problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
In the rectangular coordinate system $xOy$, the parametric equations of line $l$ are given by $\begin{cases} x=2t \\ y=-2-t \end{cases}$ (where $t$ is the parameter). In the polar coordinate system (using the same length unit as the rectangular coordinate system and with the origin $O$ as the pole and the polar axis co... | \frac{12\sqrt{5}}{5} |
Donna has $n$ boxes of doughnuts. Each box contains $13$ doughnuts.
After eating one doughnut, Donna is able to rearrange the remaining doughnuts into bags so that each bag contains $9$ doughnuts, and none are left over. What is the smallest possible value of $n$? | 7 |
Brand X soda advertises, ``We will give you 20$\%$ more soda than Brand Y for a total price that is 10$\%$ less than Brand Y's price!'' What is the ratio of the unit price of Brand X soda to the unit price of Brand Y soda? Express your answer as a common fraction. | \frac{3}{4} |
The coordinates of vertex \( C(x, y) \) of triangle \( \triangle ABC \) satisfy the inequalities \( x^{2}+y^{2} \leq 8+2y \) and \( y \geq 3 \). The side \( AB \) is on the x-axis. Given that the distances from point \( Q(0,1) \) to the lines \( AC \) and \( BC \) are both 1, find the minimum area of \( \triangle ABC \... | 6 \sqrt{2} |
In isosceles triangle $\triangle ABC$, $CA=CB=6$, $\angle ACB=120^{\circ}$, and point $M$ satisfies $\overrightarrow{BM}=2 \overrightarrow{MA}$. Determine the value of $\overrightarrow{CM} \cdot \overrightarrow{CB}$. | 12 |
In convex quadrilateral $KLMN$ side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$, side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$. Find $MO$. | 90 |
Solve for $n$: $2^n\cdot 4^n=64^{n-36}$. | 72 |
Given the arithmetic sequence $\left\{ a_n \right\}$ where each term is positive, the sum of the first $n$ terms is $S_n$. When $n \in N^*, n \geqslant 2$, it holds that $S_n = \frac{n}{n-1}\left( a_n^2 - a_1^2 \right)$. Find the value of $S_{20} - 2S_{10}$. | 50 |
If $x+\frac{1}{y}=1$ and $y+\frac{1}{z}=1$, what is the value of the product $xyz$? | -1 |
John is planning to fence a rectangular garden such that the area is at least 150 sq. ft. The length of the garden should be 20 ft longer than its width. Additionally, the total perimeter of the garden must not exceed 70 ft. What should the width, in feet, be? | -10 + 5\sqrt{10} |
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Additionally, at least one head of each dragon always tells the truth. One day, 530 dragons sat at a round table, and each of them said:
- 1st head: "To my left is a green ... | 176 |
Johnny buys 15 packs of colored pencils for his class. Each pack has a red, yellow, and green pencil inside. When he gets home he notices that 3 of the packs have two extra red pencils inside. How many red colored pencils did Johnny buy? | Johnny bought 15 packs, and every normal pack has 1 red pencil. He should have 15 * 1 = <<15*1=15>>15 red pencils.
3 of the packs have 2 extra red colored pencils, so there are 3 * 2 = <<3*2=6>>6 extra red pencils.
In total, Johnny has 15 + 6 = <<15+6=21>>21 red colored pencils.
#### 21 |
Quantities $a$ and $b$ vary inversely. When $a$ is $800$, $b$ is $0.5$. If the product of $a$ and $b$ increases by $200$ when $a$ is doubled, what is $b$ when $a$ is $1600$? | 0.375 |
Given that circle $C$ passes through points $P(0,-4)$, $Q(2,0)$, and $R(3,-1)$.
$(1)$ Find the equation of circle $C$.
