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 | Steve needed to address a letter to $2743$ Becker Road. He remembered the digits of the address, but he forgot the correct order of the digits, so he wrote them down in random order. The probability that Steve got exactly two of the four digits in their correct positions is $\tfrac m n$ where $m$ and $n$ are relatively... | 5 | 85 | 1 |
math | 1. [2] What is the smallest non-square positive integer that is the product of four prime numbers (not necessarily distinct)? | 24 | 26 | 2 |
math | 12. (10 points) Find the smallest natural number that can be expressed as the sum of four different sequences of at least two consecutive non-zero natural numbers.
---
The translation maintains the original format and line breaks as requested. | 45 | 47 | 2 |
math | The function $\mathrm{f}(\mathrm{n})$ is defined on the positive integers and takes non-negative integer values. It satisfies (1) $f(m n)=f(m)+f(n),(2) f(n)=0$ if the last digit of $n$ is 3, (3) $f(10)=0$. Find $\mathrm{f}(1985)$. | 0 | 83 | 1 |
math | Let $c$ denote the largest possible real number such that there exists a nonconstant polynomial $P$ with \[P(z^2)=P(z-c)P(z+c)\] for all $z$. Compute the sum of all values of $P(\tfrac13)$ over all nonconstant polynomials $P$ satisfying the above constraint for this $c$. | \frac{13}{23} | 77 | 9 |
math | 2. The express passenger train, two days ago in the morning, departed from Gevgelija to Skopje, with several carriages in which there was a certain number of passengers. At the station in Veles, one passenger disembarked from the first carriage, two passengers disembarked from the last carriage, and no passengers board... | 380 | 166 | 3 |
math | 11. (12 points) 1 kilogram of soybeans can be made into 3 kilograms of tofu, and 1 kilogram of soybean oil requires 6 kilograms of soybeans. Tofu is 3 yuan per kilogram, and soybean oil is 15 yuan per kilogram. A batch of soybeans totals 460 kilograms, and after being made into tofu or soybean oil and sold, 1800 yuan i... | 360 | 119 | 3 |
math | ## Task 3 - 010823
In the mess of one of our fishing fleet's ships, the crew members are talking about their ages.
The helmsman says: "I am twice as old as the youngest sailor and 6 years older than the engineer."
The 1st sailor says: "I am 4 years older than the 2nd sailor and just as many years older than the youn... | 40 | 140 | 2 |
math | G3.1 In $\triangle A B C, \angle A B C=2 \angle A C B, B C=2 A B$. If $\angle B A C=a^{\circ}$, find the value of $a$.
G3.2 Given that $x+\frac{1}{x}=\sqrt{2}, \frac{x^{2}}{x^{4}+x^{2}+1}=b$, find the value of $b$.
G3.3 If the number of positive integral root(s) of the equation $x+y+2 x y=141$ is $c$, find the value o... | 90 | 200 | 2 |
math | 3. Let $A B C$ be a triangle such that $A B=7$, and let the angle bisector of $\angle B A C$ intersect line $B C$ at $D$. If there exist points $E$ and $F$ on sides $A C$ and $B C$, respectively, such that lines $A D$ and $E F$ are parallel and divide triangle $A B C$ into three parts of equal area, determine the numbe... | 13 | 107 | 2 |
math | 3. In an equilateral triangle $\mathrm{ABC}$, the height $\mathrm{BH}$ is drawn. On the line $\mathrm{BH}$, a point $\mathrm{D}$ is marked such that $\mathrm{BD}=\mathrm{AB}$. Find $\angle \mathrm{CAD}$. | 15 | 63 | 2 |
math | In the quadrilateral $ABCD$, the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$, as well with side $AD$ an angle of $30^o$. Find the acute angle between the diagonals $AC$ and $BD$. | 80^\circ | 73 | 4 |
math | 8. (6 points) My sister is 26 years old this year, and my younger sister is 18 years old this year. When the sum of their ages was 20, my sister was $\qquad$ years old. | 14 | 51 | 2 |
math | 20. The perimeter of triangle $A B C$, in which $A B<A C$, is 7 times the side length of $B C$. The inscribed circle of the triangle touches $B C$ at $E$, and the diameter $D E$ of the circle is drawn, cutting the median from $A$ to $B C$ at $F$. Find $\frac{D F}{F E}$.
