(\angle CTB = 90^\circ\). 2. Introduce additional points and angles: - ...	[ { "content": "In the acute-angled triangle $ABC$ , the altitudes $BP$ and $CQ$ were drawn, and the point $T$ is the intersection point of the altitudes of $\\Delta PAQ$ . It turned out that $\\angle CTB = 90 {} ^ \\circ$ . Find the measure of $\\angle BAC$ . \n\n(Mikhail Plotnikov)", "role": "use...
aops_forum	A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points $1$ unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between $\frac{1}{2017}$ and $\frac{1}{2016}$ units, drawing the min...	1. Initial Setup: We start with two points on the real number line, denoted as \(0\) and \(1\), which are \(1\) unit apart. 2. Using the Midpoint Plotter: Each time we use the midpoint plotter, it will place a new point exactly halfway between two existing points. If we denote the coordinates of the poi...	[ { "content": "A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points $1$ unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between $\\frac{1}{2017}$ and $\\frac{1}{2016}$...
aops_forum	In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$ , he starts at $A$ and walks to $C$ . Then, he walks to the midpoint of side $AD$ , say $A_1$ . Then, he walks to the midpoint of side $CD$ say $C_1$ , and then the midpoint of $A_1D$ which is $A_2$ . He conti...	1. Initial Path Calculation: - The man starts at point \( A \) and walks to point \( C \). The distance \( AC \) can be calculated using the Pythagorean theorem: \[ AC = \sqrt{AB^2 + BC^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \] - Therefore, the first segment of his path is \...	[ { "content": "In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$ , he starts at $A$ and walks to $C$ . Then, he walks to the midpoint of side $AD$ , say $A_1$ . Then, he walks to the midpoint of side $CD$ say $C_1$ , and then the midpoint of $A_1D$ whic...
aops_forum	There are 100 boxers, each of them having different strengths, who participate in a tournament. Any of them fights each other only once. Several boxers form a plot. In one of their matches, they hide in their glove a horse shoe. If in a fight, only one of the boxers has a horse shoe hidden, he wins the fight; otherwise...	1. Define the problem and variables: - Let \( p \) be the number of plotters. - Let \( u \) be the number of usual boxers (those who are neither the three strongest nor plotters). - The three strongest boxers are denoted as \( s_1, s_2, \) and \( s_3 \) in decreasing order of strength. - The plotters ar...	[ { "content": "There are 100 boxers, each of them having different strengths, who participate in a tournament. Any of them fights each other only once. Several boxers form a plot. In one of their matches, they hide in their glove a horse shoe. If in a fight, only one of the boxers has a horse shoe hidden, he win...
aops_forum	<u>Round 1</u>p1. What is the smallest number equal to its cube?p2. Fhomas has $5$ red spaghetti and $5$ blue spaghetti, where spaghetti are indistinguishable except for color. In how many different ways can Fhomas eat $6$ spaghetti, one after the other? (Two ways are considered the same if the sequence o...	To determine the number of different ways Fhomas can eat 6 spaghetti, we need to consider the different combinations of red and blue spaghetti he can eat. We will use combinations to count the sequences. 1. Case 1: Eating 5 of one color and 1 of the other color - If Fhomas eats 5 red and 1 blue, the number of w...	[ { "content": "<u>Round 1</u>p1. What is the smallest number equal to its cube?p2. Fhomas has $5$ red spaghetti and $5$ blue spaghetti, where spaghetti are indistinguishable except for color. In how many different ways can Fhomas eat $6$ spaghetti, one after the other? (Two ways are considered the ...
aops_forum	At the Mountain School, Micchell is assigned a submissiveness rating of $3.0$ or $4.0$ for each class he takes. His college potential is then defined as the average of his submissiveness ratings over all classes taken. After taking 40 classes, Micchell has a college potential of $3.975$ . Unfortunately, he nee...	1. Let \( x \) be the number of additional classes Micchell needs to take. 2. The total submissiveness rating for the 40 classes he has already taken is \( 40 \times 3.975 \). 3. The total submissiveness rating for the additional \( x \) classes, assuming he receives a rating of 4.0 in each, is \( 4x \). 4. The new tot...	[ { "content": "At the Mountain School, Micchell is assigned a submissiveness rating of $3.0$ or $4.0$ for each class he takes. His college potential is then defined as the average of his submissiveness ratings over all classes taken. After taking 40 classes, Micchell has a college potential of $3.975$ ....
