problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
If $8 \tan \theta = 3 \cos \theta$ and $0 < \theta < \pi,$ then determine the value of $\sin \theta.$ | \frac{1}{3} |
Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\] | 75 |
Given the function $f(x)=2\cos^2x+2\sqrt{3}\sin x\cos x+a$, and when $x\in\left[0, \frac{\pi}{2}\right]$, the minimum value of $f(x)$ is $2$,
$(1)$ Find the value of $a$, and determine the intervals where $f(x)$ is monotonically increasing;
$(2)$ First, transform the graph of the function $y=f(x)$ by keeping the y-co... | \frac{\pi}{3} |
Given that the random variable $\xi$ follows a normal distribution $N(1,4)$, if $p(\xi > 4)=0.1$, then $p(-2 \leqslant \xi \leqslant 4)=$ _____ . | 0.8 |
Calculate:
(1) $2\log_{2}{10}+\log_{2}{0.04}$
(2) $(\log_{4}{3}+\log_{8}{3})\cdot(\log_{3}{5}+\log_{9}{5})\cdot(\log_{5}{2}+\log_{25}{2})$ | \frac{15}{8} |
James decides to bulk up. He weighs 120 kg and gains 20% of his body weight in muscle and 1 quarter that much in fat. How much does he weigh now? | He gains 120*.2=<<120*.2=24>>24 kg in muscles
So he gains 24/4=<<24/4=6>>6 kg of fat
That means he gains a total of 24+6=<<24+6=30>>30 kg of bodyweight
So his new body weight is 120+30=<<120+30=150>>150 kg
#### 150 |
On Tuesday, I worked $t+1$ hours and earned $3t-3$ dollars per hour. My friend Andrew worked $3t-5$ hours but only earned $t+2$ dollars an hour. At the end of the day, I had earned two dollars more than he had. What is the value of $t$? | 5 |
A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ | 817 |
In $\bigtriangleup ABC$, $AB = 75$, and $AC = 100$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. It is known that $\overline{BX}$ and $\overline{CX}$ have integer lengths. Calculate the length of segment $BC$. | 125 |
A toy store manager received a large order of Mr. Slinkums just in time for the holidays. The manager places $20\%$ of them on the shelves, leaving the other 120 Mr. Slinkums in storage. How many Mr. Slinkums were in this order? | 150 |
[asy] draw((0,0)--(0,2)--(2,2)--(2,0)--cycle,dot); draw((2,2)--(0,0)--(0,1)--cycle,dot); draw((0,2)--(1,0),dot); MP("B",(0,0),SW);MP("A",(0,2),NW);MP("D",(2,2),NE);MP("C",(2,0),SE); MP("E",(0,1),W);MP("F",(1,0),S);MP("H",(2/3,2/3),E);MP("I",(2/5,6/5),N); dot((1,0));dot((0,1));dot((2/3,2/3));dot((2/5,6/5)); [/asy]
If $A... | \frac{7}{15} |
Given $a_1 + a_2 = 1$, $a_2 + a_3 = 2$, $a_3 + a_4 = 3$, ..., $a_{99} + a_{100} = 99$, $a_{100} + a_1 = 100$, find the value of $a_1 + a_2 + a_3 + \ldots + a_{100}$. | 2525 |
Let $A B C D$ be a square of side length 5, and let $E$ be the midpoint of side $A B$. Let $P$ and $Q$ be the feet of perpendiculars from $B$ and $D$ to $C E$, respectively, and let $R$ be the foot of the perpendicular from $A$ to $D Q$. The segments $C E, B P, D Q$, and $A R$ partition $A B C D$ into five regions. Wha... | 5 |
Given four points \(O, A, B, C\) on a plane, such that \(OA = 4\), \(OB = 3\), \(OC = 2\), and \(\overrightarrow{OB} \cdot \overrightarrow{OC} = 3\), find the maximum value of the area \(S_{\triangle ABC}\). | 2\sqrt{7} + \frac{3\sqrt{3}}{2} |
Kristin can run three times faster than Sarith. If Kristin runs 12 times around the adult football field and Sarith runs at the same time around the children's football field that is half the distance as the other field, how many laps did Sarith go around the children's football field? | Sarith would run around the adult football field, 12 laps / 3 = <<12/3=4>>4 laps.
