task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$, and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$. There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$, which he'd like to avoid.
[center]<see attached>[/center]
Given that Mr. Ambulando wants ... | 56 | 118 | 2 |
math | Solve the following system of equations:
$$
\frac{2 x^{2}}{1+x^{2}}=y, \quad \frac{2 y^{2}}{1+y^{2}}=z, \quad \frac{2 z^{2}}{1+z^{2}}=x
$$ | 0or1 | 66 | 3 |
math | [ Sums of numerical sequences and series of differences ] Pascal's Triangle and the Binomial Theorem Complex numbers help solve tasks_
Using the expansion $(1+i)^{n}$ by the Binomial Theorem, find:
a) $C_{100}^{0}-C_{100}^{2}+C_{100}^{4}-\ldots+C_{100}^{100}$
b) $C_{99}^{1}-C_{99}^{3}+C_{99}^{5}-\ldots-C_{99}^{99}$. | -2^{50} | 132 | 6 |
math | Example 6. Given the general solution $y=C_{1} \sin 2 x+C_{2} \cos 2 x$ of the differential equation $y^{\prime \prime}+4 y=0$. What particular solutions are obtained when $C_{1}=2, C_{2}=3$? For what values of the parameters $C_{1}$ and $C_{2}$ do the particular solutions $y=\sin 2 x, y=\cos 2 x$ result? | 2\sin2x+3\cos2x,\,\sin2x\,(C_{1}=1,C_{2}=0),\,\cos2x\,(C_{1}=0,C_{2}=1) | 105 | 47 |
math | 8.187. $\sin ^{2} x \tan x+\cos ^{2} x \cot x+2 \sin x \cos x=\frac{4 \sqrt{3}}{3}$. | (-1)^{k}\frac{\pi}{6}+\frac{\pi}{2},k\inZ | 47 | 23 |
math | ## Task B-3.3.
For which natural numbers $n$ is the value of the expression $\frac{n^{2}-4 n+4}{n+1}$ an integer? | n\in{2,8} | 39 | 8 |
math | 9.1. (12 points) Two circles touch each other externally at point $K$. On their common internal tangent, point $P$ is marked such that $K P=14$. Through point $P$, two secants are drawn to the circles, with one of them intercepting a chord $A B=45$ on the first circle, and the other intercepting a chord $C D=21$ on the... | 1.75 | 134 | 4 |
math | Three, (25 points) "If $a, b, c$ are positive real numbers, then $\sqrt[3]{a b c} \leqslant \frac{a+b+c}{3}$, where the equality holds if and only if $a=b=c$." Using the above conclusion, find the maximum value of the function $y=2 x(4-x)(3-2 x)\left(0<x<\frac{3}{2}\right)$. | \frac{200}{27} | 101 | 10 |
math | 7. Find the maximum value of the function $f(x)=\sqrt{1+\sin x}+\sqrt{1-\sin x}+\sqrt{2+\sin x}+\sqrt{2-\sin x}+\sqrt{3+\sin x}+$ $\sqrt{3-\sin x}$. | 2+2\sqrt{2}+2\sqrt{3} | 63 | 15 |
math | 5.14. Calculate the limit
$$
\lim _{x \rightarrow 0} \frac{\sqrt[3]{1+\operatorname{arctan} 4 x}-\sqrt[3]{1-\operatorname{arctan} 4 x}}{\sqrt{1-\arcsin 3 x}-\sqrt{1+\operatorname{arctan} 3 x}}
$$ | -\frac{8}{9} | 88 | 7 |
math | Find the four smallest four-digit numbers that meet the following condition: by dividing by $2$, $3$, $4$, $5$ or $6$ the remainder is $ 1$. | 1021, 1081, 1141, 1201 | 39 | 22 |
math | G7.4 If $S=1+2-3-4+5+6-7-8+\ldots+1985$, find $S$. | 1 | 36 | 1 |
math | 3.220
$$
\frac{1+\cos (2 \alpha-2 \pi)+\cos (4 \alpha+2 \pi)-\cos (6 \alpha-\pi)}{\cos (2 \pi-2 \alpha)+2 \cos ^{2}(2 \alpha+\pi)-1}=2 \cos 2 \alpha
$$ | 2\cos2\alpha | 78 | 6 |
math | What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours? | 3 | 30 | 1 |
math | Source: 2017 Canadian Open Math Challenge, Problem B2
-----
There are twenty people in a room, with $a$ men and $b$ women. Each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman. The total number of handshakes is $106$. Determine the value of $a \cdot b... | 84 | 88 | 2 |
math | 11. What? Where? When? Experts and Viewers play "What, Where, When" until six wins - whoever wins six rounds first is the winner. The probability of the Experts winning in one round is 0.6, and there are no ties. Currently, the Experts are losing with a score of $3: 4$. Find the probability that the Experts will still ... | 0.4752 | 81 | 6 |
math | 685. For what base of the numeral system does the following rebus have a solution
$$
\begin{aligned}
& \text { KITO } \\
& + \text { KIOTO } \\
& \hline \text { TOKIO, }
\end{aligned}
$$
where identical letters represent identical digits, and different letters represent different digits? Find all solutions to the reb... | 4350_{7}+43050_{7}=50430_{7} | 85 | 24 |
math | In a non-isosceles triangle with side lengths $a, b, c$ being integers, the following relationship holds. What is the smallest height of the triangle?
