problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
On her previous five attempts Sarah had achieved times, in seconds, of 86, 94, 97, 88 and 96, for swimming 50 meters. After her sixth try she brought her median time down to 92 seconds. What was her time, in seconds, for her sixth attempt? | 90 |
Determine the value of $1 - 2 - 3 + 4 + 5 + 6 + 7 + 8 - 9 - 10 - \dots + 9801$, where the signs change after each perfect square and repeat every two perfect squares. | -9801 |
Triangle $AHI$ is equilateral. We know $\overline{BC}$, $\overline{DE}$ and $\overline{FG}$ are all parallel to $\overline{HI}$ and $AB = BD = DF = FH$. What is the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$? Express your answer as a common fraction.
[asy]
unitsize(0.2inch);
defaultpen(linewid... | \frac{7}{16} |
The pairwise greatest common divisors of five positive integers are $2,3,4,5,6,7,8, p, q, r$ in some order, for some positive integers $p, q, r$. Compute the minimum possible value of $p+q+r$. | 9 |
How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50? | 109 |
The sequence $\{a_n\}$ satisfies $a_1=1$, $a_2=1$, $a_{n+2}=(1+\sin^2 \frac{n\pi}{2})a_n+2\cos^2 \frac{n\pi}{2}$. Find the sum of the first $20$ terms of this sequence. | 1123 |
A point is randomly thrown on the segment [3, 8] and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \). | 4/15 |
Compute $3 \begin{pmatrix} 2 \\ -8 \end{pmatrix} - 2 \begin{pmatrix} 1 \\ -7 \end{pmatrix}$. | \begin{pmatrix} 4 \\ -10 \end{pmatrix} |
Consider a rectangle \(ABCD\) which is cut into two parts along a dashed line, resulting in two shapes that resemble the Chinese characters "凹" and "凸". Given that \(AD = 10\) cm, \(AB = 6\) cm, and \(EF = GH = 2\) cm, find the total perimeter of the two shapes formed. | 40 |
Cathy has $12 left in her wallet. Her dad sent her $25 for her weekly consumption while her mom sent her twice the amount her dad sent her. How much money does Cathy have now? | Cathy has $12 + $25 = $<<12+25=37>>37 after his father sent him money.
Her mom sent her $25 x 2 = $<<25*2=50>>50.
Therefore, Cathy has $50 + $37 = $<<50+37=87>>87.
#### 87 |
The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. What is the smallest possible difference between the two 2-digit numbers? | 14 |
If a line is perpendicular to a plane, then this line and the plane form a "perpendicular line-plane pair". In a cube, the number of "perpendicular line-plane pairs" formed by a line determined by two vertices and a plane containing four vertices is _________. | 36 |
Given that $\dfrac {\pi}{4} < \alpha < \dfrac {3\pi}{4}$ and $0 < \beta < \dfrac {\pi}{4}$, with $\cos \left( \dfrac {\pi}{4}+\alpha \right)=- \dfrac {3}{5}$ and $\sin \left( \dfrac {3\pi}{4}+\beta \right)= \dfrac {5}{13}$, find the value of $\sin(\alpha+\beta)$. | \dfrac {63}{65} |
Marty wants to paint a box. He can choose to use either blue, green, yellow, or black paint. Also, he can style the paint by painting with a brush, a roller, or a sponge. How many different combinations of color and painting method can Marty choose? | 12 |
Angle ABC is a right angle. The diagram shows four quadrilaterals, where three are squares on each side of triangle ABC, and one square is on the hypotenuse. The sum of the areas of all four squares is 500 square centimeters. What is the number of square centimeters in the area of the largest square? | \frac{500}{3} |
Allen is 25 years younger than his mother. In 3 years, the sum of their ages will be 41. What is the present age of Allen's mother? | In 3 years, the sum of their present ages will be added by 3 (from Allen) + 3 (from her mother) = <<3+3=6>>6.