So on the children's football field, she managed to run, 4 laps * 2 = <<4*2=8>>8 laps.
#### 8 |
Natalie has $26 to go shopping. She bought a jumper for $9, a T-shirt for $4, and a pair of heels for $5. How much money does Natalie have left? | The total cost of the clothes is $9 + $4 + $5 = $<<9+4+5=18>>18.
Natalie has $26 − $18 = $8 left.
#### 8 |
Let $u$ and $v$ be real numbers such that \[(u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8.\] Determine, with proof, which of the two numbers, $u$ or $v$ , is larger. | \[ v \] |
Given $$f(x)= \begin{cases} f(x+1), x < 4 \\ ( \frac {1}{2})^{x}, x\geq4\end{cases}$$, find $f(\log_{2}3)$. | \frac {1}{24} |
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose? | 27 |
Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n=\frac{n^2+n}{2}+1$, find the sum of the first 99 terms of the sequence ${\frac{1}{a_n a_{n+1}}}$, denoted as $T_{99}$. | \frac{37}{50} |
Please write an irrational number that is greater than -3 and less than -2. | -\sqrt{5} |
Tina has 12 pink pens. She has 9 fewer green pens than pink pens. Tina has 3 more blue pens than green pens. How many pens does Tina have in total? | Tina has 12 - 9 = <<12-9=3>>3 green pens.
Tina has 3 + 3 = <<3+3=6>>6 blue pens.
In total Tina has 12 + 3 + 6 = <<12+3+6=21>>21 pens
#### 21 |
If $\cos 2^{\circ} - \sin 4^{\circ} -\cos 6^{\circ} + \sin 8^{\circ} \ldots + \sin 88^{\circ}=\sec \theta - \tan \theta$ , compute $\theta$ in degrees.
*2015 CCA Math Bonanza Team Round #10* | 94 |
In a bag containing 7 apples and 1 orange, the probability of randomly picking an apple is ______, and the probability of picking an orange is ______. | \frac{1}{8} |
One-fourth of the airing time of a television program is spent on commercials. If there are 6 thirty-minute programs, how many minutes are spent on commercials for the whole duration of the 6 programs? | The total time of the programs is 6 programs * 30 minutes/program = <<6*30=180>>180 minutes
A quarter of this time is spent on commercials, so that's 180 minutes / 4 = <<180/4=45>>45 minutes
#### 45 |
What is the remainder when $7^{2010}$ is divided by $100$? | 49 |
A baker bought cooking ingredients in the supermarket. She bought 3 boxes of flour that cost $3 each box, 3 trays of eggs that cost $10 for each tray, 7 liters of milk that cost $5 each liter, and 2 boxes of baking soda that cost $3 each box. How much will she have to pay for everything? | The total cost for the flour is 3 x $3 = $<<3*3=9>>9.
The total cost of the tray of eggs is 3 x $10 = $<<3*10=30>>30.
The total cost for the liter of milk is 7 x $5 = $<<7*5=35>>35.
The total cost for the box of baking soda is 2 x $3 = $<<2*3=6>>6.
Therefore, the total cost for all the ingredients is $9 + $30 + $35 + $... |
If $f(x)=\sqrt{x-3}$, what is the smallest real number $x$ in the domain of $f(f(x))$? | 12 |
In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are on the sides $YZ$, $ZX$, and $XY$, respectively. Given that lines $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$, and that $\frac{XP}{PX'}+\frac{YP}{PY'}+\frac{ZP}{PZ'}=100$, find the product $\frac{XP}{PX'}\cdot \frac{YP}{PY'}\cdot \frac{ZP}{PZ'}$. | 102 |
Amanda had 10 notebooks. This week, she ordered 6 more and then lost 2. How many notebooks does Amanda have now? | This week, Amanda added 6 notebooks - 2 notebooks = <<6-2=4>>4 notebooks.
