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agentica-org/DeepScaleR-Preview-Dataset
|
Suppose that $\{a_n\}$ is an arithmetic sequence with $$
a_1+a_2+ \cdots +a_{100}=100 \quad \text{and} \quad
a_{101}+a_{102}+ \cdots + a_{200}=200.
$$What is the value of $a_2 - a_1$? Express your answer as a common fraction.
|
\frac{1}{100}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Create a cube $C_{1}$ with edge length 1. Take the centers of the faces and connect them to form an octahedron $O_{1}$. Take the centers of the octahedron's faces and connect them to form a new cube $C_{2}$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.
|
\frac{54+9 \sqrt{3}}{8}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A triangle ABC has vertices at points $A = (0,2)$, $B = (0,0)$, and $C = (10,0)$. A vertical line $x = a$ divides the triangle into two regions. Find the value of $a$ such that the area to the left of the line is one-third of the total area of triangle ABC.
A) $\frac{10}{3}$
B) $5$
C) $\frac{15}{4}$
D) $2$
|
\frac{10}{3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $ABCD$ be a convex quadrilateral with $AB=2$, $AD=7$, and $CD=3$ such that the bisectors of acute angles $\angle{DAB}$ and $\angle{ADC}$ intersect at the midpoint of $\overline{BC}$. Find the square of the area of $ABCD$.
|
180
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given an arithmetic sequence $\{a_n\}$ where $a_1=1$ and $a_n=70$ (for $n\geq3$), find all possible values of $n$ if the common difference is a natural number.
|
70
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Derek is deciding between two different-sized pizzas at his favorite restaurant. The menu lists a 14-inch pizza and an 18-inch pizza. Calculate the percent increase in area if Derek chooses the 18-inch pizza over the 14-inch pizza.
|
65.31\%
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \leq a+b+c+d+e \leq 10$?
|
116
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Painting the surface of a large metal ball requires 2.4 kilograms of paint. If this large metal ball is melted down to make 64 identical small metal balls, without considering any loss, the amount of paint needed to coat the surfaces of these small metal balls is \_\_\_\_\_\_ kilograms.
|
9.6
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$. What is $N$?
|
78
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle.
|
16
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A class has $25$ students. The teacher wants to stock $N$ candies, hold the Olympics and give away all $N$ candies for success in it (those who solve equally tasks should get equally, those who solve less get less, including, possibly, zero candies). At what smallest $N$ this will be possible, regardless of the number of tasks on Olympiad and the student successes?
|
600
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Points $P$ and $Q$ are both in the line segment $AB$ and on the same side of its midpoint. $P$ divides $AB$ in the ratio $2:3$,
and $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of $AB$ is:
|
70
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The integers that can be expressed as a sum of three distinct numbers chosen from the set $\{4,7,10,13, \ldots,46\}$.
|
37
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card?
|
48
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
I won a VIP trip for five to a music festival. I can bring four of my friends. I have 10 friends to choose from: 4 are musicians, and 6 are non-musicians. In how many ways can I form my music festival group so that at least one musician is in the group?
|
195
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The distance from the point $(3,0)$ to one of the asymptotes of the hyperbola $\frac{{x}^{2}}{16}-\frac{{y}^{2}}{9}=1$ is $\frac{9}{5}$.
|
\frac{9}{5}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let the set \( T = \{0, 1, \dots, 6\} \),
$$
M = \left\{\left.\frac{a_1}{7}+\frac{a_2}{7^2}+\frac{a_3}{7^3}+\frac{a_4}{7^4} \right\rvert\, a_i \in T, i=1,2,3,4\right\}.
$$
If the elements of the set \( M \) are arranged in decreasing order, what is the 2015th number?
|
\frac{386}{2401}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The number of integer solutions for the inequality \( |x| < 3 \pi \) is ( ).
|
19
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
If $C=\frac{5}{9}(F-32)$, what is $F$ when $C=20$?
|
68
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A housewife saved $2.50 in buying a dress on sale. If she spent $25 for the dress, she saved about:
|
9 \%
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the set of solutions for $x$ in the inequality $\frac{x+1}{x+2} > \frac{3x+4}{2x+9}$ when $x \neq -2, x \neq \frac{9}{2}$.
|
\frac{-9}{2} \leq x \leq -2 \cup \frac{1-\sqrt{5}}{2} < x < \frac{1+\sqrt{5}}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Evaluate \[(3a^3 - 7a^2 + a - 5)(4a - 6)\] for $a = 2$.