$$
\frac{a^{2}}{c}-(a-c)^{2}=\frac{b^{2}}{c}-(b-c)^{2}
$$ | 2.4 | 72 | 3 |
math | Task B-3.1. Solve the equation
$$
\log _{5 x-2} 2+2 \cdot \log _{5 x-2} x=\log _{5 x-2}(x+1)
$$ | 1 | 52 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{\sqrt{x-1}}{\sqrt[3]{x^{2}-1}}$ | 0 | 41 | 1 |
math | 1. Can the parameters $a, b, c$ be chosen such that for all $x$ the equality
$$
(x+a)^{2}+(2 x+b)^{2}+(2 x+c)^{2}=(3 x+1)^{2} ?
$$ | =\frac{1}{3},b==\frac{2}{3} | 59 | 16 |
math | 12. There are 2021 balls in a crate. The balls are numbered from 1 to 2021 . Erica works out the digit sum for each ball. For example, the digit sum of 2021 is 5, since $2+0+2+1=5$.
Erica notes that balls with equal digit sums have the same colour and balls with different digit sums have different colours.
How many dif... | 28 | 103 | 2 |
math | $12 \cdot 144$ Find all real numbers $p$ such that the cubic equation
$$
5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p
$$
has three roots that are all natural numbers.
(China High School Mathematics League, 1995) | 76 | 78 | 2 |
math | 10 Given the set $\Omega=\left\{(x, y) \mid x^{2}+y^{2} \leqslant 2008\right\}$, if points $P(x, y)$ and $P^{\prime}\left(x^{\prime}\right.$, $y^{\prime}$ ) satisfy $x \leqslant x^{\prime}$ and $y \geqslant y^{\prime}$, then point $P$ is said to dominate $P^{\prime}$. If a point $Q$ in set $\Omega$ satisfies: there doe... | {(x,y)\midx^{2}+y^{2}=2008,x\leqslant0,y\geqslant0} | 155 | 33 |
math | Solve over the positive real numbers the functional equation
$$
f(x y+f(x))=x f(y)+2 .
$$ | f(x)\equivx+1 | 27 | 7 |
math | 34. To calculate the product of all natural numbers from 1 to 50 inclusive, of course, it's better to use a computer. However, you can easily state the number of zeros at the end of this product without resorting to a computer. How many are there? | 12 | 59 | 2 |
math | 4. The average age of 5 basketball players currently on the court is 24 years and 6 months. If the coach's age is included in the calculation of the average, then the average age is 27 years. How old is the coach? | 39.5 | 54 | 4 |
math | Solve the following system of equations:
$$
\frac{x+y}{x^{3}+y^{3}}=\frac{2}{5}, \quad \frac{x^{2}+y^{2}}{x^{4}+y^{4}}=\frac{8}{23}
$$ | \begin{gathered}x_{1}=\frac{1}{8}(\sqrt{70}+\sqrt{30}),\quady_{1}=\frac{1}{8}(\sqrt{70}-\sqrt{30})\\x_{2}=\frac{1}{8}(\sqrt{70}-\sqrt{30}),\quady_{2}=\frac{1}{8}(\sqrt{70} | 63 | 97 |
math | ## 8. Bag of Apples
When asked how many apples are in the bag, the seller replied: "If I count them in twos, or in threes, or in fours, or in fives, or in sixes, there is always one left over. If I count them in sevens, none are left over." Determine the smallest number of apples that could be in the bag.