$(2)$ If the line $l: mx+y-1=0$ intersects circle $C$ at points $A$ and $B$, and $|AB|=4$, find the value of $m$. | \frac{4}{3} |
In the rectangular coordinate system xOy, a polar coordinate system is established with the origin O of the rectangular coordinate system as the pole and the positive semi-axis of the x-axis as the polar axis. The parametric equations of the line l are given by $$\begin{cases} x= \frac {1}{2}+ \frac {1}{2}t \\ y= \frac... | \frac{8}{3} |
Given eight students, including Abby and Bridget, are randomly assigned to the 12 spots arranged in three rows of four as shown, calculate the probability that Abby and Bridget are seated directly adjacent to each other (in the same row or same column). | \frac{17}{66} |
The graphs of four functions, labelled (2) through (5), are shown below. Note that the domain of function (3) is $$\{-5,-4,-3,-2,-1,0,1,2\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;
picture pic1,pic2,pic3,pic4;
dr... | 60 |
Solve for $z$ in the following equation: $2-iz = -1 + 3iz$.
Express your answer in standard form. | -\frac34i |
For the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ to be a decreasing function on the interval $(0, +\infty)$, then $m = \boxed{\text{answer}}$. | -1 |
Car $A$ departs from Station $J$ towards Station $Y$, while cars $B$ and $C$ depart from Station $Y$ towards Station $J$ simultaneously, and move in opposite directions towards car $A$. Car $A$ meets car $B$ first, then 20 minutes later it meets car $C$. Given the speeds of cars $A$, $B$, and $C$ are $90 \text{ km/h}$,... | 425 |
Let $x_1, x_2, ... , x_6$ be non-negative real numbers such that $x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1$, and $x_1 x_3 x_5 +x_2 x_4 x_6 \ge {\frac{1}{540}}$. Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $x_1 x_2 x_3 + x_2 x_3 x_4 +x_3 x_4 x_5 +x_4 x_5 x_6 +x_5... | 559 |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 41 |
A palindrome is a number that reads the same forwards and backwards, such as 3003. How many positive four-digit integers are palindromes? | 90 |
The solutions of $x(3x-7)=-3$ may be expressed in the form $\frac{m+\sqrt{n}}{p}$ and $\frac{m-\sqrt{n}}{p}$, where $m$, $n$, and $p$ have a greatest common divisor of 1. Find $m+n+p$. | 26 |
Find the units digit of $n$ given that $mn = 21^6$ and $m$ has a units digit of 7. | 3 |
Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}{2}$. The length of the hypotenuse is | \frac{3\sqrt{5}}{5} |
What is fifteen more than a quarter of 48? | A quarter of 48 is 48/4=<<48/4=12>>12.
The number is 12+15=<<12+15=27>>27.
#### 27 |
Find the maximum possible value of $H \cdot M \cdot M \cdot T$ over all ordered triples $(H, M, T)$ of integers such that $H \cdot M \cdot M \cdot T=H+M+M+T$. | 8 |
If $(x^2+1)(2x+1)^9 = a_0 + a_1(x+2) + a_2(x+2)^2 + \ldots + a_{11}(x+2)^{11}$, then the value of $a_0 + a_1 + \ldots + a_{11}$ is. | -2 |
Determine the value of
\[\frac{\frac{2016}{1} + \frac{2015}{2} + \frac{2014}{3} + \dots + \frac{1}{2016}}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2017}}.\] | 2017 |
The sum of two numbers is $45$. Their difference is $3$. What is the lesser of the two numbers? | 21 |
In △ABC, the sides opposite to angles A, B, C are a, b, c respectively. If acosB - bcosA = $$\frac {c}{3}$$, then the minimum value of $$\frac {acosA + bcosB}{acosB}$$ is \_\_\_\_\_\_. | \sqrt {2} |
On a straight street, there are 5 buildings numbered from left to right as 1, 2, 3, 4, 5. The k-th building has exactly k (k=1, 2, 3, 4, 5) workers from Factory A, and the distance between two adjacent buildings is 50 meters. Factory A plans to build a station on this street. To minimize the total distance all workers ... | 150 |
Three $1 imes 1 imes 1$ cubes are joined side by side. What is the surface area of the resulting prism? | 14 |
