id int64 0 66.6k | problem stringlengths 11 2.33k | solution stringlengths 35 9.02k | source stringclasses 9
values | answer stringlengths 1 378 | prompt stringlengths 802 3.12k |
|---|---|---|---|---|---|
0 | Let \( q \) be a 7th root of unity. Calculate \(\frac{q}{1+q^{2}} + \frac{q^{2}}{1+q^{4}} + \frac{q^{3}}{1+q^{6}} \). | Given \( q \) is the 7th root of unity. To solve the expression
\[
\frac{q}{1 + q^2} + \frac{q^2}{1 + q^4} + \frac{q^3}{1 + q^6},
\]
we need to consider the cases when \( q = 1 \) and \( q \neq 1 \).
1. **Case when \( q = 1 \):**
Substitute \( q = 1 \) into the expression:
\[
\frac{1}{1 + 1^2} + \frac{1^... | olympiads | -2 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
1 | Square $ABCD$ is inscribed in the region bound by the parabola $y = x^2 - 10x + 21$ and the $x$-axis. Find the area of square $ABCD$. | First, identify the axis of symmetry of the parabola $y = x^2 - 10x + 21$. Solving for the vertex,
\[x = \frac{-(-10)}{2\cdot1} = 5.\]
Let $2t$ be the side length of the square. Then the coordinates of the vertices of the square can be described as:
\begin{align*}
A &= (5 - t, 0), \\
B &= (5 + t, 0), \\
C &= (5 + t, -... | synthetic_math | 24 - 16\sqrt{5} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
2 | Itzayana is 4 inches taller than Zora, who is 8 inches shorter than Brixton. If Zara is 64 inches tall and has the same height as Brixton, and Jaxon is 170 centimeters tall, calculate the average height of the five people in inches, taking into account the necessary unit conversion. | First, let's find out the heights of Itzayana, Zora, and Brixton in inches.
We know that Zara is 64 inches tall and has the same height as Brixton. So, Brixton is also 64 inches tall.
Zora is 8 inches shorter than Brixton, so Zora's height is:
64 inches (Brixton's height) - 8 inches = 56 inches
Itzayana is 4 inches ... | orca_math | 62.2 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
3 | Given that $x$ changes from $0$ to $2 \pi$, find the range of values for $y=\sin ^{2} x-3 \sin x+2$. | Given the function \( y = \sin^2 x - 3 \sin x + 2 \), we aim to find how its values change as \( x \) varies from \( 0 \) to \( 2\pi \).
1. **Define a New Variable**:
Let \( t = \sin x \). Then the expression for \( y \) can be rewritten in terms of \( t \):
\[
y = t^2 - 3t + 2
\]
We introduce a new fun... | olympiads | 2 \rightarrow 0 \rightarrow 6 \rightarrow 2 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
4 | Find the smallest positive real number $c$, such that for all nonnegative real numbers $x$ and $y$,
\[\sqrt[3]{xy} + c |x - y| \ge \frac{x + y}{2}.\] | Let $x = a^3$ and $y = b^3$ where $a, b \ge 0$. Then the equation becomes
\[a b + c |a^3 - b^3| \ge \frac{a^3 + b^3}{2}.\]
Assuming $a = b$, both sides reduce to $a^3$, thus the inequality holds.
Now let's assume \(a < b\). Thus \(|a^3 - b^3| = b^3 - a^3\) and the equation becomes:
\[ab + c(b^3 - a^3) \ge \frac{a^3 +... | synthetic_math | \frac{1}{3} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
5 | There are two pairs $(x,y)$ of real numbers that satisfy the equation $x+y = 3xy = 12$. If the solutions for $x$ can be expressed in the form $x = \frac{a \pm b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, find the value of $a + b + c + d$. | Initially, assume $x + y = 12$; thus $y = 12 - x$. Substitute $y$ in the second equation $3xy = 12$ giving $3x(12 - x) = 12$. This simplifies to:
$$ 36x - 3x^2 = 12 \quad \Rightarrow \quad 3x^2 - 36x + 12 = 0 $$
Using the quadratic formula, $x = \frac{-(-36) \pm \sqrt{(-36)^2 - 4 \cdot 3 \cdot 12}}{2 \cdot 3}$:
$$ x =... | synthetic_math | 14 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
6 | Kevin Kangaroo starts hopping on a number line at 0 aiming to reach 2 but each time can hop only $\frac{1}{4}$ of the distance remaining. If each subsequent hop is $\frac{1}{4}$ of the remaining distance, how far has he hopped after six hops? Express your answer as a common fraction. | In each hop, Kevin hops $\frac{1}{4}$ of the remaining distance. After each hop, he has $\frac{3}{4}$ of the distance left to cover.