So, the sum of their present ages is 41 - 6 = <<41-6=35>>35.
Let x be the present age of Allen's mother and x - 25 be the present age of Allen.
So the equation using the sum of their present ages is x + x - 25 ... |
In triangle $ABC$, $AB = AC = 100$, and $BC = 56$. Circle $P$ has radius $16$ and is tangent to $\overline{AC}$ and $\overline{BC}$. Circle $Q$ is externally tangent to $P$ and is tangent to $\overline{AB}$ and $\overline{BC}$. No point of circle $Q$ lies outside of $\triangle ABC$. The radius of circle $Q$ can be expr... | 254 |
Find all solutions to
\[\sqrt[4]{47 - 2x} + \sqrt[4]{35 + 2x} = 4.\]Enter all the solutions, separated by commas. | 23,-17 |
Find the product of all values of $t$ such that $t^2 = 36$. | -36 |
Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what tim... | 11:00 |
Find the equation of the directrix of the parabola $y = \frac{x^2 - 6x + 5}{12}.$ | y = -\frac{10}{3} |
Given that $a, b \in R^{+}$, and $a + b = 1$, find the maximum value of $- \frac{1}{2a} - \frac{2}{b}$. | -\frac{9}{2} |
Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How many minutes would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time? | 120 |
An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. How many years did he live? | 79 |
Each of five, standard, six-sided dice is rolled once. What is the probability that there is at least one pair but not a three-of-a-kind (that is, there are two dice showing the same value, but no three dice show the same value)? | \frac{25}{36} |
Let \( A = (2, 0) \) and \( B = (8, 6) \). Let \( P \) be a point on the circle \( x^2 + y^2 = 8x \). Find the smallest possible value of \( AP + BP \). | 6\sqrt{2} |
Let $x_1, x_2, \ldots, x_7$ be natural numbers, and $x_1 < x_2 < \ldots < x_6 < x_7$, also $x_1 + x_2 + \ldots + x_7 = 159$, then the maximum value of $x_1 + x_2 + x_3$ is. | 61 |
If $A,B$ and $C$ are non-zero distinct digits in base $6$ such that $\overline{ABC}_6 + \overline{BCA}_6+ \overline{CAB}_6 = \overline{AAA0}_6$, find $B+C$ in base $6$. | 5 |
Compute $\displaystyle \sum_{n=2}^\infty \sum_{k=1}^{n-1} \frac{k}{2^{n+k}}$. | \frac{4}{9} |
The area of the floor in a square room is 225 square feet. The homeowners plan to cover the floor with rows of 6-inch by 6-inch tiles. How many tiles will be in each row? | 30 |
Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\frac{1}{2}$ times the exterior angle at the same vertex. Then | 2700^{\circ} |
What is the largest multiple of 7 less than 50? | 49 |
A certain shopping mall purchased a batch of daily necessities. If they are sold at a price of $5$ yuan per item, they can sell $30,000$ items per month. If they are sold at a price of $6$ yuan per item, they can sell $20,000$ items per month. It is assumed that the monthly sales quantity $y$ (items) and the price $x$ ... | 40000 |
Let $S$ be the set of all 3-digit numbers with all digits in the set $\{1,2,3,4,5,6,7\}$ (so in particular, all three digits are nonzero). For how many elements $\overline{a b c}$ of $S$ is it true that at least one of the (not necessarily distinct) 'digit cycles' $\overline{a b c}, \overline{b c a}, \overline{c a b}$ ... | 127 |
The National High School Mathematics Competition is set up as follows: the competition is divided into the first round and the second round. The first round includes 8 fill-in-the-blank questions (each worth 8 points) and 3 problem-solving questions (worth 16, 20, and 20 points respectively), with a total score of 120 ... | \frac{1}{2} |
This month, I spent 26 days exercising for 20 minutes or more, 24 days exercising 40 minutes or more, and 4 days of exercising 2 hours exactly. I never exercise for less than 20 minutes or for more than 2 hours. What is the minimum number of hours I could have exercised this month? | 22 |
In triangle $ABC$, the angle bisectors are $AD$, $BE$, and $CF$, which intersect at the incenter $I$. If $\angle ACB = 38^\circ$, then find the measure of $\angle AIE$, in degrees. | 71^\circ |
Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2020$, and $a^2 - b^2 + c^2 - d^2 = 2024$. Find the number of possible values of $a$. | 503 |
Two differentiable real functions \( f(x) \) and \( g(x) \) satisfy
\[ \frac{f^{\prime}(x)}{g^{\prime}(x)} = e^{f(x) - g(x)} \]
for all \( x \), and \( f(0) = g(2003) = 1 \). Find the largest constant \( c \) such that \( f(2003) > c \) for all such functions \( f, g \). | 1 - \ln 2 |
Let $AB$ be a diameter of a circle centered at $O$. Let $E$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $E$ and $AE$ at $C$ and $D$, respectively. If $\angle BAE = 43^\circ$, find $\angle CED$, in degrees.
