problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Let \( S = \{(x, y) \mid x, y \in \mathbb{Z}, 0 \leq x, y \leq 2016\} \). Given points \( A = (x_1, y_1), B = (x_2, y_2) \) in \( S \), define
\[ d_{2017}(A, B) = (x_1 - x_2)^2 + (y_1 - y_2)^2 \pmod{2017} \]
The points \( A = (5, 5) \), \( B = (2, 6) \), and \( C = (7, 11) \) all lie in \( S \). There is also a point ... | 1021 |
Find the largest negative integer $x$ which satisfies the congruence $34x+6\equiv 2\pmod {20}$. | -6 |
Let points $A(x_1, y_1)$ and $B(x_2, y_2)$ be on the graph of $f(x) = x^2$. The points $C$ and $D$ trisect the segment $\overline{AB}$ with $AC < CB$. A horizontal line drawn through $C$ intersects the curve at another point $E(x_3, y_3)$. Find $x_3$ if $x_1 = 1$ and $x_2 = 4$. | -2 |
Ivan wanted to buy nails. In one store, where 100 grams of nails cost 180 rubles, he couldn't buy the required amount because he was short 1430 rubles. Then he went to another store where 100 grams cost 120 rubles. He bought the required amount and received 490 rubles in change. How many kilograms of nails did Ivan buy... | 3.2 |
Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? | 12 |
Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12},$ find all possible values for $x.$
(Enter your answer as a comma-separated list.) | 4, 6 |
Determine the value of $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \dots + 12100$, where the signs change after each perfect square. | 1331000 |
Remi wants to drink more water. He has a refillable water bottle that holds 20 ounces of water. That week Remi refills the bottle 3 times a day and drinks the whole bottle each time except for twice when he accidentally spills 5 ounces the first time and 8 ounces the second time. In 7 days how many ounces of water does... | Remi drinks the same amount of water every day for a week except for when he spills some of it. His water bottle holds 20 ounces which he drinks 3 times a day, 20 x 3 = <<60=60>>60 ounces a typical day.
Over 7 days, without spilling anything, Remi drinks 60 ounces x 7 days = <<60*7=420>>420 ounces.
Except Remi spills 8... |
For how many positive integral values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$
\begin{cases}
2x>3x-3\\
3x-a>-6
\end{cases}
$$ | 3 |
A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\lbrace\frac{1}{z} \ | \ z \in R\right\rbrace$. Find the area of $S.$ | 3 \sqrt{3} + 2 \pi |
The journey from Petya's home to school takes him 20 minutes. One day, on his way to school, Petya remembered that he had forgotten a pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home for the pen and then goes to school at t... | \frac{1}{4} |
Round to the nearest hundredth: 18.4851 | 18.49 |
Timmy plans to ride a skateboard ramp that is 50 feet high. He knows he needs to go 40 mph at the start to make it all the way to the top. He measures his speed on three trial runs and goes 36, 34, and 38 mph. How much faster does he have to go than his average speed to make it up the ramp? | His total speed from the trials is 108 because 36 + 34 + 38 = <<36+34+38=108>>108
His average speed was 36 mph because 108 / 3 = <<108/3=36>>36
He needs to go 4 mph faster to make it to the top because 40 - 36 = <<40-36=4>>4.
