problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
If $θ∈[0,π]$, then the probability of $\sin (θ+ \frac {π}{3}) > \frac {1}{2}$ being true is ______. | \frac {1}{2} |
A mischievous child mounted the hour hand on the minute hand's axle and the minute hand on the hour hand's axle of a correctly functioning clock. The question is, how many times within a day does this clock display the correct time? | 22 |
$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$ , for all $ n \in N$ . Let $ r_n$ be the average of $ g_n(x)$ 's roots. If $ r_{19}\equal{}99$ , find $ r_{99}$ . | 99 |
Four cats live in the old grey house at the end of the road. Their names are Puffy, Scruffy, Buffy, and Juniper. Puffy has three times more whiskers than Juniper, but half as many as Scruffy. Buffy has the same number of whiskers as the average number of whiskers on the three other cats. If Juniper has 12 whiskers. how many whiskers does Buffy have? | If Juniper has 12 whiskers, then Puffy has 3*12=<<12*3=36>>36 whiskers.
Since Puffy has half as many whiskers and Scruffy, then Scruffy has 36*2=<<36*2=72>>72 whiskers.
Since the number of whiskers on Buffy is the average between 12, 36, and 72 whiskers, then he has (12+36+72)/3=40 whiskers.
#### 40 |
Find a ten-digit number where the first digit indicates how many times the digit 0 appears in the number, the second digit indicates how many times the digit 1 appears, and so forth, with the tenth digit indicating how many times the digit 9 appears in the number.
Generalize and solve the problem for a number system with base $n$. | 6210001000 |
Subtract $111.11$ from $333.33.$ Express the result as a decimal to the nearest hundredth. | 222.22 |
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product
\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$? | 112 |
At a particular school with 43 students, each student takes chemistry, biology, or both. The chemistry class is three times as large as the biology class, and 5 students are taking both classes. How many people are in the chemistry class? | 36 |
Rationalize the denominator of $\frac{7}{3+\sqrt{8}}$. Write the simplified expression in the form $\frac{A\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. Determine $A+B+C+D$ if the greatest common divisor of $A$, $C$, and $D$ is 1. | 10 |
What integer $n$ satisfies $0\le n<9$ and $$-1111\equiv n\pmod 9~?$$ | 5 |
The sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$ is: | 64 |
For what value of $a$ does the equation $3(2x-a) = 2(3x+12)$ have infinitely many solutions $x$? | -8 |
The ratio of the areas of a square and a circle is $\frac{250}{196}$. After rationalizing the denominator, the ratio of the side length of the square to the radius of the circle 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 $a+b+c$? | 29 |
Let $a$, $b$, and $c$ be the roots of the equation $x^3 - 2x - 5 = 0$. Find $\frac{1}{a-2} + \frac{1}{b-2} + \frac{1}{c-2}$. | 10 |
Given that point $P$ lies on the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, and $F\_1$, $F\_2$ are the foci of the ellipse with $\angle F\_1 P F\_2 = 60^{\circ}$, find the area of $\triangle F\_1 P F\_2$. | 3 \sqrt{3} |
Given \( a=\underset{2016 \uparrow}{55 \cdots 5} \), what is the remainder when \( a \) is divided by 84? | 63 |
Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. | 15 |
You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $1.02, with at least one coin of each type. How many dimes must you have? | 1 |
Calculate:<br/>$(1)(1\frac{3}{4}-\frac{3}{8}+\frac{5}{6})÷(-\frac{1}{24})$;<br/>$(2)-2^2+(-4)÷2×\frac{1}{2}+|-3|$. | -2 |
Suppose that for a positive integer \( n \), \( 2^n + 1 \) is a prime number. What remainder can this prime have when divided by 240? | 17 |
How many ordered triples $(a, b, c)$ of non-zero real numbers have the property that each number is the product of the other two? | 4 |
