problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
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In the quadrilateral \(ABCD\), the lengths of the sides \(BC\) and \(CD\) are 2 and 6, respectively. The points of intersection of the medians of triangles \(ABC\), \(BCD\), and \(ACD\) form an equilateral triangle. What is the maximum possible area of quadrilateral \(ABCD\)? If necessary, round the answer to the neare... | 29.32 |
For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1) = 1$, $c(2n) = c(n)$, and $c(2n+1) = (-1)^n c(n)$. Find the value of \[ \sum_{n=1}^{2013} c(n) c(n+2). \] | -1 |
The line $ax+2by=1$ intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$ (where $a$ and $b$ are real numbers), and $\triangle AOB$ is a right-angled triangle ($O$ is the origin). The maximum distance between point $P(a,b)$ and point $Q(0,0)$ is ______. | \sqrt{2} |
Josette bought 3 bottles of mineral water for €1.50. How much will four bottles cost? | A bottle costs 1.50 € / 3 = <<1.50/3=0.50>>0.50 €.
So, four bottles cost 0.50 € * 4 = <<0.50*4=2>>2 €.
#### 2 |
The probability that Kim has a math test today is $\frac{4}{7}$. What is the probability that Kim does not have a math test today? Express your answer as a common fraction. | \frac{3}{7} |
Given a function $f(x) = \cos x \sin \left( x + \frac{\pi}{3} \right) - \sqrt{3} \cos^2 x + \frac{\sqrt{3}}{4}$, where $x \in \mathbb{R}$,
(1) Find the smallest positive period of $f(x)$ and the interval where $f(x)$ is monotonically decreasing;
(2) Find the maximum and minimum values of $f(x)$ in the closed interval $... | -\frac{1}{2} |
What is the smallest number with three different prime factors, none of which can be less than 10? | 2431 |
Given the function $f(x)=x^{2-m}$ defined on the interval $[-3-m,m^{2}-m]$, which is an odd function, find $f(m)=$____. | -1 |
Form a three-digit number using the digits 0, 1, 2, 3. Repeating digits is not allowed.
① How many three-digit numbers can be formed?
② If the three-digit numbers from ① are sorted in ascending order, what position does 230 occupy?
③ If repeating digits is allowed, how many of the formed three-digit numbers are d... | 16 |
For how many integer values of $m$ ,
(i) $1\le m \le 5000$ (ii) $[\sqrt{m}] =[\sqrt{m+125}]$ Note: $[x]$ is the greatest integer function | 72 |
Five years ago, the sum of Sebastian's age and his sister's age was 3/4 of their father's age. How old is their father today if Sebastian is 40 years old and 10 years older than his sister? | If Sebastian is 40 years old and 10 years older than his sister, his sister is 40-10=<<40-10=30>>30 years old.
Five years ago, Sebastian was 40-5=<<40-5=35>>35 years old.
Similarly, Sebastian's sister's age was 30-5=<<30-5=25>>25 years old.
The sum of their ages five years ago was 25+35=<<25+35=60>>60
Five years ago, t... |
Given plane vectors $\vec{a}, \vec{b}, \vec{c}$ that satisfy the following conditions: $|\vec{a}| = |\vec{b}| \neq 0$, $\vec{a} \perp \vec{b}$, $|\vec{c}| = 2 \sqrt{2}$, and $|\vec{c} - \vec{a}| = 1$, determine the maximum possible value of $|\vec{a} + \vec{b} - \vec{c}|$. | 3\sqrt{2} |
Simplify: $-{-\left[-|-1|^2\right]^3}^4$. | -1 |
Given $\triangle ABC$ with $AC=1$, $\angle ABC= \frac{2\pi}{3}$, $\angle BAC=x$, let $f(x)= \overrightarrow{AB} \cdot \overrightarrow{BC}$.
