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 | 6) Let $n$ be a positive integer. Consider
$$
S=\{(x, y, z): x, y, z \in\{0,1, \cdots, n\}, x+y+z>0\}
$$
a set of points in three-dimensional space with $(n+1)^{3}-1$ points. How many planes are needed at minimum so that their union contains $S$, but does not contain $(0,0,0)$? | 3n | 101 | 2 |
math | 3. (10 points) $[a]$ represents the greatest integer not greater than $a$. Given that $\left(\left[\frac{1}{7}\right]+1\right) \times\left(\left[\frac{2}{7}\right]+1\right) \times\left(\left[\frac{3}{7}\right]+1\right) \times \cdots \times$ $\left(\left[\frac{k}{7}\right]+1\right)$ leaves a remainder of 7 when divided ... | 45 | 133 | 2 |
math | 3. (10 points) In a $3 \times 3$ grid (each cell is a $1 \times 1$ square), place two identical pieces, with at most one piece per cell. There are $\qquad$ different ways to place the pieces. (If two placements can be made to coincide by rotation, they are considered the same placement). | 10 | 76 | 2 |
math | 8. Given the sequence: $\frac{2}{3}, \frac{2}{9}, \frac{4}{9}, \frac{6}{9}, \frac{8}{9}, \frac{2}{27}, \frac{4}{27}, \cdots, \frac{26}{27}, \cdots, \frac{2}{3^{n}}, \frac{4}{3^{n}}, \cdots, \frac{3^{n}-1}{3^{n}}, \cdots$. Then $\frac{2020}{2187}$ is the \_\_\_\_ term of the sequence. | 1553 | 139 | 4 |
math | 8. At a little past 8 o'clock in the morning, two cars left the fertilizer plant one after another heading for Happiness Village. Both cars were traveling at a speed of 60 kilometers per hour. At 8:32, the distance the first car had traveled from the fertilizer plant was three times that of the second car. By 8:39, the... | 11 | 119 | 2 |
math | 8. (FRG 3) For all rational $x$ satisfying $0 \leq x<1, f$ is defined by
$$ f(x)= \begin{cases}f(2 x) / 4, & \text { for } 0 \leq x<1 / 2 \\ 3 / 4+f(2 x-1) / 4, & \text { for } 1 / 2 \leq x<1\end{cases} $$
Given that $x=0 . b_{1} b_{2} b_{3} \ldots$ is the binary representation of $x$, find $f(x)$. | f\left(0 . b_{1} b_{2} \ldots\right)=0 . b_{1} b_{1} b_{2} b_{2} \ldots | 143 | 41 |
math | Three, (50 points) Let $n$ be a positive integer. Find the number of positive integer triples $\langle a, b, c\rangle$ that satisfy the following conditions:
(i) $a b=n$;
(ii) $1 \leqslant c \leqslant b$;
(iii) The greatest common divisor of $a$, $b$, and $c$ is 1. | f(n)=n\prod_{p\midn}(1+\frac{1}{p} | 87 | 20 |
math | 140. Find the greatest common divisor of all nine-digit numbers composed of the digits $1,2,3,4,5,6,7,8,9$ (without repetition). | 9 | 41 | 1 |
math | # 5. CONDITION
A tourist goes on a hike from $A$ to $B$ and back, and completes the entire journey in 3 hours and 41 minutes. The route from $A$ to $B$ first goes uphill, then on flat ground, and finally downhill. Over what distance does the road pass on flat ground, if the tourist's speed is 4 km/h when climbing uphi... | 4 | 113 | 1 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ for which
$$
f\left(x^{2}\right)-f\left(y^{2}\right) \leq(f(x)+y)(x-f(y))
$$
for all $x, y \in \mathbb{R}$. | f(x)=x \text{ or } f(x)=-x | 72 | 14 |
math | 5. Write a given rational number as a reduced fraction, and calculate the product of the resulting numerator and denominator. How many rational numbers between 0 and 1, when processed this way, yield a product of 20!? | 128 | 47 | 3 |
math | [ Processes and operations [ Examples and counterexamples. Constructions ]
Several boxes together weigh 10 tons, and each of them weighs no more than one ton. How many three-ton trucks are definitely enough to haul this cargo?