#### 4 |
Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A horizontal line with equation $y=t$ intersects line segment $ \overline{AB} $ at $T$ and line segment $ \overline{AC} $ at $U$, forming $\triangle ATU$ with area 13.5. Compute $t$. | 2 |
A point $(x,y)$ is a distance of 12 units from the $x$-axis. It is a distance of 10 units from the point $(1,6)$. It is a distance $n$ from the origin. Given that $x>1$, what is $n$? | 15 |
Suppose that there exist nonzero complex numbers $a,$ $b,$ $c,$ and $d$ such that $k$ is a root of both the equations $ax^3 + bx^2 + cx + d = 0$ and $bx^3 + cx^2 + dx + a = 0.$ Enter all possible values of $k,$ separated by commas. | 1,-1,i,-i |
Triangle \(ABC\) has a right angle at \(B\), with \(AB = 3\) and \(BC = 4\). If \(D\) and \(E\) are points on \(AC\) and \(BC\), respectively, such that \(CD = DE = \frac{5}{3}\), find the perimeter of quadrilateral \(ABED\). | 28/3 |
What is the greatest prime factor of 99? | 11 |
Given that $\alpha$ is an angle in the second quadrant, simplify $$\frac { \sqrt {1+2\sin(5\pi-\alpha)\cos(\alpha-\pi)}}{\sin\left(\alpha - \frac {3}{2}\pi \right)- \sqrt {1-\sin^{2}\left( \frac {3}{2}\pi+\alpha\right)}}.$$ | -1 |
An ant starts at the point $(0,0)$ in the Cartesian plane. In the first minute, the ant faces towards $(1,0)$ and walks one unit. Each subsequent minute, the ant chooses an angle $\theta$ uniformly at random in the interval $\left[-90^{\circ}, 90^{\circ}\right]$, and then turns an angle of $\theta$ clockwise (negative ... | 45 |
What is the discriminant of $3x^2 - 7x - 12$? | 193 |
Calculate the lengths of the arcs of curves given by the equations in the rectangular coordinate system.
$$
y=\ln x, \sqrt{3} \leq x \leq \sqrt{15}
$$ | \frac{1}{2} \ln \frac{9}{5} + 2 |
Given that in the expansion of $\left(1+x\right)^{n}$, the coefficient of $x^{3}$ is the largest, then the sum of the coefficients of $\left(1+x\right)^{n}$ is ____. | 64 |
The diagram shows five circles of the same radius touching each other. A square is drawn so that its vertices are at the centres of the four outer circles. What is the ratio of the area of the shaded parts of the circles to the area of the unshaded parts of the circles? | 2:3 |
A point is chosen at random on the number line between 0 and 1, and this point is colored red. Another point is then chosen at random on the number line between 0 and 2, and this point is colored blue. What is the probability that the number of the blue point is greater than the number of the red point but less than th... | \frac{1}{2} |
Flat Albert and his buddy Mike are watching the game on Sunday afternoon. Albert is drinking lemonade from a two-dimensional cup which is an isosceles triangle whose height and base measure 9 cm and 6 cm; the opening of the cup corresponds to the base, which points upwards. Every minute after the game begins, the follo... | 26 |
What is the remainder when $2^{202} + 202$ is divided by $2^{101} + 2^{51} + 1$? | 201 |
In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=56^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. What is the degree measure of the acute ang... | 80 |
Amanda has taken 4 quizzes this semester and averaged a 92% score on them. The final quiz is coming up, which is worth the same as each previous quiz. What score does she need in order to get an A in the class, which requires her to average 93% over the 5 quizzes? | Amanda's total number of points from the first 4 quizzes is 368 because 4 x 92 = <<4*92=368>>368
She needs a total of 465 points because 5 x 93 = <<5*93=465>>465
She needs to score a 97 on the final quiz because 465 - 368 = <<465-368=97>>97
#### 97 |
From a deck of 32 cards which includes three colors (red, yellow, and blue) with each color having 10 cards numbered from $1$ to $10$, plus an additional two cards (a small joker and a big joker) both numbered $0$, a subset of cards is selected. The score for each card is calculated as $2^{k}$, where $k$ is the number ... | 1006009 |
In $\triangle ABC$, given that $\sin A = 10 \sin B \sin C$ and $\cos A = 10 \cos B \cos C$, what is the value of $\tan A$? | 11 |
Kendra tracks the different species of birds they spot on their birdwatching trip. On Monday they visited 5 sites and saw an average of 7 birds at each site. On Tuesday, Kendra visited 5 sites and saw an average of 5 birds at each site. On Wednesday visited 10 sites and saw an average of 8 birds at each site. On averag... | On Monday Kendra saw 5 sites x 7 birds/site = <<5*7=35>>35 different birds
On Tuesday Kendra saw 5 sites x 5 birds/site = <<5*5=25>>25 different birds
On Wednesday Kendra saw 10 sites x 8 birds/site = <<10*8=80>>80 different birds
In total Kendra saw 35 birds + 25 birds + 80 birds = <<35+25+80=140>>140 different birds
... |
The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled? | 96 |
Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$? | 8 |
The number $17!$ has a certain number of positive integer divisors. What is the probability that one of them is odd? | \frac{1}{16} |
A tetrahedron is formed using the vertices of a cube. How many such distinct tetrahedrons can be formed? | 58 |
The 3rd and 5th terms of an arithmetic sequence are 17 and 39, respectively. What is the 7th term of the same sequence? | 61 |
Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.