We roll a fair 6-sided die 5 times. What is the probability that we get an odd number in exactly 4 of the 5 rolls? | \dfrac{5}{32} |
Given that $D$ is a point on the side $AB$ of $\triangle ABC$, and $\overrightarrow{CD} = \frac{1}{3}\overrightarrow{AC} + \lambda \cdot \overrightarrow{BC}$, determine the value of the real number $\lambda$. | -\frac{4}{3} |
We define two sequences of vectors $(\mathbf{v}_n)$ and $(\mathbf{w}_n)$ as follows: First, $\mathbf{v}_0 = \begin{pmatrix} 1 \\ 3 \end{pmatrix},$ $\mathbf{w}_0 = \begin{pmatrix} 4 \\ 0 \end{pmatrix}.$ Then for all $n \ge 1,$ $\mathbf{v}_n$ is the projection of $\mathbf{w}_{n - 1}$ onto $\mathbf{v}_0,$ and $\mathbf{w}_n$ is the projection of $\mathbf{v}_n$ onto $\mathbf{w}_0.$ Find
\[\mathbf{v}_1 + \mathbf{v}_2 + \mathbf{v}_3 + \dotsb.\] | \begin{pmatrix} 4/9 \\ 4/3 \end{pmatrix} |
Given the quadratic function $f(x)=ax^{2}+(2b+1)x-a-2 (a,b \in R, a \neq 0)$ has at least one root in the interval $[3,4]$, calculate the minimum value of $a^{2}+b^{2}$. | \frac{1}{100} |
Tenisha had 40 dogs that she kept at home as pets. If 60% of them are female, and 3/4 of the female gives birth to 10 puppies each, calculate the total number of puppies that Tenisha remains with after donating 130 puppies to the church. | If Tenisha has 40 dogs, the number of female dogs is 60/100*40 = 24
3/4 of the females give birth, meaning 3/4*24 = <<3/4*24=18>>18 female dogs gives birth to puppies.
Since each dog that gave birth had 10 puppies, the number of puppies that were born is 18*10 = <<18*10=180>>180
After donating 130 puppies to the church, Tenisha remained with 180-130 = 50 puppies.
#### 50 |
A translation of the plane takes $-3 + 2i$ to $-7 - i.$ Find the complex number that the translation takes $-4 + 5i$ to. | -8 + 2i |
In a regular tetrahedron ABCD with an edge length of 2, G is the centroid of triangle BCD, and M is the midpoint of line segment AG. The surface area of the circumscribed sphere of the tetrahedron M-BCD is __________. | 6\pi |
Express $\sqrt{x} \div\sqrt{y}$ as a common fraction, given:
$\frac{ {\left( \frac{1}{2} \right)}^2 + {\left( \frac{1}{3} \right)}^2 }{ {\left( \frac{1}{4} \right)}^2 + {\left( \frac{1}{5} \right)}^2} = \frac{13x}{41y} $ | \frac{10}{3} |
In the side face $A A^{\prime} B^{\prime} B$ of a unit cube $A B C D - A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, there is a point $M$ such that its distances to the two lines $A B$ and $B^{\prime} C^{\prime}$ are equal. What is the minimum distance from a point on the trajectory of $M$ to $C^{\prime}$? | \frac{\sqrt{5}}{2} |
In one month in the Smith house, Kylie uses 3 bath towels, her 2 daughters use a total of 6 bath towels, and her husband uses a total of 3 bath towels. If the washing machine can fit 4 bath towels for one load of laundry, how many loads of laundry will the Smiths need to do to clean all of their used towels? | The total number of bath towels used in one month is 3 + 6 + 3 = <<3+6+3=12>>12 bath towels
The number of loads of laundry the Smiths need to do is 12 / 4 = <<12/4=3>>3 loads
#### 3 |
Find the minimum value of
\[(15 - x)(8 - x)(15 + x)(8 + x).\] | -6480.25 |
Connie multiplies a number by 4 and gets 200 as her result. She realizes she should have divided the number by 4 and then added 10 to get the correct answer. Find the correct value of this number. | 22.5 |
How many triangles with positive area can be formed with vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $6$, inclusive? | 6788 |
The sum of two numbers is 25 and their product is 126. What is the absolute value of the difference of the two numbers? | 11 |
At the park, Dimitri saw families riding bicycles and tricycles. Bicycles have two wheels and tricycles have three wheels. 6 adults were riding bicycles and 15 children were riding tricycles. How many wheels did Dimitri see at the park? | Dimitri saw 2 x 6 = <<2*6=12>>12 wheels on the bicycles.