1. First hop: $\frac{1}{4} \times 2 = \frac{2}{4} = \frac{1}{2}$
2. Remaining distance = $2 - \frac{1}{2} = \frac{3}{2}$.
3. Second hop: $\frac{1}{4} \times \frac{3}{2} = \frac{3}{8}$.
4... | synthetic_math | 1 \frac{297}{1024} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
7 | Given a sequence $\{a\_n\}$ that satisfies $a\_n= \begin{cases} n,&n\text{ is odd} \\ a\_{\frac{n}{2}},&n\text{ is even} \end{cases}$, let $S\_n$ be the sum of the first $n$ terms of the sequence. Calculate the value of $S\;_{2^{2016}-1}$. | Since $a\_n= \begin{cases} n,&n\text{ is odd} \\ a\_{\frac{n}{2}},&n\text{ is even} \end{cases}$, we can deduce the following:
- If $n$ is odd, then $a\_n=n$.
- If $n$ is even, then $a\_n=a\_{\frac{n}{2}}$.
Thus, $S\;_{2^{2016}-1}=(a\_1+a\_3+\dots+a\_{2^{2016}-1})+S\;_{2^{2015}-1}= \dfrac{2^{2015}×(1+2^{2016}-1)}{2}+S... | cn_k12 | S\;_{2^{2016}-1}= \dfrac{4^{2016}-1}{3} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
8 | Solve the equation
\[\sqrt{2 + \sqrt{3 + \sqrt{x}}} = \sqrt[4]{2 + \sqrt{x}}.\] | Let \( y = \sqrt[4]{2 + \sqrt{x}} \). Then, \( y^4 = 2 + \sqrt{x} \), which allows rewriting the given equation as
\[\sqrt{2 + \sqrt{3 + (\sqrt{x})}} = y.\]
Upon squaring both sides, we obtain:
\[2 + \sqrt{3 + \sqrt{x}} = y^2,\]
which yields:
\[\sqrt{3 + \sqrt{x}} = y^2 - 2.\]
Squaring again gives:
\[3 + \sqrt{x} = (y^... | synthetic_math | 2^{1/4} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
9 | From Zlatoust to Miass, "GAZ", "MAZ", and "KamAZ" started simultaneously. After reaching Miass, "KamAZ" immediately turned back and met "MAZ" at 18 km and "GAZ" at 25 km from Miass. After reaching Miass, "MAZ" also immediately turned back and met "GAZ" at 8 km from Miass. What is the distance from Zlatoust to Miass? |
We need to find the distance between Zlataust and Miass, denoted as \(x\) km. Let the speeds of the trucks be:
- GAZ: \(g\) km/h
- MAZ: \(m\) km/h
- KamAZ: \(k\) km/h
To find \(x\), we must analyze the encounters of the trucks.
1. **KamAZ** reaches Miass first, then returns and meets MAZ 18 km from Miass:
\[
\... | olympiads | 60 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
10 | If there are several piles of stones on the table, it is considered that there are many stones on the table if you can find 50 piles and number them from 1 to 50 such that the first pile has at least one stone, the second one has at least two stones, ..., the fiftieth one has at least fifty stones. Initially, there are... | ### Problem:
On a table, there are several piles of stones. It is said that there are many stones on the table if it is possible to find 50 piles and number them from 1 to 50 such that in the first pile there is at least one stone, in the second pile there are at least two stones, ..., and in the fiftieth pile, there a... | olympiads | 5099 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
11 | Let $z$ be a complex number such that $|z| = 2$. Determine the maximum value of
\[
|(z - 2)^3 (z + 1)|.
\] | Let $z = x + yi$, where $x$ and $y$ are real numbers. As $|z| = 2$, then $x^2 + y^2 = 4$. Compute
\[
|z - 2| = |(x - 2) + yi| = \sqrt{(x - 2)^2 + y^2},
\]
and
\[
|z + 1| = |(x + 1) + yi| = \sqrt{(x + 1)^2 + y^2}.
\]
Therefore,
\[
|(z - 2)^3 (z + 1)| = |(x - 2) + yi|^3 |(x + 1) + yi| = \sqrt{((x - 2)^2 + y^2)^3 ((x + 1)... | synthetic_math | 8\sqrt{5} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
12 | two trains each 170 m in length each , are running on two parallel lines in opposite directions . if one goes at the speed of 55 km / h while the other travels at 50 km / h . how long will it take for them to pass each other completely . | To find out how long it will take for the two trains to pass each other completely, we need to calculate the relative speed at which they are approaching each other and then determine the time it takes for them to cover the combined length of both trains.
Since the trains are moving in opposite directions, we add thei... | orca_math | 11.65 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
13 | Evaluate the following expression:
(2x^3 - 3a^4) / (y^2 + 4z^5) + c^4 - d^2 where x = 0.66, a = 0.1, y = 0.66, z = 0.1, c = 0.066, and d = 0.1. | First, let's substitute the given values into the expression:
(2(0.66)^3 - 3(0.1)^4) / ((0.66)^2 + 4(0.1)^5) + (0.066)^4 - (0.1)^2
Now, let's calculate each part step by step:
For the numerator:
2(0.66)^3 = 2 * (0.66 * 0.66 * 0.66) = 2 * 0.287496 = 0.574992
3(0.1)^4 = 3 * (0.1 * 0.1 * 0.1 * 0.1) = 3 * 0.0001 = 0.000... | orca_math | 1.309091916 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
14 | Consider a triangle with vertices at $(0,0)$, $(30,0)$, and $(12,20)$. 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. Calculate the volume of this pyramid. | 1. **Vertices and Midpoints**:
- Triangle vertices: \( A = (0,0), B = (30,0), C = (12,20) \)
- Midpoints: \( D = \left(\frac{30+12}{2}, \frac{0+20}{2}\right) = (21,10), E = \left(\frac{12+0}{2}, \frac{20+0}{2}\right) = (6,10), F = \left(\frac{30+0}{2}, \frac{0+0}{2}\right) = (15,0) \)
2. **Area of the Base Trian... | synthetic_math | 365.5 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
15 | Determine the number of subsets \( S \) of \(\{1,2,3, \ldots, 10\}\) with the following property: there exist integers \(a < b < c\) with \(a \in S, b \notin S, c \in S\). |
To solve this problem, we need to count the number of subsets \( S \) of \( \{1, 2, 3, \ldots, 10\} \) that satisfy the condition that there exist integers \( a < b < c \) such that \( a \in S \), \( b \notin S \), and \( c \in S \).
