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A rectangular box has a length that is 5 units more than its width. If the volume of the box is 360 cubic units, what are the dimensions of the box? | Let's denote the width of the box as $w$, the length as $l$, and the height as $h$.
Given that the length is 5 units more than the width, we have:
\[ l = w + 5 \]
The volume of a rectangular box is given by the formula:
\[ V = lwh \]
Substitute $l = w + 5$ into the volume formula:
\[ 360 = (w + 5)wh \]
Simplify the... |
In $\triangle{DEF}, DE=15, \angle{D}=45^\circ$, and $\angle{F}=60^\circ$. Let $G, I,$ and $J$ be points on the line $EF$ such that $DG\perp{EF}$, $\angle{DEI}=\angle{DFI}$, and $EJ=FJ$. Point $K$ is the midpoint of the segment $GI$, and point $L$ is on ray $DI$ such that $KL\perp{EF}$. Then, find the length of $DL$. | Since $DG \perp EF$, we know that $DG$ is the altitude of $\triangle DEF$ from point $D$.
Because $\triangle DEF$ is a 45-60-75 triangle, we can use trigonometric ratios to find $DG$.
\[ DG = DE \sin 45^\circ = 15 \cdot \frac{\sqrt{2}}{2} = \frac{15\sqrt{2}}{2} \]
Since $\angle DEI = \angle DFI$, we know that $DI$ is... |
Alex is trying to clean a polluted lake by removing 8 tons of trash every week, but 3 tons of new trash are dumped into the lake every week. If the lake is currently filled with 60 tons of trash, how many weeks will it take Alex to completely clean the lake? | Let's break down the problem. Alex removes 8 tons of trash every week, but 3 tons of new trash are added, resulting in a net removal of 8 - 3 = 5 tons of trash per week.
Since the lake is currently filled with 60 tons of trash, we need to divide 60 by the net removal rate of 5 tons per week to find out how many weeks ... |
A cylindrical barrel with a volume of 1000 cubic feet is filled with water at a rate of 1 cubic foot per minute. However, due to a hole at the bottom, the barrel leaks at a rate of 1/10 of the water level per minute. After how many minutes will the water level in the barrel first drop below 1 foot? | Let's denote the volume of water in the barrel at time $t$ (in minutes) as $V(t)$ and the water level in the barrel as $h(t)$.
Given that the volume of the barrel is 1000 cubic feet and it's being filled at a rate of 1 cubic foot per minute, but leaks at a rate of $\frac{1}{10}h(t)$ cubic feet per minute, we can set u... |
A five-digit palindrome is a number that reads the same forwards and backwards, such as 12321. If the digits 1, 2, 3, 4, and 5 are arranged to form a five-digit palindrome, what is the probability that the resulting number is divisible by 11? Express your answer as a common fraction. | A five-digit palindrome has the form $abcba$, where $a$, $b$, and $c$ are digits.
The number $abcba$ is divisible by $11$ if and only if the alternating sum of its digits is divisible by $11$. The alternating sum of the digits of $abcba$ is:
\[ a - b + c - b + a = 2a - 2b + c \]
We want $2a - 2b + c$ to be divisible ... |
Let $\alpha$ and $\beta$ be real numbers such that
$\ds 2^{\ds\sqrt {\log _2 \cos \alpha}} - \ds 2^{\ds\sqrt {\log _2 \cos \beta}} = \ds\sqrt {6^{\ds\sqrt {\log _2 (\alpha + \beta)}}-1}$.
Find the value of $\alpha+\beta$ that makes this equality possible. | First, observe that for the expression to be defined, we need $\cos \alpha > 0$, $\cos \beta > 0$, and $\alpha + \beta > 0$. We also notice that the right-hand side is non-negative.
Let $x = \sqrt{\log_2 \cos \alpha}$ and $y = \sqrt{\log_2 \cos \beta}$.
The given equation becomes $2^x - 2^y = \sqrt{6^{\sqrt{\log_2 (\... |
A quadrilateral has angles measuring 80, 90, and 120 degrees. Find the measure of the fourth angle. | The sum of the angles in a quadrilateral is 360 degrees. The three known angles add up to 80 + 90 + 120 = 170 + 120 = 290 degrees.