# | 5 | 48 | 1 |
math | Determine all non negative integers $k$ such that there is a function $f : \mathbb{N} \to \mathbb{N}$ that satisfies
\[ f^n(n) = n + k \]
for all $n \in \mathbb{N}$ | k = 0 | 58 | 5 |
math | Find all $7$-digit numbers which are multiples of $21$ and which have each digit $3$ or $7$. | \{3373377, 7373373, 7733733, 3733737, 7337337, 3777333\} | 29 | 57 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{5^{x}(2 \sin 2 x+\cos 2 x \cdot \ln 5)}{4+\ln ^{2} 5}
$$ | 5^{x}\cdot\cos2x | 48 | 9 |
math | # Problem 9.
Let $A(n)$ denote the greatest odd divisor of the number $n$. For example, $A(21)=21$, $A(72)=9, A(64)=1$. Find the sum $A(111)+A(112)+\ldots+A(218)+A(219)$. | 12045 | 80 | 5 |
math | 15. (3 points) There are four people, A, B, C, and D. It is known that the average age of A, B, and C is 1 year older than the average age of the four people, the average age of A and B is 1 year older than the average age of A, B, and C, A is 4 years older than B, and D is 17 years old. How old is A $\qquad$? | 24 | 100 | 2 |
math | 35. Let $a_{1}, a_{2}, a_{3}, \ldots$ be the sequence of all positive integers that are relatively prime to 75, where $a_{1}<a_{2}<a_{3}<\cdots$. (The first five terms of the sequence are: $a_{1}=1, a_{2}=2, a_{3}=4, a_{4}=7, a_{5}=8$.) Find the value of $a_{2008}$. | 3764 | 110 | 4 |
math | ## Task 1 - 120811
Determine all three-digit natural numbers $z$, each of which satisfies the following conditions:
(1) The cross sum of the number $z$ is 12
(2) The two-digit number formed from the tens and units digit (in that order) of the number $z$ is five times the one-digit number formed from the hundreds dig... | 525,840 | 89 | 7 |
math | 3.20. Calculate
$$
\int_{1}^{i} \frac{\ln ^{2} z}{z} d z
$$ | \frac{-i\pi^{3}}{24} | 33 | 13 |
math | 32. In an acute triangle $ABC$, the distance from vertex $A$ to the circumcenter $O$ is equal to the distance from $A$ to the orthocenter $H$. Find all possible values of $\angle A$. | 60^{\circ} | 50 | 6 |
math | 10.334. Inside a square with side $a$, a semicircle is constructed on each side as a diameter. Find the area of the rosette bounded by the arcs of the semicircles. | \frac{^{2}(\pi-2)}{2} | 46 | 14 |
math | 18. 1. 11 * Find all positive integer solutions \(x, y, z, t\) to the equation \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}+\frac{1}{t^{2}}=1\). | 2 | 68 | 1 |
math | Problem 11.1. Solve the equation
$$
\log _{a}\left(a^{2\left(x^{2}+x\right)}+a^{2}\right)=x^{2}+x+\log _{a}\left(a^{2}+1\right)
$$
where $a$ is a real number.
Emil Kolev | -2,-1,0,1 | 79 | 8 |
math | 7. $2(2004 \mathrm{CMO})$ In the convex quadrilateral $E F G H$, the vertices $E, F, G, H$ are on the sides $A B, B C, C D, D A$ of the convex quadrilateral $A B C D$, respectively, and satisfy $\frac{A E}{E B} \cdot \frac{B F}{F C} \cdot \frac{C G}{G D} \cdot \frac{D H}{H A}=1$. The points $A, B, C, D$ are on the side... | \lambda | 280 | 2 |
math | 11.13. Solve the equation
$$
3-7 \cos ^{2} x \sin x-3 \sin ^{3} x=0
$$ | \frac{\pi}{2}+2k\pi(-1)^{k}\frac{\pi}{6}+k\pi | 38 | 28 |
math | The sequence of natural numbers $1, 5, 6, 25, 26, 30, 31,...$ is made up of powers of $5$ with natural exponents or sums of powers of $5$ with different natural exponents, written in ascending order. Determine the term of the string written in position $167$. | 81281 | 77 | 5 |
math | For a given positive integer $n(\ge 2)$, find maximum positive integer $A$ such that a polynomial with integer coefficients $P(x)$ with degree $n$ that satisfies the condition below exists.
