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 | 51. A task, if worked on by person A and B together, can be completed in 8 days; if worked on by person B and C together, it can be completed in 6 days; if worked on by person C and D together, it can be completed in 12 days; then, if worked on by person A and D together, it will take $\qquad$ days to complete. | 24 | 86 | 2 |
math | 9. The numbers $1,2, \ldots, 2016$ were divided into pairs, such that the product of the numbers in each pair does not exceed some natural number $N$. What is the smallest $N$ for which this is possible? | 1017072=1008\times1009 | 56 | 18 |
math | 17. Students were given the task to write several different three-digit numbers that do not contain the digit 7 in their notation. How many such numbers can be written in total? | 648 | 37 | 3 |
math | 1. Given complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=1,\left|z_{2}\right|=2,3 z_{1}-$ $z_{2}=2+\sqrt{3} \mathrm{i}$. Then $2 z_{1}+z_{2}=$ $\qquad$ . | 3-\sqrt{3} \mathrm{i} \text{ or } -\frac{9}{7}+\frac{13 \sqrt{3}}{7} \mathrm{i} | 76 | 40 |
math | Let's calculate the value of the expression $P$, where the number of nested radicals is $n$ (>1).
$$
\begin{aligned}
& P=\sqrt{a+\sqrt{a+\ldots+\sqrt{a+\sqrt{b}}}}, \text{ where } \\
& a=\sqrt{76+24 \sqrt{10}}-2 \sqrt{19-3 \sqrt{40}} \text{ and } \\
& b=\sqrt{165+4 \sqrt{1616}}-\sqrt{165-4 \sqrt{1616}}
\end{aligned}
$... | 4 | 147 | 1 |
math | Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational. | W(x) = ax + b | 36 | 8 |
math | Compute the smallest positive integer $M$ such that there exists a positive integer $n$ such that
[list] [*] $M$ is the sum of the squares of some $n$ consecutive positive integers, and
[*] $2M$ is the sum of the squares of some $2n$ consecutive positive integers.
[/list]
[i]Proposed by Jaedon Whyte[/i] | 4250 | 83 | 6 |
math |
1. The four real solutions of the equation
$$
a x^{4}+b x^{2}+a=1
$$
form an increasing arithmetic progression. One of the solutions is also a solution of
$$
b x^{2}+a x+a=1 .
$$
Find all possible values of real parameters $a$ and $b$.
(Peter Novotný)
| (-\frac{1}{8},\frac{5}{4}),(\frac{9}{8},-\frac{5}{4}) | 85 | 29 |
math | 1. Let real numbers $a, b, c, d, e \geqslant -1$, and $a+b+c+d+e=5$. Find
$$
S=(a+b)(b+c)(c+d)(d+e)(e+a)
$$
the minimum and maximum values.
(Xiong Bin, problem contributor) | 288 | 72 | 3 |
math | 6.39. Find the first three differentials of the function $y=$ $=x^{2}-2 x+3$. | (2x-2),\quad^{2}2()^{2},\quad^{3}0 | 28 | 22 |
math | On a sphere with a radius of 2, there are three circles with a radius of 1, each touching the other two. Find the radius of a smaller circle, also located on the given sphere, that touches each of the given circles.
# | 1-\sqrt{\frac{2}{3}} | 51 | 10 |
math | What is the greatest possible number of rays in space emanating from a single point and forming obtuse angles with each other
# | 4 | 26 | 1 |
math | 16. Given in the complex plane $\triangle A B C$ has vertices $A, B, C$ corresponding to the complex numbers $3+2 \mathrm{i}, 3 \mathrm{i}, 2-\mathrm{i}$, and the moving point $P$ corresponds to the complex number $z$. If the equation $|\bar{z}|^{2}+\alpha z+\bar{\alpha} \bar{z}+\beta=0$ represents the circumcircle of ... | \alpha=-1+\mathrm{i},\beta=-3 | 115 | 12 |
math | ## Aufgabe 1 - 290831
Eine Aufgabe des bedeutenden englischen Naturwissenschaftlers Isaak Newton (1643 bis 1727) lautet:
Ein Kaufmann besaß eine gewisse Geldsumme.
Im ersten Jahr verbrauchte er davon 100 Pfund; zum Rest gewann er durch seine Arbeit ein Drittel desselben dazu.
