task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | Find positive integers $a$ and $b$, with $a>b$, such that:
$$
\frac{2}{7}=\frac{1}{a}+\frac{1}{b}
$$ | \frac{2}{7}=\frac{1}{28}+\frac{1}{4} | 42 | 22 |
math | Find all integers $n\ge 3$ for which the following statement is true:
Any arithmetic progression $a_1,\ldots ,a_n$ with $n$ terms for which $a_1+2a_2+\ldots+na_n$ is rational contains at least one rational term. | n \equiv 1 \pmod{3} | 64 | 12 |
math | Let $ \overline{AB}$ be a diameter of circle $ \omega$. Extend $ \overline{AB}$ through $ A$ to $ C$. Point $ T$ lies on $ \omega$ so that line $ CT$ is tangent to $ \omega$. Point $ P$ is the foot of the perpendicular from $ A$ to line $ CT$. Suppose $ AB \equal{} 18$, and let $ m$ denote the maximum possible length o... | 432 | 107 | 3 |
math | 4. Four elephants and eight zebras eat a ton of food daily. An elephant eats $214 \mathrm{~kg}$ more food daily than a zebra. If a zebra needs 24 minutes to eat $1 \mathrm{~kg}$ of food, how much time does it need to eat its daily amount of food? Express the obtained time in hours and minutes. | 4 | 81 | 1 |
math | 3. The boat traveled 165 km upstream against the current and then returned. On the return trip, it took 4 hours less than the trip there. Find the boat's own speed if the river's current speed is 2 km/h. | 13 | 52 | 2 |
math | # 4. CONDITION
Two two-digit numbers, written one after the other, form a four-digit number that is divisible by their product. Find these numbers. | 1734;1352 | 33 | 9 |
math | 12. Let $a, b, c$ be positive constants, and $x^{2}+y^{2}+z^{2}=1$, find
$$f(x, y, z)=\sqrt{a^{2} x^{2}+b^{2} y^{2}+c^{2} z^{2}}+\sqrt{a^{2} y^{2}+b^{2} z^{2}+c^{2} x^{2}}+\sqrt{a^{2} z^{2}+b^{2} x^{2}+c^{2} y^{2}}$$
the maximum and minimum values. (1999 China National Training Team Problem) | \sqrt{3\left(a^{2}+b^{2}+c^{2}\right)} \text{ and } a+b+c | 149 | 30 |
math | 11.7. Solve the equation $\sqrt{y \sqrt{5}}-\sqrt{x \sqrt{5}}=\sqrt{3 \sqrt{5}-5}$ in rational numbers. | (\frac{1}{2};\frac{5}{2}) | 40 | 14 |
math | 3. [5] Let $C$ be the circle of radius 12 centered at $(0,0)$. What is the length of the shortest path in the plane between $(8 \sqrt{3}, 0)$ and $(0,12 \sqrt{2})$ that does not pass through the interior of $C$ ? | 12+4\sqrt{3}+\pi | 71 | 11 |
math | 4. In the box, there are 3 white cups, 3 red cups, and 2 black cups. Sonya took out 5 cups at random. What is the probability that she took out 2 white, 2 red, and 1 black cup? (Round your answer to the nearest hundredth).
# | 0.32 | 67 | 4 |
math | 1.23 In triangle $ABC$, $AB=33$, $AC=21$, and $BC=m$, where $m$ is a positive integer. If there exists a point $D$ on $AB$ and a point $E$ on $AC$ such that $AD=DE=EC=n$, where $n$ is a positive integer, what value must $m$ take?
(Swedish Mathematical Competition, 1982) | 30 | 96 | 2 |
math | 1 Convex quadrilateral $E F G H$ has vertices $E, F, G, H$ on the sides $A B, B C, C D, D A$ of square $A B C D$, respectively. It satisfies: $\frac{A E}{E B} \cdot \frac{B F}{F C} \cdot \frac{C G}{G D} \cdot \frac{D H}{H A}=1$; and points $A, B, C, D$ are on the sides of square $E_{1} F_{1} G_{1} H_{1}$, with $E_{1} H... | \lambda | 206 | 2 |
math | 6. Given $\sqrt{2009}=\sqrt{x}+\sqrt{y}$, and $0<x<y$. Find all integer pairs $(x, y)$ that satisfy the equation. | (x, y) = (41, 1476), (164, 1025), (369, 656) | 41 | 36 |
math | (1) (20 points) There is a type of notebook originally priced at 6 yuan per book. Store A uses the following promotional method: for each purchase of 1 to 8 books, a 10% discount is applied; for 9 to 16 books, a 15% discount is applied; for 17 to 25 books, a 20% discount is applied; and for more than 25 books, a 25% di... | y=\left\{\begin{array}{ll}
4.9 x, & 11 \leqslant x \leqslant 16 ; \\
4.8 x, & 17 \leqslant x \leqslant 20 ; \\
4.5 x, & 21 \leqslant x \leqslant 40 .