$\bullet P(1), P(2), \cdots P(A)$ is a multiple of $A$
$\bullet P(0) = 0$ and the coefficient of the first term of $P(x)$ is $1... | n! | 148 | 3 |
math | Example 2 Find the values of the following expressions:
(1) $\sec 50^{\circ}+\operatorname{tg} 10$;
(2) $\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}$;
(3) $\operatorname{tg} 6^{\circ} \operatorname{tg} 42^{\circ} \operatorname{tg} 66^{\circ} \operatorname{tg} 78^{\circ}$. | \sqrt{3},-\frac{1}{2},1 | 126 | 13 |
math | ## Task 4.
Determine all pairs of natural numbers $(m, n)$ for which
$$
2^{m}=7 n^{2}+1
$$ | (,n)=(3,1)(,n)=(6,3) | 35 | 15 |
math | 16. If an internal point of some $n$-sided prism is connected to all its vertices, then we obtain $n$ quadrilateral pyramids with a common vertex at this point, the bases of which are the lateral faces of the prism. Find the ratio of the sum of the volumes of these pyramids to the volume of the given prism. | \frac{2}{3} | 74 | 7 |
math | G1.3 Let $n$ be the product 3659893456789325678 and 342973489379256 . Determine the number of digits of $n$. | 34 | 58 | 2 |
math | 5. If $\sqrt{3-a}-\sqrt{a+1}>\frac{1}{2}$ always holds, then the range of values for $a$ is . $\qquad$ | \left[-1,1-\frac{\sqrt{31}}{8}\right) | 41 | 19 |
math | Given an arbitrary angle $\alpha$, compute
$cos \alpha + cos \big( \alpha +\frac{2\pi }{3 }\big) + cos \big( \alpha +\frac{4\pi }{3 }\big)$ and $sin \alpha + sin \big( \alpha +\frac{2\pi }{3 } \big) + sin \big( \alpha +\frac{4\pi }{3 } \big)$ .
Generalize this result and justify your answer. | 0 | 110 | 3 |
math | Example 7 Find all prime pairs $(p, q)$ such that $p q \mid\left(p^{p}+q^{q}+1\right)$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | (p, q)=(2,5),(5,2) | 61 | 12 |
math | Find all positive integers $n=p_{1} p_{2} \cdots p_{k}$ which divide $\left(p_{1}+1\right)\left(p_{2}+1\right) \cdots\left(p_{k}+1\right)$, where $p_{1} p_{2} \cdots p_{k}$ is the factorization of $n$ into prime factors (not necessarily distinct).
Answer: All numbers $2^{r} 3^{s}$ where $r$ and $s$ are non-negative in... | All\\2^{r}3^{}\where\r\\\\non-negative\integers\\\leqr\leq2 | 132 | 24 |
math | ### 3.485 Find the maximum value of the expression
$$
A=\frac{1}{\sin ^{6} \alpha+\cos ^{6} \alpha} \text { for } 0 \leq \alpha \leq \frac{\pi}{2}
$$ | 4 | 63 | 1 |
math | 6. A family of beekeepers brought containers of honey to the fair with volumes of $13,15,16,17,19,21$ liters. In August, three containers were sold in full, and in September, two more, and it turned out that in August they sold twice as much honey as in September. Determine which containers were emptied in August. In y... | 21 | 90 | 2 |
math | Let $ABC$ be a triangle with $AB=10$, $AC=11$, and circumradius $6$. Points $D$ and $E$ are located on the circumcircle of $\triangle ABC$ such that $\triangle ADE$ is equilateral. Line segments $\overline{DE}$ and $\overline{BC}$ intersect at $X$. Find $\tfrac{BX}{XC}$. | \frac{8}{13} | 89 | 8 |
math | 1. Determine the number $n$, if it satisfies the following three conditions:
a) all its digits are different
b) from its digits, six different two-digit numbers can be formed
c) the sum of those two-digit numbers is $2 n$. | 198 | 53 | 3 |
math | Example 3.19. Find $d^{2} f$ :
$$
\text { 1) } f(x, y)=x^{6} y^{8}, \text { 2) } f(x, y)=x^{6}+y^{8}
$$ | ^{2}f=^{2}(x^{6}y^{8})=30x^{4}y^{8}^{2}+96x^{5}y^{7}y+56x^{6}y^{6}^{2},\quad^{2}f=^{2}(x^{6}+y^{8})=30x^{4} | 60 | 82 |
math | 12. The route from location A to location B consists only of uphill and downhill sections, with a total distance of 21 kilometers. If the speed uphill is 4 kilometers/hour, and the speed downhill is 6 kilometers/hour, the journey from A to B takes 4.25 hours, then the journey from B to A would take ( ) hours. | 4.5 | 78 | 3 |
math | ## Zadatak B-2.5.