Define $n!!$ as in the original problem. Evaluate $\sum_{i=1}^{5} \frac{(2i-1)!!}{(2i)!!}$, and express the result as a fraction in lowest terms. | \frac{437}{256} |
Tina is trying to solve the equation by completing the square: $$25x^2+30x-55 = 0.$$ She needs to rewrite the equation in the form \((ax + b)^2 = c\), where \(a\), \(b\), and \(c\) are integers and \(a > 0\). What is the value of \(a + b + c\)? | -38 |
Ponchik was having a snack at a roadside café when a bus passed by. Three pastries after the bus, a motorcycle passed by Ponchik, and three pastries after that, a car passed by. Syrupchik, who was snacking at another café on the same road, saw them in a different order: first the bus, after three pastries the car, and ... | 40 |
Alice is jogging north at a speed of 6 miles per hour, and Tom is starting 3 miles directly south of Alice, jogging north at a speed of 9 miles per hour. Moreover, assume Tom changes his path to head north directly after 10 minutes of eastward travel. How many minutes after this directional change will it take for Tom ... | 60 |
Gregor divides 2015 successively by 1, 2, 3, and so on up to and including 1000. He writes down the remainder for each division. What is the largest remainder he writes down? | 671 |
Given the complex number $z=a^{2}-1+(a+1)i (a \in \mathbb{R})$ is a purely imaginary number, find the imaginary part of $\dfrac{1}{z+a}$. | -\dfrac{2}{5} |
What is the minimum possible value for $y$ in the equation $y = x^2 + 12x + 5$? | -31 |
How many non-congruent squares can be drawn, such that their vertices are lattice points on the 5 by 5 grid of lattice points shown? [asy]
dot((0,0));dot((1,0));dot((2,0));dot((3,0));dot((4,0));
dot((0,1));dot((1,1));dot((2,1));dot((3,1));dot((4,1));
dot((0,2));dot((1,2));dot((2,2));dot((3,2));dot((4,2));
dot((0,3));do... | 8 |
Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value? | \frac{23}{108} |
Olivia earns $9 per hour. She worked 4 hours on Monday, 3 hours on Wednesday and 6 hours on Friday. How much money did Olivia make this week? | Olivia made $9 * 4 hours = $<<9*4=36>>36 on Monday.
Olivia made $9 * 3 = $<<9*3=27>>27 on Wednesday.
Olivia made $9 * 6 hours = $<<9*6=54>>54 on Friday.
For the week Olivia made $36 + $27 + $54 = $<<36+27+54=117>>117.
#### 117 |
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?
[asy]
import three; size(2inch);
currentprojection=orthographic(4,2,2);
draw((0,0,0)--(0,0,3),dashed);
draw((0,0,0)--(0,4,0),dashed);
draw((0,0,0)--(5,0,0),dashed);
draw((5,4,3... | 10 |
There are 200 students enrolled at Memorial Middle School. Seventy of the students are in band and 95 are in chorus. If only 150 students are in band and/or chorus, how many students are in both band and chorus? | 15 |
Determine the number of different ways to schedule volleyball, basketball, and table tennis competitions in 4 different gyms, given that each sport must be held in only one gym and that no more than 2 sports can take place in the same gym. | 60 |
Let the line \( p \) be the perpendicular bisector of points \( A = (20, 12) \) and \( B = (-4, 3) \). Determine the point \( C = (x, y) \) where line \( p \) meets segment \( AB \), and calculate \( 3x - 5y \). | -13.5 |
Let $S = \{1, 2,..., 8\}$ . How many ways are there to select two disjoint subsets of $S$ ? | 6561 |
For a real number $x$, define $\heartsuit(x)$ to be the average of $x$ and $x^2$. What is $\heartsuit(1)+\heartsuit(2)+\heartsuit(3)$? | 10 |
Vermont opened up 4 web pages on his web browser and found 12 ads on the first web page and twice as many ads on the second web page as the first web page. When he opened the third web page, he found 24 more ads than the number of ads on the second web page. If the fourth web page had 3/4 times as many ads as the secon... | If Vermont found 12 ads on the first web page and twice as many ads on the second web page as the first web page, he found 12*2 = <<12*2=24>>24 ads on the second web page.