The fourth angle is 360 - 290 = 70 degrees.
The answer is:
\[ \boxed{70} \] |
Let a sequence $a_1, a_2, \cdots$ satisfy the condition
$a_{n+2}=a_{n+1}a_{n}$, $a_1=2$, $a_2=3$. What value should $k$ take
for the sum of the $2^k$-th and the $(2^k+1)$-th terms to equal to $-12$? | The given sequence is $a_{n+2} = a_{n+1}a_{n}$, $a_1 = 2$, and $a_2 = 3$. Let's first find a pattern by calculating a few initial terms of the sequence.
\[ a_1 = 2 \]
\[ a_2 = 3 \]
\[ a_3 = a_2 \times a_1 = 3 \times 2 = 6 \]
\[ a_4 = a_3 \times a_2 = 6 \times 3 = 18 \]
\[ a_5 = a_4 \times a_3 = 18 \times 6 = 108 \]
\[... |
A triangular prism has six congruent square faces, each with side length 1 cm. The prism is cut by three parallel planes that are perpendicular to the bases of the prism, dividing it into four congruent pieces. The area of the cross-section of one of these pieces is 13 cm^2. What is the height of the prism? | Let the height of the prism be $h$.
The volume of the prism is the area of the base times the height:
\[ V = Ah = h \quad \text{since } A = 1 \]
The three parallel planes divide the prism into four congruent pieces, so the volume of one piece is $\frac{h}{4}$.
The area of the cross-section of one piece is $13$ cm$^2... |
Given a right triangle with legs of length $a$ and $b$ and a hypotenuse of length $c$, find the maximum value of $\frac{a+b-c}{a+b+c}$, assuming that the triangle is acute (i.e., all angles are less than 90 degrees).
(Note: This problem requires a different approach, as it involves trigonometric relationships and ineq... | To find the maximum value of $\frac{a+b-c}{a+b+c}$, we can use trigonometric relationships in the right triangle.
Let $\theta$ be the angle opposite side $a$. Then, using the definition of sine and cosine:
\[ \sin(\theta) = \frac{a}{c} \quad \text{and} \quad \cos(\theta) = \frac{b}{c} \]
Substitute these into the giv... |
Let $x,$ $y,$ and $z$ be nonnegative numbers such that $x^2 + y^2 + z^2 = 2.$ Find the maximum value of
\[4xy \sqrt{2} + 2yz \sqrt{3} + xz.\] | ## Step 1: Understand the given problem
We need to find the maximum value of the expression $4xy \sqrt{2} + 2yz \sqrt{3} + xz$ given that $x,$ $y,$ and $z$ are nonnegative numbers such that $x^2 + y^2 + z^2 = 2$.
## Step 2: Consider applying the AM-GM inequality
Since the problem involves maximizing an expression with... |
The sum of the squares of two consecutive prime numbers is equal to the square of a third prime number. Determine the sum of these three prime numbers. | Let the two consecutive prime numbers be $p$ and $p+1$.
The sum of their squares is $p^2 + (p+1)^2 = 2p^2 + 2p + 1$.
This sum is equal to the square of a third prime number, say $q$. Therefore, we have:
\[ 2p^2 + 2p + 1 = q^2 \]
Notice that $2p^2 + 2p + 1 = (2p+1)^2 - 2p$.
Now we need to find prime numbers $p$ and ... |
Let $m = 2^{7} \cdot 3^{11} \cdot 5^{15}$. What is the largest natural-number factor of $m$ that is not a multiple of 2400? | Let $f = 2^{a} \cdot 3^{b} \cdot 5^{c}$ be the largest natural-number factor of $m$ that is not a multiple of 2400.
We have that $2400 = 2^{5} \cdot 3 \cdot 5^{2}$. So we must have that either $a < 5$, $b < 1$, or $c < 2$.
The largest factor will have the largest possible values for $a$, $b$, and $c$. Since $b$ and $... |
A drawer contains 12 colored balls, with 3 of each of 4 different colors (red, blue, green, and yellow). A ball is randomly selected and removed from the drawer. If the remaining balls are then randomly paired up, what is the probability that at least one pair consists of two balls of the same color? | Let's consider the total number of ways to pair the remaining 11 balls after one ball is removed.