He saw 15 x 3 = <<15*3=45>>45 wheels on the tricycles.
He saw a total of 12 + 45 = <<12+45=57>>57 wheels.
#### 57 |
Bianca worked for 12.5 hours last weekend. Celeste worked for twice that amount of time and McClain worked 8.5 hours less than Celeste. How many minutes did the 3 people work in total? | Bianca = <<12.5=12.5>>12.5 hours
Celeste 2 * 12.5 = <<2*12.5=25>>25 hours
McClain = 25 - 8.5 = <<25-8.5=16.5>>16.5 hours
Total + 12.5 + 25 + 16.5 = <<+12.5+25+16.5=54>>54 hours
54 hours * 60 minutes = <<54*60=3240>>3240 minutes.
Bianca, Celeste and McClain worked a total of 3240 minutes last weekend.
#### 3240 |
The teacher of the summer math camp brought with him several shirts, several pairs of pants, several pairs of shoes, and two jackets for the entire summer. On each lesson, he wore pants, a shirt, and shoes, and wore a jacket for some lessons. On any two lessons, at least one element of his attire or shoes was different. It is known that if he had taken one more shirt, he could have conducted 36 more lessons; if he had taken one more pair of pants, he could have conducted 72 more lessons; if he had taken one more pair of shoes, he could have conducted 54 more lessons. What is the maximum number of lessons he could have conducted under these conditions? | 216 |
There is a peculiar computer with a button. If the current number on the screen is a multiple of 3, pressing the button will divide it by 3. If the current number is not a multiple of 3, pressing the button will multiply it by 6. Xiaoming pressed the button 6 times without looking at the screen, and the final number displayed on the computer was 12. What is the smallest possible initial number on the computer? | 27 |
Find the units digit of the decimal expansion of $\left(15 + \sqrt{220}\right)^{19} + \left(15 + \sqrt{220}\right)^{82}$. | 9 |
Given a sequence $\{a_n\}$ whose general term formula is $a_n = -n^2 + 12n - 32$, and the sum of the first $n$ terms is $S_n$, then for any $n > m$ (where $m, n \in \mathbb{N}^*$), the maximum value of $S_n - S_m$ is __. | 10 |
A chord of length √3 divides a circle of radius 1 into two arcs. R is the region bounded by the chord and the shorter arc. What is the largest area of a rectangle that can be drawn in R? | \frac{\sqrt{3}}{2} |
Matt and Blake want to watch every episode of the show The Office. There are 201 episodes. If they watch 1 episode every Monday and 2 episodes every Wednesday each week, how many weeks will it take them to watch the whole series? | Each week, they watch 1 Monday + 2 Wednesday episodes = <<1+2=3>>3 episodes a week.
If you take 201 total episodes / 3 episodes each week = <<201/3=67>>67 weeks to watch The Office series.
#### 67 |
What is the product of the digits in the base 8 representation of $6543_{10}$? | 168 |
Two distinct positive integers \( x \) and \( y \) are factors of 48. If \( x \cdot y \) is not a factor of 48, what is the smallest possible value of \( x \cdot y \)? | 32 |
If $M = 2098 \div 2$, $N = M \times 2$, and $X = M + N$, what is the value of $X$? | 3147 |
Compute
\[\frac{(1 + 17) \left( 1 + \dfrac{17}{2} \right) \left( 1 + \dfrac{17}{3} \right) \dotsm \left( 1 + \dfrac{17}{19} \right)}{(1 + 19) \left( 1 + \dfrac{19}{2} \right) \left( 1 + \dfrac{19}{3} \right) \dotsm \left( 1 + \dfrac{19}{17} \right)}.\] | 1 |
How many integers $n$ satisfy the inequality $-\frac{9\pi}{2} \leq n \leq 12\pi$? | 53 |
A geometric progression \( b_{1}, b_{2}, \ldots \) is such that \( b_{25} = 2 \tan \alpha \) and \( b_{31} = 2 \sin \alpha \) for some acute angle \( \alpha \). Find the term number \( n \) for which \( b_{n} = \sin 2\alpha \). | 37 |
Sandy earns $15 per hour. She worked 10 hours on Friday with her best friend, 6 hours on Saturday alone and 14 hours on Sunday with her other co-workers. How much money did Sandy earn in all on Friday, Saturday and Sunday? | The total number of hours Sandy worked on Friday, Saturday and Sunday was 10 + 6 + 14 = <<10+6+14=30>>30 hours.
