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$A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP < DP.$
[asy]
unitsize(0.6 inch);
draw(circle((0,0),1));
draw((-0.3,0.94)--(0.3,-0.94));
draw((-0.7,-0.7)--(0.7,-0.7));
label("$A$",(-0.3,0... | tiger_lab_math | [
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-0.023... | 0.834994 | 0.81674 | 0.018254 | <think>
Okay, so I've got this geometry problem here:
Points A, B, C, and D are on a circle, and segments AC and BD intersect at P. We’re given that AP = 8, PC = 1, and BD = 6. We need to find the length of BP, with the condition that BP is less than DP.
Hmm, let me start by visualizing the problem. The Asymptote cod... | 4 | [
{
"from": "human",
"value": "$A$, $B$, $C$, and $D$ are points on a circle, and segments $\\overline{AC}$ and $\\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP < DP.$\n\n[asy]\n\nunitsize(0.6 inch);\n\ndraw(circle((0,0),1));\ndraw((-0.3,0.94)--(0.3,-0.94));\n... |
Every bag of Dummies (a chocolate candy) contains the same number of pieces.
The Dummies in one bag can't be divided equally among $9$ kids, because after each kid gets the same (whole) number of pieces, $7$ pieces are left over.
If the Dummies in three bags are divided equally among $9$ kids, what is the smallest nu... | nvidia_math | [
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-0.0... | 0.857052 | 0.8433 | 0.013751 | <think>
First, let's understand the problem. We know that each bag of Dummies contains the same number of pieces. When we try to divide the pieces from one bag among 9 kids, we can't do so evenly, and there are 7 pieces left over. This means that the number of pieces in one bag can be expressed as 9 times some whole nu... | 22 | [
{
"from": "human",
"value": "Every bag of Dummies (a chocolate candy) contains the same number of pieces.\n\nThe Dummies in one bag can't be divided equally among $9$ kids, because after each kid gets the same (whole) number of pieces, $7$ pieces are left over.\n\nIf the Dummies in three bags are divided eq... |
In a unique international cooking contest, Emily and three friends decide to compete. They set up stations around a large circular table at equal intervals. If each participant is assigned a specific section where ingredients are placed at the start point of their section, and Emily needs to collect ingredients from ea... | tiger_math | [
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-0.0... | 0.826133 | 0.812186 | 0.013947 | <think>
First, I need to understand the setup of the problem. There are four participants, including Emily, stationed around a circular table with a circumference of 24 meters. The participants are equally spaced, which means the distance between each adjacent section is 6 meters.
Emily needs to collect ingredients fr... | 29 | [
{
"from": "human",
"value": "In a unique international cooking contest, Emily and three friends decide to compete. They set up stations around a large circular table at equal intervals. If each participant is assigned a specific section where ingredients are placed at the start point of their section, and E... |
Given a function $f(x)$, a point $(x_1,y_1)$ lies on its graph. The function is transformed into $g(x) = Af(Bx) + C$, where $A$, $B$, and $C$ are constants. Find a general formula to determine a point that must be on the graph of $g(x)$ based on the point $(x_1,y_1)$ and the constants $A$, $B$, and $C$. | nvidia_math | [
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-0.... | 0.820365 | 0.800725 | 0.01964 | <think>
Alright, so I need to figure out how a point on the graph of a function f(x) transforms when the function is transformed into g(x) = A f(Bx) + C. I'll start by recalling what each transformation does individually, and then see how they combine.
First, I know that when you replace x with Bx in the function, it ... | 14 | [
{
"from": "human",
"value": "Given a function $f(x)$, a point $(x_1,y_1)$ lies on its graph. The function is transformed into $g(x) = Af(Bx) + C$, where $A$, $B$, and $C$ are constants. Find a general formula to determine a point that must be on the graph of $g(x)$ based on the point $(x_1,y_1)$ and the con... |
In Pascal's Triangle, each number is the sum of the number just above it and to the left and the number just above it and to the right. So the middle number in Row 2 is $2$ because $1+1=2.$ What is the sum of the numbers in Row 8 of Pascal's Triangle?
\begin{tabular}{rccccccccccc}
Row 0:& & & & & & 1\\\noalign{\smalls... | tiger_math | [
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-0.... | 0.831932 | 0.817109 | 0.014823 | <think>
First, I recall that the sum of the numbers in any row \( n \) of Pascal's Triangle is \( 2^n \).