Suppose that every edge of $DABC$ has length $20$ or $45$, but no face of $DABC$ is equilateral. Then what is the surface area of $DABC$? | 40 \sqrt{1925} |
Walking is a form of exercise that falls between walking and racewalking. It is a simple and safe aerobic exercise that can enhance lung capacity and promote heart health. A sports physiologist conducted a large number of surveys on the body fat percentage ($X$) of people engaged in walking activities and found that th... | 0.03 |
Eighty percent of adults drink coffee and seventy percent drink tea. What is the smallest possible percent of adults who drink both coffee and tea? | 50\% |
In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of... | 731 |
Given that points $P$ and $Q$ are moving points on the curve $y=xe^{-2x}$ and the line $y=x+2$ respectively, find the minimum distance between points $P$ and $Q$. | \sqrt{2} |
Six small circles, each of radius $3$ units, are tangent to a large circle as shown. Each small circle also is tangent to its two neighboring small circles. What is the diameter of the large circle in units? [asy]
draw(Circle((-2,0),1));
draw(Circle((2,0),1));
draw(Circle((-1,1.73205081),1));
draw(Circle((1,1.73205081)... | 18 |
The three-digit integer $63\underline{\hphantom{0}}$ is a multiple of 3. What is the greatest possible difference between two of the possibilities for the units digit? | 9 |
A and B play a game as follows. Each throws a dice. Suppose A gets \(x\) and B gets \(y\). If \(x\) and \(y\) have the same parity, then A wins. If not, they make a list of all two-digit numbers \(ab \leq xy\) with \(1 \leq a, b \leq 6\). Then they take turns (starting with A) replacing two numbers on the list by their... | 3/4 |
Jasmine gets off of work at 4:00 pm. After that, it will take her 30 minutes to commute home, 30 minutes to grocery shop, 10 minutes to pick up the dry cleaning, 20 minutes to pick up the dog from the groomers and 90 minutes to cook dinner when she returns home. What time will she eat dinner? | Her errands include a 30 min commute and 30 min grocery shopping, 10 min for dry cleaning and 20 min to grab the dog for a total of 30+30+10+20 = <<30+30+10+20=90>>90 minutes
Her errands took 90 minutes and she has another 90 minutes of cooking for a total of 90+90 = <<90+90=180>>180 minutes of chores/errands
60 minute... |
A school organized a trip to the Expo Park for all third-grade students and rented some large buses. Initially, the plan was to have 28 people on each bus. After all the students boarded, it was found that 13 students could not get on the buses. So, they decided to have 32 people on each bus, and this resulted in 3 emp... | 125 |
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer.
[i]Laurentiu Panaitopol, Romania[/i] | (5, 3) |
There are 4 college entrance examination candidates entering the school through 2 different intelligent security gates. Each security gate can only allow 1 person to pass at a time. It is required that each security gate must have someone passing through. Then there are ______ different ways for the candidates to enter... | 72 |
Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is ... | 12 |
In how many ways can a bamboo trunk (a non-uniform natural material) of length 4 meters be cut into three parts, the lengths of which are multiples of 1 decimeter, and from which a triangle can be formed? | 171 |
Compute $\frac{6! + 7!}{5!}$ | 48 |
Let $F(z)=\dfrac{z+i}{z-i}$ for all complex numbers $z\neq i$, and let $z_n=F(z_{n-1})$ for all positive integers $n$. Given that $z_0=\dfrac{1}{137}+i$ and $z_{2002}=a+bi$, where $a$ and $b$ are real numbers, find $a+b$. | 275 |
Elsa gets 500 MB of cell phone data each month. If she spends 300 MB watching Youtube and 2/5 of what's left on Facebook, how many MB of data does she have left? | First subtract the 300 MB Elsa spent on YouTube: 500 MB - 300 MB = <<500-300=200>>200 MB.
