problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
On the sides \( AB \) and \( AD \) of the square \( ABCD \), points \( E \) and \( F \) are marked such that \( BE : EA = AF : FD = 2022 : 2023 \). The segments \( EC \) and \( FC \) intersect the diagonal of the square \( BD \) at points \( G \) and \( H \) respectively. Find \( \frac{GH}{BD} \). | \frac{12271519}{36814556} |
Roll a die twice in succession, observing the number of points facing up each time, and calculate:
(1) The probability that the sum of the two numbers is 5;
(2) The probability that at least one of the two numbers is odd;
(3) The probability that the point (x, y), with x being the number of points facing up on th... | \frac{2}{9} |
Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality $$ \left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right) $$ is valid for all posit... | 16 |
For some integers $m$ and $n$, the expression $(x+m)(x+n)$ is equal to a quadratic expression in $x$ with a constant term of -12. Which of the following cannot be a value of $m$? | 5 |
Find $b$ if $\log_{b}343=-\frac{3}{2}$. | b=\frac{1}{49} |
Majka examined multi-digit numbers in which odd and even digits alternate regularly. Those that start with an odd digit, she called "funny," and those that start with an even digit, she called "cheerful" (for example, the number 32387 is funny, the number 4529 is cheerful).
Majka created one three-digit funny number a... | 635040 |
How many 6-digit numbers have at least two zeros? | 73,314 |
Two identical cars are traveling in the same direction. The speed of one is $36 \kappa \mu / h$, and the other is catching up with a speed of $54 \mathrm{kм} / h$. It is known that the reaction time of the driver in the rear car to the stop signals of the preceding car is 2 seconds. What should be the distance between ... | 42.5 |
Find $\begin{pmatrix} 3 \\ -7 \end{pmatrix} + \begin{pmatrix} -6 \\ 11 \end{pmatrix}.$ | \begin{pmatrix} -3 \\ 4 \end{pmatrix} |
Two teachers, A and B, and four students stand in a row. (Explain the process, list the expressions, and calculate the results, expressing the results in numbers)<br/>$(1)$ The two teachers cannot be adjacent. How many ways are there to arrange them?<br/>$(2)$ A is to the left of B. How many ways are there to arrange t... | 12 |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? | 2\sqrt{61} |
Given that the height of a cylinder is $2$, and the circumferences of its two bases lie on the surface of the same sphere with a diameter of $2\sqrt{6}$, calculate the surface area of the cylinder. | (10+4\sqrt{5})\pi |
Given the parabola $C_1$: $y^{2}=4x$ with focus $F$ that coincides with the right focus of the ellipse $C_2$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and the line connecting the intersection points of the curves $C_1$ and $C_2$ passes through point $F$, determine the length of the major axis of the ell... | 2\sqrt{2}+2 |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy: $|\overrightarrow {a}| = \sqrt {2}$, $|\overrightarrow {b}| = 4$, and $\overrightarrow {a} \cdot (\overrightarrow {b} - \overrightarrow {a}) = 2$.
1. Find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$.
2. Find the m... | 2\sqrt{2} |
Given: The curve $C$ has the polar coordinate equation: $ρ=a\cos θ (a>0)$, and the line $l$ has the parametric equations: $\begin{cases}x=1+\frac{\sqrt{2}}{2}t\\y=\frac{\sqrt{2}}{2}t\end{cases}$ ($t$ is the parameter)
1. Find the Cartesian equation of the curve $C$ and the line $l$;
2. If the line $l$ is tangent to th... | a=2(\sqrt{2}-1) |
Sean adds up all the even integers from 2 to 500, inclusive. Julie adds up all the integers from 1 to 250, inclusive. What is Sean's sum divided by Julie's sum? | 2 |
Simplify the following expression: $(x^5+x^4+x+10)-(x^5+2x^4-x^3+12).$ Express your answer as a polynomial with the degrees of the terms in decreasing order. | -x^4+x^3+x-2 |
In a triangle, the base is 12 units; one of the angles at this base is $120^{\circ}$; the side opposite this angle is 28 units.
Find the third side. | 20 |
There are 30 different nuts in a bowl. If 5/6 of the nuts were eaten, how many nuts were left? | 30 x 5/6 = <<30*5/6=25>>25 nuts were eaten.