The first and the second web page had a total of 12+24 =<<12+24=36>>36
Vermont also found 24 more ads on the third web page than the second web page... |
Moving along a particular line in the Cartesian plane, when the $x$-value increases by 3 units, the $y$-value increases by 7 units. When the $x$-value increases by 9 units, by how many units will the $y$-value increase? | 21 |
The points $(0,0),(1,2),(2,1),(2,2)$ in the plane are colored red while the points $(1,0),(2,0),(0,1),(0,2)$ are colored blue. Four segments are drawn such that each one connects a red point to a blue point and each colored point is the endpoint of some segment. The smallest possible sum of the lengths of the segments ... | 305 |
Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers. | 17 |
Five socks, colored blue, brown, black, red, and purple are in a drawer. In how many different ways can we choose three socks from the drawer if the order of the socks does not matter? | 10 |
In order for Mateen to walk a kilometer(1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters? | 400 |
Let $p(x) = 2x - 7$ and $q(x) = 3x - b$. If $p(q(4)) = 7$, what is $b$? | 5 |
Mr. Smith has incurred a 2% finance charge because he was not able to pay on time for his balance worth $150. If he plans to settle his balance today, how much will Mr. Smith pay? | The finance charge amounts to $150 x 2/100 = $<<150*2/100=3>>3.
Thus, Mr. Smith will pay a total of $150 + $3 = $<<150+3=153>>153.
#### 153 |
The American Mathematics College is holding its orientation for incoming freshmen. The incoming freshman class contains fewer than $500$ people. When the freshmen are told to line up in columns of $23$, $22$ people are in the last column. When the freshmen are told to line up in columns of $21$, $14$ people are in the ... | 413 |
Nellie can eat 12 sourball candies before crying. Jacob can only manage half of that number of candies, and Lana can only do three fewer than Jacob. They had a bucket of 30 candies, and all of them ate until they cried. If they divide the remaining candies in the bucket equally, how many sourball candies will they each... | Jacob can eat 12 / 2 = <<12/2=6>>6 candies before crying.
Lana can eat 6 - 3 = <<6-3=3>>3 candies before crying.
Nellie, Jacob, and Lana all ate until they cried, so they ate 12 + 6 + 3 = <<12+6+3=21>>21 candies.
Thus, there are 30 - 21 = <<30-21=9>>9 candies left in the bucket.
After dividing the remaining candies equ... |
Let $x$ be a complex number such that $x^{2011}=1$ and $x\neq 1$. Compute the sum
\[\frac{x^2}{x-1} + \frac{x^4}{x^2-1} + \frac{x^6}{x^3-1} + \dots + \frac{x^{4020}}{x^{2010}-1}.\] | 1004 |
Given that the odd function $f(x)$ is an increasing function defined on $\mathbb{R}$, and the sequence $x_n$ is an arithmetic sequence with a common difference of 2, satisfying $f(x_8) + f(x_9) + f(x_{10}) + f(x_{11}) = 0$, then the value of $x_{2011}$ is equal to. | 4003 |
Suppose that 7 boys and 13 girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $\text{GBBGGGBGBGGGBGBGGBGG}$ we have that $S=12$. The average value of $S$ (if all possible orders of these 20 people are considered) is closest... | 9 |
When the greatest common divisor and least common multiple of two integers are multiplied, their product is 200. How many different values could be the greatest common divisor of the two integers? | 4 |
In general, if there are $d$ doors in every room (but still only 1 correct door) and $r$ rooms, the last of which leads into Bowser's level, what is the expected number of doors through which Mario will pass before he reaches Bowser's level? | \frac{d\left(d^{r}-1\right)}{d-1} |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(0,3)$, the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is ______. | \frac{6\sqrt{5}}{5} |
In the diagram, the square has a perimeter of $48$ and the triangle has a height of $48.$ If the square and the triangle have the same area, what is the value of $x?$ [asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((3,0)--(6,0)--(6,5)--cycle);
draw((5.8,0)--(5.8,.2)--(6,.2));
label("$x$",(4.5,0),S);
label("48",(6,2... | 6 |
Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be arithmetic progressions such that $a_1 = 25, b_1 = 75$, and $a_{100} + b_{100} = 100$. Find the sum of the first hundred terms of the progression $a_1 + b_1, a_2 + b_2, \ldots$ | 10,000 |
In acute triangle $ABC$ , points $D$ and $E$ are the feet of the angle bisector and altitude from $A$ respectively. Suppose that $AC - AB = 36$ and $DC - DB = 24$ . Compute $EC - EB$ . | 54 |
The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n\,$ fish for various values of $n\,$.