Now, Amanda has 10 notebooks + 4 notebooks = <<10+4=14>>14 notebooks.
#### 14 |
John went to the bookstore and purchased 20 notebooks totaling $62. Some notebooks were priced at $2 each, some at $5 each, and some at $6 each. John bought at least one notebook of each type. Let x be the number of $2 notebooks, y be the number of $5 notebooks, and z be the number of $6 notebooks. Solve for x. | 14 |
Alice has 4 sisters and 6 brothers. Given that Alice's sister Angela has S sisters and B brothers, calculate the product of S and B. | 24 |
The ratio of the areas of two squares is $\frac{32}{63}$. After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplified form $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. What is the value of the sum $a+b+c$? | 39 |
The expression \[(x+y+z)^{2006}+(x-y-z)^{2006}\]is simplified by expanding it and combining like terms. How many terms are in the simplified expression? | 1{,}008{,}016 |
Cards numbered 1 and 2 must be placed into the same envelope, and three cards are left to be placed into two remaining envelopes, find the total number of different methods. | 18 |
In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long.
Of the three equations
\begin{equation*} \label{eq:1} d-s=1, \qquad ds=1, \qquad d^2-s^2=\sqrt{5} \en... | I, II and III |
3000 bees hatch from the queen's eggs every day. If a queen loses 900 bees every day, how many total bees are in the hive (including the queen) at the end of 7 days if at the beginning the queen had 12500 bees? | 3000 new bees are born every day. In 7 days there will be 3000*7=<<3000*7=21000>>21000 new bees
If each day the queen loses 900 bees, in 7 days there will be 900*7=<<900*7=6300>>6300 fewer bees
If at the beginning the queen had 12500 bees, then at the end of 7 days she will have 12500+21000-6300=<<12500+21000-6300=2720... |
Kevin repairs phones at his job. At the beginning of the day, Kevin has 15 phones that need to be repaired. By the afternoon, Kevin has successfully repaired 3 of the 15 phones and a client has dropped off 6 more phones that need fixing. If a coworker of Kevin's offers to help him and fix half of the damaged phones, ho... | Kevin begins the day with 15 phones and repairs 3 by the afternoon, so Kevin still has to repair 15 - 3 = <<15-3=12>>12 phones.
A client drops off 6 additional phones that require fixing, increasing the total to 12 + 6 = 18 phones.
If Kevin's coworker helps Kevin fix half of the phones, then each person needs to fix 18... |
How many perfect squares are between 100 and 500? | 12 |
Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, the symmetric point $Q$ of the right focus $F(c, 0)$ with respect to the line $y = \dfrac{b}{c}x$ is on the ellipse. Find the eccentricity of the ellipse. | \dfrac{\sqrt{2}}{2} |
Given that $\overrightarrow{OA}=(-2,4)$, $\overrightarrow{OB}=(-a,2)$, $\overrightarrow{OC}=(b,0)$, $a > 0$, $b > 0$, and $O$ is the coordinate origin. If points $A$, $B$, and $C$ are collinear, find the minimum value of $\frac{1}{a}+\frac{1}{b}$. | \frac{3 + 2\sqrt{2}}{2} |
An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the least possible value of the $ 50$th term and let $ G$ be the greatest possible value of the $ 50$th term. What is the value of $ G - L$? | \frac{8080}{199} |
Let $a$ and $b$ be relatively prime positive integers such that $\dfrac ab=\dfrac1{2^1}+\dfrac2{3^2}+\dfrac3{2^3}+\dfrac4{3^4}+\dfrac5{2^5}+\dfrac6{3^6}+\cdots$, where the numerators always increase by $1$, and the denominators alternate between powers of $2$ and $3$, with exponents also increasing by $1$ for each subs... | 689 |