Thus, 30 - 25 = <<30-25=5>>5 nuts were left.
#### 5 |
Let \( a \) be the number of six-digit numbers divisible by 13 but not divisible by 17, and \( b \) be the number of six-digit numbers divisible by 17 but not divisible by 13.
Find \( a - b \). | 16290 |
Let us call a ticket with a number from 000000 to 999999 excellent if the difference between some two neighboring digits of its number is 5. Find the number of excellent tickets. | 409510 |
Jason is making a salad. The lettuce has 30 calories, the cucumber has 80 calories, and the 12 croutons have 20 calories each. How many calories are in the salad? | In total, the 12 croutons have 20 calories/crouton * 12 croutons = <<20*12=240>>240 calories.
Together, the croutons, lettuce and cucumber have 240 + 30 + 80 = <<240+30+80=350>>350 calories
#### 350 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given vectors $\overrightarrow{m}=(\sin B+\sin C, \sin A-\sin B)$ and $\overrightarrow{n}=(\sin B-\sin C, \sin(B+C))$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find the magnitude of angle $C$... | \frac{4\sqrt{3}-3}{10} |
Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m = a - b\sqrt {c},$ where $a, b,$ and $c$ are p... | 87 |
How many ways are there to put 5 balls in 3 boxes if the balls are not distinguishable but the boxes are? | 21 |
Simplify $\sqrt{245}$. | 7\sqrt{5} |
Given that $0 < α < \frac{π}{2}$ and $\frac{π}{2} < β < π$, with $\cos(α + \frac{π}{4}) = \frac{1}{3}$ and $\cos(\frac{π}{4} - \frac{β}{2}) = \frac{\sqrt{3}}{3}$,
1. Find the value of $\cos β$;
2. Find the value of $\cos(2α + β)$. | -1 |
Given that the area of $\triangle ABC$ is $\frac{1}{2}$, $AB=1$, $BC=\sqrt{2}$, determine the value of $AC$. | \sqrt{5} |
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$ | 2520 |
In rectangle $PQRS,$ $PQ=12$ and $PR=13.$ What is the area of rectangle $PQRS?$
[asy]
draw((0,0)--(12,0)--(12,5)--(0,5)--cycle,black+linewidth(1));
draw((0,5)--(12,0),black+linewidth(1));
label("$P$",(0,5),NW);
label("$Q$",(12,5),NE);
label("$R$",(12,0),SE);
label("$S$",(0,0),SW);
label("12",(0,5)--(12,5),N);
label("1... | 60 |
Let \( a \) and \( b \) be the roots of \( x^2 + 2000x + 1 = 0 \) and let \( c \) and \( d \) be the roots of \( x^2 - 2008x + 1 = 0 \). Find the value of \( (a+c)(b+c)(a-d)(b-d) \). | 32064 |
We have a standard deck of 52 cards, with 4 cards in each of 13 ranks. We call a 5-card poker hand a full house if the hand has 3 cards of one rank and 2 cards of another rank (such as 33355 or AAAKK). What is the probability that five cards chosen at random form a full house? | \frac{6}{4165} |
What is the remainder when $3x^7-x^6-7x^5+2x^3+4x^2-11$ is divided by $2x-4$? | 117 |
If $a, b, c$, and $d$ are pairwise distinct positive integers that satisfy \operatorname{lcm}(a, b, c, d)<1000$ and $a+b=c+d$, compute the largest possible value of $a+b$. | 581 |
A number of trucks with the same capacity were requested to transport cargo from one place to another. Due to road issues, each truck had to carry 0.5 tons less than planned, which required 4 additional trucks. The mass of the transported cargo was at least 55 tons but did not exceed 64 tons. How many tons of cargo wer... | 2.5 |
The figure shows three squares with non-overlapping interiors. The area of the shaded square is 1 square inch. What is the area of rectangle $ABCD$, in square inches?