Od žice duljine 4.5 metra treba napraviti šest ukrasa, tri u obliku pravilnog šesterokuta i tri u obliku jednakostraničnog trokuta. Svi šesterokuti, odnosno trokuti su međusobno sukladni. Odredite duljine stranica šesterokuta i trokuta tako da zbroj površina svih likova bude minimalan, a cijela žica ... | =10\mathrm{~},b=30\mathrm{~} | 137 | 17 |
math | 4. Find the sum of the digits of the number $\underbrace{44 \ldots 4}_{2012 \text { times }} \cdot \underbrace{99 \ldots 9}_{2012 \text { times }}$ | 18108 | 56 | 5 |
math | Let's write the polynomial
$$
S=x+3 x^{2}+6 x^{3}+10 x^{4}+\ldots+\frac{n(n+1)}{2} x^{n}
$$
in a simpler form, if $x \neq 1$. | \frac{n(n+1)x^{n+1}}{2(x-1)}-\frac{nx^{n+1}}{(x-1)^{2}}+\frac{x(x^{n}-1)}{(x-1)^{3}} | 62 | 52 |
math | Pista forgot her friend's phone number. She remembers that the first digit is 7, the fifth is 2. She knows that the number is six digits long, odd, and when divided by 3, 4, 7, 9, 11, and 13, it gives the same remainder. What is the phone number? | 720721 | 74 | 6 |
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 | Find all integers $x, y$ such that $y^{2}=x^{3}-3 x+2$. | (k^2-2,\k(k^2-3)) | 24 | 13 |
math | (14) For a positive integer $n$, let $f(n)$ be the sum of the digits in the decimal representation of the number $3 n^{2}+n+1$.
(1) Find the minimum value of $f(n)$;
(2) When $n=2 \cdot 10^{k}-1$ (where $k$ is a positive integer), find $f(n)$;
(3) Does there exist a positive integer $n$ such that $f(n)=2012$? | 3,9k-4,2012 | 112 | 11 |
math | Let $n \geqslant 2$ be an integer, and let $A_{n}$ be the set
$$ A_{n}=\left\{2^{n}-2^{k} \mid k \in \mathbb{Z}, 0 \leqslant k<n\right\} . $$
Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_{n}$. (Serbia) | (n-2) 2^{n}+1 | 104 | 11 |
math | 7. Let $x_{0}$ be the largest (real) root of the equation $x^{4}-16 x-12=0$. Evaluate $\left\lfloor 10 x_{0}\right\rfloor$. | 27 | 50 | 2 |
math | 4. To celebrate Barbara's birthday, Alberto proposes the following game: given the set of numbers $0,1, \ldots, 1024$, Barbara removes $2^{9}$ numbers from this set. In the next step, Alberto removes $2^{8}$ numbers from the remaining ones. It's Barbara's turn again, and she removes $2^{7}$ numbers from the remaining o... | 32 | 141 | 2 |
math | 51. In a box, there are 250 light bulbs, of which 100 are 100W, 50 are 60W, 50 are 25W, and 50 are 15W. Calculate the probability that the power of any randomly selected light bulb will not exceed 60W. | \frac{3}{5} | 77 | 7 |
math | 2. Find all real numbers $a$ and $b$ such that for every $x \in[-1,1]$, the inequality
$$
\left|2 x^{2}+a x+b\right| \leq 1
$$
holds. | 0,-1 | 57 | 3 |
math | Let $ n>1$ and for $ 1 \leq k \leq n$ let $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $ a_1,a_2,...,a_n$. Furthermore let $ P \equal{} P(a_1, a_2, . . . , a_n)$ be the sum of all $ p_k$ with odd values of $ k$ less than or equal to $ n$.