Since we're interested in Row 8, I'll calculate \( 2^8 \).
Calculating, \( 2^8 = 256 \).
</think>
To determine the sum of the numbers in Row 8 of Pascal's Triangle, we can use a fundamental property of Pascal's ... | 40 | [
{
"from": "human",
"value": "In Pascal's Triangle, each number is the sum of the number just above it and to the left and the number just above it and to the right. So the middle number in Row 2 is $2$ because $1+1=2.$ What is the sum of the numbers in Row 8 of Pascal's Triangle?\n\n\\begin{tabular}{rcccccc... |
The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The maximum value of $y = ax^2 + bx + c$ is 5, which occurs at $x = 3$. (2) The graph passes through the point $(0,-13)$. If the graph passes through the point $(4,m)$, then what is the value of $m$? | nvidia_math | [
0.02160654403269291,
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-0.01... | 0.837154 | 0.824631 | 0.012523 | <think>
Okay, so I need to find the value of m where the graph of the quadratic equation passes through the point (4, m). The quadratic is given as y = ax² + bx + c, and I have some information that can help me figure this out.
First, the problem says that the maximum value of y is 5, which occurs at x = 3. Since it's... | 42 | [
{
"from": "human",
"value": "The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The maximum value of $y = ax^2 + bx + c$ is 5, which occurs at $x = 3$. (2) The graph passes through the point $(0,-13)$. If the graph passes through the point $(4,m)$, then what is the value of $m$... |
"A right circular cone is inscribed in a right prism as shown, with the base of the cone tangent to (...TRUNCATED) | tiger_math | [0.020394377410411835,0.050477035343647,-0.03099273331463337,-0.046245768666267395,0.024337431415915(...TRUNCATED) | 0.830756 | 0.818047 | 0.01271 | "<think>\nOkay, so I have this problem where a right circular cone is inscribed in a right prism. Th(...TRUNCATED) | 48 | [{"from":"human","value":"A right circular cone is inscribed in a right prism as shown, with the bas(...TRUNCATED) |
"Given \\( n \\) independent and identically distributed continuous random variables \\( X_1, X_2, \(...TRUNCATED) | automath | [-0.006083749234676361,0.012807165272533894,-0.05312690511345863,-0.031233277171850204,0.02904169261(...TRUNCATED) | 0.825065 | 0.804649 | 0.020416 | "<think>\nOkay, so I have this probability problem to solve. It says that given n independent and id(...TRUNCATED) | 54 | [{"from":"human","value":"Given \\( n \\) independent and identically distributed continuous random (...TRUNCATED) |
"Given a continuous function \\( f \\) at a point \\( c \\) in the interval \\([a,b]\\), if a sequen(...TRUNCATED) | automath | [0.005868081469088793,0.044238414615392685,-0.05015231668949127,-0.021221870556473732,0.022982556372(...TRUNCATED) | 0.81169 | 0.796548 | 0.015142 | "<think>\nOkay, so I've got this question here about continuous functions and limits. Let me try to (...TRUNCATED) | 51 | [{"from":"human","value":"Given a continuous function \\( f \\) at a point \\( c \\) in the interval(...TRUNCATED) |
"For a positive integer $n,$ let\n\\[g(n) = \\frac{1}{3^n} + \\frac{1}{4^n} + \\frac{1}{5^n} + \\dot(...TRUNCATED) | nvidia_math | [0.028727248311042786,0.034835971891880035,-0.04149413853883743,-0.04476713389158249,0.0315390601754(...TRUNCATED) | 0.843631 | 0.829607 | 0.014024 | "<think>\nOkay, so I need to find the sum from n equals 1 to infinity of g(2n), where g(n) is define(...TRUNCATED) | 1 | [{"from":"human","value":"For a positive integer $n,$ let\n\\[g(n) = \\frac{1}{3^n} + \\frac{1}{4^n}(...TRUNCATED) |
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