Since the order of the pairs does not matter, we can treat this as a combination problem. The number of ways to choose 2 balls out of 11 is $\binom{11}{2}$, but this counts each pair only once. To form pairs, we need to c... |
Consider a triangle ABC with side lengths AB = 8, BC = 5, and AC = 6. Draw circles of radius 2 centered at A and C, and a circle of radius 3 centered at B. Determine the radius of the largest circle that is tangent to all three circles and completely contained within the triangle.
This problem requires a different app... | To find the radius of the largest circle that is tangent to all three circles and completely contained within the triangle, we need to find the inradius of the triangle.
The inradius of a triangle can be found using the formula
\[ r = \frac{A}{s} \]
where $A$ is the area of the triangle and $s$ is the semiperimeter.
... |
The yeast strain, S. cerevisiae, has a doubling time of 90 minutes under optimal growth conditions. If 5 yeast cells are used to start a culture grown under optimal growth conditions, how many yeast cells will there be after the culture is grown for 6 hours? | To solve this problem, we first need to calculate how many times the yeast cells will double in 6 hours. Since the doubling time is 90 minutes, we need to convert 6 hours to minutes, which is 360 minutes.
Now, we can divide 360 minutes by 90 minutes to find the number of times the yeast cells will double:
360 / 90 = 4... |
In a quadrilateral $ABCD$, $\angle ABC = 2x$ and $\angle BCD = x$. If $AC$ bisects $\angle A$ and $BD$ bisects $\angle D$, what is the measure of $\angle ABD + \angle BAC$? | We know that the sum of the angles in a quadrilateral is $360^\circ$. So, we have:
\[ \angle ABC + \angle BCD + \angle A + \angle D = 360^\circ \Rightarrow 2x + x + \angle A + \angle D = 360^\circ \Rightarrow 3x + \angle A + \angle D = 360^\circ \]
Since $AC$ bisects $\angle A$, we have:
\[ \angle BAC = \angle CAD = \... |
Let $a_n$ be a sequence defined recursively as $a_1=1$ and $a_{n+1}=2a_n+n$ for $n\geq 1$. Define $b_n=a_n-a_{n-1}$. Find the value of $n$ such that $b_n=5$. | To find the value of $n$ such that $b_n = 5$, we need to find the expression for $b_n$ in terms of $n$ and then solve for $n$.
Given the recursive sequence:
\[ a_{n+1} = 2a_n + n \quad \text{for } n \geq 1 \]
And the initial condition:
\[ a_1 = 1 \]
The sequence $b_n$ is defined as:
\[ b_n = a_n - a_{n-1} \]
We can... |
Find the sum of the squares of the numbers in the 10th row of the Pascal's Triangle.
Express the result in exponential notation, in simplest form. | ## Step 1: Understand the structure of Pascal's Triangle
Pascal's Triangle is a triangular array of the binomial coefficients where each number is the sum of the two directly above it. The rows of Pascal's Triangle are conventionally numbered starting with row 0 at the top, which contains a single number 1. For this pr... |
Find the smallest positive integer $m$ such that for each positive integer $n$, there exists an integer $k$ such that $\frac{k}{2^n}+\frac{1}{2^{m+n}}$ is an integer. | We need to find the smallest $m$ such that for any positive integer $n$, there exists an integer $k$ such that
\[ \frac{k}{2^n} + \frac{1}{2^{m+n}} \]
is an integer.
Rearranging the equation:
\[ \frac{k}{2^n} + \frac{1}{2^{m+n}} = \frac{k \cdot 2^m + 1}{2^{m+n}} \]
For the expression to be an integer, the numerator $... |
How many 4-digit numbers are there whose digits form a geometric progression? | To find the number of 4-digit numbers whose digits form a geometric progression, we can express the digits as a geometric sequence.
Let the first digit be $a$, and the common ratio be $r$. The digits can then be represented as $a$, $ar$, $ar^2$, and $ar^3$.