(3 marks)
20. The perimeter of $\triangle A B C$ ... | \frac{5}{7} | 181 | 7 |
math | 13. (6 points) Mathematician Wiener is the founder of cybernetics. At the ceremony where he was granted his Ph.D. from Harvard University, someone, curious about his youthful appearance, asked him about his age. Wiener's response was quite interesting. He said, “The cube of my age is a four-digit number, and the fourth... | 18 | 162 | 2 |
math | Florián was thinking about what bouquet he would have tied for his mom for Mother's Day. In the florist's, according to the price list, he calculated that whether he buys 5 classic gerberas or 7 mini gerberas, the bouquet, when supplemented with a decorative ribbon, would cost the same, which is 295 crowns. However, if... | 85 | 126 | 2 |
math | 1. Determine all pairs $(a, b)$ of real numbers for which each of the equations
$$
x^{2}+a x+b=0, \quad x^{2}+(2 a+1) x+2 b+1=0
$$
has two distinct real roots, and the roots of the second equation are the reciprocals of the roots of the first equation. | =-\frac{1}{3},b=-1 | 82 | 11 |
math | 5. For what value of $z$ does the function $h(z)=$ $=\sqrt{1.44+0.8(z+0.3)^{2}}$ take its minimum value? | -0.3 | 44 | 4 |
math | 1. Petrov and Vasechkin were repairing a fence. Each had to nail a certain number of boards (the same amount). Petrov nailed two nails into some boards and three nails into others. Vasechkin nailed three nails into some boards and five nails into the rest. Find out how many boards each of them nailed, given that Petrov... | 30 | 89 | 2 |
math | 2.81 The sum of the following 7 numbers is exactly 19:
$$\begin{array}{l}
a_{1}=2.56, a_{2}=2.61, a_{3}=2.65, a_{4}=2.71, a_{5}=2.79, a_{6}= \\
2.82, a_{7}=2.86 .
\end{array}$$
To approximate $a_{i}$ with integers $A_{i}$ $(1 \leqslant i \leqslant 7)$, such that the sum of $A_{i}$ is still 19, and the maximum value $M... | 61 | 188 | 2 |
math | 3. Let the function $f(x, y)$ satisfy:
(1) $f(x, x)=x$;
(2) $f(k x, k y)=k f(x, y)$;
(3) $f\left(x_{1}+x_{2}, y_{1}+y_{2}\right)=f\left(x_{1}, y_{1}\right)+f\left(x_{2}, y_{2}\right)$;
(4) $f(x, y)=f\left(y, \frac{x+y}{2}\right)$.
Then $f(x, y)=$ . $\qquad$ | \frac{x}{3}+\frac{2 y}{3} | 134 | 14 |
math | 13.318. The price of one product was reduced twice, each time by $15 \%$. For another product, which initially had the same price as the first, the price was reduced once by $x \%$. What should the number $x$ be so that after all the specified reductions, both products have the same price again? | 27.75 | 72 | 5 |
math | 2. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
\frac{a_{n+1}+a_{n}-1}{a_{n+1}-a_{n}+1}=n\left(n \in \mathbf{Z}_{+}\right) \text {, and } a_{2}=6 \text {. }
$$
Then the general term formula of the sequence $\left\{a_{n}\right\}$ is $a_{n}=$ $\qquad$ | a_{n}=n(2n-1) | 112 | 11 |
math | 1. It is known that $\operatorname{tg}(\alpha+2 \gamma)+\operatorname{tg} \alpha+\frac{5}{2} \operatorname{tg}(2 \gamma)=0, \operatorname{tg} \gamma=-\frac{1}{2}$. Find $\operatorname{ctg} \alpha$. | \frac{1}{3} | 74 | 7 |
math | 9. Let $A$ be the set of real numbers $x$ satisfying the inequality $x^{2}+x-110<0$ and $B$ be the set of real numbers $x$ satisfying the inequality $x^{2}+10 x-96<0$. Suppose that the set of integer solutions of the inequality $x^{2}+a x+b<0$ is exactly the set of integers contained in $A \cap B$. Find the maximum val... | 71 | 114 | 2 |
math | 15. Find the smallest positive real number $k$, such that the inequality
$$
a b+b c+c a+k\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geqslant 9
$$
holds for all positive real numbers $a, b, c$. | 2 | 74 | 1 |
math | 3. Let the function $f(x)=x^{3}+3 x^{2}+6 x+14$, and $f(a)=1, f(b)=19$, then $a+b=$ | -2 | 44 | 2 |
math | 12. The Zmey Gorynych has 2000 heads. A legendary hero can cut off 33, 21, 17, or 1 head with one strike of his sword, but in return, the Zmey grows 48, 0, 14, or 349 heads respectively. If all heads are cut off, no new ones grow. Can the hero defeat the Zmey? How should he act? | 17 | 101 | 2 |
math | A way to check if a number is divisible by 7 is to subtract, from the number formed by the remaining digits after removing the units digit, twice the units digit, and check if this number is divisible by 7. For example, 336 is divisible by 7, because $33 - 2 \cdot 6 = 21$ is divisible by 7, but 418 is not because $41 -... | 13 | 163 | 2 |
math | Example 4.13 Find the number of 7-combinations of the multiset $S=\{4 \cdot a, 4 \cdot b, 3 \cdot c, 3 \cdot d\}$. | 60 | 47 | 2 |
math | 9. (10 points) Congcong performs a math magic trick, writing down $1, 2, 3, 4, 5, 6, 7$ on the blackboard, and asks someone to choose 5 of these numbers. Then, the person calculates the product of these 5 numbers and tells Congcong the result, and Congcong guesses the numbers chosen. When it was Bènbèn's turn to choose... | 420 | 137 | 3 |
math | $:$ enderovv B.A.