[asy]
import graph;
unitsize(2 cm);
pair O, A, B, C, D, E;
O = (0,0);
A = ... | 47^\circ |
Given that $(1+\sin t)(1+\cos t)=5/4$ and
$(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},$
where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$
| 27 |
The graph of the rational function $\frac{p(x)}{q(x)}$ is shown below. If $q(x)$ is quadratic, $p(3)=3$, and $q(2) = 2$, find $p(x) + q(x)$.
[asy]
size(8cm);
import graph;
Label f;
f.p=fontsize(6);
//xaxis(-5,5,Ticks(f, 1.0));
//yaxis(-5,5,Ticks(f, 1.0));
draw((-5,0)--(5,0));
draw((0,-5)--(0,5));
int i;
for (i =... | x^2 |
How many two-digit primes have a ones digit of 1? | 5 |
In the sport of diving from a high platform, there is a functional relationship between the athlete's height above the water surface $h$ (m) and the time $t$ (s) after the jump: $h(t)=-4.9t^2+6.5t+10$. Determine the moment when the instantaneous velocity is $0 \text{ m/s}$. | \frac{65}{98} |
Let $P(x) = b_0 + b_1x + b_2x^2 + \dots + b_mx^m$ be a polynomial with integer coefficients, where $0 \le b_i < 5$ for all $0 \le i \le m$.
Given that $P(\sqrt{5})=23+19\sqrt{5}$, compute $P(3)$. | 132 |
Let $A B C D E F$ be a convex hexagon with the following properties. (a) $\overline{A C}$ and $\overline{A E}$ trisect $\angle B A F$. (b) $\overline{B E} \| \overline{C D}$ and $\overline{C F} \| \overline{D E}$. (c) $A B=2 A C=4 A E=8 A F$. Suppose that quadrilaterals $A C D E$ and $A D E F$ have area 2014 and 1400, ... | 7295 |
Evaluate $\lfloor-5.77\rfloor+\lceil-3.26\rceil+\lfloor15.93\rfloor+\lceil32.10\rceil$. | 39 |
Given $|m|=3$, $|n|=2$, and $m<n$, find the value of $m^2+mn+n^2$. | 19 |
John decides to start collecting art. He pays the same price for his first 3 pieces of art and the total price came to $45,000. The next piece of art was 50% more expensive than those. How much did all the art cost? | The first 3 pieces each cost 45000/3=$<<45000/3=15000>>15,000
So the next piece was 15,000*.5=$<<15000*.5=7500>>7,500 more expensive
That means it cost 15,000+7,500=$<<15000+7500=22500>>22,500
That means in total he spent 45,000+22,500=$67,500
#### 67,500 |
Chloe wants to buy a jacket that costs $45.50$. She has two $20$ bills, five quarters, a few nickels, and a pile of dimes in her wallet. What is the minimum number of dimes she needs if she also has six nickels? | 40 |
What is the smallest positive integer with exactly 12 positive integer divisors? | 288 |
Let \( a \) and \( b \) be real numbers, and consider the function \( f(x) = x^{3} + a x^{2} + b x \). If there exist three real numbers \( x_{1}, x_{2}, x_{3} \) such that \( x_{1} + 1 \leqslant x_{2} \leqslant x_{3} - 1 \), and \( f(x_{1}) = f(x_{2}) = f(x_{3}) \), find the minimum value of \( |a| + 2|b| \). | \sqrt{3} |
