problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Find the area of the region described by $x \ge 0,$ $y \ge 0,$ and
\[100 \{x\} \ge \lfloor x \rfloor + \lfloor y \rfloor.\]Note: For a real number $x,$ $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x.$ For example, $\{2.7\} = 0.7.$ | 1717 |
Find all the integer roots of
\[x^3 - 3x^2 - 13x + 15 = 0.\]Enter all the integer roots, separated by commas. | -3,1,5 |
Let the set \( M = \{1, 2, 3, \cdots, 50\} \). For any subset \( S \subseteq M \) such that for any \( x, y \in S \) with \( x \neq y \), it holds that \( x + y \neq 7k \) for any \( k \in \mathbf{N} \). If \( S_0 \) is the subset with the maximum number of elements that satisfies this condition, how many elements are ... | 23 |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=3+t\cos \alpha \\ y=1+t\sin \alpha\end{cases}$ (where $t$ is the parameter), in the polar coordinate system (with the same unit length as the Cartesian coordinate system $xOy$, and the origin $O$ as the pole, and the non-n... | 2 \sqrt {2} |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$. | \frac{9\pi - 18}{2} |
Given that the ceiling is 3 meters above the floor, the light bulb is 15 centimeters below the ceiling, Alice is 1.6 meters tall and can reach 50 centimeters above the top of her head, and a 5 centimeter thick book is placed on top of a stool to reach the light bulb, find the height of the stool in centimeters. | 70 |
It takes four painters working at the same rate $1.25$ work-days to finish a job. If only three painters are available, how many work-days will it take them to finish the job, working at the same rate? Express your answer as a mixed number. | 1\frac{2}{3} |
In the Cartesian coordinate system $(xOy)$, let the line $l: \begin{cases} x=2-t \\ y=2t \end{cases} (t \text{ is a parameter})$, and the curve $C_{1}: \begin{cases} x=2+2\cos \theta \\ y=2\sin \theta \end{cases} (\theta \text{ is a parameter})$. In the polar coordinate system with $O$ as the pole and the positive $x$-... | \frac{4\sqrt{10}}{5} |
If a number is randomly selected from the set $\left\{ \frac{1}{3}, \frac{1}{4}, 3, 4 \right\}$ and denoted as $a$, and another number is randomly selected from the set $\left\{ -1, 1, -2, 2 \right\}$ and denoted as $b$, then the probability that the graph of the function $f(x) = a^{x} + b$ ($a > 0, a \neq 1$) passes t... | \frac{3}{8} |
Three squares with a perimeter of 28cm each are combined to form a rectangle. What is the area of this rectangle? Additionally, if four such squares are combined to form a larger square, what is the perimeter of this larger square? | 56 |
Let $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\le a < b \le 20$. Find the greatest positive integer $n$ such that $2^n$ divides $K$. | 150 |
A line with a slope of $2$ passing through the right focus of the ellipse $\frac{x^2}{5} + \frac{y^2}{4} = 1$ intersects the ellipse at points $A$ and $B$. If $O$ is the origin, then the area of $\triangle OAB$ is \_\_\_\_\_\_. | \frac{5}{3} |
Given the parabola $y^{2}=2x$ with focus $F$, a line passing through $F$ intersects the parabola at points $A$ and $B$. If $|AB|= \frac{25}{12}$ and $|AF| < |BF|$, determine the value of $|AF|$. | \frac{5}{6} |
Among the four-digit numbers formed by the digits 0, 1, 2, ..., 9 without repetition, determine the number of cases where the absolute difference between the units digit and the hundreds digit equals 8. | 210 |
The entire surface of a cube with dimensions $13 \times 13 \times 13$ was painted red, and then this cube was cut into $1 \times 1 \times 1$ cubes. All faces of the $1 \times 1 \times 1$ cubes that were not painted red were painted blue. By what factor is the total area of the blue faces greater than the total area of ... | 12 |
Let the complex numbers \( z_1 \) and \( z_2 \) correspond to the points \( A \) and \( B \) on the complex plane respectively, and suppose \( \left|z_1\right| = 4 \) and \( 4z_1^2 - 2z_1z_2 + z_2^2 = 0 \). Let \( O \) be the origin. Calculate the area of triangle \( \triangle OAB \). | 8\sqrt{3} |