The numbers $a, b, c$ are such that $a^{2}(b+c)=b^{2}(a+c)=2008$ and $a \neq b$. Find the value of the expression $c^{2}(a+b)$. | 2008 | 64 | 4 |
math | 2. From the numbers $1,2,3, \cdots, 10$, if 3 numbers are randomly drawn, the probability that at least two of the numbers are consecutive positive integers is $\qquad$ . | \frac{8}{15} | 47 | 8 |
math | 6. (2006 National Winter Camp) Positive integers $a_{1}, a_{2}, \cdots$, $a_{2006}$ (which can be the same) such that $\frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}}$ are all distinct. How many different numbers are there at least among $a_{1}, a_{2}, \cdots, a_{2006}$? | 46 | 124 | 2 |
math | 45. Given an arithmetic progression consisting of four integers. It is known that the largest of these numbers is equal to the sum of the squares of the other three numbers. Find these numbers. | 2,1,0,-1 | 39 | 7 |
math | 4. In the notebook, $n$ integers are written, ordered in descending order $a_{1}>a_{2}>\ldots>a_{n}$ and having a sum of 840. It is known that the $k$-th number written in order, $a_{k}$, except for the last one when $k=n$, is $(k+1)$ times smaller than the sum of all the other written numbers. Find the maximum number ... | n_{\max}=7;a_{1}=280,a_{2}=210,a_{3}=168,a_{4}=140,a_{5}=120,a_{6}=105,a_{7}=-183 | 114 | 56 |
math | Find the total number of different integer values the function
$$
f(x)=[x]+[2 x]+\left[\frac{5 x}{3}\right]+[3 x]+[4 x]
$$
takes for real numbers \( x \) with \( 0 \leq x \leq 100 \).
Note: \([t]\) is the largest integer that does not exceed \( t \).
Answer: 734. | 734 | 95 | 3 |
math | One, (20 points) Given the equation $a x^{2}+4 x+b=0$ $(a<0)$ has two real roots $x_{1} 、 x_{2}$, and the equation $a x^{2}+3 x+b=0$ has two real roots $\alpha 、 \beta$.
(1) If $a 、 b$ are both negative integers, and $|\alpha-\beta|=1$, find the values of $a 、 b$;
(2) If $\alpha<1<\beta<2, x_{1}<x_{2}$, prove:
$-2<x_{1... | a=-1, b=-2 | 145 | 7 |
math | 4. Bag $A$ contains 2 ten-yuan bills and 3 one-yuan bills, and bag $B$ contains 4 five-yuan bills and 3 one-yuan bills. Now, two bills are randomly drawn from each bag. The probability that the sum of the remaining bills in $A$ is greater than the sum of the remaining bills in $B$ is $\qquad$ | \frac{9}{35} | 83 | 8 |
math | Define a $\text{good~word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ - some of these letters may not appear in the sequence - and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. How many seven-... | 192 | 88 | 3 |
math | L OM - I - Task 5
Find all pairs of positive integers $ x $, $ y $ satisfying the equation $ y^x = x^{50} $. | 8 | 36 | 1 |
math | Three, 13. (20 points) Given the function
$$
f(x)=a x^{2}+b x+c(a, b, c \in \mathbf{R}),
$$
when $x \in[-1,1]$, $|f(x)| \leqslant 1$.