Im zweiten Jahr verbrauchte er wiederum... | 1480 | 223 | 4 |
math | 2. Given $\odot C:(x-1)^{2}+(y-2)^{2}=25$ and the line $l:(2 a+1) x+(a+1) y=4 a+4 b(a, b \in \mathbf{R})$. If for any real number $a$, the line $l$ always intersects with $\odot C$, then the range of values for $b$ is $\qquad$. | \left[\frac{3-\sqrt{5}}{4}, \frac{3+\sqrt{5}}{4}\right] | 97 | 28 |
math | 5. Given the function $f(x)=\ln (2+3 x)-\frac{3}{2} x^{2}$, if for any $x \in\left[\frac{1}{6}, \frac{1}{3}\right]$, the inequality $|a-\ln x|+$ $\ln \left[f^{\prime}(x)+3 x\right]>0$ always holds, then the range of the real number $a$ is $\qquad$. | {\lvert\,\neq\ln\frac{1}{3}.,\in{R}} | 101 | 22 |
math | Call a positive integer $n$ $k$-pretty if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$-pretty. Let $S$ be the sum of positive integers less than $2019$ that are $20$-pretty. Find $\tfrac{S}{20}$. | 472 | 86 | 3 |
math | ## Task 5 - 100935
A triangular pyramid with vertices $A, B, C, D$ and apex $D$ has edge lengths $A B=4$ $\mathrm{cm}, A C=3 \mathrm{~cm}, B C=5 \mathrm{~cm}, B D=12 \mathrm{~cm}, C D=13 \mathrm{~cm}$, and $\angle A B D$ is a right angle.
Calculate the volume $V$ of this pyramid. | 24\mathrm{~}^{3} | 111 | 10 |
math | 16(!). Determine \(p\) so that the sum of the absolute values of the roots of the equation \(z^{2}+p z-6=0\) is equal to 5. | 1or-1 | 41 | 4 |
math | 5. Find all pairs of positive integers $(m, n)$ such that
$$
m+n-\frac{3 m n}{m+n}=\frac{2011}{3}
$$ | (,n)=(1144,377)or(377,1144) | 41 | 24 |
math | 5. Determine the largest positive integer $n$ with $n<500$ for which $6048\left(28^{n}\right)$ is a perfect cube (that is, it is equal to $m^{3}$ for some positive integer $m$ ). | 497 | 60 | 3 |
math | 4. In a joint-stock company, there are 2017 shareholders, and any 1500 of them hold a controlling stake (not less than $50 \%$ of the shares). What is the largest share of shares that one shareholder can have
# | 32.8 | 57 | 4 |
math | 98. Two cubic equations. Let $a, b, c$ be the roots of the equation $x^{3} + q x + r = 0$. Write the equation whose roots are the numbers
$$
\frac{b+c}{a^{2}}, \quad \frac{c+a}{b^{2}}, \quad \frac{a+b}{c^{2}}
$$ | rx^{3}-^{2}-1=0 | 81 | 10 |
math | \section*{Task 1 - 011041}
As announced at the XXII Congress of the CPSU, the production of means of production (i.e., raw materials, machines, equipment for industry, agriculture, and transport, etc.) in the Soviet Union will increase to 6.8 times from 1960 to 1980.
However, the production of consumer goods (goods i... | 66.7,73.1 | 161 | 9 |
math | 【Question 7】
A car has a fuel tank capacity of 50 liters, and it departs from Shanghai to Harbin, which is 2560 kilometers away, with a full tank of fuel. It is known that the car consumes 8 liters of fuel for every 100 kilometers driven, and to ensure driving safety, at least 6 liters of fuel should be kept in the tan... | 4 | 110 | 1 |
math | 11. (20 points) Let $x, y, z \in(0,1)$, and $x^{2}+y^{2}+z^{2}=1$.