Let $a$, $b$, $c$, $a+b-c$, $a+c-b$, $b+c-a$, $a+b+c$ be seven distinct prime numbers, and among $a$, $b$, $c$, the sum of two numbers is 800. Let $d$ be the difference between the largest and the smallest of these seven prime numbers. Find the maximum possible value of $d$. | 1594 |
Suppose that there exist nonzero complex numbers $a, b, c$, and $d$ such that $k$ is a root of both the equations $a x^{3}+b x^{2}+c x+d=0$ and $b x^{3}+c x^{2}+d x+a=0$. Find all possible values of $k$ (including complex values). | 1,-1, i,-i |
Given the set $M=\{x|2x^{2}-3x-2=0\}$ and the set $N=\{x|ax=1\}$. If $N \subset M$, what is the value of $a$? | \frac{1}{2} |
If
\[\tan x = \frac{2ab}{a^2 - b^2},\]where $a > b > 0$ and $0^\circ < x < 90^\circ,$ then find $\sin x$ in terms of $a$ and $b.$ | \frac{2ab}{a^2 + b^2} |
In the rectangular coordinate system xOy, the parametric equations of the curve C1 are given by $$\begin{cases} x=t\cos\alpha \\ y=1+t\sin\alpha \end{cases}$$, and the polar coordinate equation of the curve C2 with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis is ρ=2cosθ.
1. If th... | 2\sqrt{2} |
If $M = 1764 \div 4$, $N = M \div 4$, and $X = M - N$, what is the value of $X$? | 330.75 |
In $\triangle ABC$, $a=5$, $b=8$, $C=60^{\circ}$, the value of $\overrightarrow{BC}\cdot \overrightarrow{CA}$ is $\_\_\_\_\_\_$. | -20 |
Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have
\[
(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]Compute the number of distinct possible values of $c$. | 4 |
Using the digits $2, 4, 6$ to construct six-digit numbers, how many such numbers are there if no two consecutive digits in the number can both be 2 (for example, 626442 is allowed, but 226426 is not allowed)? | 448 |
Suppose we wish to divide 12 dogs into three groups, one with 4 dogs, one with 6 dogs, and one with 2 dogs. We need to form these groups such that Rex is in the 4-dog group, Buddy is in the 6-dog group, and Bella and Duke are both in the 2-dog group. How many ways can we form these groups under these conditions? | 56 |
A line $y = -2$ intersects the graph of $y = 5x^2 + 2x - 6$ at points $C$ and $D$. Find the distance between points $C$ and $D$, expressed in the form $\frac{\sqrt{p}}{q}$ where $p$ and $q$ are positive coprime integers. What is $p - q$? | 16 |
Find the positive solution to
\[\sqrt[3]{x + \sqrt[3]{x + \sqrt[3]{x + \dotsb}}} = \sqrt[3]{x \sqrt[3]{x \sqrt[3]{x \dotsm}}}.\] | \frac{3 + \sqrt{5}}{2} |
What is $2453_6 + 16432_6$? Express your answer in both base 6 and base 10. | 3881 |
A traffic light runs repeatedly through the following cycle: green for 45 seconds, then yellow for 5 seconds, then blue for 10 seconds, and finally red for 40 seconds. Peter picks a random five-second time interval to observe the light. What is the probability that the color changes while he is watching? | 0.15 |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that (i) For all $x, y \in \mathbb{R}$, $f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y)$ (ii) For all $x \in[0,1), f(0) \geq f(x)$, (iii) $-f(-1)=f(1)=1$. Find all such functions $f$. | f(x) = \lfloor x \rfloor |
Let \( q(x) = 2x - 5 \) and \( r(q(x)) = x^3 + 2x^2 - x - 4 \). Find \( r(3) \). | 88 |
A counter begins at 0 . Then, every second, the counter either increases by 1 or resets back to 0 with equal probability. The expected value of the counter after ten seconds can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$. | 103324 |
Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that
\[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\]
for all $x,y \in \mathbb{R} $. | f(x)=1-\dfrac{x^2}{2} |
A cafe has 9 indoor tables and 11 outdoor tables. Each indoor table has 10 chairs, and each outdoor table has 3 chairs. How many chairs are there in all? | There are 9 indoor tables * 10 chairs = <<9*10=90>>90 chairs.