(1) Prove: $|b| \leqslant 1$;
(2) If $f(0)=-1, f(1)=1$, find the value of $a$. | a=2 | 113 | 3 |
math | ## Problem 8'.3. (Problem for the UBM award)
Is it possible to find 100 straight lines in the plane such that there
are exactly 1998 intersecting points?
| 1998 | 44 | 4 |
math | 408. The probability distribution of a discrete two-dimensional random variable is given:
| $y$ | $x$ | | |
| :---: | :---: | :---: | :---: |
| | 3 | 10 | 12 |
| | | | |
| 4 | 0.17 | 0.13 | 0.25 |
| 5 | 0.10 | 0.30 | 0.05 |
Find the distribution laws of the components $X$ and $Y$. | \begin{pmatrix}X&3&10&12\\p&0.27&0.43&0.30\end{pmatrix}\quad\begin{pmatrix}Y&4&5\\p&0.55&0.45\end{pmatrix} | 128 | 68 |
math | \section*{Problem 6B - 111236B}
50 white and 50 black balls are to be distributed into two indistinguishable urns in such a way that neither urn remains empty and all balls are used.
How should the balls be distributed between the two urns so that the probability of drawing a white ball when one of the two urns is ch... | \frac{74}{99} | 294 | 9 |
math | 3. For the parabola $y^{2}=4 a x(a>0)$, the focus is $A$. With $B(a+4,0)$ as the center and $|A B|$ as the radius, a semicircle is drawn above the $x$-axis, intersecting the parabola at two different points $M, N$. Let $P$ be the midpoint of the segment $M N$. (I) Find the value of $|A M|+|A N|$; (II) Does there exist ... | 8 | 141 | 1 |
math | Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences. | 31 | 48 | 2 |
math | Question 239, Let $M$ be a set composed of a finite number of positive integers, and $M=U_{i=1}^{20} A_{i}=U_{i=1}^{20} B_{i}$, where $A_{i} \neq \emptyset$, $B_{i} \neq \emptyset, i=1, 2, \ldots, 20$, and for any $1 \leq i<j \leq 20$, we have $A_{i} \cap A_{j}=\emptyset, B_{i} \cap B_{j}=\emptyset$.
It is known that ... | 180 | 227 | 3 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty}\left(\frac{n+3}{n+1}\right)^{-n^{2}}$ | 0 | 42 | 1 |
math | 10. Define $S_{n}=\{0,1, \cdots, n-1\}\left(n \in \mathbf{Z}_{+}\right)$.
For a bijection $f: S_{n} \rightarrow S_{n}$, if $n \mid(a+b-c), a, b, c \in S_{n}$, then
$$
n \mid(f(a)+f(b)-f(c)) \text {. }
$$
Let the set of all solutions of $f$ be $T_{n}$. If $\left|T_{n}\right|=60$, find $n$. | 61,77,93,99,122,124,154,186,198 | 134 | 31 |
math | Example 4 Factorize $x^{5}+x+1$ over the integers.
| (x^{2}+x+1)(x^{3}-x^{2}+1) | 19 | 20 |
math | I2.1 If $\alpha, \beta$ are roots of $x^{2}-10 x+20=0$, find $a$, where $a=\frac{1}{\alpha}+\frac{1}{\beta}$.
I2.2 If $\sin \theta=a\left(0^{\circ}<\theta<90^{\circ}\right)$, and $10 \cos 2 \theta=b$, find $b$.
I2.3 The point $A(b, c)$ lies on the line $2 y=x+15$. Find $c$.
I2.4 If $x^{2}-c x+40 \equiv(x+k)^{2}+d$, fin... | \frac{1}{2},5,10,15 | 158 | 14 |
math | 5. There are 100 different cards with numbers $2,5,2^{2}, 5^{2}, \ldots, 2^{50}, 5^{50}$ (each card has exactly one number, and each number appears exactly once). In how many ways can 2 cards be chosen so that the product of the numbers on the chosen cards is a cube of an integer? | 1074 | 85 | 4 |
math | Example. Regarding the equation about $x$: $x+\frac{1}{x-1}=a+\frac{1}{a-1}$. | x_{1}=a, \mathbf{x}_{2}=\frac{\mathrm{a}}{\mathrm{a}-1} | 31 | 27 |
math | 16. Let $f(x)=k\left(x^{2}-x+1\right)-x^{4}(1-x)^{4}$. If for any $x \in[0,1]$, we have $f(x) \geqslant 0$, then the minimum value of $k$ is $\qquad$ . | \frac{1}{192} | 73 | 9 |
math | 54. How many five-digit numbers are there in which
a) the digit 5 appears exactly once?