Beth made 30 × $15 = $<<30*15=450>>450 in those 3 days.
#### 450 |
Nadine went to a garage sale and spent $56. She bought a table for $34 and 2 chairs. Each chair cost the same amount. How much did one chair cost? | Nadine spent $56 - $34 for the table = $<<56-34=22>>22 on chairs.
The cost of the chairs was equal, so one chair cost $22 / 2 = $<<22/2=11>>11.
#### 11 |
Given that $\tan \beta= \frac{4}{3}$, $\sin (\alpha+\beta)= \frac{5}{13}$, and both $\alpha$ and $\beta$ are within $(0, \pi)$, find the value of $\sin \alpha$. | \frac{63}{65} |
Emma has saved $230 in her bank account. She withdrew $60 to buy a new pair of shoes. The next week, she deposited twice as much money as she withdrew. How much is in her bank account now? | Emma had $230 - $60 = $<<230-60=170>>170 left in her bank account after she withdrew $60.
The next week, she deposited $60 x 2 = $<<60*2=120>>120.
Hence, she now has a total of $170 + $120 = $<<170+120=290>>290 in her bank account.
#### 290 |
An isosceles trapezoid has legs of length 30 cm each, two diagonals of length 40 cm each and the longer base is 50 cm. What is the trapezoid's area in sq cm? | 768 |
Given the hyperbola $x^{2}- \frac{y^{2}}{24}=1$, let the focal points be F<sub>1</sub> and F<sub>2</sub>, respectively. If P is a point on the left branch of the hyperbola such that $|PF_{1}|=\frac{3}{5}|F_{1}F_{2}|$, find the area of triangle $\triangle PF_{1}F_{2}$. | 24 |
Given the digits 0, 1, 2, 3, 4, 5, how many unique six-digit numbers greater than 300,000 can be formed where the digit in the thousand's place is less than 3? | 216 |
Ken created a care package to send to his brother, who was away at boarding school. Ken placed a box on a scale, and then he poured into the box enough jelly beans to bring the weight to 2 pounds. Then, he added enough brownies to cause the weight to triple. Next, he added another 2 pounds of jelly beans. And finally, he added enough gummy worms to double the weight once again. What was the final weight of the box of goodies, in pounds? | To the initial 2 pounds of jelly beans, he added enough brownies to cause the weight to triple, bringing the weight to 2*3=<<2*3=6>>6 pounds.
Next, he added another 2 pounds of jelly beans, bringing the weight to 6+2=<<6+2=8>>8 pounds.
And finally, he added enough gummy worms to double the weight once again, to a final weight of 8*2=<<8*2=16>>16 pounds.
#### 16 |
The line joining $(3,2)$ and $(6,0)$ divides the square shown into two parts. What fraction of the area of the square is above this line? Express your answer as a common fraction.
[asy]
draw((-2,0)--(7,0),linewidth(1),Arrows);
draw((0,-1)--(0,4),linewidth(1),Arrows);
draw((1,.25)--(1,-.25),linewidth(1));
draw((2,.25)--(2,-.25),linewidth(1));
draw((3,.25)--(3,-.25),linewidth(1));
draw((4,.25)--(4,-.25),linewidth(1));
draw((5,.25)--(5,-.25),linewidth(1));
draw((6,.25)--(6,-.25),linewidth(1));
draw((.25,1)--(-.25,1),linewidth(1));
draw((.25,2)--(-.25,2),linewidth(1));
draw((.25,3)--(-.25,3),linewidth(1));
draw((3,0)--(6,0)--(6,3)--(3,3)--(3,0)--cycle,linewidth(2));
label("$y$",(0,4),N);
label("$x$",(7,0),E);
label("$(3,0)$",(3,0),S);
label("$(6,3)$",(6,3),N);
[/asy] | \frac{2}{3} |
A hot dog stand sells 10 hot dogs every hour, each one selling for $2. How many hours does the stand need to run to make $200 in sales? | First, we find out how much in sales the stand makes each hour by multiplying the number of hot dogs sold by the sales price, finding that 2*10=<<2*10=20>>20 dollars in sales each hour.