Determine the maximum value of $f=x+y+z-x y z$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{8 \sqrt{3}}{9} | 83 | 12 |
math | a) 2 white and 2 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 4, to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, to the other white cat is 9. What cats are sitting on the edges?
b) 2 white and 3 black cats are sitting on t... | W_1 | 177 | 4 |
math | Draw a circle $k$ around the center of a regular hexagon, with a radius no larger than that of the circle tangent to the sides of the hexagon, then draw a circle around each vertex of the hexagon that touches $k$ and has a diameter no larger than the side of the hexagon. The 7 circular segments together cover a part of... | \frac{2a}{3} | 109 | 8 |
math | 10. (6 points) A few hundred years ago, Columbus discovered the New World in America, and the four digits of that year are all different, their sum equals 16. If the tens digit is increased by 1, then the tens digit is exactly 5 times the units digit, so Columbus discovered the New World in America in the year $\qquad$... | 1492 | 100 | 4 |
math | 1.44 Let $x \in N$. If a sequence of natural numbers $1=x_{0}, x_{1}, \cdots, x_{1}=x$ satisfies
$$
x_{i-1}<x_{i}, x_{i-1} \mid x_{i}, i=1,2, \cdots, l \text {, }
$$
then $\left\{x_{0}, x_{1}, x_{2}, \cdots, x_{l}\right\}$ is called a factor chain of $x$, and $l$ is the length of this factor chain. $L(x)$ and $R(x)$ r... | L(x)=k++3n,R(x)=\frac{(k++3n)!}{(k+n)!!(n!)^2} | 212 | 30 |
math | 1. Solve the equation $\left(\cos ^{2} x+\frac{1}{\cos ^{2} x}\right)^{3}+\left(\cos ^{2} y+\frac{1}{\cos ^{2} y}\right)^{3}=16 \sin z \cdot(5$ points $)$ | \pin,\pik,\frac{\pi}{2}+2\pi, | 72 | 17 |
math | 11. (20 points) Let $\triangle A B C$ be an inscribed triangle in the ellipse $\Gamma: \frac{x^{2}}{4}+y^{2}=1$, where $A$ is the intersection of the ellipse $\Gamma$ with the positive $x$-axis, and the product of the slopes of lines $A B$ and $A C$ is $-\frac{1}{4}$. Let $G$ be the centroid of $\triangle A B C$. Find ... | [\frac{2\sqrt{13}+4}{3},\frac{16}{3}) | 125 | 23 |
math | [ Degree of a vertex ] $[\underline{\text { Pigeonhole Principle (restated). }}]$
For what $n>1$ can it happen that in a company of $n+1$ girls and $n$ boys, all girls are acquainted with a different number of boys, while all boys are acquainted with the same number of girls? | 2m+1foranyoddn>1 | 74 | 10 |
math | 2ag * Find all natural numbers $x$, such that $x^{2}$ is a twelve-digit number of the form: $2525 * * * * * * 89$ (the six “*”s represent six unknown digits, which are not necessarily the same). | 502517,502533,502567,502583 | 60 | 27 |
math | Three, (20 points) Place a real number in each cell of a $4 \times 4$ grid paper,
such that the sum of the four numbers in each row, each column, and each diagonal equals a constant $k$. Find the sum of the numbers in the four corners of this $4 \times 4$ grid paper. | k | 73 | 1 |
math | 6. Find the largest ten-digit number of the form $\overline{a_{9} a_{8} a_{7} a_{6} a_{5} a_{4} a_{3} a_{2} a_{1} a_{0}}$, possessing the following property: the digit equal to $\mathrm{a}_{\mathrm{i}}$ appears in its representation exactly $\mathrm{a}_{9-\mathrm{i}}$ times (for example, the digit equal to $\mathrm{a}_... | 8888228888 | 132 | 10 |
math | Let $k$ be a strictly positive integer. Find all strictly positive integers $n$ such that $3^{k}$ divides $2^{n}-1$
## - Solutions of the course exercises - | 2\times3^{k-1}dividesn | 41 | 12 |
math | 14. 5. 11 $\star \star$ Find the positive integer $n$ that satisfies $133^{5}+110^{5}+84^{5}+27^{5}=n^{5}$. | 144 | 54 | 3 |
math | 1. Call a positive integer a hussel number if:
(1) All digits are not equal to 0.
(2) The number is divisible by 11.
(3) The number is divisible by 12. If you rearrange the digits in any other random order, you always get a number that is divisible by 12.
How many 5-digit hussel numbers are there? | 2 | 86 | 1 |
math | Example 12 (2002 Hunan Provincial Competition Question) Let the two roots of the quadratic equation $2 x^{2}-t x-2=0$ with respect to $x$ be $\alpha, \beta(\alpha<\beta$.