b) the digit 5 appears no more than once?
c) the digit 5 appears at least once? | 37512 | 46 | 5 |
math | Let the sequence $(a_n)_{n \in \mathbb{N}}$, where $\mathbb{N}$ denote the set of natural numbers, is given with $a_1=2$ and $a_{n+1}$ $=$ $a_n^2$ $-$ $a_n+1$. Find the minimum real number $L$, such that for every $k$ $\in$ $\mathbb{N}$
\begin{align*} \sum_{i=1}^k \frac{1}{a_i} < L \end{align*} | 1 | 122 | 1 |
math | $1 \cdot 60$ Find the smallest positive integer $n>1$, such that the arithmetic mean of $1^{2}, 2^{2}, 3^{2}, \cdots, n^{2}$ is a perfect square. | 337 | 52 | 3 |
math | Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$. | 4 | 57 | 1 |
math | 3. (BUL) Solve the equation $\cos ^{n} x-\sin ^{n} x=1$, where $n$ is a given positive integer. | {\pi\mid\in\mathbb{Z}}forevenn{2\pi,2\pi-\pi/2\mid\in\mathbb{Z}}foroddn | 36 | 40 |
math | 2. Let $a$, $b$, and $c$ be the lengths of the sides of $\triangle ABC$, and suppose they satisfy $a^{2}+b^{2}=m c^{2}$. If $\frac{\cot C}{\cot A+\cot B}=$ 999, then $m=$ . $\qquad$ | 1999 | 73 | 4 |
math | How to determine the function $\ln z$ for a complex argument $z$?
# | w_{k}=\ln|z|+i(\varphi+2k\pi)(k\in{Z}) | 18 | 26 |
math | Solve the following equation:
$$
\frac{1}{\sqrt{2+x}-\sqrt{2-x}}+\frac{1}{\sqrt{2+x}+\sqrt{2-x}}=1
$$ | 2 | 46 | 1 |
math |
1. Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation:
$$
a !+b !=2^{n}
$$
Notation: $k !=1 \times 2 \times \cdots \times k$, for example: $1 !=1$, and $4 !=1 \times 2 \times 3 \times 4=24$.
| (1,1,1),(2,2,2),(2,3,3),(3,2,3) | 85 | 25 |
math | 10. There are three right-angled triangles containing a
$30^{\circ}$ angle, they are of different sizes,
but they have one side equal. Then, in the order of the areas of these three triangles from largest to smallest, the ratio of their hypotenuses is | 2: \frac{2 \sqrt{3}}{3}: 1 | 61 | 16 |
math | For which positive number quadruples $x, y, u, v$ does the following system of equations hold?
$$
x+y=u v x y=u+v x y u v=16
$$ | u=v=2 | 42 | 4 |
math | 3. (i) Find all prime numbers for which -3 is a quadratic residue;
(ii) Find all prime numbers for which ±3 is a quadratic residue;
(iii) Find all prime numbers for which ±3 is a quadratic non-residue;
(iv) Find all prime numbers for which 3 is a quadratic residue and -3 is a quadratic non-residue;
(v) Find all prime n... | 100^{2}-3=13 \cdot 769 ; 150^{2}+3=3 \cdot 13 \cdot 577 | 130 | 39 |
math | 【Question 6】
Using the digits $0, 1, 2, 3, 4, 5$, many positive integers can be formed. Arranging them in ascending order gives $1, 2, 3, 4, 5, 10$, 11, 12, 13, $\cdots$. What is the position of 2015 in this sequence? | 443 | 90 | 3 |
math | ## Task Condition
Find the point $M^{\prime}$ symmetric to the point $M$ with respect to the line.