Then we divide the sales goal for the day by the sales per hour, finding that the stand needs to be open for 200/20= <<200/20=10>>10 hours to hit its sales goal.
#### 10 |
What percent of the positive integers less than or equal to $100$ have no remainders when divided by $5?$ | 20 |
In a triangle configuration, each row consists of increasing multiples of 3 unit rods. The number of connectors in a triangle always forms an additional row than the rods, with connectors enclosing each by doubling the requirements of connecting joints from the previous triangle. How many total pieces are required to build a four-row triangle? | 60 |
The bowling alley has 30 red bowling balls. There are 6 more green bowling balls than red. How many bowling balls are there in total? | There are 30 + 6 = <<30+6=36>>36 green bowling balls
In total there are 30 + 36 = <<30+36=66>>66 bowling balls
#### 66 |
What is the area, in square inches, of a right triangle with a 24-inch leg and a 25-inch hypotenuse? | 84 |
Marsha has two numbers, $a$ and $b$. When she divides $a$ by 70 she gets a remainder of 64. When she divides $b$ by 105 she gets a remainder of 99. What remainder does she get when she divides $a+b$ by 35? | 23 |
Given the function $f(x)=\sin \frac {x}{2}\cos \frac {x}{2}+\cos ^{2} \frac {x}{2}-1$.
$(1)$ Find the smallest positive period of the function $f(x)$ and the interval where it is monotonically decreasing;
$(2)$ Find the minimum value of the function $f(x)$ on the interval $\left[ \frac {\pi}{4}, \frac {3\pi}{2}\right]$. | - \frac { \sqrt {2}+1}{2} |
Given an arithmetic sequence $\{a_n\}$, the sum of its first $n$ terms is $S_n$. It is known that $S_8 \leq 6$ and $S_{11} \geq 27$. Determine the minimum value of $S_{19}$. | 133 |
A reflection takes $\begin{pmatrix} -1 \\ 7 \end{pmatrix}$ to $\begin{pmatrix} 5 \\ -5 \end{pmatrix}.$ Which vector does the reflection take $\begin{pmatrix} -4 \\ 3 \end{pmatrix}$ to? | \begin{pmatrix} 0 \\ -5 \end{pmatrix} |
If the equation $x^{2}+(k^{2}-4)x+k-1=0$ has two roots that are opposite numbers, solve for $k$. | -2 |
What is the value of $x$ if \begin{align*}x &= y+5,\\
y &= z+10,\\
z &= w+20,\\
\text{and }\qquad w &= 80?
\end{align*} | 115 |
Compute
\[\frac{\tan^2 20^\circ - \sin^2 20^\circ}{\tan^2 20^\circ \sin^2 20^\circ}.\] | 1 |
An ice cream cone has radius 1 inch and height 4 inches, What is the number of inches in the radius of a sphere of ice cream which has the same volume as the cone? | 1 |
Given the vectors $\overrightarrow{m}=(2\sin \omega x, \cos ^{2}\omega x-\sin ^{2}\omega x)$ and $\overrightarrow{n}=( \sqrt {3}\cos \omega x,1)$, where $\omega > 0$ and $x\in R$. If the minimum positive period of the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$ is $\pi$,
(I) Find the value of $\omega$.