Since the number is 4 digits long, the first digit $a$ must ... |
Find the largest possible number of marbles in Carlos's collection if he can divide them into groups of 3, 5, or 7, with 2 marbles left over in each case. | Let $N$ be the total number of marbles. We know that $N$ leaves a remainder of $2$ when divided by $3$, $5$, and $7$.
Therefore, we can write the following equations:
\[ N = 3a + 2 \]
\[ N = 5b + 2 \]
\[ N = 7c + 2 \]
where $a$, $b$, and $c$ are integers.
We can rewrite the equations as:
\[ N - 2 = 3a \]
\[ N - 2 = ... |
In a round robin tournament, 8 teams are competing in a series of matches, with each team playing against every other team exactly once. In how many ways can the teams be scheduled to play, such that no team plays two matches in a row, and no team plays the same opponent twice? | There are 8 teams, and each team plays against every other team exactly once. Since each match involves two teams, we can treat the scheduling as arranging the 8 teams in a sequence such that each team plays against its adjacent teams in the sequence.
To ensure that no team plays two matches in a row, we need to arran... |
The polynomial $g(x)$ satisfies $g(x - 1) - g(x) = -3x + 2.$ Find the constant term of $g(x).$ | We are given $g(x - 1) - g(x) = -3x + 2$.
Let's find $g(x - 2) - g(x - 1)$:
\[ g(x - 2) - g(x - 1) = -3(x - 1) + 2 \Rightarrow g(x - 2) - g(x - 1) = -3x + 3 + 2 \Rightarrow g(x - 2) - g(x - 1) = -3x + 5 \]
And $g(x - 3) - g(x - 2)$:
\[ g(x - 3) - g(x - 2) = -3(x - 2) + 2 \Rightarrow g(x - 3) - g(x - 2) = -3x + 6 + 2 ... |
Find the values of $a$, $b$, and $c$ such that the quadratic expression $ax^2+bx+c$ can be factored as $(x-d)(x^2+ex+f)$, where $d$, $e$, and $f$ are constants, and $a+b+c=0$. | The factored form of the quadratic expression is given as $(x-d)(x^2+ex+f)$. Expanding this expression gives:
\[ ax^2 + bx + c = (x-d)(x^2+ex+f) = x^3 + (e-d)x^2 + (f-ed)x - fd \]
We can see that $ax^2 + bx + c$ and $x^3 + (e-d)x^2 + (f-ed)x - fd$ are polynomials of different degrees, so they can't be equal. However, ... |
Let $x$ and $y$ be two positive integers that satisfy the equation:
$$x^2 - 4xy + y^2 = 63$$
Find the maximum possible value of $x + y$. | Given the equation $x^2 - 4xy + y^2 = 63$, we can rewrite it as:
\[ (x - 2y)^2 - y^2 = 63 \]
To simplify, let $a = x - 2y$, so the equation becomes:
\[ a^2 - y^2 = 63 \]
Factoring the left side as a difference of squares:
\[ (a - y)(a + y) = 63 \]
Since $x$ and $y$ are positive integers, $a$ and $y$ must also be int... |
Determine the largest positive integer $n$ for which there exist real numbers $a_1,a_2,\ldots,a_n$ such that
\begin{align*}
a_1^2+a_2^2+\ldots+a_n^2&=42,\\
(a_1+a_2+\ldots+a_n)(1^2+2^2+\ldots+n^2)&=2022.
\end{align*} | We can rewrite the two given equations as:
\[ \sum_{i=1}^{n} a_i^2 = 42 \]
\[ \left( \sum_{i=1}^{n} a_i \right) \left( \sum_{i=1}^{n} i^2 \right) = 2022 \]
From the second equation, we can find $\sum_{i=1}^{n} i^2$:
\[ \sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6} \]
Substitute this into the second equation:
\[ \le... |
In an acute triangle $ABC$, we have $\angle CAB = \angle ACB$ and $\angle BAC = \frac{\pi}{2}$. We let $D$ and $E$ be the feet of the perpendiculars drawn from $A$ to $BC$ and $B$ to $AC$, respectively. Given that $BD = 4$ and $DC = 9$, find $AE^2$. | Since $\angle BAC = \frac{\pi}{2}$, we know that $\triangle ABC$ is a right triangle. Because $\angle CAB = \angle ACB$, $\triangle ABC$ is an isosceles right triangle. This means that $AB = AC$.