Then multiply 200 MB by 2/5th to find out how much data Elsa spent on Facebook: 200 MB * 2/5 = <<200*2/5=80>>80 MB.
Finally, subtract the data Elsa spent on Facebook from the 200 MB: 200 MB - 80 MB = <<200-80=120>>120 MB.
#### 120 |
The sum of the first four terms of an arithmetic progression, as well as the sum of the first nine terms, are natural numbers. Additionally, the first term \( b_{1} \) of this progression satisfies the inequality \( b_{1} \leq \frac{3}{4} \). What is the greatest possible value of \( b_{1} \)? | 11/15 |
James spends 30 minutes twice a day on meditation. How many hours a week does he spend meditating? | Each session is 30/60=<<30/60=.5>>.5 hours
So she spends .5*2=<<.5*2=1>>1 hour a day
So she spends 1*7=<<1*7=7>>7 hours a week
#### 7 |
Given a tetrahedron P-ABC, if PA, PB, and PC are mutually perpendicular, and PA=2, PB=PC=1, then the radius of the inscribed sphere of the tetrahedron P-ABC is \_\_\_\_\_\_. | \frac {1}{4} |
Let $N$ be the number of ways in which the letters in "HMMTHMMTHMMTHMMTHMMTHMMT" ("HMMT" repeated six times) can be rearranged so that each letter is adjacent to another copy of the same letter. For example, "MMMMMMTTTTTTHHHHHHHHHHHH" satisfies this property, but "HMMMMMTTTTTTHHHHHHHHHHHM" does not. Estimate $N$. An es... | 78556 |
How many distinct products can you obtain by multiplying two or more distinct elements from the set $\{1, 2, 3, 5, 7, 11\}$? | 26 |
If three, standard, 6-faced dice are rolled, what is the probability that the sum of the face up integers is 16? | \frac{1}{36} |
Mrs. Carlton gives out penalty points whenever her students misbehave. They get 5 points for interrupting, 10 points for insulting their classmates, and 25 points for throwing things. If they get 100 points, they have to go to the office. Jerry already interrupted twice and insulted his classmates 4 times. How many tim... | For interrupting, Jerry got 5 points per interruption * 2 interruptions = <<5*2=10>>10 points
For insulting, he got 10 points per insult * 4 insults = <<10*4=40>>40 points
To get to the 100 limit points, Jerry has 100 points - 10 points - 40 points = <<100-10-40=50>>50 points left
He still has 50 points / 25 points per... |
What is the smallest positive integer \( n \) such that \( 5n \equiv 105 \pmod{24} \)? | 21 |
Ignatius owns 4 bicycles. A friend of his owns different types of cycles, which have three times are many tires as Ignatius's bikes have. He has one unicycle, a tricycle, and the rest are bikes. How many bicycles does the friend own? | Ignatius has 8 tires because 2 x 4 = <<2*4=8>>8
His friend has 24 tires because 3 x 8 = <<3*8=24>>24
There are 20 bicycle wheels because 24-1-3 = <<24-1-3=20>>20
He has 10 bicycles because 20 / 2 = <<20/2=10>>10
#### 10 |
Unlucky Emelya was given several metal balls. He broke the 3 largest ones (their mass was 35% of the total mass of all the balls), then lost the 3 smallest ones, and brought home the remaining balls (their mass was \( \frac{8}{13} \) of the unbroken ones). How many balls was Emelya given? | 10 |
Suppose the probability distribution of the random variable $X$ is given by $P\left(X=\frac{k}{5}\right)=ak$, where $k=1,2,3,4,5$.
(1) Find the value of $a$.
(2) Calculate $P\left(X \geq \frac{3}{5}\right)$.