How ... | 2 | 155 | 3 |
math | 4.1. Point $O$ is the intersection of the altitudes of an acute-angled triangle $ABC$. Find $OC$, if $AB=4$ and $\sin \angle C=\frac{5}{13}$. | 9.6 | 49 | 3 |
math | Five. (25 points) Given the system of equations in $x$ and $y$
$$
\left\{\begin{array}{l}
x^{2}-y^{2}=p, \\
3 x y+p(x-y)=p^{2}
\end{array}\right.
$$
has integer solutions $(x, y)$. Find the prime number $p$ that satisfies the condition. | 3 | 84 | 1 |
math | Example 13 Given that the function $f(x)$ is defined on $[0,1]$. And it satisfies the following conditions: (1) $f(1)=3$; (2) $f(x) \geqslant 2$; (3) If $x_{1} \geqslant 0, x_{2} \geqslant 0, x_{1}+x_{2} \leqslant 1$, then $f\left(x_{1}+x_{2}\right) \geqslant f\left(x_{1}\right)+f\left(x_{2}\right)-2$. (1) Find the max... | f(x)<2x+2 | 292 | 7 |
math | 11. The number of real solutions to the equation $\left(x^{2006}+1\right)\left(1+x^{2}+x^{4}+\cdots+\right.$ $\left.x^{2004}\right)=2006 x^{2005}$ is $\qquad$ | 1 | 70 | 1 |
math | Task 7. On a circle, there are 25 non-overlapping arcs, and on each of them, two arbitrary prime numbers are written. The sum of the numbers on each arc is not less than the product of the numbers on the arc following it in a clockwise direction. What can the sum of all the numbers be? | 100 | 68 | 3 |
math | 1st CaMO 1969 Problem 2 If x is a real number not less than 1, which is larger: √(x+1) - √x or √x - √(x-1)? Solution | \sqrt{x}-\sqrt{x-1}>\sqrt{x+1}-\sqrt{x} | 52 | 20 |
math | 8. For the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, the right vertex is $A$, the upper vertex is $B$, the left focus is $F$, and $\angle A B F=$ $90^{\circ}$. Then the eccentricity of the ellipse is . $\qquad$ | \frac{\sqrt{5}-1}{2} | 85 | 11 |
math | 34. Find the number of pairs of positive integers $(x, y)$ are there which satisfy the equation $2 x+3 y=2007$. | 334 | 34 | 3 |
math | 13. In a regular tetrahedron $S-ABC$, the dihedral angle between two adjacent lateral faces is $2 \alpha$, and the distance from the center of the base $O$ to a lateral edge is 1. Find $V_{\text {S }}$. | \frac{9\tan^{3}\alpha}{4\sqrt{3\tan^{2}\alpha-1}} | 61 | 25 |
math | 10.2. Solve the equation:
$1+\frac{3}{x+3}\left(1+\frac{2}{x+2}\left(1+\frac{1}{x+1}\right)\right)=x$. | 2 | 50 | 1 |
math | 30. [14] Alice has an equilateral triangle $A B C$ of area 1. Put $D$ on $B C, E$ on $C A$, and $F$ on $A B$, with $B D=D C, C E=2 E A$, and $2 A F=F B$. Note that $A D, B E$, and $C F$ pass through a single point $M$. What is the area of triangle $E M C$ ? | \frac{1}{6} | 104 | 7 |
math | 2. In an $8 \mathrm{~kg}$ mixture, there is $65 \%$ alcohol and 35 \% water. How much water needs to be added so that the diluted mixture has 45 \% water? | 1454.54 | 48 | 7 |
math | 2. How many positive numbers are there among the first 100 terms of the sequence: $\sin 1^{\circ}, \sin 10^{\circ}, \sin 100^{\circ}, \sin 1000^{\circ}, \ldots ?$ | 3 | 63 | 1 |
math | 2. For integers $x, y, z$ it holds that $x^{2}+y-z=10, x^{2}-y+z=22$. Find the smallest possible value of the expression $x^{2}+y^{2}+z^{2}$. | 34 | 60 | 2 |