\end{array}\right.} | 291 | 93 |
math | 3. Let the function be
$$
f(x)=x^{3}+a x^{2}+b x+c \quad (x \in \mathbf{R}),
$$
where $a, b, c$ are distinct non-zero integers, and
$$
f(a)=a^{3}, f(b)=b^{3} .
$$
Then $a+b+c=$ $\qquad$ | 18 | 85 | 2 |
math | Find all natural triples $(a,b,c)$, such that:
$a - )\,a \le b \le c$
$b - )\,(a,b,c) = 1$
$c - )\,\left. {{a^2}b} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{b^2}c} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{c^2}a} \right|{a^3} + {b^3} + {c^3}$. | (1, 1, 1) | 138 | 10 |
math | 249 Find the range of the real number $\lambda$ such that the inequality
$$
\frac{1}{\sqrt{1+x}}+\frac{1}{\sqrt{1+y}}+\frac{1}{\sqrt{1+z}} \leqslant \frac{3}{\sqrt{1+\lambda}}
$$
holds for any positive real numbers $x, y, z$ satisfying $x y z=\lambda^{3}$. | \left(0, \frac{5}{4}\right] | 96 | 14 |
math | # 6. Variant 1
A doll maker makes one doll in 1 hour 45 minutes. After every three dolls made, the master has to rest for half an hour. Ten dolls are needed for gifts. At what time the next day (specify hours and minutes) will the order be completed if the master started making dolls at 10:00 AM and worked through the... | 5:00 | 84 | 4 |
math | ## Task B-4.3.
For which real parameters $b$ and $c$ is the natural domain of the function $f(x)=\frac{1}{\sqrt{-10 x^{2}+b x+c}}$ equal to the set $A=\left\{x \in \mathbb{R} \left\lvert\, \log _{\frac{1}{2}} \frac{2 x-1}{2-3 x}>0\right.\right\}$? | b=11,=-3 | 107 | 7 |
math | Three, (16 points) Let the equation $x^{2}-|2 x-1|-4=0$. Find the sum of all roots that satisfy the equation.
| 2-\sqrt{6} | 36 | 6 |
math | 5. Find all positive integers $n$ such that the ternary polynomial
$$
\begin{array}{l}
P_{n}(x, y, z) \\
=(x-y)^{2 n}(y-z)^{2 n}+(y-z)^{2 n}(z-x)^{2 n}+ \\
(z-x)^{2 n}(x-y)^{2 n}
\end{array}
$$
divides the ternary polynomial
$$
\begin{array}{l}
Q_{n}(x, y, z) \\
=\left[(x-y)^{2 n}+(y-z)^{2 n}+(z-x)^{2 n}\right]^{2 n}... | n=1 | 156 | 3 |
math | 9.063. $4^{\frac{1}{x}-1}-2^{\frac{1}{x}-2}-3 \leq 0$. | x\in(-\infty;0)\cup[\frac{1}{2};\infty) | 36 | 22 |
math | 6-175 Let $f$ be a function from $R \rightarrow R$, and
(1) For any $x, y \in R$,
$$
f(x)+f(y)+1 \geqslant f(x+y) \geqslant f(x)+f(y) ;
$$
(2) For any $x \in[0,1)$, $f(0) \geqslant f(x)$;
(3) $-f(-1)=f(1)=1$.