(3) Find $P\left(\frac{1}{10} < X \leq \frac{7}{10}\right)$. | \frac{1}{3} |
For a polynomial $p(x),$ define its munificence as the maximum value of $|p(x)|$ on the interval $-1 \le x \le 1.$ For example, the munificence of the polynomial $p(x) = -x^2 + 3x - 17$ is 21, since the maximum value of $|-x^2 + 3x - 17|$ for $-1 \le x \le 1$ is 21, occurring at $x = -1.$
Find the smallest possible m... | \frac{1}{2} |
An isosceles triangle has its vertex at $(0,5)$ and a base between points $(3,5)$ and $(13,5)$. The two equal sides are each 10 units long. If the third vertex (top vertex) is in the first quadrant, what is the y-coordinate? | 5 + 5\sqrt{3} |
There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number? | 12 |
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} |
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to ... | \frac{3}{4} |
Lisa and Carly go shopping together. Lisa spends $40 on t-shirts then spends half of this amount on jeans and twice this amount on coats. Carly spends only a quarter as much as Lisa on t-shirts but spends 3 times as much on jeans and a quarter of the amount Lisa spent on coats. In dollars, how much did Lisa and Carly s... | Lisa spends $40 on t-shirts / 2 = $<<40/2=20>>20 on jeans.
She also spends $40 on t-shirts * 2 = $<<40*2=80>>80 on coats.
So Lisa has spent a total of 40 + 20 + 80 = $<<40+20+80=140>>140.
Carly spends $40 / 4 = $<<40/4=10>>10 on t-shirts.
She also spends $20 per pair of jeans * 3 = $<<20*3=60>>60 on jeans.
She then als... |
Triangle $\vartriangle ABC$ has circumcenter $O$ and orthocenter $H$ . Let $D$ be the foot of the altitude from $A$ to $BC$ , and suppose $AD = 12$ . If $BD = \frac14 BC$ and $OH \parallel BC$ , compute $AB^2$ .
. | 160 |
Let $a,b,c$ be positive integers such that $a,b,c,a+b-c,a+c-b,b+c-a,a+b+c$ are $7$ distinct primes. The sum of two of $a,b,c$ is $800$ . If $d$ be the difference of the largest prime and the least prime among those $7$ primes, find the maximum value of $d$ . | 1594 |
In triangle $ABC,$ $D$ is on $\overline{AB}$ such that $AD:DB = 3:2,$ and $E$ is on $\overline{BC}$ such that $BE:EC = 3:2.$ If lines $DE$ and $AC$ intersect at $F,$ then find $\frac{DE}{EF}.$ | \frac{1}{2} |
Given that $x$ and $y$ are distinct nonzero real numbers such that $x+\frac{2}{x} = y + \frac{2}{y}$, what is $xy$? | 2 |
Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$ , define $b_m=\min\{n: a_n \ge m\}$ , that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$ . If $a_{19}=85$ , determine the maximum value of $a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}$ . | \boxed{1700} |
A $\textit{palindrome}$ is a positive integer which reads the same forward and backward, like $12321$ or $4884$.
How many $4$-digit palindromes are there? | 90 |
If $a$ and $b$ are two positive numbers, and the three numbers $a$, $b$, and $-4$ can be rearranged to form both an arithmetic sequence and a geometric sequence, then the value of $a+b$ is __________. | 10 |
When skipping rope, the midpoint of the rope can be considered to move along the same circle. If Xiaoguang takes 0.5 seconds to complete a "single skip" and 0.6 seconds to complete a "double skip," what is the ratio of the speed of the midpoint of the rope during a "single skip" to the speed during a "double skip"?
(N... | 3/5 |
A certain pharmaceutical company has developed a new drug to treat a certain disease, with a cure rate of $p$. The drug is now used to treat $10$ patients, and the number of patients cured is denoted as $X$.
$(1)$ If $X=8$, two patients are randomly selected from these $10$ people for drug interviews. Find the distri... | \frac{90}{11} |
Given positive real numbers \(a, b, c\) satisfy \(2(a+b)=ab\) and \(a+b+c=abc\), find the maximum value of \(c\). | \frac{8}{15} |
A man asks: "How old are you, my son?"