|
-14
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the smallest positive integer with exactly 16 positive divisors?
|
120
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The scores (in points) of the 15 participants in the final round of a math competition are as follows: $56$, $70$, $91$, $98$, $79$, $80$, $81$, $83$, $84$, $86$, $88$, $90$, $72$, $94$, $78$. What is the $80$th percentile of these 15 scores?
|
90.5
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A bivalent metal element is used in a chemical reaction. When 3.5g of the metal is added into 50g of a dilute hydrochloric acid solution with a mass percentage of 18.25%, there is some metal leftover after the reaction finishes. When 2.5g of the metal is added into the same mass and mass percentage of dilute hydrochloric acid, the reaction is complete, after which more of the metal can still be reacted. Determine the relative atomic mass of the metal.
|
24
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.
|
9
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
How many diagonals can be drawn for a hexagon?
|
9
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given an arithmetic sequence $\{a_n\}$ with the common difference $d$ being an integer, and $a_k=k^2+2$, $a_{2k}=(k+2)^2$, where $k$ is a constant and $k\in \mathbb{N}^*$
$(1)$ Find $k$ and $a_n$
$(2)$ Let $a_1 > 1$, the sum of the first $n$ terms of $\{a_n\}$ is $S_n$, the first term of the geometric sequence $\{b_n\}$ is $l$, the common ratio is $q(q > 0)$, and the sum of the first $n$ terms is $T_n$. If there exists a positive integer $m$, such that $\frac{S_2}{S_m}=T_3$, find $q$．
|
\frac{\sqrt{13}-1}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The number $n$ is a three-digit integer and is the product of two distinct prime factors $x$ and $10x+y$, where $x$ and $y$ are each less than 10, with no restrictions on $y$ being prime. What is the largest possible value of $n$?
|
553
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given a set of data pairs (3,y_{1}), (5,y_{2}), (7,y_{3}), (12,y_{4}), (13,y_{5}) corresponding to variables x and y, the linear regression equation obtained is \hat{y} = \frac{1}{2}x + 20. Calculate the value of \sum\limits_{i=1}^{5}y_{i}.
|
120
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Simplify $(2^5+7^3)(2^3-(-2)^2)^8$.
|
24576000
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
King Qi and Tian Ji are competing in a horse race. Tian Ji's top horse is better than King Qi's middle horse, worse than King Qi's top horse; Tian Ji's middle horse is better than King Qi's bottom horse, worse than King Qi's middle horse; Tian Ji's bottom horse is worse than King Qi's bottom horse. Now, each side sends one top, one middle, and one bottom horse, forming 3 groups for separate races. The side that wins 2 or more races wins. If both sides do not know the order of the opponent's horses, the probability of Tian Ji winning is ____; if it is known that Tian Ji's top horse and King Qi's middle horse are in the same group, the probability of Tian Ji winning is ____.
|
\frac{1}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A gardener plans to enclose a rectangular garden with 480 feet of fencing. However, one side of the garden will be twice as long as another side. What is the maximum area of this garden?
|
12800
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
$ABCDEFGH$ shown below is a cube. Find $\sin \angle GAC$.
[asy]
import three;
triple A,B,C,D,EE,F,G,H;
A = (0,0,0);
B = (1,0,0);
C = (1,1,0);
D= (0,1,0);
EE = (0,0,1);
F = B+EE;
G = C + EE;
H = D + EE;
draw(B--C--D);
draw(B--A--D,dashed);
draw(EE--F--G--H--EE);
draw(A--EE,dashed);
draw(B--F);
draw(C--G);
draw(D--H);
label("$A$",A,S);
label("$B$",B,W);
label("$C$",C,S);
label("$D$",D,E);
label("$E$",EE,N);
label("$F$",F,W);
label("$G$",G,SW);
label("$H$",H,E);
[/asy]
|
\frac{\sqrt{3}}{3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The gas tank in Catherine's car is $\frac{1}{8}$ full. When 30 litres of gas are added, the tank becomes $\frac{3}{4}$ full. If the gas costs Catherine $\$ 1.38$ per litre, how much will it cost her to fill the remaining quarter of the tank?
|
\$16.56
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A thin diverging lens with an optical power of $D_{p} = -6$ diopters is illuminated by a beam of light with a diameter $d_{1} = 10$ cm. On a screen positioned parallel to the lens, a light spot with a diameter $d_{2} = 20$ cm is observed. After replacing the thin diverging lens with a thin converging lens, the size of the spot on the screen remains unchanged. Determine the optical power $D_{c}$ of the converging lens.
|
18
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the smallest positive integer \( n \) such that the mean of the squares of the first \( n \) natural numbers (\( n > 1 \)) is an integer.