The sides of triangle \(ABC\) are divided by points \(M, N\), and \(P\) such that \(AM : MB = BN : NC = CP : PA = 1 : 4\). Find the ratio of the area of the triangle bounded by lines \(AN, BP\), and \(CM\) to the area of triangle \(ABC\). | 3/7 |
A residential building has a construction cost of 250 yuan per square meter. Considering a useful life of 50 years and an annual interest rate of 5%, what monthly rent per square meter is required to recoup the entire investment? | 1.14 |
Define an operation between sets A and B: $A*B = \{x | x = x_1 + x_2, \text{ where } x_1 \in A, x_2 \in B\}$. If $A = \{1, 2, 3\}$ and $B = \{1, 2\}$, then the sum of all elements in $A*B$ is ____. | 14 |
Suppose that a real number $x$ satisfies \[\sqrt{49-x^2}-\sqrt{25-x^2}=3.\]What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$? | 8 |
Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and $S_{n}=n^{2}+n+1$. In the positive geometric sequence ${b_n}$, $b_3=a_2$, $b_4=a_4$. Find:
1. The general term formulas for ${a_n}$ and ${b_n}$;
2. If $c_n$ is defined as $c_n=\begin{cases} a_{n},(n\text{ is odd}) \\ b_{n},(n\text{ is even}... | 733 |
How many integers between $1000$ and $9999$ have four distinct digits? | 4536 |
Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[\frac{\sqrt{(x^2 + y^2)(3x^2 + y^2)}}{xy}.\] | 1 + \sqrt{3} |
Mia sells four burritos and five empanadas for $\$$4.00 and she sells six burritos and three empanadas for $\$$4.50. Assuming a fixed price per item, what is the cost, in dollars, of five burritos and seven empanadas? Express your answer as a decimal to the nearest hundredth. | 5.25 |
Find the area in square feet of a square with a perimeter of 32ft. | 64 |
The vertices and midpoints of the sides of a regular decagon (thus a total of 20 points marked) are noted.
How many triangles can be formed with vertices at the marked points? | 1130 |
Simplify $3\cdot\frac{11}{4}\cdot \frac{16}{-55}$. | -\frac{12}{5} |
Given that $x$ is a multiple of $46200$, determine the greatest common divisor of $f(x) = (3x + 5)(5x + 3)(11x + 6)(x + 11)$ and $x$. | 990 |
What is the minimum number of sides of a regular polygon that approximates the area of its circumscribed circle with an error of less than 1 per thousand (0.1%)? | 82 |
Given that the eccentricities of a confocal ellipse and a hyperbola are \( e_1 \) and \( e_2 \), respectively, and the length of the minor axis of the ellipse is twice the length of the imaginary axis of the hyperbola, find the maximum value of \( \frac{1}{e_1} + \frac{1}{e_2} \). | 5/2 |
A two-digit positive integer is said to be $cuddly$ if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly? | 1 |
A four-dimensional rectangular hyper-box has side lengths $W$, $X$, $Y$, and $Z$. It has "faces" (three-dimensional volumes) whose measures are $60$, $80$, $120$, $60$, $80$, $120$ cubic units. What is $W$ + $X$ + $Y$ + $Z$?