$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\ \hline \text{number of contestants who caught} \ n \ \text{fi... | 943 |
John eats a bag of chips for dinner and then eats twice as many after dinner. How many bags of chips did he eat? | He ate 2 * 1 = <<2*1=2>>2 bags after dinner
So he ate 2 + 1 = <<2+1=3>>3 bags of chips
#### 3 |
Equilateral $\triangle ABC$ is inscribed in a circle of radius $2$. Extend $\overline{AB}$ through $B$ to point $D$ so that $AD=13,$ and extend $\overline{AC}$ through $C$ to point $E$ so that $AE = 11.$ Through $D,$ draw a line $l_1$ parallel to $\overline{AE},$ and through $E,$ draw a line $l_2$ parallel to $\overlin... | 865 |
The diagonal of a square is $\sqrt{2}$ inches long. How many inches long is the side of the square? | 1 |
The function $g : \mathbb{R} \to \mathbb{R}$ satisfies
\[g(x) + 3g(1 - x) = 2x^2 + 1\]for all $x.$ Find $g(5).$ | -9 |
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\frac{n}{11}$ is also a palindrome? | \frac{11}{30} |
A positive integer $a$ is input into a machine. If $a$ is odd, the output is $a+3$. If $a$ is even, the output is $a+5$. This process can be repeated using each successive output as the next input. If the input is $a=15$ and the machine is used 51 times, what is the final output? | 218 |
An orphaned kitten was only 4 inches when he was found. In the next two weeks, he doubled in length, and by 4 months old, he had double in length again. What is its current length? | In two weeks, the kitten measured 4 x 2 = <<4*2=8>>8 inches.
4 months later, the kitten measured 8 x 2 = <<8*2=16>>16 inches.