## Result: ... | 301 | 94 | 3 |
math | 5. Let $P(x)$ be the polynomial of minimal degree such that $P(k)=720 k /\left(k^{2}-1\right)$ for $k \in\{2,3,4,5\}$. Find the value of $P(6)$. | 48 | 60 | 2 |
math | Task 4. Find the smallest natural $m$, for which there exists such a natural $n$ that the sets of the last 2014 digits in the decimal representation of the numbers $a=2015^{3 m+1}$ and $b=2015^{6 n+2}$ are the same, and $a<b$. | 671 | 76 | 3 |
math | 3. Find the greatest and least values of the function $y=3 x^{4}-6 x^{2}+4$ on the interval $[-1 ; 3]$. | y(-1)=y(1)=1,y(3)=193 | 38 | 16 |
math | Restore the acute triangle $ABC$ given the vertex $A$, the foot of the altitude drawn from the vertex $B$ and the center of the circle circumscribed around triangle $BHC$ (point $H$ is the orthocenter of triangle $ABC$). | ABC | 56 | 2 |
math | Example 1. When $a$ takes what values, the equation $\left(a^{2}-1\right) x^{2}-$ $6(3 a-1) x+72=0$ has two distinct positive integer roots? | a=2 | 51 | 3 |
math | 4. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Calculate: $\sum_{k=0}^{2019}\left[\frac{4^{k}}{5}\right]=$ $\qquad$ | \frac{4^{2020}-1}{15}-1010 | 51 | 19 |
math | Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations. | 5 | 61 | 1 |
math | 5. On one side of the road, some power poles are arranged at equal distances. Xiao Ming rides his bike along the road. He took 3 minutes to ride from the 1st power pole to the 10th power pole. At this speed, in another 3 minutes, Xiao Ming can ride to the $\qquad$th power pole. | 19 | 74 | 2 |
math | Find all positive integers $n<1000$ such that the cube of the sum of the digits of $n$ equals $n^{2}$. | 1, 27 | 33 | 5 |
math | 1. At what speed does $\mathrm{E} \min \left(\sigma_{2 n}, 2 n\right)$ tend to $\infty$ as $n \rightarrow \infty$? (Here $\sigma_{2 n}=\min \left\{1 \leqslant k \leqslant 2 n: S_{k}=0\right\}$ and we assume $\sigma_{2 n}=\infty$ (or $\left.\sigma_{2 n}=2 n\right)$, if $S_{k} \neq 0$ for all $1 \leqslant k \leqslant 2 n... | 4\sqrt{n\pi^{-1}} | 143 | 9 |
math | 4. Given $P_{1}, P_{2}, \cdots, P_{100}$ as 100 points on a plane, satisfying that no three points are collinear. For any three of these points, if their indices are in increasing order and they form a clockwise orientation, then the triangle with these three points as vertices is called "clockwise". Question: Is it po... | 2017 | 100 | 4 |
math | Calculate
$$
\frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\ldots+\frac{1}{2013 \times 2014}
$$
## - Inequalities -
For common inequalities, see the handouts from previous sessions, as well as the Animath course written by Pierre Bornsztein. | \frac{2013}{2014} | 91 | 13 |
math | 4. (7 points) On the board, 50 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 50 minutes? | 1225 | 69 | 4 |
math | 5. In the "6 out of 45" lottery, a participant makes a bet by selecting any 6 numbers from 1 to 45 (the order of selection does not matter, but all numbers must be different). During the draw, a random winning combination of 6 numbers is determined. A "jackpot" is the event where a participant, by making a bet, guesses... | 0.0000123 | 125 | 9 |
math | 4. Solve the equation $\left(x^{\log _{2} x}-1\right)^{2}=1$. | x_{1}=\frac{1}{2}x_{2}=2 | 26 | 16 |
math | Example 11 Given $0<a<1$, and
$$
\left[a+\frac{1}{30}\right]+\left[a+\frac{2}{30}\right]+\cdots+\left[a+\frac{29}{30}\right]=18 \text {. }
$$
Then $[10 a$ ] equals $\qquad$