(Note: The mean of the squares of \( n \) numbers \( a_1, a_2, \cdots, a_n \) is given by \( \sqrt{\frac{a_{1}^2 + a_{2}^2 + \cdots + a_{n}^2}{n}} \).)
(Note: Fifteenth American Mathematical Olympiad, 1986)
|
337
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In right triangle $PQR$, we have $\angle Q = \angle R$ and $PR = 6\sqrt{2}$. What is the area of $\triangle PQR$?
|
36
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Ivan Tsarevich is fighting the Dragon Gorynych on the Kalinov Bridge. The Dragon has 198 heads. With one swing of his sword, Ivan Tsarevich can cut off five heads. However, new heads immediately grow back, the number of which is equal to the remainder when the number of heads left after Ivan's swing is divided by 9. If the remaining number of heads is divisible by 9, no new heads grow. If the Dragon has five or fewer heads before the swing, Ivan Tsarevich can kill the Dragon with one swing. How many sword swings does Ivan Tsarevich need to defeat the Dragon Gorynych?
|
40
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
From point \( A \), two rays are drawn intersecting a given circle: one at points \( B \) and \( C \), and the other at points \( D \) and \( E \). It is known that \( AB = 7 \), \( BC = 7 \), and \( AD = 10 \). Determine \( DE \).
|
0.2
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
When \( N \) takes all values from 1, 2, 3, ..., to 2015, how many numbers of the form \( 3^n + n^3 \) are divisible by 7?
|
288
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Out of 8 shots, 3 hit the target, and we are interested in the total number of ways in which exactly 2 hits are consecutive.
|
30
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
There are 7 volunteers, among which 3 people only speak Russian, and 4 people speak both Russian and English. From these, 4 people are to be selected to serve as translators for the opening ceremony of the "Belt and Road" summit, with 2 people serving as English translators and 2 people serving as Russian translators. There are a total of \_\_\_\_\_\_ different ways to select them.
|
60
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Solve for $\log_{3} \sqrt{27} + \lg 25 + \lg 4 + 7^{\log_{7} 2} + (-9.8)^{0} = \_\_\_\_\_\_\_\_\_\_\_$.
|
\frac{13}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Carl and Dina each arrive at a café at a random time between 1:00 PM and 1:45 PM. Each stays for 30 minutes before leaving. What is the probability that Carl and Dina meet each other at the café?
|
\frac{8}{9}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Shuai Shuai memorized more than one hundred words in seven days. The number of words memorized in the first three days is $20\%$ less than the number of words memorized in the last four days, and the number of words memorized in the first four days is $20\%$ more than the number of words memorized in the last three days. How many words did Shuai Shuai memorize in total over the seven days?
|
198
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$ .
|
463
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is $15\mbox{cm}^2$ and the area of face $ABD$ is $12 \mbox { cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\mbox{cm}^3$.
|
20
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A smaller regular tetrahedron is formed by joining the midpoints of the edges of a larger regular tetrahedron. Determine the ratio of the volume of the smaller tetrahedron to the volume of the larger tetrahedron.
|
\frac{1}{8}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
How many ways are there to cut a 1 by 1 square into 8 congruent polygonal pieces such that all of the interior angles for each piece are either 45 or 90 degrees? Two ways are considered distinct if they require cutting the square in different locations. In particular, rotations and reflections are considered distinct.
|
54
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A student, Ellie, was supposed to calculate $x-y-z$, but due to a misunderstanding, she computed $x-(y+z)$ and obtained 18. The actual answer should have been 6. What is the value of $x-y$?
|
12
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $ABCDEF$ be a regular hexagon with each side length $s$. Points $G$, $H$, $I$, $J$, $K$, and $L$ are the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The segments $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ form another hexagon inside $ABCDEF$. Find the ratio of the area of this inner hexagon to the area of hexagon $ABCDEF$, expressed as a fraction in its simplest form.