[asy]size(100);
pair A = (0,0), D = (3,0),C = (3,2),B = (0,2);
draw(A--B--C--D--cycle);
draw(A--(1,0)--(1,1)--(0,1)--cycle);
filldraw(B--(1,2)--(1,1)--(... | 6 |
A right circular cone sits on a table, pointing up. The cross-section triangle, perpendicular to the base, has a vertex angle of 60 degrees. The diameter of the cone's base is $12\sqrt{3}$ inches. A sphere is placed inside the cone so that it is tangent to the sides of the cone and sits on the table. What is the volume... | 288\pi |
Given that the function $g(x)$ satisfies
\[ g(x + g(x)) = 5g(x) \]
for all $x$, and $g(1) = 5$. Find $g(26)$. | 125 |
Isabel has some money in her piggy bank. She spent half the amount and bought a toy. She then spent half of the remaining money and bought her brother a book. If she has $51 left, how much money, in dollars, did she have at first? | After buying the toy, Isabel has 51*2=<<51*2=102>>102 dollars left.
Isabel had 102*2=<<102*2=204>>204 dollars at first.
#### 204 |
A rectangle has a perimeter of 80 inches and an area greater than 240 square inches. How many non-congruent rectangles meet these criteria? | 13 |
Find the largest positive integer $n$ such that $\sigma(n) = 28$ , where $\sigma(n)$ is the sum of the divisors of $n$ , including $n$ . | 12 |
Tire repair for each tire costs $7, plus another 50 cents for sales tax. If Juan needs all 4 tires repaired, how much is the final cost? | Juan pays 4*7 = <<4*7=28>>28 dollars for all four tire repairs.
Juan pays 0.50*4 = <<0.50*4=2>>2 dollars in sales tax for all four tire repairs.
Juan pays a total of 28+2 = <<28+2=30>>30 dollars.
#### 30 |
Find the focus of the parabola $x = -\frac{1}{12} y^2.$ | (-3,0) |
Real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2000} \left| x_{k} - x_{k+1} \right| = 2001 \). Let \( y_{k} = \frac{1}{k} \left( x_{1} + x_{2} + \cdots + x_{k} \right) \) for \( k = 1, 2, \cdots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} | y_{k} - y_{k+1} | \). (2001 Sha... | 2000 |
Five brothers divided their father's inheritance equally. The inheritance included three houses. Since it was not possible to split the houses, the three older brothers took the houses, and the younger brothers were given money: each of the three older brothers paid $2,000. How much did one house cost in dollars? | 3000 |
Let $k$ and $s$ be positive integers such that $s<(2k + 1)^2$. Initially, one cell out of an $n \times n$ grid is coloured green. On each turn, we pick some green cell $c$ and colour green some $s$ out of the $(2k + 1)^2$ cells in the $(2k + 1) \times (2k + 1)$ square centred at $c$. No cell may be coloured green twice... | {3k^2+2k} |
A scientist begins an experiment with a cell culture that starts with some integer number of identical cells. After the first second, one of the cells dies, and every two seconds from there another cell will die (so one cell dies every odd-numbered second from the starting time). Furthermore, after exactly 60 seconds, ... | 61 |
You use a lock with four dials, each of which is set to a number between 0 and 9 (inclusive). You can never remember your code, so normally you just leave the lock with each dial one higher than the correct value. Unfortunately, last night someone changed all the values to 5. All you remember about your code is that no... | 10 |
In the diagram, $AB$ and $CD$ are straight lines. What is the value of $x?$ [asy]
draw((0,0)--(12,0));
draw((0,5)--(12,5));
draw((3,0)--(5,5)--(9,0));
label("$60^\circ$",(5,4.5),W);
label("$50^\circ$",(5.5,4.5),E);
label("$A$",(0,5),W);
label("$C$",(0,0),W);
label("$B$",(12,5),E);
label("$D$",(12,0),E);
label("$120^\ci... | 50 |
What is the remainder when 1,493,824 is divided by 4? | 0 |
In $\triangle ABC$, it is known that $AB=2$, $AC=3$, and $A=60^{\circ}$.
$(1)$ Find the length of $BC$;
$(2)$ Find the value of $\sin 2C$. | \frac{4\sqrt{3}}{7} |
Andrew's father buys a package of 100 masks. Andrew lives with his 2 parents and 2 siblings. All members of Andrew's family change masks every 4 days. How many days will it take to finish the pack of masks? | Andrew's family consists of 1 + 2 + 2 = <<1+2+2=5>>5 people.