Since $BD = 4$ and $DC = 9$, we have $BC = BD + DC = 4 + 9 = 13$.
From the Pythagorean theorem, we can find $AB = AC$:
\[ ... |
Let $S$ be the set of all positive integers $n$ such that $n^2+an+2a^2$ is the square of an integer for some non-zero integer $a$. Determine the least possible value of the sum of all elements in $S$. | Consider the equation $n^2 + an + 2a^2 = m^2$.
We can rewrite this as $n^2 + an + \frac{a^2}{4} = m^2 - \frac{7a^2}{4}$ by completing the square on the left-hand side.
Thus, $\left(n + \frac{a}{2}\right)^2 = m^2 - \frac{7a^2}{4}$.
Rearrange to get $m^2 - \left(n + \frac{a}{2}\right)^2 = \frac{7a^2}{4}$.
We can fact... |
Let $f(x) = \frac{x^2 - x - 42}{x^2 + 1}$. Find constants $A$ and $B$ such that $\frac{3x + 5}{f(x)} = \frac{A}{x - 7} + \frac{B}{x + 6}$ for all $x \neq 7, -6$. Express your answer in the form $(A, B)$. | To find constants $A$ and $B$ such that $\frac{3x + 5}{f(x)} = \frac{A}{x - 7} + \frac{B}{x + 6}$, we need to use partial fraction decomposition.
First, factor the denominator of $f(x)$:
\[ x^2 - x - 42 = (x - 7)(x + 6) \]
Then, rewrite the original expression:
\[ \frac{3x + 5}{f(x)} = \frac{3x + 5}{\frac{x^2 - x - 4... |
Let $a \nabla b = \sqrt{a} - b^2$. What is the value of $(7^{\sqrt{5 \nabla 2}}) \nabla (3^{2\nabla 4})$? | First, we need to evaluate the expression $5 \nabla 2$.
\[ 5 \nabla 2 = \sqrt{5} - 2^2 = \sqrt{5} - 4 \]
Next, we use this result as the exponent for $7$:
\[ 7^{\sqrt{5 \nabla 2}} = 7^{\sqrt{5} - 4} \]
Then, we evaluate $2 \nabla 4$:
\[ 2 \nabla 4 = \sqrt{2} - 4^2 = \sqrt{2} - 16 \]
And use this result as the expone... |
What is the largest positive integer $k$ such that $\frac{k}{k+201}$ is equal to a repeating decimal with a cycle length of 3 digits? | Let's find the largest positive integer $k$ such that $\frac{k}{k+201}$ is equal to a repeating decimal with a cycle length of 3 digits.
If the fraction is equal to a repeating decimal with a cycle length of 3 digits, then the denominator $k+201$ must be a divisor of $999$.
The prime factorization of $999$ is $3^3 \c... |
Ava has been training for a marathon and has been tracking her progress over the past 6 weeks. She has been running 3 times a week, and each run has been a different distance, ranging from 3 to 10 miles. After each run, she calculates her average distance for the week, and she has noticed that it has always been a whol... | To solve this problem, we need to work backward from the information given and use the fact that Ava's average distance for each week is always a whole number.
Let's denote the distances of Ava's three runs in the 5th week as $x$, $y$, and $z$ miles, respectively.
We know that Ava's total distance for the 6th week wa... |
Find the value of $x$, if
$$\max\{x-1, 2x+3\} + \min\{x-1, 2x+3\} = x^2 + 4.$$ | First, let's analyze the given equation:
\[ \max\{x-1, 2x+3\} + \min\{x-1, 2x+3\} = x^2 + 4 \]
We have two cases:
Case 1: $x - 1 \ge 2x + 3$.
Simplify the inequality:
\[ x - 1 \ge 2x + 3 \Rightarrow -x \ge 4 \Rightarrow x \le -4 \]
In this case, $\max\{x-1, 2x+3\} = x-1$ and $\min\{x-1, 2x+3\} = 2x+3$.