Find all functions $f$ that satisfy the conditions. | f(x)=[x] | 123 | 6 |
math | 934. Three classmates bought 13 pies, with Kostya buying half as many as Tolya, and Volodya buying more than Kostya but less than Tolya. How many pies did each of them buy? | 3,4,6 | 51 | 5 |
math | 1. Tomorrow, the Janssen family is going on a trip by car, and they have a beautiful route in mind for it. The youngest member of the family notes that their planned stop in Germany is exactly halfway along the route. Father responds: "When we cross the border after 150 kilometers tomorrow, our stop will be just one-fi... | 400 | 92 | 3 |
math | 25.16. Calculate $\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+n}-n\right)$. | \frac{1}{2} | 33 | 7 |
math | 2. Rationalize and simplify the fraction: $\frac{9+2 \sqrt{2}}{4+\sqrt{162}}$ | \frac{\sqrt{2}}{2} | 30 | 10 |
math | 5. Nika and Tim were playing cards. An undecided outcome was not possible. They had agreed in advance that the winner of each game would receive more points than the loser and that the loser would receive a positive number of points. They determined how many points the winner would receive after each game and how many ... | 8 | 169 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{1-\sin \left(\frac{x}{2}\right)}{\pi-x}$ | 0 | 41 | 1 |
math | Example 1. Find the equations of the tangent and normal to the parabola $y=x^{2}+4 x+2$ at the point $A(1,7)$. | 6x+1,\;x+6y-43=0 | 40 | 15 |
math | 11. Let $A$ be a set composed of any 100 distinct positive integers, and let
$$
B=\left\{\left.\frac{a}{b} \right\rvert\, a, b \in A \text { and } a \neq b\right\},
$$
$f(A)$ denotes the number of elements in set $B$. Then the sum of the maximum and minimum values of $f(A)$ is $\qquad$ | 10098 | 99 | 5 |
math | How many integers divide either $2018$ or $2019$? Note: $673$ and $1009$ are both prime.
[i]2019 CCA Math Bonanza Lightning Round #1.1[/i] | 7 | 57 | 1 |
math | 1. Let $S$ be the set of all integers $n$ such that $\frac{8 n^{3}-96 n^{2}+360 n-400}{2 n-7}$ is an integer. Find the value of $\sum_{n \in S}|n|$. | 50 | 65 | 2 |
math | ## Exercise 11.
1) Alice wants to color the integers between 2 and 8 (inclusive) using $k$ colors. She wishes that if $m$ and $n$ are integers between 2 and 8 such that $m$ is a multiple of $n$ and $m \neq n$, then $m$ and $n$ are of different colors. Determine the smallest integer $k$ for which Alice can color the in... | 4 | 223 | 1 |
math | Example 1 The numbers 1447, 1005, and 1231 have certain common points, that is, each number is a four-digit number starting with 1, and in each four-digit number, exactly two digits are the same. How many such four-digit numbers are there? | 432 | 66 | 3 |
math | 5. In the spatial quadrilateral $ABCD$, $AB=2, BC=$ $3, CD=4, DA=5$. Then $\overrightarrow{AC} \cdot \overrightarrow{BD}=$ $\qquad$ | 7 | 49 | 1 |
math | The price of a product has increased by 40%. By what percentage do we need to reduce our consumption of this product if we can only spend 12% more money on its purchase? | 20 | 40 | 2 |
math | 3. In 2018, Pavel will be as many years old as the sum of the digits of the year he was born. How old is Pavel? | 10 | 34 | 2 |
math | 8. The real value range of the function $f(x)=\sqrt{\cos ^{2} x-\frac{3}{4}}+\sin x$ is | [-\frac{1}{2},\frac{\sqrt{2}}{2}] | 34 | 18 |
math | 2A. It is known that $a^{x}=b^{y}=c^{z}=30^{w}$ and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{w}$, where $a, b, c$ are natural numbers, and $x, y, z, w$ are real numbers. Determine the sum $a+b+c$. | 10 | 89 | 2 |
math | 3. Let the chord $P Q$ of the parabola $y^{2}=x$ be perpendicularly bisected by the line $l: y=k(x-1)+1(k \in \mathbf{Z})$. Then the length of the chord $P Q$ is $\qquad$ . | \sqrt{10} | 66 | 6 |
math | [The triangle formed by the bases of two altitudes and a vertex]
The side of the triangle is $\sqrt{2}$, and the angles adjacent to it are $75^{\circ}$ and $60^{\circ}$.