There are 11 outdoor tables * 3 chairs = <<11*3=33>>33 chairs.
The total number of chairs is 90 chairs + 33 chairs = <<90+33=123>>123 chairs.
#### 123 |
Given that the cube root of \( m \) is a number in the form \( n + r \), where \( n \) is a positive integer and \( r \) is a positive real number less than \(\frac{1}{1000}\). When \( m \) is the smallest positive integer satisfying the above condition, find the value of \( n \). | 19 |
Bucky earns money each weekend catching and selling fish. He wants to save up for a new video game, which costs $60. Last weekend he earned $35. He can earn $5 from trout and $4 from blue-gill. He caught five fish this Sunday. If 60% were trout, and the rest were blue-gill, how much more does he need to save before he ... | He is $25 short for the game because 60 - 35 = <<60-35=25>>25
He caught 3 trout because 5 x .6 = <<5*.6=3>>3
He caught 2 blue-gill because 5 - 3 = <<5-3=2>>2
He earned $15 from the trout because 3 x 5 = <<3*5=15>>15
He earned $8 from the blue-gill because 2 x 4 = <<2*4=8>>8
He earned $23 total because 15 + 8 = <<15+8=2... |
Du Chin bakes 200 meat pies in a day. He sells them for $20 each and uses 3/5 of the sales to buy ingredients to make meat pies for sale on the next day. How much money does Du Chin remain with after setting aside the money for buying ingredients? | From the 200 meat pies, Du Chin makes 200*$20 = $<<200*20=4000>>4000 after the sales.
The amount of money he uses to buy ingredients to make pies for sale on the next day is 3/5*$4000 = $<<3/5*4000=2400>>2400
After setting aside money for buying ingredients, Du Chin remains with $4000-$2400 = $<<4000-2400=1600>>1600
##... |
What is $2343_6+15325_6$? Express your answer in base $6$. | 22112_6 |
Regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are all equal. In how many ways can this be done?
[asy]
p... | 1152 |
The house number. A person mentioned that his friend's house is located on a long street (where the houses on the side of the street with his friend's house are numbered consecutively: $1, 2, 3, \ldots$), and that the sum of the house numbers from the beginning of the street to his friend's house matches the sum of the... | 204 |
In a right triangle, one of the acute angles $\alpha$ satisfies
\[\tan \frac{\alpha}{2} = \frac{1}{\sqrt[3]{2}}.\]Let $\theta$ be the angle between the median and the angle bisector drawn from this acute angle. Find $\tan \theta.$ | \frac{1}{2} |
How many whole numbers between $200$ and $500$ contain the digit $3$? | 138 |
Let \( a \) and \( b \) be integers such that \( ab = 144 \). Find the minimum value of \( a + b \). | -145 |
There are 7 balls of each of the three colors: red, blue, and yellow. When randomly selecting 3 balls with different numbers, determine the total number of ways to pick such that the 3 balls are of different colors and their numbers are not consecutive. | 60 |
When two fair dice are rolled, the probability that the sum of the numbers facing up is $5$ equals \_\_\_\_\_\_． | \frac{1}{9} |
Denny, an Instagram influencer, has 100000 followers and gets 1000 new followers every day. How many followers will he have if 20000 people unfollowed him in a year? | The total number of new followers after a year is 365*1000 = <<365*1000=365000>>365000 followers.
His total number of followers in a year is 100000+365000 = <<100000+365000=465000>>465000 followers.
If 20000 people unfollowed him, the total number of followers remaining is 465000-20000 = 445000 followers.