1. **Calculate the Total Number of Subsets**:
The set \( \{1, 2, 3, \ldots, 10\}... | olympiads | 968 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
16 | Prove that angle B is a right angle.
Draw diagonals FB, BD, and DF. The three triangles FAB, BCD, and DEF are congruent by two sides and the included angle. From the congruence of the triangles, it follows that FB = BD = DF. Thus, triangle FBD is equilateral, and its angles are each equal to \(60^\circ\). All the ment... |
1. **Define the variables and set the problem:**
Given that \( a, b, c, d \) are four distinct integers, we need to understand the relationships these integers have with their complements relative to 8, and use this relationship to determine their sum.
2. **Complement transformation:**
Since \( a, b, c, \) and ... | olympiads | 32 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
17 | From the sequence of natural numbers \(1, 2, 3, 4, 5, \ldots\), remove each multiple of 4 and 7, but retain any multiple of 5. What is the 1995th remaining number? | 1. We need to determine the 1995th number in the sequence of natural numbers after removing all multiples of 4 and 7 but keeping all multiples of 5.
2. Identify and calculate the smallest common multiple (LCM) for 4 and 7, which is 28.
3. Next, consider how these removals affect the natural number series up to a certai... | olympiads | 2795 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
18 | Consider a right pyramid with a regular hexagonal base where each side of the base measures 8 cm. The peak of this pyramid is 15 cm above the center of its base. On top of this pyramid, there is another smaller inverted right pyramid with a pentagonal base where each side measures 5 cm and its peak is 10 cm below the p... | 1. **Calculate the length of the hexagonal base pyramid's edges:**
The inradius \( r_1 \) of the hexagon can be calculated as:
\[ r_1 = \frac{s}{2} \cdot \frac{1}{\tan(\pi/6)} = \frac{8}{2} \cdot \frac{1}{\tan(\pi/6)} \approx 4.62 \text{ cm} \]
Each slant height \( l_1 \) of the hexagonal pyramid:
\[ l_1 = ... | synthetic_math | 220 \text{ cm} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
19 | From a point outside a circle, two tangent lines and a secant line are drawn such that the points of tangency and the points where the secant intersects the circle are the vertices of a certain trapezoid. Find the ratio of the bases of the trapezoid, given that the angle between the tangents is $60^{\circ}$. |
Let's solve the given problem step-by-step. Here, we are given a point outside a circle from which two tangents and a secant are drawn, with the points of tangency and intersection forming a trapezoid. The angle between the tangents is given as $60^\circ$, and we need to find the ratio of the bases of the trapezoid.
... | olympiads | \frac{3 - \sqrt{5}}{2} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
20 | Fill the numbers 1 to 6 into the 6 circles shown in the figure, such that the sum of the numbers in the circles on each straight line is equal (the figure has 3 straight lines with 3 circles each, and 2 straight lines with 2 circles each). Find the two-digit number $\overline{\mathrm{AB}}=$ . | ### Problem:
Fill the circles with the numbers from 1 to 6 such that the sum of the numbers on each straight line is equal. The diagram has three straight lines with three circles each, and two straight lines with two circles each. Then find the two-digit number $\overline{AB}$ created by placing the numbers $A$ and $B... | olympiads | 63 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
21 | Given that 2.68 x 0.74 = a, find the value of b = a^2 + cos(a) without using a calculator. | To find the value of b = a^2 + cos(a) without using a calculator, we need to first calculate the value of a, which is the product of 2.68 and 0.74. Since we're not using a calculator, we'll have to do this multiplication by hand.