$M(2 ; 1 ; 0)$
$\frac{x-2}{0}=\frac{y+1.5}{-1}=\frac{z+0.5}{1}$ | M^{\}(2;-2;-3) | 69 | 10 |
math | 12. Let the function $f(x)$ be a differentiable function defined on $(-\infty, 0)$, with its derivative being $f^{\prime}(x)$, and it satisfies $2 f(x)+x f^{\prime}(x)>x^{2}$. Then the solution set of the inequality $(x+2017)^{2} f(x+2017)-f(-1)>0$ is $\qquad$ . | (-\infty,-2018) | 99 | 10 |
math | 2. (3 points) Calculate: $99 \times \frac{5}{8}-0.625 \times 68+6.25 \times 0.1=$ $\qquad$ | 20 | 46 | 2 |
math | Fix positive integers $k$ and $n$.Derive a simple expression involving Fibonacci numbers for the number of sequences $(T_1,T_2,\ldots,T_k)$ of subsets $T_i$ of $[n]$ such that $T_1\subseteq T_2\supseteq T_3\subseteq T_4\supseteq\ldots$.
[color=#008000]Moderator says: and the original source for this one is Richard Sta... | F_{k+2}^n | 134 | 8 |
math | 3. Given a positive integer $n \geqslant 2$. It is known that non-negative real numbers $a_{1}$, $a_{2}, \cdots, a_{n}$ satisfy $a_{1}+a_{2}+\cdots+a_{n}=1$. Find
$$
a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}+\sqrt{a_{1} a_{2} \cdots a_{n}}
$$
the maximum value.
(Zhang Lei provided the problem) | \frac{9}{8}whenn=2;1whenn\geqslant3 | 123 | 21 |
math | 4.055. The first term of a certain infinite geometric progression with a common ratio $|q|<1$ is 1, and its sum is $S$. A new infinite geometric progression is formed from the squares of the terms of this progression. Find its sum. | \frac{S^{2}}{2S-1} | 58 | 13 |
math | Are there positive integers $a, b, c$, such that the numbers $a^2bc+2, b^2ca+2, c^2ab+2$ be perfect squares? | \text{No} | 42 | 5 |
math | 14. Solve the inequality $\log _{2}\left(x^{12}+3 x^{10}+5 x^{8}+3 x^{6}+1\right)<1+\log _{2}\left(x^{4}+1\right)$. | x\in(-\sqrt{\frac{\sqrt{5}-1}{2}},\sqrt{\frac{\sqrt{5}-1}{2}}) | 60 | 31 |
math | 11. How many four-digit numbers greater than 5000 can be formed from the digits $0,1,2$. $3,4,5,6,7,8,9$ if only the digit 4 may be repeated? | 2645 | 53 | 4 |
math | 1. Find the sum of the numbers $1-2+3-4+5-6+\ldots+2013-2014$ and $1+2-3+4-5+6-\ldots-2013+2014$. | 2 | 61 | 1 |
math | 2. 79 In a game, scoring is as follows: answering an easy question earns 3 points, and answering a difficult question earns 7 points. Among the integers that cannot be the total score of a player, find the maximum value. | 11 | 51 | 2 |
math | 43. On a bookshelf, there are ten different books, three of which are on mathematics. In how many ways can all these books be arranged in a row so that the mathematics books are together? | 241920 | 42 | 6 |
math | 8. Let $\left(1+x+x^{2}\right)^{150}=\sum_{k=0}^{300} c_{k} x^{k}$, where $c_{0}$, $c_{1}, \cdots, c_{300}$ are constants. Then $\sum_{k=0}^{100} c_{3 k}=$ $\qquad$ . | 3^{149} | 89 | 6 |
math | 3. If a two-digit number $\bar{x}$ and a three-digit number $\overline{3 y z}$ have a product of 29,400, then, $x+y+z=$ $\qquad$ | 18 | 47 | 2 |
math | Task 2. (10 points)
Find the denominator of the fraction $\frac{100!}{28^{20}}$ after it has been reduced to its simplest form.
(The expression 100! is equal to the product of the first 100 natural numbers: $100!=1 \cdot 2 \cdot 3 \cdot 4 \cdot \ldots \cdot 100$.$) .$ | 2401 | 95 | 4 |
math | ## Problem A1
Find all positive integers $n<1000$ such that the cube of the sum of the digits of $n$ equals $n^{2}$.
| 1,27 | 38 | 4 |
math | 2. In the Cartesian coordinate system, a line segment $A B$ of length 1 moves on the $x$-axis (point $A$ is to the left of $B$), point $P(0,1)$ is connected to $A$ by a line, and point $Q(1,2)$ is connected to $B$ by a line. Then the equation of the trajectory of the intersection point $R$ of lines $P A$ and $Q B$ is $... | y(x-2)=-2 | 107 | 7 |
math | 415. Calculate $\sin 210^{\circ}$. | -\frac{1}{2} | 16 | 7 |
math | ## Task 3 - 210613
The three students Bianka, Heike, and Kerstin harvested white cabbage in the school garden, a total of 128 cabbages. Bianka harvested exactly 8 more cabbages than Heike, and Kerstin harvested exactly 5 fewer cabbages than Bianka.