Find the segment connecting the bases of the altitudes drawn from the vertices of these angles. | 1 | 68 | 1 |
math | 7、A book's page numbers are consecutive natural numbers, $1,2,3, \cdots$, when these page numbers are added together, one page number is added twice, resulting in the incorrect sum of 1997, then the page number that was added twice is $\qquad$ | 44 | 63 | 2 |
math | Example 8 Solve the system of equations in the set of real numbers
$$\left\{\begin{array}{l}
2 x+3 y+z=13 \\
4 x^{2}+9 y^{2}+z^{2}-2 x+15 y+3 z=82
\end{array}\right.$$ | x=3, y=1, z=4 | 72 | 11 |
math | Compute the number of five-digit positive integers whose digits have exactly $30$ distinct permutations (the permutations do not necessarily have to be valid five-digit integers).
[i]Proposed by David Sun[/i] | 9720 | 42 | 4 |
math | 7. [5] Simplify the product
$$
\prod_{m=1}^{100} \prod_{n=1}^{100} \frac{x^{n+m}+x^{n+m+2}+x^{2 n+1}+x^{2 m+1}}{x^{2 n}+2 x^{n+m}+x^{2 m}}
$$
Express your answer in terms of $x$. | x^{9900}(\frac{1+x^{100}}{2})^{2} | 97 | 23 |
math | 6. Let $M=\{1,2,3, \cdots, 1995\}, A$ be a subset of $M$ and satisfy the condition: if $x \in A$, then $15 x \notin A$. Then the maximum number of elements in $A$ is $\qquad$ . | 1870 | 70 | 4 |
math | 8・ 129 Let the sequence $\left\{x_{n}\right\}$ satisfy $x_{1}=5$, and
$$
x_{n+1}=x_{n}^{2}-2, n=1,2, \cdots
$$
Find: $\lim _{n \rightarrow \infty} \frac{x_{n+1}}{x_{1} x_{2} \cdots x_{n}}$. | \sqrt{21} | 97 | 6 |
math | 7. (10 points) On the board, 33 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 33 minutes? | 528 | 70 | 3 |
math | Example 15 Find the general term formula of the Fibonacci sequence $a_{0}=1, a_{1}=1, a_{n+2}=a_{n+1}+a_{n}(n \geqslant 1)$. | a_{n}=\frac{1}{\sqrt{5}}[(\frac{1+\sqrt{5}}{2})^{n+1}-(\frac{1-\sqrt{5}}{2})^{n+1}] | 53 | 50 |
math | 11.1. Solve the equation $\cos ^{2}(\sqrt{2} x)-\sin ^{2} x=1$. | 0 | 31 | 1 |
math | 2. After a reduction of $12.5 \%$ the price of the gaming console is 2044 kn. What was the price of the gaming console before the reduction? | 2336 | 39 | 4 |
math | Let $A, B, C$ and $D$ be four points not lying in the same plane. A plane is drawn through the centroid of triangle $ABC$ and is parallel to the lines $AB$ and $CD$. In what ratio does this plane divide the median drawn to the side $CD$ of triangle $ACD$? | 1:2 | 70 | 3 |
math | 2. A production team in a factory is tasked with producing a batch of parts. When each worker works at their original position, it takes 9 hours to complete the production task. If the positions of workers A and B are swapped, with the production efficiency of other workers remaining unchanged, the task can be complete... | 108 | 140 | 3 |
math | Example 2 Let $S=\{1,2,3,4\}, n$ terms of the sequence: $a_{1}, a_{2}, \cdots, a_{n}$ have the following property, for any non-empty subset $B$ of $S$ (the number of elements in $B$ is denoted as $|B|$), there are adjacent $\mid B$ | terms in the sequence that exactly form the set $B$, find the minimum value of $n$.
(1... | 8 | 114 | 1 |
math | Let ABC be a scalene triangle and AM is the median relative to side BC. The diameter circumference AM intersects for the second time the side AB and AC at points P and Q, respectively, both different from A. Assuming that PQ is parallel to BC, determine the angle measurement <BAC.