Perry wants to buy a new modern bedroom set for $2000. He received $200.00 in gift cards over the holidays that he can use toward the set. The store is currently offering the entire set at 15% off. If he signs up for the store credit card and uses it to buy the set, he will save an additional 10% off on the discount... | The bedroom set is $2000 and it’s currently 15% off so 2000*.15 = $<<2000*.15=300.00>>300.00 off
So the new price of the set is 2000-300 = $<<2000-300=1700.00>>1,700.00
He signs up for the credit card so that’s 10% off so .10*1700 = $<<10*.01*1700=170.00>>170.00
Now the set is 1700-170 = $<<1700-170=1530.00>>1,530.00
H... |
We write the equation on the board:
$$
(x-1)(x-2) \ldots(x-2016) = (x-1)(x-2) \ldots(x-2016)
$$
We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimum number of factors that must be erased to achieve this? | 2016 |
Given a sequence $\{a_n\}$ that satisfies: $a_{n+2}=\begin{cases} 2a_{n}+1 & (n=2k-1, k\in \mathbb{N}^*) \\ (-1)^{\frac{n}{2}} \cdot n & (n=2k, k\in \mathbb{N}^*) \end{cases}$, with $a_{1}=1$ and $a_{2}=2$, determine the maximum value of $n$ for which $S_n \leqslant 2046$. | 19 |
On the sides \( AB, BC \), and \( AC \) of triangle \( ABC \), points \( M, N, \) and \( K \) are taken respectively so that \( AM:MB = 2:3 \), \( AK:KC = 2:1 \), and \( BN:NC = 1:2 \). In what ratio does the line \( MK \) divide the segment \( AN \)? | 6:7 |
Compute the integral: \(\int_{0}^{\pi / 2}\left(\sin ^{2}(\sin x) + \cos ^{2}(\cos x)\right) \,dx\). | \frac{\pi}{4} |
Given a sequence $\{a_{n}\}$ such that $a_{2}=2a_{1}=4$, and $a_{n+1}-b_{n}=2a_{n}$, where $\{b_{n}\}$ is an arithmetic sequence with a common difference of $-1$.
$(1)$ Investigation: Determine whether the sequence $\{a_{n}-n\}$ is an arithmetic sequence or a geometric sequence, and provide a justification.
$(2)$ F... | 12 |
In a convex quadrilateral inscribed around a circle, the products of opposite sides are equal. The angle between a side and one of the diagonals is $20^{\circ}$. Find the angle between this side and the other diagonal. | 70 |
A student recorded the exact percentage frequency distribution for a set of measurements, as shown below.
However, the student neglected to indicate $N$, the total number of measurements. What is the smallest possible value of $N$?
\begin{tabular}{c c}\text{measured value}&\text{percent frequency}\\ \hline 0 & 12.5\\ 1... | 8 |
The number of geese in a flock increases so that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$. If the populations in the years $1994$, $1995$, and $1997$ were $39$, $60$, and $123$, respectively, then the population in $1996$ was | 84 |
A designer has 3 fabric colors he may use for a dress: red, green, and blue. Four different patterns are available for the dress. If each dress design requires exactly one color and one pattern, how many different dress designs are possible? | 12 |
How many consecutive "0"s are there at the end of the product \(5 \times 10 \times 15 \times 20 \times \cdots \times 2010 \times 2015\)? | 398 |
If $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ and $\mathbf{d}$ are unit vectors, find the largest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{a} - \mathbf{d}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{d}\|^2 + \|\mathbf{c} - \mathbf{d}\|^2.