#### 445000 |
A band has 72 members who will all be marching during halftime. They need to march in rows with the same number of students per row. If there must be between 5 and 20 students per row, in how many possible row-lengths can the band be arranged? | 5 |
Consider $\triangle \natural\flat\sharp$ . Let $\flat\sharp$ , $\sharp\natural$ and $\natural\flat$ be the answers to problems $4$ , $5$ , and $6$ , respectively. If the incircle of $\triangle \natural\flat\sharp$ touches $\natural\flat$ at $\odot$ , find $\flat\odot$ .
*Proposed by Evan Chen* | 2.5 |
From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$ . The point $ M \in (AE$ is such that $ M$ external to $ ABC$ , $ \angle AMB \equal{} 20 ^\circ$ and $ \angle AMC \equal{} 30 ^ \circ$ . What is the measure of the angle $ \angle... | 20 |
Circles of radius $3$ and $6$ are externally tangent to each other and are internally tangent to a circle of radius $9$. The circle of radius $9$ has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.
[asy] pointpen = black; pathpen = black + linewidth(0.7);... | 224 |
Ross takes 17 breaths per minute by inhaling 5/9 liter of air to his lungs. What is the volume of air inhaled in 24 hours? | We have 60 minutes per hour.
So, in 24 hours we have 60 * 24 = <<60*24=1440>>1440 minutes.
So, the volume of air Ross inhaled in 24 hours is: 17 breaths/minute * 5/9 L/breath * 1440 minutes = 122400 / 9 = <<17*5/9*1440=13600>>13600 liters.
The volume of air inhaled is 13600 liters.
#### 13600 |
The graph of $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $\frac{y}{x}$ over all points $(x,y)$ on the ellipse. What is the value of $a+b$? | \frac{7}{2} |
A triple of integers \((a, b, c)\) satisfies \(a+b c=2017\) and \(b+c a=8\). Find all possible values of \(c\). | -6,0,2,8 |
Find
\[\sum_{n = 1}^\infty \frac{2^n}{1 + 2^n + 2^{n + 1} + 2^{2n + 1}}.\] | \frac{1}{3} |
Evaluate $$64^{1/2}\cdot27^{-1/3}\cdot16^{1/4}.$$ | \frac{16}{3} |
How many degrees are there in the measure of angle $P?$
[asy]
size (5cm,5cm);
pair A,B,C,D,E;
A=(0,1.1);
B=(4.5,0);
C=(6.4,1.7);
D=(4.2,5);
E=(0.5,4.2);
draw (A--B--C--D--E--A,linewidth(1));
label("$P$",A,SW);
label("$128^\circ$",shift(0,0.6)*B);
label("$92^\circ$",C,W);
label("$113^\circ$",shift(-0.3,-0.5)*D);
la... | 96^\circ |
Calculate \( \left[6 \frac{3}{5}-\left(8.5-\frac{1}{3}\right) \div 3.5\right] \times\left(2 \frac{5}{18}+\frac{11}{12}\right) = \) | \frac{368}{27} |
For any real number $\alpha$, define $$\operatorname{sign}(\alpha)= \begin{cases}+1 & \text { if } \alpha>0 \\ 0 & \text { if } \alpha=0 \\ -1 & \text { if } \alpha<0\end{cases}$$ How many triples $(x, y, z) \in \mathbb{R}^{3}$ satisfy the following system of equations $$\begin{aligned} & x=2018-2019 \cdot \operatornam... | 3 |
Evaluate the sum of $1001101_2$ and $111000_2$, and then add the decimal equivalent of $1010_2$. Write your final answer in base $10$. | 143 |
Let $C$ be the coefficient of $x^2$ in the expansion of the product $(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).$ Find $|C|.$ | 588 |
An 18-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Exactly ten slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms? | 2 |
[help me] Let m and n denote the number of digits in $2^{2007}$ and $5^{2007}$ when expressed in base 10. What is the sum m + n? | 2008 |
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1? | \frac{\sqrt{2}}{2} |
In 12 years, Janice will be 8 times as old as she was 2 years ago. How old is Janice now? | 4 |
What is the greatest common divisor of all the numbers $7^{n+2} + 8^{2n+1}$ for $n \in \mathbb{N}$? | 57 |
If I have a $5 \times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 |
Consider an unusual biased coin, with probability $p$ of landing heads, probability $q \leq p$ of landing tails, and probability \frac{1}{6}$ of landing on its side (i.e. on neither face). It is known that if this coin is flipped twice, the likelihood that both flips will have the same result is \frac{1}{2}$. Find $p$. | \frac{2}{3} |
Alfonso earns $6 each day walking his aunt’s dog. He is saving to buy a mountain bike helmet for $340. Currently, he already has $40. If he walks his aunt's dog 5 days a week, in how many weeks should Alfonso work to buy his mountain bike? | Alfonso has to work for the $340 - $40 = $<<300=300>>300.