Any solution without trigonometry? | \angle BAC = 90^\circ | 67 | 11 |
math | 7. For a regular quadrilateral pyramid $P-ABCD$ with base and lateral edge lengths all being $a$, $M$ and $N$ are moving points on the base edges $CD$ and $CB$ respectively, and $CM = CN$. When the volume of the tetrahedron $P-AMN$ is maximized, the angle between line $PA$ and plane $PMN$ is $\qquad$ | \frac{\pi}{4} | 91 | 7 |
math | Given the equation \[ y^4 \plus{} 4y^2x \minus{} 11y^2 \plus{} 4xy \minus{} 8y \plus{} 8x^2 \minus{} 40x \plus{} 52 \equal{} 0,\] find all real solutions. | (1, 2) \text{ and } (2.5, -1) | 71 | 19 |
math | Example 3.16. Find $u_{x x}^{\prime \prime}, u_{y y}^{\prime \prime}, u_{x y}^{\prime \prime}, u_{y x}^{\prime \prime}, u_{x y z}^{\prime \prime \prime}, u_{x y^{2} z}^{I V}$, if $u=e^{x y z}$. | u_{xy^{2}z}^{IV}=e^{xyz}(x^{3}y^{2}z^{3}+5x^{2}yz^{2}+4xz) | 91 | 41 |
math | An arithmetic sequence is a sequence of numbers in which the difference between each number and the one preceding it is always the same; this difference is called the common difference. (For example, 2, 8, 14, 20, 26, 32 is an arithmetic sequence with a common difference of 6.)
Bolek and Lolek each had their own arith... | 12 | 185 | 2 |
math | Example 4 Find all positive integers $a, b$ such that
$$(a, b)+9[a, b]+9(a+b)=7 a b .$$ | (4,38), (38,4), (4,4) | 35 | 17 |
math | ## Task A-1.8. (20 points)
Determine all natural numbers $n$ for which $n^{2}-440$ is a perfect square. | 111,57,27,21 | 37 | 12 |
math | Let $ ABC$ be an isosceles triangle with $ \left|AB\right| \equal{} \left|AC\right| \equal{} 10$ and $ \left|BC\right| \equal{} 12$. $ P$ and $ R$ are points on $ \left[BC\right]$ such that $ \left|BP\right| \equal{} \left|RC\right| \equal{} 3$. $ S$ and $ T$ are midpoints of $ \left[AB\right]$ and $ \left[AC\right]$, ... | \frac{10\sqrt{13}}{13} | 298 | 15 |
math | The lines with the two equations below intersect at the point $(2,-3)$.
$$
\begin{array}{r}
\left(a^{2}+1\right) x-2 b y=4 \\
(1-a) x+b y=9
\end{array}
$$
What are the possible ordered pairs $(a, b)$ ? | (4,-5)\text{}(-2,-1) | 74 | 12 |
math | 3. Two people simultaneously step onto an escalator from opposite ends, which is moving downward at a speed of $u=1.5 \mathrm{~m} / \mathrm{s}$. The person moving downward has a speed of $v=3 \mathrm{~m} / \mathrm{s}$ relative to the escalator, while the person moving upward has a speed of $2 v / 3$ relative to the esc... | 10\mathrm{~} | 125 | 7 |
math | Five. (15 points) Given a function $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any real numbers $x, y, z$ we have
$$
\begin{array}{l}
\frac{1}{2} f(x y)+\frac{1}{2} f(x z)-f(x) f(y z) \geqslant \frac{1}{4} . \\
\text { Find }[1 \times f(1)]+[2 f(2)]+\cdots+[2011 f(2011)]
\end{array}
$$
where $[a]$ denotes th... | 1011030 | 154 | 7 |
math | 1. Let $S=x^{2}+y^{2}-2(x+y)$, where $x, y$ satisfy $\log _{2} x+\log _{2} y=1$, then the minimum value of $S$ is | 4-4\sqrt{2} | 52 | 8 |
math | Let $c$ be a real number. If the inequality
$$f(c)\cdot f(-c)\ge f(a)$$
holds for all $f(x)=x^2-2ax+b$ where $a$ and $b$ are arbitrary real numbers, find all possible values of $c$. | c = \pm \frac{1}{2} | 64 | 12 |
math | ## Task A-1.2.
Determine all pairs of integers $(m, n)$ such that
$$
n^{2}-6 n=m^{2}+m-10
$$ | (1,4),(1,2),(-2,2),(-2,4) | 41 | 19 |
math | Initially 255, set 2009 can be decomposed into the sum of squares of four positive integers, among which, the ratio of two numbers is $\frac{5}{14}$, and the ratio of the other two numbers is $\frac{1}{2}$. Write down this decomposition. | 2009=30^{2}+28^{2}+15^{2}+10^{2}, \\ 2009=42^{2}+15^{2}+4^{2}+2^{2} | 66 | 56 |
math | 10. In the Cartesian coordinate system, let point $A(0,4)$, $B(3,8)$. If point $P(x, 0)$ makes $\angle A P B$ maximum, then $x=$ $\qquad$ | 5 \sqrt{2}-3 | 53 | 7 |
math | 6. What is the minimum value that the function $F(x ; y)=6 y+8 x-9$ can take, given that $x^{2}+y^{2}+25=10(x+y)$.