(II) In $\triangle ABC$, if $f(B)=-2$, $BC= \sqrt {3}$, and $\sin B= \sqrt {3}\sin A$, find the value of $\overrightarrow{BA}\cdot \overrightarrow{BC}$. | -\frac{3}{2} |
If $f(x)=ax+b$ and $f^{-1}(x)=bx+a$ with $a$ and $b$ real, what is the value of $a+b$? | -2 |
Given that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n = -a_n - \left(\frac{1}{2}\right)^{n-1} + 2$, and $(1) b_n = 2^n a_n$, find the general term formula for $\{b_n\}$. Also, $(2)$ find the maximum term of $\{a_n\}$. | \frac{1}{2} |
The chicken crossed the road to get to the other side twice for the thrill of it. The first time, it had to dodge 23 speeding cars. The second time, a person tried to catch it and accidentally pulled out twice as many feathers as the number of cars the chicken had dodged. The chicken had 5263 feathers before its thrill-seeking road crossings. How many feathers did it have afterward? | The chicken lost 23 * 2 = <<23*2=46>>46 feathers on its second road crossing.
Thus, it had 5263 - 46 = <<5263-46=5217>>5217 feathers after crossing the road twice.
#### 5217 |
Calculate the number of multiplication and addition operations needed to compute the value of the polynomial $f(x) = 3x^6 + 4x^5 + 5x^4 + 6x^3 + 7x^2 + 8x + 1$ at $x = 0.7$ using the Horner's method. | 12 |
In triangle $\triangle ABC$, $\overrightarrow{BC}=\sqrt{3}\overrightarrow{BD}$, $AD\bot AB$, $|{\overrightarrow{AD}}|=1$, then $\overrightarrow{AC}•\overrightarrow{AD}=\_\_\_\_\_\_$. | \sqrt{3} |
Suppose $f(x)$ is a rational function such that $3f\left(\dfrac{1}{x}\right)+\dfrac{2f(x)}{x}=x^2$ for $x\neq 0$. Find $f(-2)$. | \frac{67}{20} |
For a pyramid S-ABCD, each vertex is colored with one color, and the two ends of the same edge are colored differently. If there are exactly 5 colors available, calculate the number of different coloring methods. | 420 |
A number $x$ is equal to $7\cdot24\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | 588 |
In a convex quadrilateral \(ABCD\), the midpoint of side \(AD\) is marked as point \(M\). Segments \(BM\) and \(AC\) intersect at point \(O\). It is known that \(\angle ABM = 55^\circ\), \(\angle AMB = 70^\circ\), \(\angle BOC = 80^\circ\), and \(\angle ADC = 60^\circ\). How many degrees is \(\angle BCA\)? | 35 |
When a right triangle is rotated about one leg, the volume of the cone produced is $800\pi \;\textrm{ cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\pi \;\textrm{ cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?
| 26 |
In a square piece of grid paper containing an integer number of cells, a hole in the shape of a square, also consisting of an integer number of cells, was cut out. How many cells did the large square contain if, after cutting out the hole, 209 cells remained? | 225 |
55% of Toby's friends are boys and the rest are girls. If he has 33 friends who are boys, how many friends does he have who are girls? | Toby has 60 friends because 33 / .55 = <<33/.55=60>>60
45% of his friends are girls because 100 - 55 = <<100-55=45>>45
He has 27 friends who are girls because 60 x .45 = <<27=27>>27
#### 27 |
Several island inhabitants gather in a hut, with some belonging to the Ah tribe and the rest to the Uh tribe. Ah tribe members always tell the truth, while Uh tribe members always lie. One inhabitant said, "There are no more than 16 of us in the hut," and then added, "All of us are from the Uh tribe." Another said, "There are no more than 17 of us in the hut," and then noted, "Some of us are from the Ah tribe." A third person said, "There are five of us in the hut," and looking around, added, "There are at least three Uh tribe members among us." How many Ah tribe members are in the hut? | 15 |
In triangle \(A B C, A B=6, B C=7\) and \(C A=8\). Let \(D, E, F\) be the midpoints of sides \(B C\), \(A C, A B\), respectively. Also let \(O_{A}, O_{B}, O_{C}\) be the circumcenters of triangles \(A F D, B D E\), and \(C E F\), respectively. Find the area of triangle \(O_{A} O_{B} O_{C}\). | \frac{21 \sqrt{15}}{16} |