$(2009$, Beijing Mathematical Competition (Grade 8)) | 6 | 91 | 1 |
math | . A lighthouse emits a yellow signal every 15 seconds and a red signal every 28 seconds. The yellow signal is seen 2 seconds after midnight and the red signal 8 seconds after midnight. At what time will we see both signals emitted at the same time for the first time | 92 | 60 | 2 |
math | Consider coins with positive real denominations not exceeding 1 . Find the smallest $C>0$ such that the following holds: if we are given any 100 such coins with total value 50 , then we can always split them into two stacks of 50 coins each such that the absolute difference between the total values of the two stacks is... | \frac{50}{51} | 78 | 9 |
math | Let's calculate as simply as possible the value of
$$
a^{3}+b^{3}+3\left(a^{3} b+a b^{3}\right)+6\left(a^{3} b^{2}+a^{2} b^{3}\right)
$$
if $a+b=1$. | 1 | 69 | 1 |
math | (1) Given the sets $A=\left\{y \mid y=x^{2}+2 x-3\right\}, B=\left\{y \left\lvert\, y=x+\frac{1}{x}\right., x<\right.$ 0), then $A \cap B=$ . $\qquad$ | [-4,-2] | 73 | 5 |
math | [ Mathematical logic (other).]
Lilac. In a vase, there is a bouquet of 7 white and blue lilac branches. It is known that 1) at least one branch is white, 2) out of any two branches, at least one is blue. How many white branches and how many blue branches are in the bouquet?
# | 1 | 73 | 1 |
math | Determine all pairs $(m, n)$ of positive integers such that $2^{m}+1=n^{2}$ | (3,3) | 25 | 5 |
math | 106. Easy Division. By dividing the number 8101265822784 by 8, you will see that the answer can be obtained simply by moving the 8 from the beginning to the end of the number!
Could you find a number starting with 7 that can be divided by 7 in such a simple way? | 7101449275362318840579 | 76 | 22 |
math | ## Task Condition
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$$
\left\{\begin{array}{l}
x=\ln t \\
y=\operatorname{arctg} t
\end{array}\right.
$$ | \frac{\cdot(1-^2)}{(1+^2)^2} | 65 | 18 |
math | Find the smallest possible $\alpha\in \mathbb{R}$ such that if $P(x)=ax^2+bx+c$ satisfies $|P(x)|\leq1 $ for $x\in [0,1]$ , then we also have $|P'(0)|\leq \alpha$. | \alpha = 8 | 66 | 6 |
math | In triangle $ABC$ with $AB < AC$, let $H$ be the orthocenter and $O$ be the circumcenter. Given that the midpoint of $OH$ lies on $BC$, $BC = 1$, and the perimeter of $ABC$ is 6, find the area of $ABC$. | \frac{6}{7} | 66 | 7 |
math | 6. The general term formula of the sequence $\left\{a_{n}\right\}$ is $a_{n}=\tan n \cdot \tan (n-1)$, and for any positive integer $n$, there is $a_{1}+a_{2}+\cdots+a_{n}=A \tan n+B n$ holds, then $A=$ $\qquad$ ,$B=$ $\qquad$ | A=\frac{1}{\tan1},B=-1 | 90 | 13 |
math | Example 3-18 For the problem
$$
\begin{array}{c}
x_{1}+x_{2}+x_{3}=15 \\
0 \leqslant x_{1} \leqslant 5, \quad 0 \leqslant x_{2} \leqslant 6, \quad 0 \leqslant x_{3} \leqslant 7
\end{array}
$$
Find the number of integer solutions. | 10 | 110 | 2 |
math | 14. Given a four-person challenge group composed of two male students, Jia and Yi, and two female students, Bing and Ding, participating in a knowledge-based question challenge activity organized by a TV station. The activity consists of four rounds. The probabilities of male students passing the first to fourth rounds... | \frac{16}{15} | 186 | 9 |
math | 13. (15 points) In $\triangle A B C$, $a$, $b$, and $c$ are the sides opposite to $\angle A$, $\angle B$, and $\angle C$ respectively, with $b=1$, and
$\cos C+(2 a+c) \cos B=0$.