\] | 14 |
Compute
\[\cos^2 0^\circ + \cos^2 1^\circ + \cos^2 2^\circ + \dots + \cos^2 90^\circ.\] | \frac{91}{2} |
Find the greatest natural number $n$ such that $n\leq 2008$ and $(1^2+2^2+3^2+\cdots + n^2)\left[(n+1)^2+(n+2)^2+(n+3)^2+\cdots + (2n)^2\right]$ is a perfect square.
| 1921 |
Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. | \frac{1}{8} |
How many edges does a square-based pyramid have? | 8 |
Vasya and Petya live in the mountains and like to visit each other. They ascend the mountain at a speed of 3 km/h and descend at a speed of 6 km/h (there are no flat sections of the road). Vasya calculated that it takes him 2 hours and 30 minutes to go to Petya, and 3 hours and 30 minutes to return. What is the distanc... | 12 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. It is known that $a^{2}+c^{2}=ac+b^{2}$, $b= \sqrt{3}$, and $a\geqslant c$. The minimum value of $2a-c$ is ______. | \sqrt{3} |
Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Amal now rolls a fair twelve-sided die marked with the numbers 1 through 12. What is the probability that the product of the two rolls is a multiple of 4? | \frac{7}{16} |
Given a sequence $\{a_n\}$ satisfying $a_1=0$, for any $k\in N^*$, $a_{2k-1}$, $a_{2k}$, $a_{2k+1}$ form an arithmetic sequence with a common difference of $k$. If $b_n= \dfrac {(2n+1)^{2}}{a_{2n+1}}$, calculate the sum of the first $10$ terms of the sequence $\{b_n\}$. | \dfrac {450}{11} |
Evie is collecting seashells while at the beach. Each day she collects her favorite 10 shells. At the end of 6 days, she gives 2 shells to her brother. How many shells does she have left? | Multiply the number of shells collected each day by 6, to get the total number of seashells Evie has collected: 10 x 6 = <<10*6=60>>60 seashells
Subtract the 2 shells she gave her brother, to get the number of shells she has left: 60 - 2 = <<60-2=58>>58 seashells
#### 58 |
Many of the students in M. Gamache's class brought a skateboard or a bicycle to school yesterday. The ratio of the number of skateboards to the number of bicycles was $7:4$. There were 12 more skateboards than bicycles. How many skateboards and bicycles were there in total? | 44 |
A function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies: $f(0)=0$ and $$\left|f\left((n+1) 2^{k}\right)-f\left(n 2^{k}\right)\right| \leq 1$$ for all integers $k \geq 0$ and $n$. What is the maximum possible value of $f(2019)$? | 4 |
Eve wants to buy her 3 nieces cooking gear that's made for kids. The hand mitts cost $14.00 and the apron is $16.00. A set of 3 cooking utensils is $10.00 and a small knife is twice the amount of the utensils. The store is offering a 25% off sale on all cooking gear. How much will Eve spend on the gifts? | A small knife is twice the amount of the utensils that costs $10.00 so the knife costs 2*10 = $<<2*10=20.00>>20.00
All total, each little kit will costs $14.00 for mitts, $16.00 for an apron, $10.00 for utensils and $20.00 for a knife for a total of 14+16+10+20 = $<<14+16+10+20=60.00>>60.00
Each kit costs $60.00 and th... |
Ray has 175 cents in nickels. Ray gives 30 cents to Peter, and he gives twice as many cents to Randi as he gave to Peter. How many more nickels does Randi have than Peter? | Ray gave 30*2 = <<30*2=60>>60 cents to Randi.
Randi has 60/5 = <<60/5=12>>12 nickels.
Peter has 30/5 = <<30/5=6>>6 nickels.
Randi has 12-6 = <<12-6=6>>6 more nickels than Peter.