How many cabbages did each of the three girls harvest in total? | Bianka:47,Heike:39,Kerstin:42 | 88 | 18 |
math | The positive integers $v, w, x, y$, and $z$ satisfy the equation \[v + \frac{1}{w + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}} = \frac{222}{155}.\] Compute $10^4 v + 10^3 w + 10^2 x + 10 y + z$. | 12354 | 95 | 5 |
math | 1. Find all integer pairs $(x, y)$ that satisfy the equation $x^{2}-2 x y+126 y^{2}=2009$. (Supplied by Zhang Pengcheng) | (1,4),(7,4),(-1,-4),(-7,-4) | 44 | 19 |
math | On a $4 \times 4$ board, the numbers from 1 to 16 must be placed in the cells without repetition, such that the sum of the numbers in each row, column, and diagonal is the same. We call this sum the Magic Sum.
a) What is the Magic Sum of this board?
b) If the sum of the cells marked with $X$ in the board below is 34, ... | 5 | 208 | 1 |
math | 10.1. Solve the equation $\left(x^{4}+x+1\right)(\sqrt[3]{80}-\sqrt[3]{0.01})=2(\sqrt[3]{5.12}+\sqrt[3]{0.03375})$. | x_{1}=0,x_{2}=-1 | 65 | 11 |
math | 1. Alice and the White Rabbit left the Rabbit's house together at noon and went to the Duchess's reception. Having walked halfway, the Rabbit remembered that he had forgotten his gloves and fan, and ran home for them at twice the speed he had walked with Alice. Grabbing the gloves and fan, he ran to the Duchess (at the... | 12:40 | 123 | 5 |
math | A geometric series is one where the ratio between each two consecutive terms is constant (ex. 3,6,12,24,...). The fifth term of a geometric series is 5!, and the sixth term is 6!. What is the fourth term? | 20 | 55 | 2 |
math | 3. Find the smallest possible value of the expression
$$
\left(\frac{x y}{z}+\frac{z x}{y}+\frac{y z}{x}\right)\left(\frac{x}{y z}+\frac{y}{z x}+\frac{z}{x y}\right)
$$
where $x, y, z$ are non-zero real numbers. | 9 | 82 | 1 |
math | 3. In the number $2 * 0 * 1 * 6 * 07 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,2,4,5,6,7$ (digits can repeat) so that the resulting 11-digit number is divisible by 75. In how many ways can this be done? | 432 | 81 | 3 |
math | 1. Given the sequence $\left\{x_{n}\right\}$ satisfies $x_{n}=\frac{n}{n+2016}$. If $x_{2016}=x_{m} x_{n}$, then a solution for the positive integers $m, n$ is $\{m, n\}$ $=$ $\qquad$ . | \{4032,6048\} | 79 | 13 |
math | Example 3. Calculate $\int_{4}^{9}\left(\frac{2 x}{5}+\frac{1}{2 \sqrt{x}}\right) d x$. | 14 | 38 | 2 |
math | 6. Let $n$ be the smallest positive integer satisfying the following conditions: (1) $n$ is a multiple of 75; (2) $n$ has exactly 75 positive divisors (including 1 and itself). Find $\frac{n}{75}$. | 432 | 60 | 3 |
math | 9.1. Chords $A A^{\prime}, B B^{\prime}$, and $C C^{\prime}$ of a sphere intersect at a common point $S$. Find the sum $S A^{\prime}+S B^{\prime}+S C^{\prime}$, if $A S=6, B S=3, C S=2$, and the volumes of pyramids $S A B C$ and $S A^{\prime} B^{\prime} C^{\prime}$ are in the ratio $2: 9$. If the answer is not an integ... | 18 | 136 | 2 |
math | 3. Let $[x]$ denote the greatest integer not greater than the real number $x$. The number of real roots of the equation $\lg ^{2} x-[\lg x]-2=0$ is $\qquad$ .
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 3 | 75 | 1 |
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