$(1)$ Find the analytical expression of $f(x)$ and indicate its domain;
$(2)$ Let $g(x)=6mf(x)+1$ $(m < 0)$, if the range of $g(x)$ is $\left[- \frac{3}{2},1\right)$, find the va... | - \frac{5}{2} |
Find the integer $n$, $0 \le n \le 9$, that satisfies \[n \equiv -2187 \pmod{10}.\] | 3 |
Given the function $f(x)=2x^{3}-ax^{2}+1$ $(a\in\mathbb{R})$ has exactly one zero in the interval $(0,+\infty)$, find the sum of the maximum and minimum values of $f(x)$ on the interval $[-1,1]$. | -3 |
Round $54.\overline{54}$ to the nearest hundredth. | 54.55 |
To popularize knowledge of fire safety, a certain school organized a competition on related knowledge. The competition is divided into two rounds, and each participant must participate in both rounds. If a participant wins in both rounds, they are considered to have won the competition. It is known that in the first ro... | \frac{223}{300} |
On Tuesday, a fruit vendor sold 2.5 dozen lemons and 5 dozens avocados. What is the total number of fruits that the fruit vendor sold? | Since 1 dozen is equal to 12, then the vendor sold 2.5 x 12 = <<2.5*12=30>>30 lemons.
While he sold 5 x 12 = <<5*12=60>>60 avocados.
So, the fruit vendor sold a total of 30 + 60 = <<30+60=90>>90 fruits.
#### 90 |
What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock? | 60 |
Given an arithmetic sequence $\{a_n\}$, its sum of the first $n$ terms is $S_n$. It is known that $a_2=2$, $S_5=15$, and $b_n=\frac{1}{a_{n+1}^2-1}$. Find the sum of the first 10 terms of the sequence $\{b_n\}$. | \frac {175}{264} |
Find the greatest root of $f(x) = 15x^4-13x^2+2$. | \frac{\sqrt{6}}{3} |
Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation. | (0,9) |
Let $A_{1} A_{2} \ldots A_{19}$ be a regular nonadecagon. Lines $A_{1} A_{5}$ and $A_{3} A_{4}$ meet at $X$. Compute $\angle A_{7} X A_{5}$. | \frac{1170^{\circ}}{19} |
Consider five-dimensional Cartesian space $\mathbb{R}^{5}=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \mid x_{i} \in \mathbb{R}\right\}$ and consider the hyperplanes with the following equations: - $x_{i}=x_{j}$ for every $1 \leq i<j \leq 5$; - $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=-1$ - $x_{1}+x_{2}+x_{3}+x_{4}+x_{... | 480 |
Calculate:<br/>$(1)2(\sqrt{3}-\sqrt{5})+3(\sqrt{3}+\sqrt{5})$;<br/>$(2)-{1}^{2}-|1-\sqrt{3}|+\sqrt[3]{8}-(-3)×\sqrt{9}$. | 11 - \sqrt{3} |
James wants to learn to become a chess grandmaster. It takes 2 hours to learn the rules. It then takes him 49 times that long to get a level of proficiency to start playing in local tournaments. After that, he devotes his life to chess and spends 100 times as much as the combined time to get proficient to becoming a... | It takes him 2*49=<<2*49=98>>98 hours to go from knowing how to play to proficient
So he spends 98+2=<<98+2=100>>100 hours combined on those
So it takes him 100*100=<<100*100=10000>>10,000 hours to become a master from proficient
So the total time was 100+10000=<<100+10000=10100>>10,100 hours
#### 10,100 |
In $\triangle ABC$, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively. Given that a = 3, cosC = $- \frac{1}{15}$, and 5sin(B + C) = 3sin(A + C).
(1) Find the length of side c.