Find (1) $\angle B$;
(2) the maximum value of $S_{\triangle A B C}$. | \frac{\sqrt{3}}{12} | 94 | 11 |
math | (7) Given $A=\left\{x \mid x^{2}-1=0\right\}, B=\left\{y \mid y^{2}-2 a y+b=0, y \in \mathbf{R}\right\}$, if $B \subseteq$ $A$, and $B \neq \varnothing$, find the values of real numbers $a$ and $b$. | {\begin{pmatrix}=1,\\b=1\end{pmatrix}. | 88 | 18 |
math | 8. (3 points) In the multiplication equation GrassGreen・FlowerRed $=$ SpringBright, the Chinese characters represent non-zero digits, and different characters represent different digits. Therefore, the smallest four-digit number represented by SpringBright is $\qquad$
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 4396 | 74 | 4 |
math | 11. Given $m \in\{11,13,15,17,19\}, n \in\{2000,2001, \cdots, 2019\}$, the probability that the unit digit of $m^{n}$ is 1 is | \frac{2}{5} | 69 | 7 |
math | 2.13. Find the mass of the cylindrical surface $x^{2}+$ $+y^{2}=R^{2}$, bounded by the planes $z=0, z=H$, if at each point the surface density is inversely proportional to the square of the distance from it to the origin. | 2k\pi\operatorname{arctg}\frac{H}{R} | 65 | 18 |
math | 4. In $\triangle A B C$, the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively, $\angle A B C=120^{\circ}$, the angle bisector of $\angle A B C$ intersects $A C$ at point $D$, and $B D=1$. Then the minimum value of $4 a+c$ is $\qquad$ | 9 | 92 | 1 |
math | Problem 7. There is a certain number of identical plastic bags that can be placed inside each other. If all the other bags end up inside one of the bags, we will call this situation a "bag of bags." Calculate the number of ways to form a "bag of bags" from 10 bags.
Explanation. Denote the bag with parentheses.
If we ... | 719 | 160 | 3 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$
f\left(x^{2}-y^{2}\right)=(x-y)(f(x)+f(y))
$$
## Solution of exercises | f(x)=cx | 64 | 4 |
math | 1. On a plane, an overlapping square and a circle are drawn. Together they occupy an area of 2018 cm². The area of intersection is 137 cm². The area of the circle is 1371 cm². What is the perimeter of the square? | 112 | 63 | 3 |
math | 1. Determine all real numbers $c$ for which the equation
$$
\left(c^{2}+c-8\right)(x+2)-8|x-c+2|=c|x+c+14|
$$
has infinitely many solutions in the set of integers. | =0or=-4 | 58 | 5 |
math | 2. Given $\theta=\arctan \frac{5}{12}$, then the principal value of the argument of the complex number $z=\frac{\cos 2 \theta+i \sin 2 \theta}{239+i}$ is $\qquad$ | \frac{\pi}{4} | 57 | 7 |
math | 3. If $z$ is a complex number with a non-zero real part, then the minimum value of $\frac{\operatorname{Re}\left(z^{4}\right)}{(\operatorname{Re}(z))^{4}}$ is $\qquad$ | -8 | 55 | 2 |
math | 10.5. Currently, there are coins of 1, 2, 5, and 10 rubles. Indicate all monetary amounts that can be paid with both an even and an odd number of coins. (You can use identical coins.) | Anyamountofmoneygreaterthan1rublecanbepaidwitheitheranevenoroddof | 54 | 21 |
math | A soccer coach named $C$ does a header drill with two players $A$ and $B$, but they all forgot to put sunscreen on their foreheads. They solve this issue by dunking the ball into a vat of sunscreen before starting the drill. Coach $C$ heads the ball to $A$, who heads the ball back to $C$, who then heads the ball to $... | \frac{10}{19} | 163 | 9 |
math | 2. Find the fraction of the form $\frac{n}{23}$ that is closest to the fraction $\frac{37}{57}$ ( $n$ - integer). | \frac{15}{23} | 37 | 9 |
math | Determine all triples $(a, b, c)$ of positive integers for which
$$
a !+b !=2^{c !}
$$
holds.