|
\frac{3}{4}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A jar contains 8 red balls and 2 blue balls. Every minute, a ball is randomly removed. The probability that there exists a time during this process where there are more blue balls than red balls in the jar can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Compute $100 a+b$.
|
209
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Express this sum as a common fraction: $.\overline{8} + .\overline{2}$
|
\frac{10}{9}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Four boys, four girls, and a coach are positioned on a circular track. Each girl is diametrically opposite to one of the boys. The length of the track is 50 meters. On the coach's signal, they all run towards the coach by the shortest path along the track. What is the total distance run by all the children together?
|
100
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
There are births in West Northland every 6 hours, deaths every 2 days, and a net immigration every 3 days. Calculate the approximate annual increase in population.
|
1400
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
(1) Calculate the value of $(\frac{2}{3})^{0}+3\times(\frac{9}{4})^{{-\frac{1}{2}}}+(\log 4+\log 25)$.
(2) Given $\alpha \in (0,\frac{\pi }{2})$, and $2\sin^{2}\alpha - \sin \alpha \cdot \cos \alpha - 3\cos^{2}\alpha = 0$, find the value of $\frac{\sin \left( \alpha + \frac{\pi }{4} \right)}{\sin 2\alpha + \cos 2\alpha + 1}$.
(3) There are three cards, marked with $1$ and $2$, $1$ and $3$, and $2$ and $3$, respectively. Three people, A, B, and C, each take a card. A looks at B's card and says, "The number that my card and B's card have in common is not $2$." B looks at C's card and says, "The number that my card and C's card have in common is not $1$." C says, "The sum of the numbers on my card is not $5$." What is the number on A's card?
(4) Given $f\left( x \right)=x-\frac{1}{x+1}$ and $g\left( x \right)={{x}^{2}}-2ax+4$, for any ${{x}_{1}}\in \left[ 0,1 \right]$, there exists ${{x}_{2}}\in \left[ 1,2 \right]$ such that $f\left( {{x}_{1}} \right)\geqslant g\left( {{x}_{2}} \right)$. Find the minimum value of the real number $a$.
|
\frac{9}{4}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The average score of 60 students is 72. After disqualifying two students whose scores are 85 and 90, calculate the new average score for the remaining class.
|
71.47
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
There are 3 boys and 4 girls, all lined up in a row. How many ways are there for the following situations?
- $(1)$ Person A is neither at the middle nor at the ends;
- $(2)$ Persons A and B must be at the two ends;
- $(3)$ Boys and girls alternate.
|
144
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In triangle $ABC,$ $\angle C = \frac{\pi}{2}.$ Find
\[\arctan \left( \frac{a}{b + c} \right) + \arctan \left( \frac{b}{a + c} \right).\]
|
\frac{\pi}{4}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A convex 2019-gon \(A_{1}A_{2}\ldots A_{2019}\) is cut into smaller pieces along its 2019 diagonals of the form \(A_{i}A_{i+3}\) for \(1 \leq i \leq 2019\), where \(A_{2020}=A_{1}, A_{2021}=A_{2}\), and \(A_{2022}=A_{3}\). What is the least possible number of resulting pieces?
|
5049
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
There are $5$ people arranged in a row. Among them, persons A and B must be adjacent, and neither of them can be adjacent to person D. How many different arrangements are there?
|
36
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with $x_1$ , $x_2$ , $\cdots$ , $x_n$ are distinct postive integers. Find the maximum value of $n$ .
*Proposed by Le Duc Minh*
|
64
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The numbers assigned to 100 athletes range from 1 to 100. If each athlete writes down the largest odd factor of their number on a blackboard, what is the sum of all the numbers written by the athletes?
|
3344
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $M$ be the number of ways to write $3050$ in the form $3050 = b_3 \cdot 10^3 + b_2 \cdot 10^2 + b_1 \cdot 10 + b_0$, where the $b_i$'s are integers, and $0 \le b_i \le 99$. Find $M$.
|
306
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let the line \( p \) be the perpendicular bisector of points \( A = (20, 12) \) and \( B = (-4, 3) \). Determine the point \( C = (x, y) \) where line \( p \) meets segment \( AB \), and calculate \( 3x - 5y \).