(2) Find the value of sin(B - $\frac{\pi}{3}$). | \frac{2\sqrt{14} - 5\sqrt{3}}{18} |
Given $\cos \left(\alpha- \frac {\beta}{2}\right)=- \frac {1}{9}$ and $\sin \left( \frac {\alpha}{2}-\beta\right)= \frac {2}{3}$, with $0 < \beta < \frac {\pi}{2} < \alpha < \pi$, find $\sin \frac {\alpha+\beta}{2}=$ ______. | \frac {22}{27} |
A small square is entirely contained in a larger square, as shown. The side length of the small square is 3 units and the side length of the larger square is 7 units. What is the number of square units in the area of the black region?
[asy]
fill((0,0)--(21,0)--(21,21)--(0,21)--cycle,black);
fill((9,4)--(9,13)--(18,13)... | 40 |
The sum of the digits in the product of $\overline{A A A A A A A A A} \times \overline{B B B B B B B B B}$. | 81 |
Simplify the following expression: $(9x^9+7x^8+4x^7) + (x^{11}+x^9+2x^7+3x^3+5x+8).$ Express your answer as a polynomial with the degrees of the terms in decreasing order. | x^{11}+10x^9+7x^8+6x^7+3x^3+5x+8 |
Find
\[\binom{100}{0} - \binom{100}{1} + \binom{100}{2} - \dots + \binom{100}{100}.\] | 0 |
Oliver has $40 and 200 quarters. If he gives his sister $5 and 120 quarters, how much money in total is he left with? | The 200 quarters equals 200*$0.25 = $<<200*0.25=50>>50.
Oliver has a total of $40+$50 = $<<40+50=90>>90.
He gives his sister 120*$0.25 = $<<120*0.25=30>>30 in total from the quarters.
The total amount he gave to his sister is $30+$5= $<<30+5=35>>35.
Oliver is left with $90-$35 = $<<90-35=55>>55.
#### 55 |
(1) Given the hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, its right vertex is $A$, and a circle $A$ with center $A$ and radius $b$ intersects one of the asymptotes of the hyperbola $C$ at points $M$ and $N$. If $\angle MAN = 60^{\circ}$, then the eccentricity of $C$ is ______.
(2... | \dfrac{4}{5} |
Bill buys a stock that decreases by $20\%$ on the first day, and then on the second day the stock increases by $30\%$ of its value at the end of the first day. What was the overall percent increase in Bill's stock over the two days? | 4 |
A bicycle trip is 30 km long. Ari rides at an average speed of 20 km/h. Bri rides at an average speed of 15 km/h. If Ari and Bri begin at the same time, how many minutes after Ari finishes the trip will Bri finish? | 30 |
Find $x+y+z$ when $$ a_1x+a_2y+a_3z= a $$ $$ b_1x+b_2y+b_3z=b $$ $$ c_1x+c_2y+c_3z=c $$ Given that $$ a_1\left(b_2c_3-b_3c_2\right)-a_2\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c_2-b_2c_1\right)=9 $$ $$ a\left(b_2c_3-b_3c_2\right)-a_2\left(bc_3-b_3c\right)+a_3\left(bc_2-b_2c\right)=17 $$ $$ a_1\left(bc... | 16/9 |
For how many ordered pairs of positive integers $(a, b)$ such that $a \le 50$ is it true that $x^2 - ax + b$ has integer roots? | 625 |
Felicity and Adhira took separate trips. Felicity used 5 less gallons of gas than four times the number of gallons that Adhira used for her trip. Together the girls used 30 gallons of gas. How many gallons did Felicity use? | Let A = Adhira's gallons
Felicity = 4A - 5
5A - 5 = 30
5A = 35
A = <<7=7>>7
Felicity used 23 gallons of gas for her trip.
#### 23 |
Mike gets paid 100 dollars a week. He decides to spend half of that at an arcade. He spends 10 dollars at the arcade on food and uses the rest on arcade tokens. He can play for 1 hour for $8. How many minutes can he play? | He spent 100*.5=$<<100*.5=50>>50 at the arcade.