Each of Andrew's family members uses 100 masks / 5 = <<100/5=20>>20 masks.
With everyone wearing a mask every 4 days, the masks will last 20 masks * 4 days/mask = <<20*4=80>>80 days.
#### 80 |
Solve for $r$: \[\frac{r-45}{2} = \frac{3-2r}{5}.\] | \frac{77}{3} |
Given that $15^{-1} \equiv 31 \pmod{53}$, find $38^{-1} \pmod{53}$, as a residue modulo 53. | 22 |
Ten adults went to a ball game with eleven children. Adult tickets are $8 each and the total bill was $124. How many dollars is one child's ticket? | Adults = 10 * 8 = $<<10*8=80>>80
Children = 124 - 80 = $<<124-80=44>>44
Child = 44/11 = $<<44/11=4>>4
Each child's ticket is $<<4=4>>4.
#### 4 |
Transform the following expression into a product: 447. \(\sin 75^{\circ} + \sin 15^{\circ}\). | \frac{\sqrt{6}}{2} |
Let $f(x)$ be a monic quartic polynomial such that $f(-2)=-4$, $f(1)=-1$, $f(-3)=-9$, and $f(5)=-25$. Find $f(2)$. | -64 |
Solve the following equations using appropriate methods:
(1) $x^2=49$;
(2) $(2x+3)^2=4(2x+3)$;
(3) $2x^2+4x-3=0$ (using the formula method);
(4) $(x+8)(x+1)=-12$. | -5 |
Find the area bounded by the graph of \( y = \arcsin(\cos x) \) and the \( x \)-axis on the interval \( \left[0, 2\pi\right] \). | \frac{\pi^2}{2} |
For a finite sequence \( B = (b_1, b_2, \dots, b_{50}) \) of numbers, the Cesaro sum is defined as
\[
\frac{S_1 + \cdots + S_{50}}{50},
\]
where \( S_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq 50 \).
If the Cesaro sum of the 50-term sequence \( (b_1, \dots, b_{50}) \) is 500, what is the Cesaro sum of the 51-term ... | 492 |
The stem and leaf plot represents the heights, in inches, of the players on the Pine Ridge Middle School boys' basketball team. Calculate the mean height of the players on the team. (Note: $5|3$ represents 53 inches.)
Height of the Players on the Basketball Team (inches)
$4|8$
$5|0\;1\;4\;6\;7\;7\;9$
$6|0\;3\;4\;5\... | 61.44 |
Calculate the arc lengths of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\frac{1}{2} \cos t-\frac{1}{4} \cos 2 t \\
y=\frac{1}{2} \sin t-\frac{1}{4} \sin 2 t
\end{array}\right. \\
& \frac{\pi}{2} \leq t \leq \frac{2 \pi}{3}
\end{aligned}
$$ | \sqrt{2} - 1 |
The graphs of $y=\log_3 x$, $y=\log_x 3$, $y=\log_{\frac{1}{3}} x$, and $y=\log_x \dfrac{1}{3}$ are plotted on the same set of axes. How many points in the plane with positive $x$-coordinates lie on two or more of the graphs? | 3 |
In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. What is the perimeter of the hexagon? | 12\sqrt{3} |
Given a pyramid P-ABCD whose base ABCD is a rectangle with side lengths AB = 2 and BC = 1, the vertex P is equidistant from all the vertices A, B, C, and D, and ∠APB = 90°. Calculate the volume of the pyramid. | \frac{\sqrt{5}}{3} |
In a kingdom of animals, tigers always tell the truth, foxes always lie, and monkeys sometimes tell the truth and sometimes lie. There are 100 animals of each kind, divided into 100 groups, with each group containing exactly 2 animals of one kind and 1 animal of another kind. After grouping, Kung Fu Panda asked each an... | 76 |
Given a regular tetrahedron $S-ABC$ with a base that is an equilateral triangle of side length 1 and side edges of length 2. If a plane passing through line $AB$ divides the tetrahedron's volume into two equal parts, the cosine of the dihedral angle between the plane and the base is: | $\frac{2 \sqrt{15}}{15}$ |
A regular octagon's perimeter is given as $P=16\sqrt{2}$. If $R_i$ denotes the midpoint of side $V_iV_{i+1}$ (with $V_8V_1$ as the last side), calculate the area of the quadrilateral $R_1R_3R_5R_7$.