#### 16 |
Let $ ABC$ be an isosceles triangle with $ \left|AB\right| \equal{} \left|AC\right| \equal{} 10$ and $ \left|BC\right| \equal{} 12$ . $ P$ and $ R$ are points on $ \left[BC\right]$ such that $ \left|BP\right| \equal{} \left|RC\right| \equal{} 3$ . $ S$ and $ T$ are midpoints of $ \left[AB\right]$ and ... | $ \frac {10\sqrt {13} }{13} $ |
For what values of $j$ does the equation $(2x+7)(x-5) = -43 + jx$ have exactly one real solution? Express your answer as a list of numbers, separated by commas. | 5,\,-11 |
Three marbles are randomly selected, without replacement, from a bag containing two red, two blue and two green marbles. What is the probability that one marble of each color is selected? Express your answer as a common fraction. | \frac{2}{5} |
What is the smallest four-digit number that is divisible by $35$? | 1015 |
If $m>0$ and the points $(m,3)$ and $(1,m)$ lie on a line with slope $m$, then $m=$ | \sqrt{3} |
Evaluate \[ \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n + 1}. \] | 1 |
Using the side lengths 2, 3, 5, 7, and 11, how many different triangles with exactly two equal sides can be formed? | 14 |
Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is: | 79 |
Given that the random variable $\xi$ follows a normal distribution $N(4, 6^2)$, and $P(\xi \leq 5) = 0.89$, find the probability $P(\xi \leq 3)$. | 0.11 |
Find the midsegment (median) of an isosceles trapezoid, if its diagonal is 25 and its height is 15. | 20 |
Find the minimum possible value of
\[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4},\]
given that $a,b,c,d,$ are nonnegative real numbers such that $a+b+c+d=4$ . | \[\frac{1}{2}\] |
A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minim... | \[ 98 \] |
Let \(ABCD\) be an isosceles trapezoid with \(AB=1, BC=DA=5, CD=7\). Let \(P\) be the intersection of diagonals \(AC\) and \(BD\), and let \(Q\) be the foot of the altitude from \(D\) to \(BC\). Let \(PQ\) intersect \(AB\) at \(R\). Compute \(\sin \angle RPD\). | \frac{4}{5} |
Shift the graph of the function $y=\cos(\frac{π}{2}-2x)$ by an amount corresponding to the difference between the arguments of $y=\sin(2x-\frac{π}{4})$ and $y=\cos(\frac{π}{2}-2x)$. | \frac{\pi}{8} |
A king gets a crown made that costs $20,000. He tips the person 10%. How much did the king pay after the tip? | The tip was 20000*.1=$<<20000*.1=2000>>2000
So the total cost was 2000+20000=$<<2000+20000=22000>>22000
#### 22000 |
Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$ Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are $(-28,784),$ $(-2,4),$ and $(13,169),$ find the sum of the distances from the focus of $\mathcal{P}$ to all four of the int... | 1247 |
There are twice as many cows in Devonshire as there are hearts on a standard deck of 52 playing cards. If there are 4 hearts on a card, calculate the total cost of the cows when they are sold at $200 each. | If there are 4 hearts on a card, a standard deck of 52 playing cards will have 208 hearts.
There are twice as many cows in Devonshire as there are hearts on a standard deck of 52 playing cards, meaning there are 2*208 = <<2*208=416>>416 cows in Devonshire.
The total cost of the cows when they are sold at $200 each is 4... |
What is the smallest whole number $b$ such that 62 can be expressed in base $b$ using only three digits? | 4 |
Determine the value of the following expressions:
$(1)(2 \frac{7}{9})^{0.5}+0.1^{-2}+(2 \frac{10}{27})\,^{- \frac{2}{3}}-3π^{0}+ \frac{37}{48}$;
$(2)(-3 \frac{3}{8})\,^{- \frac{2}{3}}+(0.002)\,^{- \frac{1}{2}}-10( \sqrt{5}-2)^{-1}+( \sqrt{2}- \sqrt{3})^{0}$. | - \frac{167}{9} |
Two boards, one 5 inches wide and the other 7 inches wide, are nailed together to form an X. The angle at which they cross is 45 degrees. If this structure is painted and the boards are later separated, what is the area of the unpainted region on the five-inch board? Assume the holes caused by the nails are negligible. | 35\sqrt{2} |
Train 109 T departs from Beijing at 19:33 and arrives in Shanghai the next day at 10:26; train 1461 departs from Beijing at 11:58 and arrives in Shanghai the next day at 8:01. How many minutes are the running times of these two trains different? | 310 |
We have a triangle $\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude of $\triangle ABC$. If $AC = 10,$ $BK = 7$, and $BC = 13,$ then what is the area of $\triangle ABC$? | 52 |
**p1.** Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$ . Given that $\angle ABC$ is a right angle, determine the length of $AC$ .**p2.** Suppose $m$ and $n$ are integers such that $m^2+n^2 = 65$ . Find the largest possible value of $m-n$ .**p3.** Six middle school students are sitting in a circ... | 31 |
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