There are 3 different pairs of shoes in a shoe cabinet. If one shoe is picked at random from the left shoe set of 6 shoes, and then another shoe is picked at random from the right shoe set of 6 shoes, calculate the probability that the two shoes form a pair. | \frac{1}{3} |
Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) that satisfies the condition: for any real numbers \( x \) and \( y \),
\[ f(2x) + f(2y) = f(x+y) f(x-y) \]
and given that \( f(\pi) = 0 \) and \( f(x) \) is not identically zero, determine the period of \( f(x) \). | 4\pi |
$a,b,c$ are distinct real roots of $x^3-3x+1=0$. $a^8+b^8+c^8$ is | 186 |
Page collects fancy shoes and has 80 pairs in her closet. She decides to donate 30% of her collection that she no longer wears. After dropping off her donation, she treats herself and buys 6 more pairs to add to her collection. How many shoes does she have now? | She donates 30% of her 80 pairs of shoes so that’s .30*80 = <<30*.01*80=24>>24 pair of shoes she donates
She had 80 shoes in her collection and donates 24 so that leaves 80-24 = <<80-24=56>>56 pairs of shoes
She now has 56 pairs of shoes and buys 6 more pairs so her collection is now 56+6 = <<56+6=62>>62 pairs of shoes
#### 62 |
Anna is reading a 31-chapter textbook, but she skips all the chapters that are divisible by 3. If it takes her 20 minutes to read each chapter, how many hours does she spend reading total? | First divide 32 by 3, ignoring the remainder, to find how many chapters in the book are divisible by 3: 31 / 3 = 10.33..., which means 10 chapters are divisible by 3
Then subtract that number from the total number of chapters to find how many chapters she reads: 31 chapters - 10 chapters = <<31-10=21>>21 chapters
Then multiply the number chapters by the time to read each chapter to find the total time she spends reading: 21 chapters * 20 minutes/chapter = <<21*20=420>>420 minutes
Then divide that time by the number of minutes per movie to find the total time spent reading in hours: 420 minutes / 60 minutes/hour = <<420/60=7>>7 hours
#### 7 |
For how many integers \( n \) is \(\frac{2n^3 - 12n^2 - 2n + 12}{n^2 + 5n - 6}\) equal to an integer? | 32 |
$\Delta ABC$ is isosceles with $AC = BC$. If $m\angle C = 40^{\circ}$, what is the number of degrees in $m\angle CBD$? [asy] pair A,B,C,D,E;
C = dir(65); B = C + dir(-65); D = (1.5,0); E = (2,0);
draw(B--C--A--E); dot(D);
label("$A$",A,S); label("$B$",B,S); label("$D$",D,S); label("$C$",C,N);
[/asy] | 110 |
Of the 24 students in class, one-third are in the after-school chess program, and half of those students will be absent from school on Friday for the regional chess tournament. How many of the students in this class are going to that tournament? | There are 24/3=<<24/3=8>>8 students in the program.
And 8/2=<<8/2=4>>4 go to the tournament.
#### 4 |
A $4\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums? | 4 |
Three of the vertices of parallelogram $ABCD$ are $A = (3,-1,2),$ $B = (1,2,-4),$ and $C = (-1,1,2).$ Find the coordinates of $D.$ | (1,-2,8) |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | \sqrt{55} |
Define the function $f(x)=\frac{b}{2x-3}$. If $f(2)=f^{-1}(b+1)$, find the product of all possible values of $b$. | -\frac{3}{2} |
Given a cube \( ABCD A_1 B_1 C_1 D_1 \) with an edge length of 1. A sphere passes through vertices \( A \) and \( C \) and the midpoints \( F \) and \( E \) of edges \( B_1 C_1 \) and \( C_1 D_1 \) respectively. Find the radius \( R \) of this sphere. | \frac{\sqrt{41}}{8} |
A function $f(x)$ defined on $R$ satisfies $f(x+1) = 2f(x)$. When $x \in (-1,0]$, $f(x) = x^{3}$. Find $f(\frac{21}{2})$. | -256 |
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