A) $8$
B) $4\sqrt{2}$
C) $10$
D) $8 + 4\sqrt{2}$
E) $12$ | 8 + 4\sqrt{2} |
Regular decagon $P_1 P_2 \dotsb P_{10}$ is drawn in the coordinate plane with $P_1$ at $(1,0)$ and $P_6$ at $(3,0).$ If $P_n$ is the point $(x_n,y_n),$ compute the numerical value of the product
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{10} + y_{10} i).\] | 1023 |
A rectangular piece of cardboard was cut along its diagonal. On one of the obtained pieces, two cuts were made parallel to the two shorter sides, at the midpoints of those sides. In the end, a rectangle with a perimeter of $129 \mathrm{~cm}$ remained. The given drawing indicates the sequence of cuts.
What was the peri... | 258 |
Let set $\mathcal{A}$ be a 70-element subset of $\{1,2,3,\ldots,120\}$, and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$. | 3501 |
Placing no more than one X in each small square, what is the greatest number of X's that can be put on the grid shown without getting three X's in a row vertically, horizontally, or diagonally?
[asy] for(int a=0; a<4; ++a) { draw((a,0)--(a,3)); } for(int b=0; b<4; ++b) { draw((0,b)--(3,b)); } [/asy] | 6 |
Let $a_1 , a_2 , \dots$ be a sequence for which $a_1=2$ , $a_2=3$, and $a_n=\frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$? | 3 |
Let $a, b, c$ be positive real numbers such that $a \leq b \leq c \leq 2 a$. Find the maximum possible value of $$\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$$ | \frac{7}{2} |
There are two ways of choosing six different numbers from the list \( 1,2,3,4,5,6,7,8,9 \) so that the product of the six numbers is a perfect square. Suppose that these two perfect squares are \( m^{2} \) and \( n^{2} \), with \( m \) and \( n \) positive integers and \( m \neq n \). What is the value of \( m+n \)? | 108 |
The inclination angle of the line $x+ \sqrt {3}y+c=0$ is \_\_\_\_\_\_. | \frac{5\pi}{6} |
Quarter circles of radius 1' form a pattern as shown below. What is the area, in square feet, of the shaded region in a 3-foot length of this pattern? The quarter circles' flat sides face outward, alternating top and bottom along the length. | \frac{3}{4}\pi |
Bill decides to bring donuts to work for a meeting that day. He buys a box of donuts that has 50 in total in them. He eats 2 on the ride in because he's hungry. When he gets to the office, the secretary takes another 4 out of the box when he's not looking. Lastly, right before the meeting Bill sets the box down on ... | Bill eats 2 donuts out of a box of 50 for 50-2=<<50-2=48>>48 donuts remaining.
The secretary takes 4 more donuts out of the box, leaving 44 donuts remaining.
Bills coworkers take half of the remaining donuts, leaving 44/2= <<44/2=22>>22 donuts remaining.
#### 22 |
Find the largest value of the parameter \(a\) for which the equation:
\[ (|x-2| + 2a)^2 - 3(|x-2| + 2a) + 4a(3-4a) = 0 \]
has exactly three solutions. Indicate the largest value in the answer. | 0.5 |
Mr. Jones has 6 children. Assuming that the gender of each child is determined independently and with equal likelihood of male and female, what is the probability that Mr. Jones has more sons than daughters or more daughters than sons? | \dfrac{11}{16} |
Find the number of permutations \(a_1, a_2, \ldots, a_{10}\) of the numbers \(1, 2, \ldots, 10\) such that \(a_{i+1}\) is not less than \(a_i - 1\) for \(i = 1, 2, \ldots, 9\). | 512 |
Vivian plays 10 Spotify songs every day. Her best friend Clara plays 2 fewer songs each day. If in June they didn't play any song during the weekends only, and there were 8 weekend days in June, what's the total number of songs they both listened to in that month? | June has 30 days, so if they didn't play any songs on weekends, then they played 30 days - 8 days = <<30-8=22>>22 days.
If Vivian played 10 songs each day, the total number of songs she played for the month is 10 songs/day * 22 days = <<10*22=220>>220 songs.