# | 11 | 50 | 2 |
math | 14. Concept. In Anchuria, there are $K$ laws and $N$ ministers. The probability that a randomly chosen minister knows a randomly chosen law is $p$. One day, the ministers gathered for a council to write the Concept. If at least one minister knows a law, then this law will be considered in the Concept; otherwise, this l... | )C_{K}^{M}(1-(1-p)^{N})^{M}(1-p)^{N\cdot(K-M)};b)K(1-(1-p)^{N}) | 113 | 42 |
math | ## Task Condition
Find the derivative.
$$
y=x-\ln \left(1+e^{x}\right)-2 e^{-\frac{x}{2}} \cdot \operatorname{arctg} e^{\frac{x}{2}}-\left(\operatorname{arctg} e^{\frac{x}{2}}\right)^{2}
$$ | \frac{\operatorname{arctg}e^{x/2}}{e^{x/2}\cdot(1+e^{x})} | 77 | 32 |
math | Find the largest possible value of the expression $\left|\sqrt{x^{2}+4 x+8}-\sqrt{x^{2}+8 x+17}\right|$ where $x$ is a real number. | \sqrt{5} | 46 | 5 |
math | ## 9. Balls
In a box, there are only red and blue balls. If five red balls were drawn from the box, one seventh of the remaining balls in the box would be red. If ten blue balls were drawn from the box instead of five red ones, one fifth of the remaining balls in the box would be red. How many balls are there in the b... | 110 | 90 | 3 |
math | The value of
$$\left(1-\frac{1}{2^2-1}\right)\left(1-\frac{1}{2^3-1}\right)\left(1-\frac{1}{2^4-1}\right)\dots\left(1-\frac{1}{2^{29}-1}\right)$$
can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $2m - n.$ | 1 | 109 | 1 |
math | 3. Now, take five integers, each time take two different numbers to add, you can get ten sums, then add the ten sums, the sum obtained is 2020, then the average of the original five numbers is $\qquad$ . | 101 | 53 | 3 |
math | \section*{Problem 1 - 111241}
All real numbers \(x\) are to be specified for which the expression
\[
\frac{2 x}{|x-3|-5}+\frac{1}{x+2}
\]
exists, and among these, all \(x\) are to be determined that satisfy the following inequality (2):
\[
\frac{2 x}{|x-3|-5}+\frac{1}{x+2} \geq 1
\] | x\in(-2,-\frac{1}{3}]\cup(8,\infty) | 111 | 21 |
math | 7.1. Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \cdot 2 \cdot \ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a solution is a triplet $(n, k, l)$; if soluti... | 10 | 110 | 2 |
math | 6. During an early morning drive to work, Simon encountered $n$ sets of traffic lights, each set being red, amber or green as he approached it. He noticed that consecutive lights were never the same colour.
Given that he saw at least two red lights, find a simplified expression, in terms of $n$, for the number of possi... | 3\times2^{n-1}-4n+2 | 78 | 13 |
math | 84. Equation containing radicals. Solve the equation
$$
(6 x+28)^{\frac{1}{3}}-(6 x-28)^{\frac{1}{3}}=2
$$ | \6 | 45 | 2 |
math | 11. The sequence $\left\{a_{n}\right\}$ satisfies -
$$
\begin{array}{l}
a_{1}=8, a_{2}=26, \\
a_{n}=a_{n-1}+a_{n-2}+a_{n-1} a_{n-2} .
\end{array}
$$
Then $a_{10}=$ | 3^{144}-1 | 87 | 7 |
math | ## Task Condition
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=3 p+q$
$b=p-2 q$
$|p|=4$
$|q|=1$
$\widehat{(\widehat{p}, q})=\frac{\pi}{4}$ | 14\sqrt{2} | 67 | 7 |
math | 1. In a math test, $N<40$ people participate. The passing score is set at 65. The test results are as follows: the average score of all participants is 66, that of the promoted is 71, and that of the failed is 56. However, due to an error in the formulation of a question, all scores are increased by 5. At this point, t... | 12,24,36 | 160 | 8 |
math | Shapovalov A.V.