math | The integer 72 is the first of three consecutive integers 72, 73, and 74, that can each be expressed as the sum of the squares of two positive integers. The integers 72, 288, and 800 are the first three members of an infinite increasing sequence of integers with the above property. Find a function that generates the se... | 1800, 3528, 6272 | 88 | 18 |
math | 10. Given the sequence $\left\{a_{n}\right\}$ defined by $a_{1}=\frac{2}{3}, a_{n+1}=a_{n}^{2}+a_{n-1}^{2}+\cdots+a_{1}^{2}\left(n \in \mathbf{N}^{*}\right)$. If for any $n$ $\in \mathbf{N}^{*}, \frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\cdots+\frac{1}{a_{n}+1}<M$ always holds, find the minimum value of $M$. | \frac{57}{20} | 149 | 9 |
math | 20. The surname of a Russian writer consists of six letters. It is known that the numbers indicating the positions of these letters in the Russian alphabet are in the following ratios: the first is equal to the third; the second is equal to the fourth; the fifth is 9 more than the first; the sixth is 2 less than the su... | GOGOL | 112 | 3 |
math | Find all positive integers $a$ for which the equation $7an -3n! = 2020$ has a positive integer solution $n$.
(Richard Henner) | a = 289 \text{ or } a = 68 | 40 | 16 |
math | Example 6 Given that $\alpha^{2005}+\beta^{2005}$ can be expressed as a bivariate polynomial in terms of $\alpha+\beta$ and $\alpha \beta$, find the sum of the coefficients of this polynomial.
(2005 Western Olympiad Problem) | 1 | 63 | 1 |
math | 7. Given a regular tetrahedron $P-ABC$ with the side length of the base being 6 and the side length of the lateral edges being $\sqrt{21}$. Then the radius of the inscribed sphere of the tetrahedron is $\qquad$. | 1 | 59 | 1 |
math | 5. Determine the number of ways to represent a natural number $n$ as the sum of several (two or more) natural numbers, where the order matters. (For example, for $n=4$, we have the following possibilities: $3+1, 2+2, 1+3, 2+1+1$, $1+2+1, 1+1+2, 1+1+1+1$, i.e., a total of 7 desired ways.)
## Fourth grade - B category | 2^{n-1}-1 | 111 | 7 |
math | Example 6, Fold a rectangle $A B C D$ with length and width of 4 and 3 respectively along the diagonal $A C$ to form a right dihedral angle, find the dihedral angle $D-A B-C$. | \arccos \frac{9}{\sqrt{481}} | 52 | 16 |
math | 10. All prime numbers $p$ that make $\frac{p(p+1)+2}{2}$ a perfect square are $\qquad$ . | 2 \text{ or } 5 | 32 | 8 |
math | 5. If the edges $AB, BC$ of the quadrilateral pyramid $P-ABCD$ are both $\sqrt{2}$, and the lengths of the other edges are all 1, then the volume of the quadrilateral pyramid is $\qquad$ . | \frac{\sqrt{2}}{6} | 55 | 10 |
math | Pick out three numbers from $0,1,\cdots,9$, their sum is an even number and not less than $10$. We have________different ways to pick numbers. | 51 | 38 | 2 |
math | Task solved by cyrix
# Task 2 - 261222
Determine all triples $(p, q, r)$ of prime numbers that satisfy the following conditions (1), (2):
(1) In the sequence of all prime numbers, $p, q, r$ are consecutive prime numbers in this order.