Clara played 2 fewer songs, which is 10 songs/day - 2 songs/d... |
Given real numbers \(a\) and \(b\) that satisfy \(0 \leqslant a, b \leqslant 8\) and \(b^2 = 16 + a^2\), find the sum of the maximum and minimum values of \(b - a\). | 12 - 4\sqrt{3} |
Given $a\ln a=be^{b}$, where $b > 0$, find the maximum value of $\frac{b}{{{a^2}}}$ | \frac{1}{2e} |
A freight train travels 1 mile in 1 minute 30 seconds. At this rate, how many miles will the train travel in 1 hour? | 40 |
Darren has borrowed $100$ clams from Ethan at a $10\%$ simple daily interest. Meanwhile, Fergie has borrowed $150$ clams from Gertie at a $5\%$ simple daily interest. In how many days will Darren and Fergie owe the same amounts, assuming that they will not make any repayments in that time period? | 20\text{ days} |
Find the quadratic polynomial $p(x)$ such that $p(-7) = 0,$ $p(4) = 0,$ and $p(5) = -36.$ | -3x^2 - 9x + 84 |
Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly one letter is repeated exactly once, but digits cannot be repeated? [asy]
size(150);
draw((0,0)--(0,5)--(10,5)--(10,0)--cycle);
label("\Huge{CHIC ... | 8,\!424,\!000 |
In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{3}$, both $j$ and $k$ are positive integers. What is the sum of all possible values for $k$? | 22 |
The sides of an isosceles triangle are $\cos x,$ $\cos x,$ and $\cos 7x,$ and its vertex angle is $2x.$ (All angle measurements are in degrees.) Enter all possible values of $x,$ separated by commas. | 10^\circ, 50^\circ, 54^\circ |
Given the sequence 2, $\frac{5}{3}$, $\frac{3}{2}$, $\frac{7}{5}$, $\frac{4}{3}$, ..., then $\frac{17}{15}$ is the \_\_\_\_\_ term in this sequence. | 14 |
The area of a square plot of land is 325 square meters. What is the perimeter of the square, in meters? Express your answer in simplest radical form. | 20\sqrt{13} |
Find all real numbers $k$ such that
\[\left\| k \begin{pmatrix} 2 \\ -3 \end{pmatrix} - \begin{pmatrix} 4 \\ 7 \end{pmatrix} \right\| = 2 \sqrt{13}.\]Enter all the solutions, separated by commas. | -1 |
If an integer is divisible by $6$ and the sum of its last two digits is $15$, then what is the product of its last two digits? | 54 |
Given bed A has 600 plants, bed B has 500 plants, bed C has 400 plants, beds A and B share 60 plants, beds A and C share 80 plants, beds B and C share 40 plants, and beds A, B, and C share 20 plants collectively, calculate the total number of unique plants when considering just beds A, B, and C. | 1340 |
Find the shortest distance between the point $(5,10)$ and the parabola given by the equation $x = \frac{y^2}{3}.$ | \sqrt{53} |
Milo's parents tell him that he can win cash rewards for good grades. He will get $5 times the average grade he gets. He gets three 2s, four 3s, a 4, and a 5. How much cash does he get? | His total score is 27 because (3 x 2) + (4 x 3) + 4 + 5 = <<3*2+4*3+4+5=27>>27
His total number of grades is 9 because 3 + 4 +1 +1 = <<3+4+1+1=9>>9
His average grade is a 3 because 27 / 9 = <<27/9=3>>3
He earns $15 because 5 x 3 = <<5*3=15>>15
#### 15 |
A club consists initially of 20 total members, which includes eight leaders. Each year, all the current leaders leave the club, and each remaining member recruits three new members. Afterwards, eight new leaders are elected from outside. How many total members will the club have after 4 years? | 980 |
Let $a>0$ and $b>0,$ and define two operations:
$$a \nabla b = \dfrac{a + b}{1 + ab}$$
$$a \Delta b = \dfrac{a - b}{1 - ab}$$
Calculate $3 \nabla 4$ and $3 \Delta 4$. | \frac{1}{11} |
The Smith family has 4 sons and 3 daughters. In how many ways can they be seated in a row of 7 chairs such that at least 2 boys are next to each other? | 4896 |
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