(1) If $x_{1} 、 x_{2}$ are two different points in the interval $[\alpha, \beta]$, prove: $4 x^{2} x_{2}-t\left(x_{1}+x_{2}\right)-4<0$;
(2) Let $f(x... | 4 | 205 | 1 |
math | 1.4.4 * Given the function $f(x)=\log \frac{1}{2}\left(\frac{\sqrt{a^{2}+1}-a}{a}\right)^{x}$ is a decreasing function on $\mathbf{R}$. Find the range of real number $a$. | 0<<\frac{\sqrt{3}}{3} | 65 | 12 |
math | 3. Find all natural solutions to the inequality
$$
\frac{4}{5}+\frac{4}{45}+\frac{4}{117}+\cdots+\frac{4}{16 n^{2}-8 n-3}>n-5
$$
In your answer, write the sum of all found solutions. | 15 | 73 | 2 |
math | ## Task A-2.2.
Determine all triples of prime numbers whose sum of squares decreased by 1 is equal to the square of some natural number. | (3,2,2),(2,3,2),(2,2,3) | 33 | 19 |
math | 2 Answer the following two questions and justify your answers:
(1) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+5^{2012}$ ?
(2) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+\cdots+2011^{2012}+2012^{2012}$ ? | 0 | 132 | 1 |
math | The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 231361 | 60 | 6 |
math | Example 2. Find the maximum and minimum values of the function $y=\frac{x^{2}+x-1}{x^{2}+x+1}$. | y_{\mathrm{min}}=-\frac{5}{3} | 36 | 15 |
math | Find all integers $n$ such that $n(n+1)$ is a perfect square. | n=0orn=-1 | 19 | 6 |
math | If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that? | 53 | 53 | 2 |
math | For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$, and $g(2020) = \text{the digit sum of }12_{\text{8}} = 3$. Let... | 151 | 176 | 3 |
math | One, (20 points) Given that $a$, $b$, and $c$ satisfy the system of equations
$$
\left\{\begin{array}{l}
a+b=8, \\
a b-c^{2}+8 \sqrt{2} c=48 .
\end{array}\right.
$$
Try to find the roots of the equation $b x^{2}+c x-a=0$. | x_{1}=\frac{-\sqrt{2}+\sqrt{6}}{2}, x_{2}=-\frac{\sqrt{2}+\sqrt{6}}{2} | 91 | 40 |
math | 4. (8 points) Today is January 30th, we start by writing down 130; the rule for writing the next number is: if the number just written is even, divide it by 2 and add 2 to write it down, if the number just written is odd, multiply it by 2 and subtract 2 to write it down. Thus, we get: $130, 67, 132, 68 \cdots$, so the ... | 6 | 120 | 1 |
math | Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find... | 507 | 107 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1}\left(\frac{2 x-1}{x}\right)^{1 /(\sqrt[5]{x}-1)}
$$ | e^5 | 49 | 3 |
math | 3. We will call a pair of numbers magical if the numbers in the pair add up to a multiple of 7. What is the maximum number of magical pairs of adjacent numbers that can be obtained by writing down all the numbers from 1 to 30 in a row in some order? | 26 | 60 | 2 |
math | 1. Given two groups of numerical sequences, each consisting of 15 arithmetic progressions containing 10 terms each. The first terms of the progressions in the first group are $1, 2, 3, \ldots, 15$, and their differences are respectively $2, 4, 6, \ldots, 30$. The second group of progressions has the same first terms $1... | \frac{160}{151} | 155 | 11 |
math | 1. Let $a, b, c, d$ be positive numbers such that $\frac{1}{a^{3}}=\frac{512}{b^{3}}=\frac{125}{c^{3}}=\frac{d}{(a+b+c)^{3}}$. Find $d$.