**A)** 200
**B)** 250
**C)** 300
**D)** 318.5
**E)** 400 | 318.5 |
Find all triples $(a, b, c)$ of real numbers such that
$$ a^2 + ab + c = 0, $$
$$b^2 + bc + a = 0, $$
$$c^2 + ca + b = 0.$$ | (0, 0, 0)\left(-\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}\right) |
Compute $\cos 150^\circ$. | -\frac{\sqrt{3}}{2} |
The complete graph of $y=f(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)
What is the sum of the $x$-coordinates of all points where $f(x) = x+1$? | 3 |
Given an arithmetic sequence $\left\{a_{n}\right\}$ with the first term $a_{1}>0$, and the following conditions:
$$
a_{2013} + a_{2014} > 0, \quad a_{2013} a_{2014} < 0,
$$
find the largest natural number $n$ for which the sum of the first $n$ terms, $S_{n}>0$, holds true. | 4026 |
Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$.
Find the minimum value and the maximum value of the product $abcd$. | 0\frac{1}{16} |
The leak in Jerry's roof drips 3 drops a minute into the pot he put under it. Each drop is 20 ml, and the pot holds 3 liters. How long will it take the pot to be full? | First find the pot's volume in ml: 3 liters * 1000 ml/liter = <<3*1000=3000>>3000 ml
Then find the number of ml that enter the pot per minute: 3 drops/minute * 20 ml/drop = <<3*20=60>>60 ml/minute
Then divide the volume of the pot by the volume that enters it every minute: 3000 ml / 60 ml/minute = <<3000/60=50>>50 minu... |
Rationalize the denominator of $\displaystyle \frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{7}}$, and write your answer in the form \[
\frac{A\sqrt{2} + B\sqrt{3} + C\sqrt{7} + D\sqrt{E}}{F},
\]where everything is in simplest radical form and the fraction is in lowest terms, and $F$ is positive. What is $A + B + C + D + E + F$? | 57 |
A man decides to try and do everything off his bucket list while he is still young and fit enough to finish it all. One of his goals was to climb the seven summits. He realizes first he has to get in shape to do it and it takes him 2 years of working out to get healthy enough to take up mountain climbing. He then sp... | He spent 2*2=<<2*2=4>>4 years learning to climb
He spends 5*7=<<5*7=35>>35 months climbing all 7 mountains
After a 13 month break, he starts diving so that is 35+13=<<35+13=48>>48 months
That means it is 48/12=<<48/12=4>>4 years
So in total it all took 2+4+4+2=<<2+4+4+2=12>>12 years
#### 12 |
A knight begins on the lower-left square of a standard chessboard. How many squares could the knight end up at after exactly 2009 legal knight's moves? | 32 |
Square $EFGH$ has a side length of $40$. Point $Q$ lies inside the square such that $EQ = 15$ and $FQ = 34$. The centroids of $\triangle{EFQ}$, $\triangle{FGQ}$, $\triangle{GHQ}$, and $\triangle{HEQ}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral? | \frac{1600}{9} |
Let $p(x)=x^{2}-x+1$. Let $\alpha$ be a root of $p(p(p(p(x))))$. Find the value of $(p(\alpha)-1) p(\alpha) p(p(\alpha)) p(p(p(\alpha)))$ | -1 |
There are 30 students in Ms. Leech's class. Twice as many girls as boys are in the class. There are 10 boys in the class and each boy today brought 5 cups for science activity class as Ms. Leech had instructed. If the total number of cups brought by the students in the class is 90, how many cups did each girl bring? | If there are 30 students in Ms. Leech's class, and the number of boys is 10, the number of girls in Ms. Leech class is 30-10=<<30-10=20>>20
If each boy brought 5 cups for the science activity, the total number of cups brought by the boys is 5*10=<<5*10=50>>50 cups.