#### 6 |
Evaluate the first three digits to the right of the decimal point in the decimal representation of $\left(10^{987} + 1\right)^{8/3}$ using the Binomial Expansion. | 666 |
When Tanya excluded all numbers from 1 to 333 that are divisible by 3 but not by 7, and all numbers that are divisible by 7 but not by 3, she ended up with 215 numbers. Did she solve the problem correctly? | 205 |
The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r$, where $p$, $q$, and $r$ are positive integers, and $q$ is not divisible by the square of... | 554 |
Let \( M = \{1, 2, \cdots, 17\} \). If there exist four distinct numbers \( a, b, c, d \in M \) such that \( a + b \equiv c + d \pmod{17} \), then \( \{a, b\} \) and \( \{c, d\} \) are called a balanced pair of the set \( M \). Find the number of balanced pairs in the set \( M \). | 476 |
A group of 300 athletes spent Saturday night at Ultimate Fitness Camp. The next morning, for 4 hours straight, they left the camp at a rate of 28 athletes per hour. Over the next 7 hours, a new group of athletes trickled into the camp at a rate of 15 athletes per hour. Immediately thereafter, the camp gate was shut and... | The number that left the camp on Saturday morning was 4*28 = <<4*28=112>>112 athletes
This means that in the camp there remained 300-112 = <<300-112=188>>188 athletes.
The new group trickling in on Sunday morning comprised 7*15 = <<7*15=105>>105 athletes
The number that slept at the camp on Sunday night is therefore 18... |
Given that $a, b, c$ are positive real numbers and $\log _{a} b+\log _{b} c+\log _{c} a=0$, find the value of $\left(\log _{a} b\right)^{3}+\left(\log _{b} c\right)^{3}+\left(\log _{c} a\right)^{3}$. | 3 |
Given a pyramid with a vertex and base ABCD, each vertex is painted with one color, ensuring that two vertices on the same edge are of different colors. There are 5 different colors available. Calculate the total number of distinct coloring methods. (Answer with a number) | 420 |
Let $f(x) = \frac{x+1}{x-1}$. Then for $x^2 \neq 1$, $f(-x)$ is | \frac{1}{f(x)} |
Compute $\arcsin \frac{1}{\sqrt{2}}.$ Express your answer in radians. | \frac{\pi}{4} |
If the length of a rectangle is increased by $15\%$ and the width is increased by $25\%$, by what percent is the area increased? | 43.75\% |
In rectangle $EFGH$, we have $E=(1,1)$, $F=(101,21)$, and $H=(3,y)$ for some integer $y$. What is the area of rectangle $EFGH$?
A) 520
B) 1040
C) 2080
D) 2600 | 1040 |
Given the triangular pyramid P-ABC, PA is perpendicular to the base ABC, AB=2, AC=AP, BC is perpendicular to CA. If the surface area of the circumscribed sphere of the triangular pyramid P-ABC is $5\pi$, find the value of BC. | \sqrt{3} |
Find the number of positive integers \(n \le 500\) that can be expressed in the form
\[
\lfloor x \rfloor + \lfloor 3x \rfloor + \lfloor 4x \rfloor = n
\]
for some real number \(x\). | 248 |
Rhombus $ABCD$ has $\angle BAD < 90^\circ.$ There is a point $P$ on the incircle of the rhombus such that the distances from $P$ to the lines $DA,AB,$ and $BC$ are $9,$ $5,$ and $16,$ respectively. Find the perimeter of $ABCD.$
Diagram
[asy] /* Made by MRENTHUSIASM; inspired by Math Jams. */ size(300); pair A, B, C, D,... | 125 |
Calculate 8 divided by $\frac{1}{8}.$ | 64 |
Tim takes his 3 children trick or treating. They are out for 4 hours. Each hour they visited 5 houses. Each house gives 3 treats per kid. How many treats do his children get in total? | They visit 4*5=<<4*5=20>>20 houses
That means each child get 20*3=<<20*3=60>>60 treats
So in total they get 60*3=<<60*3=180>>180 treats
#### 180 |
Jenny collects cans and bottles to take down to the recycling center. Each bottle weighs 6 ounces and each can weighs 2 ounces. Jenny can carry a total of 100 ounces. She collects 20 cans and as many bottles as she can carry. If she gets paid 10 cents per bottle and 3 cents per can, how much money does she make (in cen... | First, the weight of 20 cans is 20 cans * 2 oz per can = <<20*2=40>>40 oz.