Substitute ... |
Consider the set of all convex quadrilaterals whose vertices are lattice points in the plane and whose side lengths are all less than or equal to $6$ units. Find the smallest possible area of such a quadrilateral. | We want to find the smallest possible area of a convex quadrilateral with vertices as lattice points and side lengths not exceeding $6$ units.
First, consider a triangle with the smallest possible area under these conditions. The smallest triangle with lattice point vertices would be a right triangle with legs of leng... |
What is the largest base-10 integer that can be represented as a three-digit palindrome $aba$ in base $c$, where $a$, $b$, and $c$ are positive integers and $c$ is larger than $b$? | To find the largest base-10 integer that can be represented as a three-digit palindrome $aba$ in base $c$, we should consider the largest possible values for $a$, $b$, and $c$.
First, let's express the three-digit palindrome $aba$ in base $c$ as a base-10 integer. The value of $aba$ in base $c$ is:
\[ aba = ac^2 + bc ... |
An ellipse with center $O$ and major axis $\overline{OM}$ is inscribed in triangle $PQR$, where $OP\perp PQ$, $OQ\perp QR$, and $OR\perp RP$. Given that $PQ=35$, $QR=85$, and $RP=145$, find the area of the ellipse.
(Note: You can use the fact that the area of an ellipse is given by $A=\pi ab$, where $a$ and $b$ are th... | We start by noticing that the points $P$, $Q$, and $R$ lie on the ellipse because they are the vertices of the triangle, and the ellipse is inscribed in the triangle.
Since the center of the ellipse is $O$, and the major axis is $\overline{OM}$, we can assume that the equation of the ellipse is $\frac{x^2}{a^2} + \fra... |
A right triangle with a height of 12 inches is formed by cutting a triangular section from the center of a right isosceles triangle. The area of the top part of the isosceles triangle that was removed is 18 square inches, and the area of the bottom part that remains is 108 square inches. What is the height of the origi... | Let the height of the original isosceles triangle be $h$.
The area of the top part of the isosceles triangle that was removed is 18 square inches. This can be expressed as:
\[ \frac{1}{2} \times (h - 12) \times (h - 12) = 18 \Rightarrow (h - 12)^2 = 36 \Rightarrow h - 12 = \pm 6 \Rightarrow h = 18 \text{ or } h = 6 \]... |
In a regular hexagon, a smaller regular hexagon is inscribed so that the vertices of the smaller hexagon divide each side of the larger hexagon into segments with a ratio of 2:1. The area of the larger hexagon is 120 square centimeters. What is the area of the smaller hexagon? | To solve this problem, we can use the properties of regular hexagons and the ratio given between the sides of the larger and smaller hexagons.
A regular hexagon can be divided into 6 equilateral triangles.
The area of each equilateral triangle can be found using the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} \times... |
Suppose that $x^2$ is directly proportional to $y$. If $x=3$ when $y=4$, find the value of $x$ when $y=9$. Express your answer in simplest radical form. | Since $x^2$ is directly proportional to $y$, we have
\[ x^2 = ky \]
When $x = 3$ and $y = 4$, we can solve for $k$:
\[ 3^2 = k(4) \Rightarrow 9 = 4k \Rightarrow k = 9 : 4 \Rightarrow k = \frac{9}{4} \]
So the equation becomes:
\[ x^2 = \frac{9}{4}y \]
When $y = 9$, we have:
\[ x^2 = \frac{9}{4}(9) = \frac{81}{4} \]
... |
A biologist is studying the behavior of two species of birds, one of which is known to build its nest with a specific type of twig with a probability of 0.8, while the other species builds its nest with the same type of twig with a probability of 0.4. The biologist observes a nest and finds that it contains two twigs o... | Let's denote the species that builds its nest with the specific type of twig with a probability of 0.8 as species A, and the other species as species B.
We are given that the biologist observes a nest and finds that it contains two twigs of the specific type. This means that the nest is either built by species A or sp... |
Find the remainder when the product $$(x^2+1)(x^2+2)(x^2+3)(x^2+4)$$ is divided by $x^4+1$. | The key idea here is to use polynomial long division or synthetic division to simplify the problem. However, we can also simplify the product using the fact that $x^4 + 1$ can be factored as $(x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1)$.