In a week, he earns $6 x 5 = $<<6*5=30>>30.
Thus, he needs to work $300/$30 = <<300/30=10>>10 weeks to buy the bike.
#### 10 |
Tom decided to go on a trip. During this trip, he needs to cross a lake, which in one direction takes 4 hours. During this time, Tom needs an assistant to help him with the crossing. Hiring an assistant costs $10 per hour. How much would Tom have to pay for help with crossing the lake back and forth? | Crossing the lake two times (back and forth) would take 4 * 2 = <<4*2=8>>8 hours.
So Tom would need to pay an assistant 8 * 10 = $<<8*10=80>>80.
#### 80 |
For each positive integer $n$ , find the number of $n$ -digit positive integers that satisfy both of the following conditions:
$\bullet$ no two consecutive digits are equal, and
$\bullet$ the last digit is a prime. | \[
\frac{2}{5}\left(9^n + (-1)^{n+1}\right)
\] |
Suppose that $PQRS TUVW$ is a regular octagon. There are 70 ways in which four of its sides can be chosen at random. If four of its sides are chosen at random and each of these sides is extended infinitely in both directions, what is the probability that they will meet to form a quadrilateral that contains the octagon? | \frac{19}{35} |
The average of $a, b$ and $c$ is 16. The average of $c, d$ and $e$ is 26. The average of $a, b, c, d$, and $e$ is 20. What is the value of $c$? | 26 |
Let \( X = \{1, 2, \ldots, 98\} \). Call a subset of \( X \) good if it satisfies the following conditions:
1. It has 10 elements.
2. If it is partitioned in any way into two subsets of 5 elements each, then one subset has an element coprime to each of the other 4, and the other subset has an element which is not copri... | 50 |
John is working as an IT specialist. He repairs broken computers. One day he had to fix 20 computers. 20% of them were unfixable, and 40% of them needed to wait a few days for spare parts to come. The rest John was able to fix right away. How many computers John was able to fix right away? | From all the computers, 100% - 40% - 20% = 40% got fixed right away.
So John was able to fix 40/100 * 20 = <<40/100*20=8>>8 computers.
#### 8 |
Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly 50 of her first 100 shots? | \(\frac{1}{99}\) |
Given that the equation \(2x^3 - 7x^2 + 7x + p = 0\) has three distinct roots, and these roots form a geometric progression. Find \(p\) and solve this equation. | -2 |
For real numbers $a$ and $b$ , define $$ f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}. $$ Find the smallest possible value of the expression $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$ | 4 \sqrt{2018} |
Calculate the sum of the geometric series $3 - \left(\frac{3}{4}\right) + \left(\frac{3}{4}\right)^2 - \left(\frac{3}{4}\right)^3 + \dots$. Express your answer as a common fraction. | \frac{12}{7} |
Evaluate $\sum_{i=1}^{\infty} \frac{(i+1)(i+2)(i+3)}{(-2)^{i}}$. | 96 |
Let $x$ and $y$ be real numbers satisfying $x^4y^5+y^4x^5=810$ and $x^3y^6+y^3x^6=945$. Evaluate $2x^3+(xy)^3+2y^3$. | 89 |
Given that $n$ is an integer and $0 < 4n <30$, what is the sum of all possible integer values of $n$? | 28 |
Morgan goes to the drive-thru and orders his lunch. He gets a hamburger for $4, onion rings for $2 and a smoothie for $3. If he pays with a $20 bill, how much change does he receive? | Morgan’s lunch cost $4 + $2 + $3 = $<<4+2+3=9>>9.