|
-13.5
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let be the set $ \mathcal{C} =\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f''\bigg|_{[0,1]} \right.\quad\exists x_1,x_2\in [0,1]\quad x_1\neq x_2\wedge \left( f\left(
x_1 \right) = f\left( x_2 \right) =0\vee f\left(
x_1 \right) = f'\left( x_1 \right) = 0\right) \wedge f''<1 \right\} , $ and $ f^*\in\mathcal{C} $ such that $ \int_0^1\left| f^*(x) \right| dx =\sup_{f\in\mathcal{C}} \int_0^1\left| f(x) \right| dx . $ Find $ \int_0^1\left| f^*(x) \right| dx $ and describe $ f^*. $
|
1/12
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Find the greatest common divisor of 4,004 and 10,010.
|
2002
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
For each pair of distinct natural numbers \(a\) and \(b\), not exceeding 20, Petya drew the line \( y = ax + b \) on the board. That is, he drew the lines \( y = x + 2, y = x + 3, \ldots, y = x + 20, y = 2x + 1, y = 2x + 3, \ldots, y = 2x + 20, \ldots, y = 3x + 1, y = 3x + 2, y = 3x + 4, \ldots, y = 3x + 20, \ldots, y = 20x + 1, \ldots, y = 20x + 19 \). Vasia drew a circle of radius 1 with center at the origin on the same board. How many of Petya’s lines intersect Vasia’s circle?
|
190
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $x$ be a real number such that $x^{3}+4 x=8$. Determine the value of $x^{7}+64 x^{2}$.
|
128
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A club has increased its membership to 12 members and needs to elect a president, vice president, secretary, and treasurer. Additionally, they want to appoint two different advisory board members. Each member can hold only one position. In how many ways can these positions be filled?
|
665,280
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In a convex quadrilateral \(ABCD\), \(\overrightarrow{BC} = 2 \overrightarrow{AD}\). Point \(P\) is a point in the plane of the quadrilateral such that \(\overrightarrow{PA} + 2020 \overrightarrow{PB} + \overrightarrow{PC} + 2020 \overrightarrow{PD} = \mathbf{0}\). Let \(s\) and \(t\) be the areas of quadrilateral \(ABCD\) and triangle \(PAB\), respectively. Then \(\frac{t}{s} =\) ______.
|
337/2021
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A nickel is placed on a table. The number of nickels which can be placed around it, each tangent to it and to two others is:
|
6
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the minimum value of the function \( y = \sin^4 x + \cos^4 x + \sec^4 x + \csc^4 x \)?
|
8.5
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Teams A and B each have 7 players who will compete in a Go tournament in a predetermined order. The match starts with player 1 from each team competing against each other. The loser is eliminated, and the winner next competes against the loser’s teammate. This process continues until all players of one team are eliminated, and the other team wins. Determine the total number of possible sequences of matches.
|
3432
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Determine the smallest integral value of $n$ such that the quadratic equation
\[3x(nx+3)-2x^2-9=0\]
has no real roots.
A) -2
B) -1
C) 0
D) 1
|
-1
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
On an island, there are knights who always tell the truth and liars who always lie. At the main celebration, 100 islanders sat around a large round table. Half of the attendees said the phrase: "both my neighbors are liars," while the remaining said: "among my neighbors, there is exactly one liar." What is the maximum number of knights that can sit at this table?
|
67
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Among the six-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, calculate the number of the numbers that are divisible by 2.
|
312
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$.
[asy]size(160); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1]; draw(C1); draw(C2); draw(O2--O1); dot(O1); dot(O2); draw(Q--R); label("$Q$",Q,NW); label("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);[/asy]
|
130
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given the function $f(x)=x^{3}- \frac {3}{2}x^{2}+ \frac {3}{4}x+ \frac {1}{8}$, find the value of $\sum\limits_{k=1}^{2016}f( \frac {k}{2017})$.
|
504
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Each of $b_1, b_2, \dots, b_{150}$ is equal to $2$ or $-2$. Find the minimum positive value of
\[\sum_{1 \le i < j \le 150} b_i b_j.\]
|
38
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
A function $f$ from the integers to the integers is defined as follows:
\[f(n) = \left\{
\begin{array}{cl}
n + 3 & \text{if $n$ is odd}, \\
n/2 & \text{if $n$ is even}.