(2) The number $s=p^{2}+q^{2}+r^{2}$ is a prime number. | (3,5,7) | 96 | 7 |
math | 12. The expansion of $(a+b+c)^{10}$, after combining like terms, has
$\qquad$ terms. | 66 | 29 | 2 |
math | Adults made up $\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived. | 154 | 69 | 3 |
math | Problem 3. At a training, there were 225 children and 105 balls. The children were divided into groups with an equal number of children, and the trainers distributed the balls to each group so that each group received an equal number of balls. How many groups were formed and how many balls were distributed to each? How... | 4 | 77 | 1 |
math | Cynthia and Lynnelle are collaborating on a problem set. Over a $24$-hour period, Cynthia and Lynnelle each independently pick a random, contiguous $6$-hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time. | \frac{4}{9} | 66 | 8 |
math | 313. Find the mathematical expectation of a random variable $X$, uniformly distributed in the interval (a,b). | (+b)/2 | 24 | 4 |
math | 3. If a square pyramid with a base edge length of 2 is inscribed with a sphere of radius $\frac{1}{2}$, then the volume of this square pyramid is .. $\qquad$ | \frac{16}{9} | 43 | 8 |
math | Example 2 Given that $f(x)$ is an $n(>0)$-degree polynomial of $x$, and for any real number $x$, it satisfies $8 f\left(x^{3}\right)-x^{6} f(2 x)-$ $2 f\left(x^{2}\right)+12=0$
(1). Find $f(x)$. | f(x)=x^{3}-2 | 81 | 8 |
math | Find the number of squares in the sequence given by $ a_0\equal{}91$ and $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$ for $ n \ge 0.$ | 0 | 54 | 1 |
math | 7.4. Masha and Misha set out to meet each other simultaneously, each from their own house, and met one kilometer from Masha's house. Another time, they again set out to meet each other simultaneously, each from their own house, but Masha walked twice as fast, and Misha walked twice as slow as the previous time. This ti... | 3 | 101 | 1 |
math | 12. To promote the development of freshwater fish farming in a certain area, the government controls the price within an appropriate range and provides subsidies for freshwater fish farming. Let the market price of freshwater fish be $x$ yuan/kg, and the government subsidy be 1 yuan/kg. According to market research, wh... | 1 | 251 | 1 |
math | 10. Let $a_{1}, a_{2}, a_{3}, a_{4}$ be 4 rational numbers such that
$$
\left\{a_{i} a_{j} \mid 1 \leqslant i<j \leqslant 4\right\}=\left\{-24,-2,-\frac{3}{2},-\frac{1}{8}, 1,3\right\},
$$
Find the value of $a_{1}+a_{2}+a_{3}+a_{4}$. | \frac{9}{4} | 123 | 7 |
math | Many years ago, a teacher who did not want to give a lecture ordered his students to calculate the sum of the numbers from 1 to 100. A very clever student named Gauss found a very simple way to accomplish the task by discovering the formula:
$$
1+2+3+\ldots+n=\frac{n(n+1)}{2}
$$
Since this story is from a long time a... | 166000 | 221 | 6 |
math | Task 2. The ratio of the areas of the sides of a cuboid is 2:3:5. Determine the ratio of the lengths of the edges of the cuboid. | 10:6:15 | 38 | 7 |
math | Let $N$ be the greatest integer multiple of $36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by $1000$. | 640 | 46 | 3 |
math | 10.18. Provide an example of a polynomial $P(x)$ that is divisible by $x^{2}+1$, and at the same time $P(x)-1$ is divisible by $x^{3}+1$. | P(x)=-\frac{1}{2}(x^{2}+1)(x^{2}+x-1) | 50 | 27 |
math | 1. The smallest natural number whose square ends with three fours is 38, since $38^{2}=1444$. What is the next smallest natural number with this property? | 462 | 40 | 3 |
math | Example 1 Let the positive integer $n$ be a multiple of 75, and have exactly 75 positive divisors (including 1 and itself). Find the minimum value of $n$. | 32400 | 42 | 5 |
math | 1. Given vectors $\overrightarrow{A P}=(1, \sqrt{3}), \overrightarrow{P B}=(-\sqrt{3}, 1)$, then the angle between vector $\overrightarrow{A P}$ and $\overrightarrow{A B}$ is $\qquad$ | \frac{\pi}{4} | 61 | 7 |
math | SG. 2 Let $[x]$ be the largest integer not greater than $x$. If $B=[10+\sqrt{10+\sqrt{10+\sqrt{10+\cdots}}}]$, find the value of $B$. | 13 | 53 | 2 |
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