(1 mark)設 $a 、 b 、 c 、 d$ 為正數, 使得 $\frac{1}{a^{3}}=\frac{512}{b^{3}}=\frac{125}{c^{3}}=\frac{d}{(a+b+c)^{3}}$ 。求 $d$ 。 | 2744 | 143 | 4 |
math | ## Task $5 / 85$
Given a circle with center $M$, radius $r$, and a chord $A B=s$. By how much must $A B$ be extended beyond $B$ so that the tangent from the endpoint $E$ of the extension to the circle has length $t$? | 0.5(\sqrt{^{2}+4^{2}}-) | 65 | 15 |
math | 8. In jar A, there are 6 balls, of which 4 are red and 2 are white; in jar B, there are 4 balls, of which 3 are red and 1 is white; in jar C, there are 5 balls, of which 2 are red and 3 are white. One ball is drawn from each of the jars A, B, and C. Find:
(1) the probability that exactly 2 of the balls are white;
(2) t... | \frac{1}{3} | 117 | 7 |
math | Example 1 Let $f(x)$ be a monotonic function defined on the set of real numbers $\mathbf{R}$. Solve the functional equation $f(x) \cdot f(y)=f(x+y)$. | f(x)=^{x}(>0,\neq1) | 45 | 13 |
math | 10. Given $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \in \mathbf{N}$ and satisfy $a_{1}+a_{2}+a_{3}=b_{1}+b_{2}+b_{3}$
$$
\begin{array}{c}
a_{1} a_{2} a_{3}=b_{1} b_{2} b_{3} \\
a_{1} a_{2}+a_{1} a_{3}+a_{2} a_{3} \neq b_{1} b_{2}+b_{1} b_{3}+b_{2} b_{3}
\end{array}
$$
then the minimum value of the maximum number among... | 8 | 204 | 1 |
math | Example 9 Given $x^{2}-y^{2}=16$. Find the minimum and maximum values of the function
$$
f(x, y)=\frac{1}{x^{2}}+\frac{y}{8 x}+1
$$ | \frac{9}{8} \text{ and } \frac{7}{8} | 54 | 19 |
math | $6 \cdot 140$ Find all functions $f: Q \rightarrow Q$ (where $Q$ is the set of rational numbers) satisfying $f(1)=2$ and
$f(x y) \equiv f(x) f(y)-f(x+y)+1, x, y \in Q$. | f(x)=x+1 | 67 | 6 |
math | 123. In reverse order. What nine-digit number, when multiplied by 123456 789, gives a product where the nine least significant digits are 9,8,7,6,5,4,3, 2,1 (in that exact order)? | 989010989 | 63 | 9 |
math | 65. In triangle $ABC$ with angle $\widehat{ABC}=60^{\circ}$, the bisector of angle $A$ intersects $BC$ at point $M$. On side $AC$, a point $K$ is taken such that $\widehat{AM} K=30^{\circ}$. Find $\widehat{OKC}$, where $O$ is the center of the circumcircle of triangle $AMC$. | 30 | 95 | 2 |
math | 7. Given that the equations of the asymptotes of a hyperbola are $y= \pm \frac{2}{3} x$, and it passes through the point $(3,4)$, the equation of this hyperbola is $\qquad$ . | \frac{y^{2}}{12}-\frac{x^{2}}{27}=1 | 55 | 22 |
math | Let's find a three-digit number where the sum of the digits is equal to the difference between the number formed by the first two digits and the number formed by the last two digits. | 209,428,647,866,214,433,652,871 | 37 | 31 |
math | 11. A triangle $\triangle A B C$ is inscribed in a circle of radius 1 , with $\angle B A C=60^{\circ}$. Altitudes $A D$ and $B E$ of $\triangle A B C$ intersect at $H$. Find the smallest possible value of the length of the segment $A H$. | 1 | 74 | 1 |
math | ## Task 1 - 010831
In a circle, the plan for the production of bricks (Plan: 1,350,000 pieces) was fulfilled by 100.1 percent in one quarter. An inspection of the enterprises showed that two enterprises, which were supposed to produce 150,000 and 290,000 bricks according to the plan, only fulfilled the plan by 80.0 pe... | 105.3 | 156 | 5 |
math | (7) Let $\left(1+x-x^{2}\right)^{10}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{20} x^{20}$, then $a_{0}+$ $a_{1}+2 a_{2}+3 a_{3}+\cdots+20 a_{20}=$ $\qquad$ | -9 | 91 | 2 |
math | Ryan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and he chooses whether to weight heads or tails more with equal probability. Brice flips his modified coin twice and it lands up heads both times. The probability that the coin lands up heads on th... | 8 | 110 | 3 |
math | 2. Find all right-angled triangles with integer side lengths, in which the hypotenuse is one unit longer than one of the legs. | all\triangles\with\legs\2k+1\\2k(k+1)\\hypotenuse\2k^2+2k+1,\where\k\is\any\natural\ | 29 | 45 |
math | 2. You have a list of real numbers, whose sum is 40 . If you replace every number $x$ on the list by $1-x$, the sum of the new numbers will be 20 . If instead you had replaced every number $x$ by $1+x$, what would the sum then be? | 100 | 67 | 3 |
math | ## Subject IV (7 points)
Consider the set of numbers formed only from the digits 1 and 2, with at most 2016 digits.
a) Calculate how many numbers have 5 digits.
b) Calculate how many numbers are in the set.
c) Show that there exists a number in the set that is divisible by \(3^6\).