(Walther Janous)
Answer. The only solutions are $(1,1,1)$ and $(2,2,2)$. | (1,1,1)(2,2,2) | 61 | 13 |
math | 1. In the plane 2014 lines are arranged in three groups of parallel lines. What is the largest number of triangles formed by the lines (each side of the triangle lies on one of the lines). | 672\cdot671^{2} | 44 | 11 |
math | 1. Given real numbers $a, b, c$ satisfy
$$
\begin{array}{l}
a+b+c=1, \\
\frac{1}{a+b-c}+\frac{1}{b+c-a}+\frac{1}{c+a-b}=1 .
\end{array}
$$
Then $a b c=$ . $\qquad$ | 0 | 77 | 1 |
math | 13.397 Two trains depart simultaneously from A and B towards each other and meet at a distance of $p$ km from B. After $t$ hours from the meeting, the second train, having passed point A, was $q$ km away from it, while the first train, having passed point B, was at a distance from the second train that was twice the di... | 3p-q | 99 | 3 |
math | 10. It is known that $\frac{7}{13}+\sin \phi=\cos \phi$ for some real $\phi$. What is $\sin 2 \phi$ ? Answer: $\frac{120}{169}$. | \frac{120}{169} | 54 | 11 |
math | Solve the following equations and verify the correctness of the obtained roots:
а) $\frac{3}{1-6 x}=\frac{2}{6 x+1}-\frac{8+9 x}{36 x^{2}-1}$,
b) $\frac{3}{1-z^{2}}=\frac{2}{(1+z)^{2}}-\frac{5}{(1-z)^{2}}$. | \frac{1}{3},-\frac{3}{7} | 90 | 14 |
math | 9. Solve the equation $\sin x-\cos x-3 \sin 2 x+1=0$.
Solution: Let $t=\sin x-\cos x$. Since $t^{2}=(\sin x-\cos x)^{2}=$ $1-\sin 2 x$, we get the equation $t-3\left(1-t^{2}\right)+1=0$, or $3 t^{2}+t-2=0$. The roots of this equation are $t=-1 ; \frac{2}{3}$. Substituting the found values into the formula $\sin 2 x=1-... | \frac{\pik}{2};\frac{(-1)^{k}}{2}\arcsin\frac{5}{9}+\frac{\pik}{2},k\in\mathbb{Z} | 214 | 47 |
math | 1. Let $\log _{2} x=m \in Z, m>0, \log _{6} y=n \in Z, n>0$.
Then $x=2^{m}, y=6^{n}$. As a result, we have
$$
\text { GCD }(x, y)=\text { GCD }\left(2^{m}, 6^{n}\right)=\text { GCD }\left(2^{m}, 2^{n} \cdot 3^{n}\right)=8=2^{3} .