2.68
x 0.74
--------
1888 (2.68 x 4, with two decimal places to be adjusted)
1340 (2.68 x... | orca_math | 2.96535 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
22 | An item's price goes through several increases and decreases:
1. It changes from $150 to $120
2. Then it increases from $120 to $130
3. Later, it decreases from $130 to $100
4. Finally, it increases from $100 to $80
Calculate the percentage changes for each of these price changes and rank them in order of their magn... | To calculate the percentage change for each price change, we use the formula:
Percentage Change = [(New Price - Old Price) / Old Price] * 100
1. The price changes from $150 to $120:
Percentage Change = [($120 - $150) / $150] * 100
Percentage Change = [(-$30) / $150] * 100
Percentage Change = -0.2 * 100
Percentage Cha... | orca_math | -20\% | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
23 | Let $\mathbf{B} =\begin{pmatrix} 1 & 4 \\ -1 & 3 \end{pmatrix}.$ Then there exist scalars $r$ and $s$ such that
\[\mathbf{B}^6 = r \mathbf{B} + s \mathbf{I}.\] Enter the ordered pair $(r,s).$ | First, compute $\mathbf{B}^2$:
\begin{align*}
\mathbf{B}^2 &= \begin{pmatrix} 1 & 4 \\ -1 & 3 \end{pmatrix} \begin{pmatrix} 1 & 4 \\ -1 & 3 \end{pmatrix} \\
&= \begin{pmatrix} 1\cdot1 + 4\cdot(-1) & 1\cdot4 + 4\cdot3 \\ -1\cdot1 + 3\cdot(-1) & -1\cdot4 + 3\cdot3 \end{pmatrix} \\
&= \begin{pmatrix} 1 - 4 & 4 + 12 \\ -1 ... | synthetic_math | (3995,1558) | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
24 | A particle moves so that it is at \((3t + 10, 6t - 16)\) at time \(t\). After undergoing a 45-degree counterclockwise rotation about the origin, find the speed of the rotated particle. | Let's first find the speed of the particle before the rotation. At time \(t = k\), the particle is at
\[
(3k + 10, 6k - 16).
\]
At time \(t = k + 1\), the particle is at
\[
(3(k + 1) + 10, 6(k + 1) - 16).
\]
These simplify to
\[
(3k + 13, 6k - 10).
\]
The change in the \(x\)-coordinate is \(3k + 13 - (3k + 10) = 3\), a... | synthetic_math | 3\sqrt{10} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
25 | Given a scalene acute triangle \(ABC\). Altitudes \(BB_1\) and \(CC_1\) are drawn, intersecting at point \(H\). Circles \(\omega_1\) and \(\omega_2\) with centers at \(H\) and \(C\) respectively touch line \(AB\). From point \(A\), tangents are drawn to \(\omega_1\) and \(\omega_2\) different from \(AB\). Let \(D\) and... |
1. Consider the isosceles acute-angled triangle \( ABC \) with heights \( B B_1 \) and \( C C_1 \) intersecting at the orthocenter \( H \). There are two circles, \(\omega_1\) with center \( H \) and \(\omega_2\) with center \( C \), both tangent to the line \( AB \).
2. Tangents are drawn from point \( A \) to \(\om... | olympiads | 180^\circ | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
26 | Let \( S = \{1, 2, 3, \ldots, 280\} \). Find the minimal natural number \( n \) such that in any \( n \)-element subset of \( S \) there are five numbers that are pairwise relatively prime. |
1. **Define the Set**:
Let \( S = \{1, 2, 3, \ldots, 280\} \).
2. **Define \( A_m \)**:
Let \( A_m \) be the subset of \( S \) consisting of elements divisible by \( m \).
3. **Use Inclusion-Exclusion Principle**:
Using the inclusion-exclusion principle, we compute the number of elements in \( S \) that are... | olympiads | 217 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
27 | Two polynomials $x^3 + Cx^2 + 15 = 0$ and $x^3 + Dx + 35 = 0$ share a single common root. Find the value of this common root if it is also known that the product of the roots of the first polynomial is $-15$ and of the second polynomial is $-35$. | Let the common root between $x^3 + Cx^2 + 15 = 0$ and $x^3 + Dx + 35 = 0$ be $p$. Let the other roots of the first polynomial be $q$ and $r$ and the other roots of the second polynomial be $s$ and $t$.
Using Vieta's formulas, for the first polynomial:
\[
p + q + r = -C \quad (1),
\]
\[
pqr = -15 \quad (2).
\]
For the ... | synthetic_math | p = \sqrt[3]{525} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
28 | Real numbers $x$ and $y$ satisfy the equations:
1. $x^2 + y^2 = 8x + 4y - 20$
2. $x + y = 6$ | For the first equation, we rewrite it as:
\[x^2 - 8x + y^2 - 4y + 20 = 0.\]
Completing the square:
\[(x - 4)^2 - 16 + (y - 2)^2 - 4 + 20 = 0,\]
\[(x - 4)^2 + (y - 2)^2 = 0.\]
This implies that \(x = 4\) and \(y = 2\).
Now, we verify this solution with the second equation:
\[x + y = 6.\]
Substituting the values \(x = 4... | synthetic_math | 6 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
29 | When some fraction of the votes on a certain resolution have been counted, 3/4 of those counted are in favor of the resolution. The fraction x of the remaining votes must be against the resolution so that the total count will result in a vote of 2 to 1 against the resolution. The fraction x is 0.7857142857142856. What ... | Let's denote the total number of votes as V and the number of votes counted so far as C. We know that 3/4 of the votes counted are in favor of the resolution, so the number of votes in favor is (3/4)C. The number of votes against the resolution from the counted votes is then C - (3/4)C = (1/4)C.