He uses 50-10=$<<50-10=40>>40 on tokens
He can play 40/8=<<40/8=5>>5 hours
There are 60 minutes in an hour so he can play 5*60=<<5*60=300>>300 minutes
#### 300 |
A $7 \times 7$ board is either empty or contains an invisible $2 \times 2$ ship placed "by the cells." You are allowed to place detectors in some cells of the board and then activate them all at once. An activated detector signals if its cell is occupied by the ship. What is the minimum number of detectors needed to gu... | 16 |
Travel along the alley clockwise.
In 1 hour of walking, the pedestrian walked 6 kilometers and did not reach point $B$ (a whole $2 \pi - 6$ km!), so the third option is clearly longer than the first and can be excluded.
In the first case, when moving along the alley, they would need to cover a distance of 6 km, and i... | 0.21 |
Two ferries cross a river with constant speeds, turning at the shores without losing time. They start simultaneously from opposite shores and meet for the first time 700 feet from one shore. They continue to the shores, return, and meet for the second time 400 feet from the opposite shore. Determine the width of the ri... | 1400 |
A church has 100 members who've volunteered to bake cookies for the homeless in their local town. If each member baked 10 sheets of cookies, and each sheet has 16 cookies, calculate the total number of cookies the church members baked? | A sheet has 16 cookies, so if each church member baked 10 sheets, they each baked 10 sheets/person * 16 cookies/sheet = <<10*16=160>>160 cookies/person
The total number of church members who decided to volunteer is 100, so in total, they baked 160 cookies/person * 100 people = <<100*160=16000>>16000 cookies
#### 16000 |
A non-increasing sequence of 100 non-negative reals has the sum of the first two terms at most 100 and the sum of the remaining terms at most 100. What is the largest possible value for the sum of the squares of the terms? | 10000 |
Let \( A, B, C \) be points on the same plane with \( \angle ACB = 120^\circ \). There is a sequence of circles \( \omega_0, \omega_1, \omega_2, \ldots \) on the same plane (with corresponding radii \( r_0, r_1, r_2, \ldots \) where \( r_0 > r_1 > r_2 > \cdots \)) such that each circle is tangent to both segments \( CA... | \frac{3}{2} + \sqrt{3} |
Place each of the digits 6, 7, 8 and 9 in exactly one square to make the smallest possible product. What is this product? [asy]draw((0,.5)--(10,.5),linewidth(1));
draw((4,1)--(6,1)--(6,3)--(4,3)--(4,1),linewidth(1));
draw((7,1)--(9,1)--(9,3)--(7,3)--(7,1),linewidth(1));
draw((7,4)--(9,4)--(9,6)--(7,6)--(7,4),linewidth(... | 5372 |
A six place number is formed by repeating a three place number; for example, $256256$ or $678678$, etc. Any number of this form is always exactly divisible by: | 1001 |
Cameron guides tour groups in a museum. He usually answers two questions per tourist. Today, he did four tours. The early morning first group was only 6 people. The following group was a busy group of 11. The third group had 8 people, but one was inquisitive and asked three times as many questions as usual. The last gr... | Cameron answered 2 * 6 = <<2*6=12>>12 questions for the first group.
He answered 2 * 11 = <<2*11=22>>22 questions for the second group.
The third group had 8 - 1 = 7 tourists that asked 2 questions each.
The third group also had 1 tourist that asked 2 * 3 = <<1*2*3=6>>6 questions.
Thus, the third group asked 7 * 2 + 6 ... |
John is very unfit and decides to work up to doing a push-up. He trains 5 days a week for them and starts with wall push-ups. He adds 1 rep a day and once he gets to 15 reps he will start training high elevation push-ups. and then low elevation push-ups, and finally floor push-ups. How many weeks will it take him to... | He needs to do 15*3=<<15*3=45>>45 progressions
That will take 45/5=<<45/5=9>>9 weeks
#### 9 |
The least common multiple of $x$ and $y$ is $18$, and the least common multiple of $y$ and $z$ is $20$. Determine the least possible value of the least common multiple of $x$ and $z$. | 90 |
Viggo's age was 10 years more than twice his younger brother's age when his brother was 2. If his younger brother is currently 10 years old, what's the sum of theirs ages? | Twice Viggo's younger brother's age when his brother was 2 is 2*2 = <<2*2=4>>4 years.