$$
Case 1. $m \geq n$. Then $n=3, \quad y=6^{3}=216$,
GCD $\left(\log _{2} x, 3\right)... | 8^{k},k=1,2,\ldots;216\text | 225 | 18 |
math | Example 2 In the sequence $\left\{a_{n}\right\}$,
$$
a_{1}=2, a_{n+1}=4 a_{n}-3 n+1\left(n \in \mathbf{N}_{+}\right) \text {. }
$$
Find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=4^{n-1}+n | 82 | 12 |
math | Task B-3.5. Two planes touch the sphere at points $A$ and $B$. If the radius of the sphere is 20 $\mathrm{cm}$ and $|A B|=10 \mathrm{~cm}$, determine the sine of the angle between these planes. | \frac{\sqrt{15}}{8} | 61 | 11 |
math | 6. In a three-digit number, the digit in the hundreds place is 2 more than the digit in the units place. Find the difference between this number and the number formed by the same digits but in reverse order. | 198 | 45 | 3 |
math | Example 7 Let the roots of the odd-degree real-coefficient equation $f(x)=a_{0} x^{n}+a_{1} x^{n-1}+\cdots+a_{n-1} x+a_{n}=0$ all lie on the unit circle, and $-a_{n}=a_{0} \neq 0$, find $a_{0}+a_{1}+\cdots+a_{n}$. | 0 | 96 | 1 |
math | 11. In a school of 100 students, there are 63 basketball enthusiasts and 75 football enthusiasts, but not every student necessarily has both of these interests, so the minimum and maximum values for the number of students who love both basketball and football are $\qquad$. | 38,63 | 60 | 5 |
math | Let $A_{1}, \ldots, A_{2022}$ be the vertices of a regular 2022-gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the plane i... | 22 | 191 | 2 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; y ; 0)$
$B(0 ; 5 ;-9)$
$C(-1 ; 0 ; 5)$ | A(0;8;0) | 60 | 8 |
math | 39th Putnam 1978 Problem B5 Find the real polynomial p(x) of degree 4 with largest possible coefficient of x 4 such that p( [-1, 1] ) ∈ [0, 1]. | 4x^4-4x^2+1 | 50 | 11 |
math | How many natural divisors does 121 have? How many natural divisors does 1000 have? How many natural divisors does 1000000000 have? | 3,16,100 | 44 | 8 |
math | 5. From the set of numbers $\{1,2,3, \ldots, 200\}$, one number is randomly selected. Calculate the probability that the following random event will occur $A=\{$ A number that is not divisible by 6 is selected \}. | \frac{167}{200} | 59 | 11 |
math | 29. In 5 lottery tickets, there is 1 winning ticket. 5 people draw 1 ticket each in a predetermined order to decide who gets the winning ticket. Would the probability of drawing the winning ticket be the same for the first drawer and the later drawers (later drawers do not know the results of the earlier drawers)? | \frac{1}{5} | 68 | 7 |
math | Example 4 Let $z=\frac{\frac{\sin t}{\sqrt{2}}+\mathrm{i} \cos t}{\sin t-\mathrm{i} \frac{\cos t}{\sqrt{2}}}$, find the range of $|z|$. | \frac{1}{\sqrt{2}}\leqslant|z|\leqslant\sqrt{2} | 56 | 27 |
math | 7. At the initial moment, a positive integer $N$ is written on the blackboard. In each step, Misha can choose a positive integer $a>1$ that is already written on the blackboard, erase it, and write down all its positive divisors except itself. It is known that after several steps, there are exactly $N^{2}$ numbers on t... | 1 | 89 | 1 |
math | 17. Mingming's mother found an interesting phenomenon while shopping. Every time she paid, the amount of money in her wallet was exactly 5 times the amount she paid. After settling the bill twice, she still had 320 yuan left in her wallet. How much money did she have in her wallet before shopping? | 500 | 67 | 3 |
math | Problem 5.6. A three-digit number and two two-digit numbers are written on the board. The sum of the numbers that contain a seven in their notation is 208. The sum of the numbers that contain a three in their notation is 76. Find the sum of all three numbers. | 247 | 64 | 3 |
math | 6. The parabola $y=a x^{2}+b x+c$ has vertex at $\left(\frac{1}{4},-\frac{9}{8}\right)$. If $a>0$ and $a+b+c$ is an integer, find the minimum possible value of $a$.
The parabola $y=a x^{2}+b x+c$ has its vertex at $\left(\frac{1}{4},-\frac{9}{8}\right)$. If $a>0$ and $a+b+c$ is an integer, find the minimum possible va... | \frac{2}{9} | 130 | 7 |
math | Example 1. If $p, q$ are both natural numbers, and the two roots of the equation $p x^{2}-$ $q x+1985=0$ are both prime numbers, then what is the value of $12 p^{2}+q$? (85 Beijing Mathematics Competition Question) | 414 | 70 | 3 |
math | 58 The top and bottom faces of a regular quadrilateral frustum have side lengths of positive integers $a$ and $b$ ($a<b$), and the height is 3. The lateral area is equal to the sum of the areas of the top and bottom faces. Then the volume of this regular quadrilateral frustum is | 208 | 68 | 3 |
math | 4. Let $f(x)=a x^{2}+b x+c$ be a quadratic trinomial with integer coefficients. If integers $m, n$ satisfy $f(m)-f(n)=1$. Then $|m-n|=$ $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 1 | 80 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.