We are told that the f... | orca_math | 38.57\% | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
30 | Find the smallest positive integer whose cube ends in $544$. | We require $n^3 \equiv 544 \pmod{1000}$, which requires $n^3 \equiv 0 \pmod 8$ because 544 is divisible by 8, and $n^3 \equiv 544 \pmod{125}$. The last digit of $n$ must be 4 to satisfy the condition that $n^3 \equiv 4 \pmod{5}$ (since $4^3 = 64 \equiv 4 \pmod{5}$). Setting $n = 5a + 4$, expanding and substituting, we ... | synthetic_math | 69 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
31 | Pipe P can fill a tank in 6 hours, pipe Q in some hours, and pipe R in 24 hours. If all the pipes are open, the tank will be filled in 3.4285714285714284 hours. In how many hours can pipe Q fill the tank? | Let's denote the rate at which pipe Q can fill the tank as \( \frac{1}{x} \) tanks per hour, where \( x \) is the number of hours it takes for pipe Q to fill the tank on its own.
The rate at which pipe P can fill the tank is \( \frac{1}{6} \) tanks per hour, since it can fill the tank in 6 hours.
The rate at which pi... | orca_math | 8 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
32 | Find the number of eight-digit numbers for which the product of the digits equals 7000. The answer must be given as an integer. |
To find the number of eight-digit numbers whose digits multiply to 7000, we start by examining the prime factorization of 7000:
\[
7000 = 7 \cdot 2^3 \cdot 5^3
\]
By breaking down these prime factors, we can deduce which digits (ranging from 0 to 9) can be used and in what quantity.
We consider the following possib... | olympiads | 5600 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
33 | In a 5-week period, an airplane pilot flew the following miles and stopovers per week:
- Week 1:
- Monday: 1134 miles with a 3-hour stopover
- Wednesday: 1475 miles with a 2.5-hour stopover
- Friday: 1290 miles with a 1-hour stopover
- Week 2:
- Tuesday: 1550 miles with a 1.5-hour stopover
- Thursday: 1340 ... | First, let's calculate the total miles flown by the pilot over the 5-week period by adding up the miles flown each day:
Week 1:
- Monday: 1134 miles
- Wednesday: 1475 miles
- Friday: 1290 miles
Week 2:
- Tuesday: 1550 miles
- Thursday: 1340 miles
- Saturday: 1444 miles
Week 3:
- Sunday: 1200 miles
- Tuesday: 1000 mi... | orca_math | 2:25 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
34 | Find the largest integer less than 2012 all of whose divisors have at most two 1's in their binary representations. |
1. **Identify Constraints of Good Numbers:**
Define a number as "good" if all its positive divisors have at most two 1's in their binary representations. If \( p \) is an odd prime divisor of a good number, then \( p \) must be of the form \( 2^k + 1 \).
2. **List of Appropriate Primes:**
The only such primes l... | olympiads | 1536 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
35 | Find the smallest positive integer $M$ such that among the numbers $M$, $M+1$, $M+2$, and $M+3$, one is divisible by $2^3$, one by $3^2$, one by $5^2$, and one by $7^2$. | Begin by searching for multiples of $7^2 = 49$. The first multiple is 49 itself. We check the divisibility of $49, 50, 51, 52$:
- $49 = 7^2$
- $50 = 2 \times 5^2$
- $51 = 3 \times 17$ (not divisible by $3^2$ or $2^3$)
- $52 = 2^2 \times 13$ (not divisible by $3^2$ or $2^3$)
Next, try the next multiple of $7^2$, which ... | synthetic_math | 196 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
36 | The average height of 35 boys in a class was calculated as 181 cm. It was later found that the height of one of the boys in the class was wrongly written as 166 cm whereas his actual height was a different value. The actual average height of the boys in the class is 179 cm. What is the actual height of the boy whose he... | Let's denote the actual height of the boy whose height was wrongly written as \( x \) cm.
The incorrect total height for all 35 boys was calculated as:
\( 35 \times 181 = 6335 \) cm
The correct total height for all 35 boys should be:
\( 35 \times 179 = 6275 \) cm
The difference between the incorrect and correct tota... | orca_math | 226 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
37 | Ashok and Pyarelal invested money together in a business. The share of Ashok's capital is 1/9 of that of Pyarelal. Due to market fluctuations, Ashok experienced a loss of 12% on his investment, while Pyarelal's loss was 9% of his investment. If their total combined loss amounted to Rs. 2,100, what is the loss incurred ... | Let's denote Pyarelal's investment as P. According to the information given, Ashok's investment is 1/9 of Pyarelal's investment, so Ashok's investment is P/9.
The total loss is the sum of the losses of both Ashok and Pyarelal, which is Rs. 2,100.
Ashok's loss is 12% of his investment, so his loss is 0.12 * (P/9).
Py... | orca_math | 1,829.32 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
38 | ( 0.96 ) ( power 3 ) - ( 0.1 ) ( power 3 ) / ( 0.96 ) ( power 2 ) + 0.096 + ( 0.1 ) ( power 2 ) is : | To solve this expression, we need to follow the order of operations, which is parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). This is often remembered by the acronym PEMDAS.