If he pays with a $20 bill, he will receive change of $20 - $9 = $<<20-9=11>>11.
#### 11 |
All the positive integers greater than 1 are arranged in five columns (A, B, C, D, E) as shown. Continuing the pattern, in what column will the integer 800 be written?
[asy]
label("A",(0,0),N);
label("B",(10,0),N);
label("C",(20,0),N);
label("D",(30,0),N);
label("E",(40,0),N);
label("Row 1",(-10,-7),W);
label("2",(10,... | \text{B} |
If \(\lceil{\sqrt{x}}\rceil=20\), how many possible integer values of \(x\) are there? | 39 |
Points $A(3,5)$ and $B(7,10)$ are the endpoints of a diameter of a circle graphed in a coordinate plane. How many square units are in the area of the circle? Express your answer in terms of $\pi$. | \frac{41\pi}{4} |
In the coordinate plane, points whose x-coordinates and y-coordinates are both integers are called lattice points. For any natural number \( n \), connect the origin \( O \) with the point \( A_n(n, n+3) \). Let \( f(n) \) denote the number of lattice points on the line segment \( OA_n \) excluding the endpoints. Find ... | 1326 |
Given a positive integer $ n$, for all positive integers $ a_1, a_2, \cdots, a_n$ that satisfy $ a_1 \equal{} 1$, $ a_{i \plus{} 1} \leq a_i \plus{} 1$, find $ \displaystyle \sum_{i \equal{} 1}^{n} a_1a_2 \cdots a_i$. | (2n-1)!! |
Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$? | 8181 |
Chenny bought 9 plates at $2 each. She also bought spoons at $1.50 each. How many spoons did Chenny buy if she paid a total of $24 for the plates and spoon? | The cost of 9 plates is $2 x 9 = $<<2*9=18>>18.
So, Chenny paid $24 - $18 = $<<24-18=6>>6 for the spoons.
Therefore, she bought $6/$1.50 = <<6/1.5=4>>4 spoons.
#### 4 |
Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all four slips bear the same number. Let $q$ be the probability that two of the slips be... | 162 |
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 |
The ratio of the interior angles of two regular polygons with sides of unit length is $3: 2$. How many such pairs are there? | 3 |
Compute
$\frac{1622^2-1615^2}{1629^2-1608^2}$. | \frac{1}{3} |
A newspaper subscription costs $10/month. If you purchase an annual subscription you obtain a 20% discount on your total bill. How much do you end up paying if you go with the annual subscription? | A year has 12 months. If 1 month costs $10, then a year will cost $12 * 10 = $<<12*10=120>>120 per year without an annual subscription
If the annual subscription has a 20% discount then you would pay $120 * (1-20%) = $<<120*(1-20*.01)=96>>96 for the annual subscription
#### 96 |
Given that points $E$ and $F$ are on the same side of diameter $\overline{GH}$ in circle $P$, $\angle GPE = 60^\circ$, and $\angle FPH = 90^\circ$, find the ratio of the area of the smaller sector $PEF$ to the area of the circle. | \frac{1}{12} |
Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is $CF$? | 30 |
What is the value of $\left(1 - \frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right) \dotsm \left(1-\frac{1}{50}\right)$? Express your answer as a common fraction. | \frac{1}{50} |
Billy and Tiffany are having a contest to see how can run the most miles in a week. On Sunday, Monday, and Tuesday, Billy runs 1 mile each day and Tiffany runs 2 miles each day. On Wednesday, Thursday, and Friday, Billy runs 1 mile each day and Tiffany runs a 1/3 of a mile each day. On Saturday Tiffany assumes she's go... | Billy has already run 6 miles because 1 plus 1 plus 1 plus 1 plus 1 plus plus 1 equals 6
Tiffany has already run 7 miles because 2 plus 2 plus 2 plus 1/3 plus 1/3 plus 1/3 equals <<2+2+2+1/3+1/3+1/3=7>>7
Billy has to run 1 mile to tie Tiffany because 7 minus 6 equals 1.
#### 1 |
How many three digit numbers are there? | 900 |
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