Let's simplify the expression by multiplying the first two and last two factors se... |
Find the point on the circle x^2 + y^2 = 25 that is equidistant from the points (3, 0) and (0, 4). | We can find the point on the circle $x^2 + y^2 = 25$ that is equidistant from the points $(3, 0)$ and $(0, 4)$ by using the distance formula. Let $(x, y)$ be the point on the circle that we're looking for.
The distance between $(x, y)$ and $(3, 0)$ is:
\[ \sqrt{(x - 3)^2 + y^2} \]
The distance between $(x, y)$ and $(... |
Positive integers $p$, $q$, and $r$ are chosen so that $p<q<r$, and the system of equations \[
3x + 2y = 2017\quad\text{and}\quad y = |x-p| + |x-q| + |x-r|
\]has exactly one solution. What is the minimum value of $r$? | The system of equations given is:
\[ 3x + 2y = 2017 \quad \text{and} \quad y = |x - p| + |x - q| + |x - r| \]
We want the minimum value of $r$ such that this system has exactly one solution. To approach this, let's first understand the nature of the function $y = |x - p| + |x - q| + |x - r|$.
This function represents... |
The sum of the smallest one-digit composite number and the largest two-digit composite number is 105. Determine the sum of their prime factors. | To solve this problem, we need to identify the smallest one-digit composite number and the largest two-digit composite number that add up to 105.
The smallest one-digit composite number is 4, as it is the smallest number that is not prime (it can be divided by 2).
The largest two-digit composite number is 100, as it ... |
A deck of 52 cards is shuffled and 10 cards are drawn at random. What is the probability that at least 3 consecutive cards have the same suit? | Let's break this problem down into manageable steps.
1. The total number of ways to choose 10 cards out of 52:
\[ \text{Total combinations} = \binom{52}{10} \]
2. To find the number of combinations where at least 3 consecutive cards have the same suit, we'll use the complementary counting principle. We will first fin... |
The parametric equations of a curve are given by $x = 2 \cos t + \sin 2t$ and $y = t^2 - 3.$ Find the explicit equation of the curve in the form $y = f(x).$ | To find the explicit equation of the curve, we need to eliminate the parameter $t$ from the parametric equations.
Given equations:
\[ x = 2 \cos t + \sin 2t \]
\[ y = t^2 - 3 \]
First, let's simplify the equation for $x$ using the double angle identity $\sin 2t = 2 \sin t \cos t$:
\[ x = 2 \cos t + 2 \sin t \cos t \]... |
Find the sum of all the integers $x$ satisfying $\left \lceil \frac{x}{5} \right \rceil - \left \lfloor \frac{x}{7} \right \rfloor = 1$, where $\lceil x \rceil$ denotes the smallest integer greater than or equal to $x$ and $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$. | We know that $x$ must be an integer.
1. We examine the ceiling function $\left \lceil \frac{x}{5} \right \rceil$.
For $0 \leq x \leq 4$, $\left \lceil \frac{x}{5} \right \rceil = 1$. For $5 \leq x \leq 9$, $\left \lceil \frac{x}{5} \right \rceil = 2$. For $10 \leq x \leq 14$, $\left \lceil \frac{x}{5} \right \rcei... |
A circle is inscribed in a regular hexagon. If the radius of the circle is $5$ cm, find the area of the hexagon. | To find the area of the hexagon, we need to consider the properties of the regular hexagon and the circle inscribed within it.
A regular hexagon can be divided into 6 equilateral triangles. Since the circle is inscribed in the hexagon, the radius of the circle is equal to the apothem of the hexagon (the distance from ... |
Evaluate $\lceil\sqrt[3]{100}\rceil$. | We need to evaluate the expression $\lceil\sqrt[3]{100}\rceil$. This means we first need to find the cube root of 100 and then take the ceiling of that value.
The cube root of 100 is:
\[ \sqrt[3]{100} \approx 4.64 \]
The ceiling function rounds up to the nearest whole number, so:
\[ \lceil\sqrt[3]{100}\rceil = \lceil... |
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