Let's solve it step by step:
1. Calculate the exponents:
(0.96)^... | orca_math | 0.989651 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
39 | Find a three-digit number, knowing that the number of its tens is the geometric mean of the hundreds and units digits. If the hundreds and units digits are swapped and the new number is subtracted from the original number, the difference will be 297. |
To find the three-digit number based on the given conditions, let's set up the problem step-by-step:
1. **Define the number**:
Let the number be represented as $\overline{xyz}$, where $x$, $y$, and $z$ are the digits of the number, $x$ is the hundreds digit, $y$ is the tens digit, and $z$ is the units digit.
2. *... | olympiads | 300 \text{ or } 421 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
40 | $A, B, C, D,$ and $E$ are collinear in that order such that $AB = BC = 1, CD = 3,$ and $DE = 8$. If $P$ can be any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$? | Let $P$ be a point on line $AE$, and let the coordinates of $A$ be $0$, $B$ be $1$, $C$ be $2$, $D$ be $5$, and $E$ be $13$. The distance $AP = r$, $BP = |r - 1|$, $CP = |r - 2|$, $DP = |r - 5|$, $EP = |r - 13|$. Squaring these distances, we get:
\[ AP^2 + BP^2 + CP^2 + DP^2 + EP^2 = r^2 + (r - 1)^2 + (r - 2)^2 + (r - ... | synthetic_math | 88.2 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
41 | Calculate the product of \(1101_2 \cdot 111_2\) and express your result in base 2. | We will multiply two binary numbers similar to the decimal multiplication method:
\[
\begin{array}{r}
\quad & & 1 & 1 & 0 & 1_2 \\
\times & & & 1 & 1 & 1_2 \\
\hline
\quad & & 1 & 1 & 0 & 1_2 \\
\quad & & 1 & 1 & 0 & 1_2 & (shift \, left \, by \, 1) \\
+ & 1 & 1 & 0 & 1 & 0 & 0_2 & (shift \, lef... | synthetic_math | 1011110_2 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
42 | During 1 week in September, the temperature fluctuated severely, and various weather factors were recorded in addition to the daily temperatures. On Sunday, the temperature started at 40 degrees, with a humidity of 50% and wind speed of 15 mph. Monday had a temperature of 50 degrees, 60% humidity, and 10 mph winds. On ... | To calculate the average weather conditions for the week, we will need to sum up the values for each condition and then divide by the number of days, which is 7.
**Average Temperature:**
We will calculate the average daily temperature first and then find the average for the week. The average daily temperature is calcu... | orca_math | 14.29 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
43 | One angle of a parallelogram is 150 degrees, and two consecutive sides have lengths of 10 inches and 18 inches. What is the area of the parallelogram? Express your answer in simplest radical form. | Given a parallelogram where one angle measures 150 degrees, the adjacent angle will measure $180^\circ - 150^\circ = 30^\circ$ due to the property of parallelograms where consecutive angles are supplementary.
Consider parallelogram $ABCD$ with $\overline{AB} = 18$ inches and $\overline{BC} = 10$ inches, and angle $ABC... | synthetic_math | 60\sqrt{3} \text{ square inches} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
44 | In an isosceles right triangle $ABC$ with $AB = AC = 10$, a point $P$ is placed randomly inside the triangle. What is the probability that the area of triangle $PBC$ is less than one third of the area of triangle $ABC$? | Let $h$ be the distance from $P$ to line $BC$. The area of triangle $ABC$ is $\frac{1}{2} \times 10 \times 10 = 50$. The area of triangle $PBC$ is $\frac{1}{2} \times h \times BC$. For the area of $PBC$ to be less than one third of $ABC$, we need $\frac{1}{2} \times h \times 10 < \frac{1}{3} \times 50$, which simplifie... | synthetic_math | \frac{2222}{5000} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
45 | In a two-day problem-solving competition, Gamma and Delta participated and attempted questions worth a total of 500 points. On the first day, Gamma scored 180 points out of 280 points attempted, and on the second day, he scored 120 points out of 220 points attempted. Delta, who also divided his attempts across the two ... | Let $\frac{x}{y}$ and $\frac{z}{w}$ denote Delta's scores and attempts on day 1 and day 2 respectively, with conditions:
- $0 < \frac{x}{y} < \frac{9}{14}$, and $0 < \frac{z}{w}< \frac{6}{11}$.
- $\frac{14}{9}x < y$ and $\frac{11}{6}z < w$.
- Adding, we have $y + w = 500$.
Thus, $\frac{14}{9}x + \frac{11}{6}z < 500$. ... | synthetic_math | \frac{409}{500} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
46 | Madeline has 5 boxes with 24 crayons in each. She noticed that 5/8 of the crayons in the 2 boxes were not yet used. In the 2 other boxes, only 2/3 of the crayons were used while the last box was not entirely used. How many unused crayons did Madeline have? | Let's calculate the number of unused crayons step by step.
First, we find out how many crayons are in 5 boxes:
5 boxes * 24 crayons/box = 120 crayons
Now, let's find out how many crayons were not used in the first 2 boxes:
5/8 of the crayons in each of the 2 boxes were not used, so:
2 boxes * 24 crayons/box = 48 cray... | orca_math | 47 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
47 | A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F,$ in such a way that t... | To solve this problem, let's break it down step by step, following the given solution closely but with more detailed explanations.