To this question, I answered:
"If my father were seven times as old as I was eight years ago, then one quarter of my father's current age would surely be fourteen years now. Please calculate from this how many years weigh upon my shoulders!" | 16 |
A two-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that according to regulations, a stove should not be located closer than 50 cm to the main gas pipe. | 1/16 |
If a whole number $n$ is not prime, then the whole number $n-2$ is not prime. A value of $n$ which shows this statement to be false is | 9 |
Given a triangular pyramid \( P-ABC \) with a base that is an equilateral triangle of side length \( 4 \sqrt{3} \), and with \( PA=3 \), \( PB=4 \), and \( PC=5 \). If \( O \) is the center of the triangle \( ABC \), find the length of \( PO \). | \frac{\sqrt{6}}{3} |
Find all positive integers $n$ such that:
\[ \dfrac{n^3+3}{n^2+7} \]
is a positive integer. | 2 \text{ and } 5 |
Let $x,$ $y,$ $z$ be nonzero real numbers such that $x + y + z = 0,$ and $xy + xz + yz \neq 0.$ Find all possible values of
\[\frac{x^5 + y^5 + z^5}{xyz (xy + xz + yz)}.\]Enter all possible values, separated by commas. | -5 |
There is a heads up coin on every integer of the number line. Lucky is initially standing on the zero point of the number line facing in the positive direction. Lucky performs the following procedure: he looks at the coin (or lack thereof) underneath him, and then, - If the coin is heads up, Lucky flips it to tails up,... | 6098 |
The integer $n > 9$ is a root of the quadratic equation $x^2 - ax + b=0$. In this equation, the representation of $a$ in the base-$n$ system is $19$. Determine the base-$n$ representation of $b$. | 90_n |
Jo adds up all the positive integers from 1 to 50. Kate does a similar thing with the first 50 positive integers; however, she first rounds every integer to its nearest multiple of 10 (rounding 5s up) and then adds the 50 values. What is the positive difference between Jo's sum and Kate's sum? | 25 |
Find all 6-digit multiples of 22 of the form $5d5,\!22e$ where $d$ and $e$ are digits. What is the maximum value of $d$? | 8 |
A deck of fifty-two cards consists of four $1$'s, four $2$'s, ..., four $13$'s. Two matching pairs (two sets of two cards with the same number) are removed from the deck. After removing these cards, find the probability, represented as a fraction $m/n$ in simplest form, where $m$ and $n$ are relatively prime, that two ... | 299 |
Compute
\[\sum_{n = 1}^\infty \frac{1}{n(n + 2)}.\] | \frac{3}{4} |
The left focus of the hyperbola $C$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \, (a > 0, b > 0)$ is $F$. If the symmetric point $A$ of $F$ with respect to the line $\sqrt{3}x + y = 0$ is a point on the hyperbola $C$, then the eccentricity of the hyperbola $C$ is \_\_\_\_\_\_. | \sqrt{3} + 1 |
The lines $-2x + y = k$ and $0.5x + y = 14$ intersect when $x = -8.4$. What is the value of $k$? | 35 |
Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) between 80 and 100. The second jogger complete... | 3705 |
Let \[f(x) = \left\{
\begin{array}{cl}
ax+3 & \text{ if }x>0, \\
ab & \text{ if }x=0, \\
bx+c & \text{ if }x<0.
\end{array}
\right.\]If $f(2)=5$, $f(0)=5$, and $f(-2)=-10$, and $a$, $b$, and $c$ are nonnegative integers, then what is $a+b+c$? | 6 |
The slope angle of the tangent line to the curve $f\left(x\right)=- \frac{ \sqrt{3}}{3}{x}^{3}+2$ at $x=1$ is $\tan^{-1}\left( \frac{f'\left(1\right)}{\mid f'\left(1\right) \mid} \right)$, where $f'\left(x\right)$ is the derivative of $f\left(x\right)$. | \frac{2\pi}{3} |
Let the coefficient of \( x^{1992} \) in the power series \( (1 + x)^{\alpha} = 1 + \alpha x + \dots \) be \( C(\alpha) \). Find \( \int_{0}^{1} C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \, dy \). | 1992 |
The perimeter of an isosceles right triangle is $2p$. Its area is: | $(3-2\sqrt{2})p^2$ |
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