Thus, Jenny has the capacity to carry 100 oz - 40 oz = <<100-40=60>>60 oz of bottles.
Jenny can carry 60 oz / 6 oz per bottle = <<60/6=10>>10 bottles.
Jenny earns 10 bottles * 10 cents per bottle = <<10*10=100>>100 cents for the bottles.
Jenny e... |
The hyperbola $tx^{2}-y^{2}-1=0$ has asymptotes that are perpendicular to the line $2x+y+1=0$. Find the eccentricity of this hyperbola. | \frac{\sqrt{5}}{2} |
What percent of square $ABCD$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));
xaxis(0,5,Ticks(1.0,NoZero));
yaxis(0,5,Ticks(1.0,NoZero));
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle);
fill((2,0)--(3,0)--(3,3)--(0,3)--(0,2)--(2,2)--cycle);
fill((4,0)--(5,0)--(5,5)--(0... | 60 |
In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$?
(The top row of Pascal's Triangle has only a single $1$ and is the $0$th row.) | 62 |
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? | 9 |
A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S}\cup\{1\}$ is $13$ less than the mean of $\mathcal{S}$, and the mean of $\mathcal{S}\cup\{2001\}$ is $27$ more than the mean of $\mathcal{S}$. Find the mean of $\mathcal{S}$. | 651 |
A rectangular prism has dimensions 8 inches by 2 inches by 32 inches. If a cube has the same volume as the prism, what is the surface area of the cube, in square inches? | 384 |
Delete all perfect squares from the sequence of positive integers $1, 2, 3, \cdots$. Find the 2003rd term of the new sequence. | 2048 |
An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$ and the other focus is at $(d,7).$ Compute $d.$ | \frac{49}{3} |
Give an example of a number $x$ for which the equation $\sin 2017 x - \operatorname{tg} 2016 x = \cos 2015 x$ holds. Justify your answer. | \frac{\pi}{4} |
Let $F=\log\dfrac{1+x}{1-x}$. Find a new function $G$ by replacing each $x$ in $F$ by $\dfrac{3x+x^3}{1+3x^2}$, and simplify.
The simplified expression $G$ is equal to: | 3F |
The numbers $1-10$ are written in a circle randomly. Find the expected number of numbers which are at least 2 larger than an adjacent number. | \frac{17}{3} |
Let \( M \) be a subset of the set \(\{1, 2, 3, \cdots, 15\}\), and suppose that the product of any three different elements in \( M \) is not a perfect square. Let \( |M| \) denote the number of elements in the set \( M \). Find the maximum value of \( |M| \). | 10 |
Given that $\sin \alpha \cos \alpha = \frac{1}{8}$, and $\alpha$ is an angle in the third quadrant. Find $\frac{1 - \cos^2 \alpha}{\cos(\frac{3\pi}{2} - \alpha) + \cos \alpha} + \frac{\sin(\alpha - \frac{7\pi}{2}) + \sin(2017\pi - \alpha)}{\tan^2 \alpha - 1}$. | \frac{\sqrt{5}}{2} |
Given a point $p$ and a line segment $l$, let $d(p, l)$ be the distance between them. Let $A, B$, and $C$ be points in the plane such that $A B=6, B C=8, A C=10$. What is the area of the region in the $(x, y)$-plane formed by the ordered pairs $(x, y)$ such that there exists a point $P$ inside triangle $A B C$ with $d(... | \frac{288}{5} |
Four points $B,$ $A,$ $E,$ and $L$ are on a straight line, as shown. The point $G$ is off the line so that $\angle BAG = 120^\circ$ and $\angle GEL = 80^\circ.$ If the reflex angle at $G$ is $x^\circ,$ then what does $x$ equal?
[asy]
draw((0,0)--(30,0),black+linewidth(1));
draw((10,0)--(17,20)--(15,0),black+linewidth(... | 340 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.