If Viggo's age was 10 more than twice his younger brother's age when his brother was 2, Viggo was 10+4 = 14 years old.
Viggo is 14 years - 2 years = <<14-2=12>>12 years older than his brother
Since Viggo's brother is currently 10 and ... |
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$ . (Note: $n$ is written in the usual base ten notation.) | 9999 |
What is the sum of all of the two-digit primes that are greater than 12 but less than 99 and are still prime when their two digits are interchanged? | 418 |
When $0.42\overline{153}$ is expressed as a fraction in the form $\frac{x}{99900}$, what is the value of $x$? | 42111 |
What is the coefficient of $x^4$ in the expansion of $(1-2x^2)^5$? | 40 |
Given the line $y=-x+1$ and the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, they intersect at points $A$ and $B$. $OA \perp OB$, where $O$ is the origin. If the eccentricity of the ellipse $e \in [\frac{1}{2}, \frac{\sqrt{3}}{2}]$, find the maximum value of $a$. | \frac{\sqrt{10}}{2} |
Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8)=8$ and $\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x))=3$? | 10 |
On the board we write a series of $n$ numbers, where $n \geq 40$ , and each one of them is equal to either $1$ or $-1$ , such that the following conditions both hold:
(i) The sum of every $40$ consecutive numbers is equal to $0$ .
(ii) The sum of every $42$ consecutive numbers is not equal to $0$ .
We den... | 20 |
Find the value of $a$ so that the lines described by
\[\begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix} + t \begin{pmatrix} a \\ -2 \\ 1 \end{pmatrix}\]and
\[\begin{pmatrix} 1 \\ -3/2 \\ -5 \end{pmatrix} + u \begin{pmatrix} 1 \\ 3/2 \\ 2 \end{pmatrix}\]are perpendicular. | 1 |
Alice wants to write down a list of prime numbers less than 100, using each of the digits 1, 2, 3, 4, and 5 once and no other digits. Which prime number must be in her list? | 41 |
John buys 3 boxes of burritos. He gives away a 3rd of them to his friend. Each box has 20 burritos. He eats 3 burritos per day for 10 days. How many burritos does he have left? | He gave 3/3=<<3/3=1>>1 box away
So he has 3-1=<<3-1=2>>2 boxes
So he got 2*20=<<2*20=40>>40 burritos
He ate 3*10=<<3*10=30>>30 burritos
So he has 40-30=<<40-30=10>>10 burritos left
#### 10 |
In a division problem, the dividend is 12, and the divisor is a natural number less than 12. What is the sum of all possible different remainders? | 15 |
At the "Economics and Law" congress, a "Best of the Best" tournament was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. During one match, participants had to ask each other questions within a limited time and record correct answers. Each participant played with each other... | 105 |
Given four one-inch squares are placed with their bases on a line. The second square from the left is lifted out and rotated 30 degrees before reinserting it such that it just touches the adjacent square on its right. Determine the distance in inches from point B, the highest point of the rotated square, to the line on... | \frac{2 + \sqrt{3}}{4} |
Let $P(x) = b_0 + b_1x + \dots + b_nx^n$ be a polynomial with integer coefficients, and $0 \le b_i < 5$ for all $0 \le i \le n$.