The number of cups that the girls brought is 90-50=<<9... |
Two numbers $90$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors? | 9 |
Let $(x, y)$ be a pair of real numbers satisfying $$56x+33y=\frac{-y}{x^{2}+y^{2}}, \quad \text { and } \quad 33x-56y=\frac{x}{x^{2}+y^{2}}$$ Determine the value of $|x|+|y|$. | \frac{11}{65} |
When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 72$. | \frac{1}{24} |
Each bank teller has 10 rolls of coins. Each roll has 25 coins. How many coins do four bank tellers have in all? | Each bank teller has 10 x 25 = <<10*25=250>>250 coins.
Thus, four bank tellers have 250 x 4 = <<250*4=1000>>1000 coins.
#### 1000 |
Given a runner who is 30 years old, and the maximum heart rate is found by subtracting the runner's age from 220, determine the adjusted target heart rate by calculating 70% of the maximum heart rate and then applying a 10% increase. | 146 |
What value should the real number $m$ take so that the point representing the complex number $z=(m^2-8m+15)+(m^2-5m-14)i$ in the complex plane
(Ⅰ) lies in the fourth quadrant;
(Ⅱ) lies on the line $y=x$. | \frac{29}{3} |
Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other? | \frac{1}{3} |
The volume of a regular triangular pyramid, whose lateral face is inclined at an angle of $45^{\circ}$ to the base, is $9 \mathrm{~cm}^{3}$. Find the total surface area of the pyramid. | 9 \sqrt{3} (1 + \sqrt{2}) |
The deli has four kinds of bread, six kinds of meat, and five kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Ham, chicken, cheddar cheese, and white bread are each offered at the deli. If Al never orders a sandwich with a ham/cheddar cheese combination nor a sandwic... | 111 |
Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 5$. Find $r$. | \frac{\sqrt{15}}{4} |
Given the triangle $\triangle ABC$ is an isosceles right triangle with $\angle ABC = 90^\circ$, and sides $\overline{AB}$ and $\overline{BC}$ are each tangent to a circle at points $B$ and $C$ respectively, with the circle having center $O$ and located inside the triangle, calculate the fraction of the area of $\triang... | 1 - \frac{\pi}{2} |
Given that the terminal side of angle $\alpha$ passes through the point $P(\sqrt{3}, m)$ ($m \neq 0$), and $\cos\alpha = \frac{m}{6}$, then $\sin\alpha = \_\_\_\_\_\_$. | \frac{\sqrt{3}}{2} |
Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \in [-3, -1]$, $f(x) = -(x+2)^2$, and when $x \in [-1, 3)$, $f(x) = x$. Calculate the value of $f(1) + f(2) + f(3) + \ldots + f(2015)$. | 336 |
Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 11? | \frac{11}{12} |
A student named Zhang has a set of 6 questions to choose from, with 4 categorized as type A and 2 as type B. Zhang randomly selects 2 questions to solve.
(1) What is the probability that both selected questions are type A?
(2) What is the probability that the selected questions are not of the same type? | \frac{8}{15} |
Simplify
\[\cos \frac{2 \pi}{13} + \cos \frac{6 \pi}{13} + \cos \frac{8 \pi}{13}.\] | \frac{\sqrt{13} - 1}{4} |
How many integers $-15 \leq n \leq 15$ satisfy $(n-3)(n+5)(n+9) < 0$? | 13 |
How many positive integers less than $201$ are multiples of either $4$ or $9$, but not both at once? | 62 |
There are four people in a room. For every two people, there is a $50 \%$ chance that they are friends. Two people are connected if they are friends, or a third person is friends with both of them, or they have different friends who are friends of each other. What is the probability that every pair of people in this ro... | \frac{19}{32} |
Given a convex quadrilateral \(ABCD\) with \(X\) as the midpoint of the diagonal \(AC\), it turns out that \(CD \parallel BX\). Find \(AD\) if it is known that \(BX = 3\), \(BC = 7\), and \(CD = 6\). | 14 |
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