Note:
All subjects are mandator... | 32 | 100 | 2 |
math | I2.4 If $d=1-2+3-4+\ldots-c$, find the value of $d$. | -50 | 27 | 3 |
math | Determine all real polynomials $P$ such that $P\left(X^{2}\right)=\left(X^{2}+1\right) P(X)$ | (X^{2}-1) | 35 | 6 |
math | 37. Let $a_{1}, a_{2}, \ldots, a_{2005}$ be real numbers such that
$$
\begin{array}{l}
a_{1} \cdot 1+a_{2} \cdot 2+a_{3} \cdot 3+\cdots+a_{2005} \cdot 2005=0 \\
a_{1} \cdot 1^{2}+a_{2} \cdot 2^{2}+a_{3} \cdot 3^{2}+\cdots+a_{2005} \cdot 2005^{2}=0 \\
a_{1} \cdot 1^{3}+a_{2} \cdot 2^{3}+a_{3} \cdot 3^{3}+\cdots+a_{2005}... | \frac{1}{2004!} | 359 | 11 |
math | 2. Determine all natural numbers $n$ such that each of the digits $0,1,2, \ldots$, 9, appears exactly once in the representations of the numbers $n^{3}$ or $n^{4}$, but not in both numbers simultaneously.
Example. If only the digits $1,2,7$ and 8 need to be used, then the number 3 is a solution. Each of the digits $1,... | 18 | 130 | 2 |
math | 6. Given the sequence $\left\{a_{n}\right\}$ satisfies the recurrence relation
$$
a_{n+1}=2 a_{n}+2^{n}-1\left(n \in \mathbf{N}_{+}\right) \text {, }
$$
and $\left\{\frac{a_{n}+\lambda}{2^{n}}\right\}$ is an arithmetic sequence. Then the value of $\lambda$ is $\qquad$ | -1 | 101 | 2 |
math | 8. The number 197719771977 is represented as
$$
a_{0}+a_{1} \cdot 10^{1}+a_{2} \cdot 10^{2}+\ldots+a_{11} \cdot 10^{11}
$$
where $a_{0}, a_{1}, \ldots, a_{11}$ are non-negative integers, the sum of which does not exceed 72. Find these numbers. | 1,9,7,7,1,9,7,7,1,9,7,7 | 111 | 23 |
math | 5. The number of 4-digit numbers whose sum of digits equals 12 is. $\qquad$ | 342 | 23 | 3 |
math | 3. Given that for any $m \in\left[\frac{1}{2}, 3\right]$, we have $x^{2}+m x+4>2 m+4 x$.
Then the range of values for $x$ is | x>2 \text{ or } x<-1 | 55 | 11 |
math | 4. Given positive integers $a, b, c$ satisfying
$$
1<a<b<c, a+b+c=111, b^{2}=a c \text {. }
$$
then $b=$ . $\qquad$ | 36 | 50 | 2 |
math | Example 12. Find $\int\left(2^{3 x}-1\right)^{2} \cdot 4^{x} d x$. | (2^{8x-3}-\frac{1}{5}\cdot2^{5x+1}+2^{2x-1})\cdot\frac{1}{\ln2}+C | 33 | 44 |
math | 2. If the difference between the maximum and minimum elements of the real number set $\{1,2,3, x\}$ equals the sum of all elements in the set, then the value of $x$ is $\qquad$ . | -\frac{3}{2} | 50 | 7 |
math | In trapezoid $A B C D$, the bases $A D=12$ and $B C=8$ are given. On the extension of side $B C$, a point $M$ is chosen such that $C M=2.4$.
In what ratio does the line $A M$ divide the area of trapezoid $A B C D$? | 1:1 | 82 | 3 |
math | 886. Find the mass of the arc $A B$ of the curve $y=\ln x$, if at each point of it the linear density is proportional to the square of the abscissa of the point; $x_{A}=1, x_{B}=3$. | \frac{k}{3}(10\sqrt{10}-2\sqrt{2}) | 59 | 20 |
math | ## Task 2 - 260832
Determine all pairs $(p ; q)$ of prime numbers that satisfy the following conditions (1), (2), (3)!