**Step 1: Calculate the Volume and Surface Area of the Original Cone**
The volume of a right circular cone is given by $V = \frac{1}{3} \pi r^2 h$. For our cone, with a radius ($r$) of 3 ... | math | 512 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
48 | Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $? | To solve for the number of complex numbers $z$ such that $\text{Im}(z) > 0$ and both the real and imaginary parts of $f(z) = z^2 + iz + 1$ are integers with absolute value at most $10$, we proceed as follows:
1. **Express $f(z)$ in terms of $c = a + bi$**:
- We have $f(z) = z^2 + iz + 1 = c = a + bi$, where we seek... | math | 399 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
49 | Find the number of integers $ c$ such that $ \minus{}2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 \plus{} c$ is a multiple of $ 2^{2007}$ . | To solve the problem, we need to find the number of integers \( c \) such that \( -2007 \leq c \leq 2007 \) and there exists an integer \( x \) such that \( x^2 + c \) is a multiple of \( 2^{2007} \). This means we need \( x^2 + c \equiv 0 \pmod{2^{2007}} \), or equivalently, \( x^2 \equiv -c \pmod{2^{2007}} \).
1. **... | aops_forum | 670 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
50 | Appending three digits at the end of 2007, one obtains an integer \(N\) of seven digits. In order to get \(N\) to be the minimal number which is divisible by 3, 5, and 7 simultaneously, what are the three digits that one would append? | To solve this problem, we need to find the three digits to append to the number $2007$ such that the resulting seven-digit number $N$ is divisible by $3$, $5$, and $7$. Since these numbers are pairwise coprime, their least common multiple (LCM) is their product:
\[
\text{LCM}(3, 5, 7) = 3 \times 5 \times 7 = 105
\]
T... | olympiads | 075 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
51 | Sean adds up all the squares of even integers from 2 to 300, inclusive. Julie adds up all the squares of odd integers from 1 to 299, inclusive. What is Sean's sum divided by Julie's sum? | First, calculate Sean's sum:
Sean sums the squares of even integers from 2 to 300. This can be written as:
\[ S_{Sean} = 2^2 + 4^2 + \cdots + 300^2 \]
This is a sum of squares of an arithmetic series where each term is an even number given by \(2n\) where \( n = 1, 2, \ldots, 150 \). Therefore,
\[ S_{Sean} = \sum_{n=1... | synthetic_math | 1.0099 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
52 | Determine the value of the expression:
\[3003 + \frac{1}{3} \left( 3002 + \frac{1}{6} \left( 3001 + \frac{1}{9} \left( 3000 + \dots + \frac{1}{3 \times 1000} \cdot 3 \right) \right) \dotsb \right).\] | Let
\[ S = 3003 + \frac{1}{3} \left( 3002 + \frac{1}{6} \left( 3001 + \frac{1}{9} \left( 3000 + \dots + \frac{1}{3 \times 1000} \cdot 3 \right) \right) \dotsb \right) \]
which can be expressed as
\[ S = 3003 + \frac{3002}{3} + \frac{3001}{3 \times 2} + \frac{3000}{3 \times 3} + \dots + \frac{3}{3 \times 1000}. \]
Now... | synthetic_math | 3002.5 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
53 | The sum of $4m$ different positive even numbers and $n$ different positive odd numbers is 1987. For all such values of $m$ and $n$, what is the maximum value of $3m + 4n$? Please prove your conclusion. | Given \( m \) even, \( n \) odd, and the sum of these integers is 1987. We need to determine the maximum value of \( 3m + 4n \).
1. **Set up the equation:**
The problem states that:
\[
\sum_{i=1}^{m}a_i + \sum_{j=1}^{n}b_j = 1987
\]
where \( a_i \) are distinct positive even numbers, and \( b_j \) are... | olympiads | 221 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
54 | Calculate the definite integral:
$$
\int_{0}^{2} \frac{(4 \sqrt{2-x}-\sqrt{x+2}) d x}{(\sqrt{x+2}+4 \sqrt{2-x})(x+2)^{2}}
$$ | 1. Initially, we have the given integral:
\[
\int_{0}^{2} \frac{4 \sqrt{2-x} - \sqrt{x+2}}{(\sqrt{x+2} + 4 \sqrt{2-x})(x+2)^{2}} \, dx
\]
2. To simplify the integrand, let us introduce the substitution:
\[
t = \sqrt{\frac{2-x}{x+2}}
\]
3. Then,
\[
t^2 = \frac{2-x}{x+2}
\]
implies
\[
... | olympiads | \frac{1}{2} \ln 5 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
55 | Let \(a,\) \(b,\) \(c\) be positive real numbers. Find the smallest possible value of
\[10a^3 + 15b^3 + 50c^3 + \frac{1}{5abc}.\] | By AM-GM, we apply it to the cubic terms:
\[10a^3 + 15b^3 + 50c^3 \geq 3 \sqrt[3]{10a^3 \cdot 15b^3 \cdot 50c^3} = 90abc.\]
Next, apply AM-GM to \(90abc + \frac{1}{5abc}\):
\[90abc + \frac{1}{5abc} \geq 2 \sqrt{90abc \cdot \frac{1}{5abc}} = 6\sqrt{18}.\]
Equality holds when \(10a^3 = 15b^3 = 50c^3\) and \(90abc = 5\sqr... | synthetic_math | 6\sqrt{18} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
56 | Find the smallest natural number \( n \) such that if the set \(\{1, 2, \ldots, n\}\) is arbitrarily divided into two nonintersecting subsets, then one of the subsets contains three distinct numbers such that the product of two of them equals the third. |
1. **Problem Restatement:**
We need to find the smallest natural number \( n \) such that if the set \( \{1, 2, \ldots, n\} \) is arbitrarily divided into two nonintersecting subsets, then one of those subsets contains three distinct numbers such that the product of two of them equals the third.