A banker learned that among identical-looking coins, one is counterfeit (lighter). He asked an expert to identify this coin using a balance scale without weights, and required that each coin participate in weighings no more than twice. What is the maximum number of coins the banker can have so that the... | 2n^2+1 | 80 | 6 |
math | C2
(a) Find the distance from the point $(1,0)$ to the line connecting the origin and the point $(0,1)$.
(b) Find the distance from the point $(1,0)$ to the line connecting the origin and the point $(1,1)$.
(c) Find the distance from the point $(1,0,0)$ to the line connecting the origin and the point $(1,1,1)$. | \frac{\sqrt{2}}{2} | 91 | 10 |
math | 29.29. Calculate the volume $V$ of a sphere of radius $R$ using a definite integral. | \frac{4\piR^{3}}{3} | 25 | 13 |
math | 2. Given in $\triangle A B C$, $\overrightarrow{A B} \cdot \overrightarrow{B C}=2 \overrightarrow{B C} \cdot \overrightarrow{C A}=4 \overrightarrow{C A} \cdot \overrightarrow{A B}$, then the cosine value of the largest angle in $\triangle A B C$ is | \frac{\sqrt{15}}{15} | 77 | 12 |
math | 1. Given the right triangle $\triangle A B C$ with side lengths of 3, 4, and 5. If its incircle is removed, the remaining area is $\qquad$ . | 6-\pi | 42 | 3 |
math | 10.221. Two sides of a triangle are 6 and $8 \mathrm{~cm}$. The medians drawn to these sides are perpendicular to each other. Find the third side of the triangle. | 2\sqrt{5}\mathrm{~} | 46 | 10 |
math | Rubaanov I.S.
A polynomial $P(x)$ of degree $n$ has $n$ distinct real roots. What is the maximum number of its coefficients that can be zero?
# | \frac{n}{2}forevenn,\frac{n+1}{2}foroddn | 40 | 20 |
math | ## Task Condition
Find the derivative.
$y=-\frac{1}{2} \cdot \ln \left(\tanh \frac{x}{2}\right)-\frac{\cosh x}{2 \sinh^{2} x}$ | \frac{1}{\sinh^3x} | 51 | 12 |
math | 2. Given the set $M=\{1,99,-1,0,25,-36, -91,19,-2,11\}$, let the non-empty subsets of $M$ be $M_{i}(i=1,2, \cdots, 1023)$. If the product of all elements in each $M_{i}$ is $m_{i}$, then $\sum_{i=1}^{1023} m_{i}=$ $\qquad$ . | -1 | 114 | 2 |
math | 1. (6 points) Calculate: $4.165 \times 4.8 + 4.165 \times 6.7 - 4.165 \div \frac{2}{3}=$ | 41.65 | 50 | 5 |
math | 4. Given that $a, b, c$ are the lengths of the three sides of $\triangle ABC$, and satisfy the conditions
$$
\frac{2 a^{2}}{1+a^{2}}=b, \frac{2 b^{2}}{1+b^{2}}=c, \frac{2 c^{2}}{1+c^{2}}=a \text {. }
$$
Then the area of $\triangle ABC$ is . $\qquad$ | \frac{\sqrt{3}}{4} | 100 | 10 |
math | 334. Solve the system of equations:
$$
\left\{\begin{array}{l}
x^{2}+y z=y+z \\
y^{2}+x z=x+z \\
z^{2}+x y=x+y
\end{array}\right.
$$ | (0;0;0),(1;1;1),(-1;1;1),(1;-1;1),(1;1;-1) | 60 | 32 |
math | Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write (and submit) before he has written half of the total problems? | 65 | 52 | 4 |
math | Compute the sum of all positive integers whose digits form either a strictly increasing or a strictly decreasing sequence. | 25617208995 | 20 | 11 |
math | In Miroslav's kingdom, the cobbler Matěj used to go not only to sing but also to eat and drink well. For one gold piece, he got a whole goose and one jug of wine. Then, however, they increased the prices by $20 \%$, and for a gold piece, he got only half a jug of wine and a whole goose. It is said that after the full m... | 0.96 | 119 | 4 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.