\end{array}
\right.\]Suppose $k$ is odd and $f(f(f(k))) = 27.$ Find $k.$
|
105
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given \( f(x)=\frac{2x+3}{x-1} \), the graph of the function \( y=g(x) \) is symmetric with the graph of the function \( y=f^{-1}(x+1) \) with respect to the line \( y=x \). Find \( g(3) \).
|
\frac{7}{2}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
If $y=f(x)=\frac{x+2}{x-1}$, then it is incorrect to say:
|
$f(1)=0$
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the value of the expression $[ a-(b-c) ] - [(a-b) - c ]$ when $a = 17$, $b=21$ and $c=5$?
|
10
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Let $a,$ $b,$ $c$ be integers such that
\[\mathbf{A} = \frac{1}{5} \begin{pmatrix} -3 & a \\ b & c \end{pmatrix}\]and $\mathbf{A}^2 = \mathbf{I}.$ Find the largest possible value of $a + b + c.$
|
20
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The coefficient of the $x$ term in the expansion of $(x^{2}-x-2)^{3}$ is what value?
|
-12
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
There are four different passwords, $A$, $B$, $C$, and $D$, used by an intelligence station. Each week, one of the passwords is used, and each week it is randomly chosen with equal probability from the three passwords not used in the previous week. Given that the password used in the first week is $A$, find the probability that the password used in the seventh week is also $A$ (expressed as a simplified fraction).
|
61/243
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
$C$ is a point on the extension of diameter $A B$, $C D$ is a tangent, and the angle $A D C$ is $110^{\circ}$. Find the angular measure of arc $B D$.
|
40
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given the ratio of women to men is $7$ to $5$, and the average age of women is $30$ years and the average age of men is $35$ years, determine the average age of the community.
|
32\frac{1}{12}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Suppose that on a parabola with vertex $V$ and a focus $F$ there exists a point $A$ such that $AF=20$ and $AV=21$. What is the sum of all possible values of the length $FV?$
|
\frac{40}{3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
The area of the ground plane of a truncated cone $K$ is four times as large as the surface of the top surface. A sphere $B$ is circumscribed in $K$ , that is to say that $B$ touches both the top surface and the base and the sides. Calculate ratio volume $B :$ Volume $K$ .
|
9/14
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In the polar coordinate system, the equation of curve C is $\rho^2= \frac{3}{1+2\sin^2\theta}$. Point R is at $(2\sqrt{2}, \frac{\pi}{4})$.
P is a moving point on curve C, and side PQ of rectangle PQRS, with PR as its diagonal, is perpendicular to the polar axis. Find the maximum and minimum values of the perimeter of rectangle PQRS and the polar angle of point P when these values occur.
|
\frac{\pi}{6}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given points $A(\cos\alpha, \sin\alpha)$ and $B(\cos\beta, \sin\beta)$, where $\alpha, \beta$ are acute angles, and that $|AB| = \frac{\sqrt{10}}{5}$:
(1) Find the value of $\cos(\alpha - \beta)$;
(2) If $\tan \frac{\alpha}{2} = \frac{1}{2}$, find the values of $\cos\alpha$ and $\cos\beta$.
|
\frac{24}{25}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
What is the 100th letter in the pattern ABCABCABC...?
|
A
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer "Yes" or "No" to the question: "Are there more knights on the island than liars?". They answered in turn such that everyone else could hear. Knights told the truth, liars lied. Each follower gave the same answer as the majority of those who answered before them, and if "Yes" and "No" answers were equal, they gave either answer. It turned out that the number of "Yes" answers was exactly 1009. What is the maximum number of followers that could have been among the islanders?
|
1009
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
(a) A natural number $n$ is less than 120. What is the largest remainder that the number 209 can give when divided by $n$?
(b) A natural number $n$ is less than 90. What is the largest remainder that the number 209 can give when divided by $n$?
|
69
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Given that $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $(\overrightarrow{a}+\overrightarrow{b})\perp\overrightarrow{a}$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
|
\frac{2\pi}{3}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\sin C + \sin(B - A) = \sqrt{2} \sin 2A$, and $A \neq \frac{\pi}{2}$.
(I) Find the range of values for angle $A$;
(II) If $a = 1$, the area of $\triangle ABC$ is $S = \frac{\sqrt{3} + 1}{4}$, and $C$ is an obtuse angle, find the measure of angle $A$.
|
\frac{\pi}{6}
|
|
agentica-org/DeepScaleR-Preview-Dataset
|
Through the vertices \( A, C, D \) of the parallelogram \( ABCD \) with sides \( AB = 7 \) and \( AD = 4 \), a circle is drawn that intersects the line \( BD \) at point \( E \), and \( DE = 13 \). Find the length of diagonal \( BD \).
|
15
|
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