2. **Initial Insig... | olympiads | 96 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
57 | Eight teams play a soccer tournament where each team plays every other team exactly once. There are no ties; each team has a $50\%$ chance of winning any game, and the outcomes are independent. Winners earn a point, losers get none. Team $X$ defeats team $Y$ in the first game. Calculate the probability that team $X$ fi... | Since team $X$ already leads team $Y$ by one point after the first game, we focus on the remaining games:
- Each team plays $7$ games (against the remaining teams).
- The probability that $X$ and $Y$ tie in their remaining games is calculated by the binomial theorem.
We calculate the number of ways $X$ and $Y$ can win... | synthetic_math | \frac{561}{1024} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
58 | Consider a round table with 5 seats. There are no restrictions on how many men or women should be seated. Let $f \geq 0$ denote the number of people sitting next to at least one female and $m \geq 0$ denote the number of people sitting next to at least one male. Determine the number of possible ordered pairs $(f,m)$. | Let's consider different scenarios based on how many women (W) and men (M) are around the table:
1. **0 Women, 5 Men**: Everyone is next to only men, so $(f, m) = (0, 5)$.
2. **1 Woman, 4 Men**: The woman's neighbors are next to both genders, others are next to only men, so $(f, m) = (2, 5)$.
3. **2 Women Together, 3 ... | synthetic_math | 7 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
59 | The angles of a triangle (measured in radians) are \(\alpha, \beta,\) and \(\gamma\), its semiperimeter is \(s\), and the radius of its circumcircle is \(R\). Prove that:
$$
(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha) \leq 4\left(\frac{\pi}{\sqrt{3}}\right)^{3} \frac{R}{s}
$$ |
1. **Simplifying the left-hand side of the inequality:**
The expression given on the left-hand side is:
\[
(\alpha + \beta)(\beta + \gamma)(\gamma + \alpha)
\]
We can use the arithmetic-geometric mean inequality, which states:
\[
\sqrt[3]{a \cdot b \cdot c} \leq \frac{a + b + c}{3}
\]
In th... | olympiads | \frac{\sin \alpha + \sin \beta + \sin \gamma}{3} \leq \sin \left( \frac{\alpha + \beta + \gamma}{3} \right)=\frac{\sqrt{3}}{2} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
60 | If 3 cards from a standard deck are selected randomly, what is the probability that either two queens or at least 1 jack occurs? (There are 4 Jacks, 4 Queens, and 52 total cards in a standard deck.) | Like in the original problem, it's impossible to have both two queens and at least 1 jack in a selection of 3 cards, making the events exclusive. We'll evaluate each case separately.
**Case 1: Two Queens.**
The probability that the first card is a queen is $\frac{4}{52}$. After drawing one queen, 3 queens and 51 cards... | synthetic_math | \frac{217}{882} | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
61 |
Calculate the limit of the numerical sequence:
$$\lim_{n \rightarrow \infty} \frac{\sqrt{(n^{5}+1)(n^{2}-1)} - n \sqrt{n(n^{4}+1)}}{n}$$ |
We need to compute the limit:
\[
\lim_{n \rightarrow \infty} \frac{\sqrt{(n^5 + 1)(n^2 - 1)} - n \sqrt{n(n^4 + 1)}}{n}
\]
1. Start by rationalizing the numerator:
\[
\lim_{n \rightarrow \infty} \frac{\sqrt{(n^5 + 1)(n^2 - 1)} - n \sqrt{n(n^4 + 1)}}{n} = \lim_{n \rightarrow \infty} \frac{\left(\sqrt{(n^5 + 1)(n^2 - 1)... | olympiads | -\infty | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
62 | Find the product of all positive integer values of $c$ such that $4x^2 + 20x + c = 0$ has two real roots. | For a quadratic equation to have two real roots, its discriminant must be positive. The given quadratic equation is $4x^2 + 20x + c = 0$.
1. Compute the discriminant $\Delta$:
\[
\Delta = b^2 - 4ac = 20^2 - 4 \cdot 4 \cdot c = 400 - 16c
\]
2. Set the discriminant greater than zero to find valid values of $c$:
... | synthetic_math | 620448401733239439360000 | [INST]You will be given a mathematical problem. Carefully analyze it before providing a well-structured response.
<thinking>
First, analyze the problem in depth and outline your approach.
This section should capture your reasoning, including any abstract thoughts or potential strategies.
Feel fre... |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 4