aops_forum	p1. Sujay sees a shooting star go across the night sky, and took a picture of it. The shooting star consists of a star body, which is bounded by four quarter-circle arcs, and a triangular tail. Suppose $AB = 2$ , $AC = 4$ . Let the area of the shooting star be $X$ . If $6X = a-b\pi$ for positive integers $a, ...	1. First, we need to determine the total number of distinct arrangements of the letters in "DISCUSSIONS". The word "DISCUSSIONS" has 11 letters where the letter 'S' appears 4 times, 'I' appears 2 times, and the other letters appear once each. The formula for the number of distinct permutations of a multiset is given by...	[ { "content": "p1. Sujay sees a shooting star go across the night sky, and took a picture of it. The shooting star consists of a star body, which is bounded by four quarter-circle arcs, and a triangular tail. Suppose $AB = 2$ , $AC = 4$ . Let the area of the shooting star be $X$ . If $6X = a-b\\pi$ for ...
aops_forum	Indonesia Regional <details><summary>MO</summary>also know as provincial level, is a qualifying round for National Math Olympiad</details> Year 2019 <details><summary>Part A</summary>Part B consists of 5 essay / proof problems, posted [here](https://artofproblemsolving.com/community/c4h2671394p23150636)</details> T...	To solve the problem, we need to find the value of \((r-s)^2\) given that \(r\), \(s\), and \(1\) are the roots of the cubic equation \(x^3 - 2x + c = 0\). 1. Identify the value of \(c\): Since \(1\) is a root of the equation, we substitute \(x = 1\) into the equation: \[ 1^3 - 2 \cdot 1 + c = 0 \implies ...	[ { "content": "Indonesia Regional <details><summary>MO</summary>also know as provincial level, is a qualifying round for National Math Olympiad</details> \nYear 2019 <details><summary>Part A</summary>Part B consists of 5 essay / proof problems, posted [here](https://artofproblemsolving.com/community/c4h2671394...
aops_forum	The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at $(5, 1)$ and put poison in two pipes, one along the line $y=x$ and one along the line $x=7$ . However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest...	1. Reflect the line \( x = 7 \) over the line \( y = x \): - The reflection of the line \( x = 7 \) over the line \( y = x \) is the line \( y = 7 \). 2. Reflect the point \((7, 1)\) about the line \( y = x \): - The reflection of the point \((7, 1)\) about the line \( y = x \) is \((1, 7)\). 3. **Calcu...	[ { "content": "The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at $(5, 1)$ and put poison in two pipes, one along the line $y=x$ and one along the line $x=7$ . However, they need to get the job done quickly before Captain Hammer catches the...
aops_forum	Let $ABC$ be a triangle and $\ell$ be a line parallel to $BC$ that passes through vertex $A$ . Draw two circles congruent to the circle inscribed in triangle $ABC$ and tangent to line $\ell$ , $AB$ and $BC$ (see picture). Lines $DE$ and $FG$ intersect at point $P$ . Prove that $P$ lies on $BC$ i...	1. Setup and Definitions: - Let \( \triangle ABC \) be a triangle with \( \ell \) as a line parallel to \( BC \) passing through vertex \( A \). - Draw two circles congruent to the incircle of \( \triangle ABC \) and tangent to \( \ell \), \( AB \), and \( AC \). Let these circles be denoted as \( \omega_1 \)...	[ { "content": "Let $ABC$ be a triangle and $\\ell$ be a line parallel to $BC$ that passes through vertex $A$ . Draw two circles congruent to the circle inscribed in triangle $ABC$ and tangent to line $\\ell$ , $AB$ and $BC$ (see picture). Lines $DE$ and $FG$ intersect at point $P$ . Prove tha...
aops_forum	Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrila...	1. We start with the given quartic equation: \[ r^4z^4 + (10r^6 - 2r^2)z^2 - 16r^5z + (9r^8 + 10r^4 + 1) = 0 \] 2. We factor the quartic polynomial. Notice that the constant term can be factored as: \[ 9r^8 + 10r^4 + 1 = (9r^4 + 1)(r^4 + 1) \] 3. Assume the quartic can be factored into two quadratic...	[ { "content": "Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \\[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \\] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate\r\nand the imaginary part as the y-coordinate). Show that the a...