(1) The difference $q-p$ is greater than 0 and less than 10.
(2) The sum $p+q$ is the square of a natural number $n$.
(3) If you add the number $n$ to the sum of $p$ and $q$, you get 42. | (17,19) | 110 | 7 |
math | ## Task Condition
Find the $n$-th order derivative.
$y=\sqrt{x}$ | y^{(n)}=(-1)^{n-1}\cdot\frac{\prod_{k=1}^{n-1}(2k-1)}{2^{(n+1)}}\cdotx^{-\frac{2n-1}{2}} | 20 | 55 |
math | 35.1. How many natural numbers $n$ exist such that
$$
100<\sqrt{n}<101 ?
$$
$$
\text { (8-9 grades) }
$$ | 200 | 45 | 3 |
math | # Problem No. 6 (10 points)
A pot was filled with $2 \pi$ liters of water, taken at a temperature of $t=0{ }^{\circ} C$, and brought to a boil in 10 minutes. After that, without removing the pot from the stove, ice at a temperature of $t=0{ }^{\circ} \mathrm{C}$ was added. The water began to boil again only after 15 m... | 1.68 | 205 | 4 |
math | In trapezoid $A B C D$, $A D / / B C, \frac{A D}{B C}=\frac{1}{2}$, point $M$ is on side $A B$, such that $\frac{A M}{M B}=\frac{3}{2}$, point $N$ is on side $C D$: such that line segment $M N$ divides the trapezoid into two parts with an area ratio of $3: 1$. Find $\frac{C N}{N D}$. | \frac{29}{3} | 116 | 8 |
math | Example 2 Find the range of the function $y=\sqrt{2 x-3}-\sqrt{x-2}$. | \left[\frac{\sqrt{2}}{2},+\infty\right) | 26 | 18 |
math | Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$ | (3, 223, 9) | 44 | 12 |
math | 4. In quadrilateral $A B C D, \angle D A C=98^{\circ}, \angle D B C=82^{\circ}, \angle B C D=70^{\circ}$, and $B C=A D$. Find $\angle A C D$. | 28 | 62 | 2 |
math | 5. (6 points) A canteen bought 500 kilograms of rice and 200 kilograms of flour. After eating for some time, it was found that the amount of rice and flour consumed was the same, and the remaining rice was exactly 7 times the remaining flour. How many kilograms of rice and flour were consumed each?
| 150 | 73 | 3 |
math | 8.084. $\cos 5 x+\cos 7 x=\cos (\pi+6 x)$.
8.084. $\cos 5 x+\cos 7 x=\cos (\pi+6 x)$. | x_{1}=\frac{\pi}{12}(2n+1),\quadx_{2}=\\frac{2}{3}\pi+2\pik,\quadn,k\inZ | 50 | 43 |
math | Example 4 (Euler's $\varphi$ function calculation formula)
Euler's $\varphi$ function value at $n$, $\varphi(n)$, is defined as the number of natural numbers that are coprime to $n$ in the set $\{1,2, \cdots, n\}$. Suppose $n$ has the standard factorization
$$n=p_{1}^{z_{1}} \cdots p_{s}^{\alpha_{s}},$$
where $p_{1}, ... | 16 | 545 | 2 |
math | 42nd Swedish 2002 Problem 5 The reals a, b satisfy a 3 - 3a 2 + 5a - 17 = 0, b 3 - 3b 2 + 5b + 11 = 0. Find a+b. | 2 | 65 | 1 |
math | 5. Let any real numbers $x_{0}>x_{1}>x_{2}>x_{3}>0$, to make $\log _{\frac{x_{0}}{x_{1}}} 1993+$ $\log _{\frac{x_{1}}{x_{2}}} 1993+\log _{\frac{x_{2}}{x_{3}}} 1993 \geqslant k \log _{\frac{x_{0}}{x_{3}}} 1993$ always hold, then the maximum value of $k$ is | 9 | 123 | 1 |
math | [Pairing and Grouping; Bijections] $[$ Evenness and Oddness $]$
Given a 29-digit number $X=\overline{a_{1}} \overline{-} \overline{-} \overline{29}\left(0 \leq a_{k} \leq 9, a_{1} \neq 0\right)$. It is known that for every $k$ the digit $a_{k}$ appears in the representation of this number $a_{30-k}$ times (for example... | 201 | 153 | 3 |
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