Given that $P(\sqrt{5}) = 40 + 31\sqrt{5}$, compute $P(3)$. | 381 |
Ms. Alice can grade 296 papers in 8 hours. How many papers can she grade in 11 hours? | First you must recognize that the problem involves finding the rate of how fast Ms. Alice can grade papers. To find how many papers she can grade in 11 hours, it would be useful to first find out how many papers she can grade in one hour. To do so, divide 296 papers by 8 hours, 296/8, to get 37. Therefore, Ms. Alice ca... |
Given a convex quadrilateral \(ABCD\) with \(\angle C = 57^{\circ}\), \(\sin \angle A + \sin \angle B = \sqrt{2}\), and \(\cos \angle A + \cos \angle B = 2 - \sqrt{2}\), find the measure of angle \(D\) in degrees. | 168 |
In $\triangle ABC$, side $a = \sqrt{3}$, side $b = \sqrt{3}$, and side $c > 3$. Let $x$ be the largest number such that the magnitude, in degrees, of the angle opposite side $c$ exceeds $x$. Then $x$ equals: | 120^{\circ} |
From 50 products, 10 are selected for inspection. The total number of items is \_\_\_\_\_\_\_, and the sample size is \_\_\_\_\_\_. | 10 |
James has a room that is 13 feet by 18 feet. He increases each dimension by 2 feet. He then builds 3 more rooms of equal size and 1 room of twice that size. How much area does he have? | He increases the length to 13+2=<<13+2=15>>15 feet
He increases the width to 18+2=<<18+2=20>>20 feet
So the rooms are 15*20=<<15*20=300>>300 square feet
So he has 1+3=<<1+3=4>>4 rooms of this size
So he has 4*300=<<4*300=1200>>1200 square feet of rooms this size
He also has one room of size 300*2=<<300*2=600>>600 squar... |
There is a unique two-digit positive integer $t$ for which the last two digits of $13 \cdot t$ are $26$. | 62 |
There are two red, two black, two white, and a positive but unknown number of blue socks in a drawer. It is empirically determined that if two socks are taken from the drawer without replacement, the probability they are of the same color is $\frac{1}{5}$. How many blue socks are there in the drawer? | 4 |
Compute $\arccos (\sin 3)$. All functions are in radians. | 3 - \frac{\pi}{2} |
When $p(x) = Ax^5 + Bx^3 + Cx + 4$ is divided by $x - 3,$ the remainder is 11. Find the remainder when $p(x)$ is divided by $x + 3.$ | -3 |
A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle that has the endpoints of its diameter on the vertices of the rectangle. What percent larger is the area of the large semicircles than the area of the small semicircles? Express your answer to the nearest whole number. | 178\% |
Square \(ABCD\) has sides of length 14. A circle is drawn through \(A\) and \(D\) so that it is tangent to \(BC\). What is the radius of the circle? | 8.75 |
Four brothers have together forty-eight Kwanzas. If the first brother's money were increased by three Kwanzas, if the second brother's money were decreased by three Kwanzas, if the third brother's money were triplicated and if the last brother's money were reduced by a third, then all brothers would have the same quant... | 6, 12, 3, 27 |
A bowling ball cannot weigh more than 16 pounds and must have a diameter of $8 \frac{1}{2}$ inches. How many square inches are in the surface area of a bowling ball before the finger holes are drilled? Express your answer as a common fraction in terms of $\pi$. | \frac{289\pi}{4} |
For natural numbers \\(m\\) greater than or equal to \\(2\\) and their powers of \\(n\\), the following decomposition formula is given:
\\(2^{2}=1+3\\) \\(3^{2}=1+3+5\\) \\(4^{2}=1+3+5+7\\) \\(…\\)
\\(2^{3}=3+5\\) \\(3^{3}=7+9+11\\) \\(…\\)
\\(2^{4}=7+9\\) \\(…\\)
Following this pattern, the third nu... | 125 |
Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one vertex on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$? | \sqrt{95} |
Let $S$ be the set of $10$-tuples of non-negative integers that have sum $2019$. For any tuple in $S$, if one of the numbers in the tuple is $\geq 9$, then we can subtract $9$ from it, and add $1$ to the remaining numbers in the tuple. Call thus one operation. If for $A,B\in S$ we can get from $A$ to $B$ in finitely ma... | 10^8 |
Find all possible positive integers represented in decimal as $13 x y 45 z$, which are divisible by 792, where $x, y, z$ are unknown digits. | 1380456 |
A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a $2 / 3$ chance of catching each individual error still in the article. After 3 days,... | \frac{416}{729} |
There are four distinct positive integers $a,b,c,d$ less than $8$ which are invertible modulo $8$. Find the remainder when $(abc+abd+acd+bcd)(abcd)^{-1}$ is divided by $8$. | 0 |
Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([3, 5]\). Find \(\underbrace{f(f(\ldots f}_{2017}\left(\frac{7+\sqrt{15}}{2}\right)) \ldots)\). Round the answer to hundredths if necessary. | 1.56 |
Let $\mathrm {P}$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have a positive imaginary part, and suppose that $\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})$, where $0<r$ and $0\leq \theta <360$. Find $\theta$. | 276 |
Consider a cube PQRSTUVW with a side length s. Let M and N be the midpoints of edges PU and RW, and let K be the midpoint of QT. Find the ratio of the area of triangle MNK to the area of one of the faces of the cube. | \frac{1}{4} |
The shape of a bridge arch is a parabola. It is known that the width of the parabolic arch is 8 meters, and the area of the parabolic arch is 160 square meters. Then, the height of the parabolic arch is | 30 |
Jessica is six years older than Claire. In two years, Claire will be 20 years old. How old is Jessica now? | Claire's age now is 20 - 2 = <<20-2=18>>18 years old.
Being 6 years older than Claire, Jessica is 18 + 6 = <<6+18=24>>24 years old.
#### 24 |
Let $d$ and $e$ denote the solutions of $3x^2+10x-25=0$. Find $(d-e)^2$. | \frac{400}{9} |
What is the largest power of 2 by which the number \(10^{10} - 2^{10}\) is divisible? | 13 |
In the line $4x+7y+c=0$, the sum of the $x$- and $y$- intercepts is $22$. Find $c$. | -56 |
How many subsets $S$ of the set $\{1,2, \ldots, 10\}$ satisfy the property that, for all $i \in[1,9]$, either $i$ or $i+1$ (or both) is in $S$? | 144 |
Given the lines $l_{1}$: $x+ay-a+2=0$ and $l_{2}$: $2ax+(a+3)y+a-5=0$.
$(1)$ When $a=1$, find the coordinates of the intersection point of lines $l_{1}$ and $l_{2}$.
$(2)$ If $l_{1}$ is parallel to $l_{2}$, find the value of $a$. | a = \frac{3}{2} |
How many whole numbers lie in the interval between $\frac{5}{3}$ and $2\pi$ ? | 5 |
Given that $-1 - 4\sqrt{2}$ is a root of the equation \[x^3 + ax^2 + bx + 31 = 0\]and that $a$ and $b$ are rational numbers, compute $a.$ | 1 |
There are 3 rods with several golden disks of different sizes placed on them. Initially, 5 disks are arranged on the leftmost rod (A) in descending order of size. According to the rule that only one disk can be moved at a time and a larger disk can never be placed on top of a smaller one, the goal is to move all 5 disk... | 31 |
This weekend's football game matched the Seattle Seahawks with the Denver Broncos. The final score was 37-23, with the Seattle Seahawks taking the win. If a touchdown is worth 7 points and a field goal is worth 3 points, how many touchdowns did the Seattle Seahawks score if they scored 3 field goals during the game? | The Seattle Seahawks scored 3 field goals during the length of the game and each field goal is worth 3 points, so 3 * 3=<<3*3=9>>9 points were made by the Seattle Seahawks from field goals.
The Seattle Seahawks won so their total points would be 37 with 9 of those points accrued through the scoring of field goals, whic... |
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