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 | 2. $[4]$ If $x^{x}=2012^{2012^{2013}}$, find $x$. | 2012^{2012} | 32 | 10 |
math | 3. Find the smallest number that gives a remainder of 1 when divided by 2, and a remainder of 2 when divided by 3. | 5 | 31 | 1 |
math | What is the difference between the largest possible three-digit positive integer with no repeated digits and the smallest possible three-digit positive integer with no repeated digits? | 885 | 29 | 3 |
math | $$
\begin{array}{l}
\text { 2. If } f(g(x))=\sin 2 x, \\
g(x)=\tan \frac{x}{2}(0<x<\pi),
\end{array}
$$
then $f\left(\frac{\sqrt{2}}{2}\right)=$ $\qquad$ | \frac{4 \sqrt{2}}{9} | 74 | 12 |
math | 1. When $n=1,2, \cdots, 2006$, the sum of the lengths of the segments cut off by the x-axis for all quadratic functions $y=n(n+1) x^{2}-(2 n+1) x+1$ is $\qquad$ . | \frac{2006}{2007} | 65 | 13 |
math | I am thinking of a non-negative number in the form of a fraction with an integer numerator and a denominator of 12. When I write it as a decimal number, it will have one digit before and one digit after the decimal point, and both of these digits will be non-zero. There are multiple numbers that have both of these prop... | 8.5 | 101 | 3 |
math | 9. Given that $\alpha, \beta$ are the two roots of the quadratic equation $2 x^{2}-t x-2=0$ with respect to $x$, and $\alpha<\beta$, if the function $f(x)=$ $\frac{4 x-t}{x^{2}+1}$.
(1) Find the value of $\frac{f(\alpha)-f(\beta)}{\alpha-\beta}$;
(2) For any positive numbers $\lambda_{1}, \lambda_{2}$, prove that $\lef... | 2|\alpha-\beta| | 190 | 6 |
math | 7. Given $a, b \in \mathbf{R}^{+}$ and $\frac{\sin ^{4} x}{a}+\frac{\cos ^{4} x}{b}=\frac{1}{a+b}$, then $\frac{\sin ^{8} x}{a^{3}}+\frac{\cos ^{8} x}{b^{3}}=$ $\qquad$ | \frac{1}{(+b)^{3}} | 86 | 11 |
math | 7.029. $0.25^{\log _{2} \sqrt{x+3}-0.5 \log _{2}\left(x^{2}-9\right)}=\sqrt{2(7-x)}$. | 5 | 51 | 1 |
math | 8. (10 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=27 \\
y^{2}+y z+z^{2}=25 \\
z^{2}+x z+x^{2}=52
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 30 | 102 | 2 |
math | Let $A=1989^{1990}-1988^{1990}, \quad B=1989^{1989}-1988^{1989}$. What is the greatest common divisor of $A$ and $B$? | 1 | 64 | 1 |
math | ## Task 2 - 211212
Determine all ordered pairs $(x ; y)$ of non-zero real numbers $x, y$ that satisfy the following system of equations:
$$
\begin{aligned}
(x+y)^{2}+3(x+y) & =4 \\
\frac{1}{x}+\frac{1}{y} & =-\frac{1}{6}
\end{aligned}
$$ | (3,-2)(-2,3) | 92 | 10 |
math | 7. (6 points) The triples of numbers $x, y, z$ and $a, b, c$ satisfy the system of equations
$$
\left\{\begin{array}{c}
12 x^{2}+4 y^{2}+9 z^{2}=20 \\
9 a^{2}+b^{2}+3 c^{2}=25
\end{array}\right.
$$
Within what limits can the expression $2cx-2ay+bz$ vary? | [-\frac{10}{3}\sqrt{5};\frac{10}{3}\sqrt{5}] | 109 | 25 |
math | Three. (25 points) Let $n$ be an integer. If there exist integers $x, y, z$ satisfying
$$
n=x^{3}+y^{3}+z^{3}-3 x y z \text {, }
$$
then $n$ is said to have property $P$.
(1) Determine whether $1, 2, 3$ have property $P$;
(2) Among the 2014 consecutive integers $1, 2, \cdots, 2014$, how many do not have property $P$? | 448 | 127 | 3 |
math | Four, among the 10-digit numbers where each digit is different, how many are multiples of 11111? Prove your conclusion.
In the 10-digit numbers where each digit is different, how many are multiples of 11111? Prove your conclusion. | 3456 | 62 | 4 |
math | 2. 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 $\frac{n}{75}$.
(Eighth American Mathematical Invitational) | 432 | 67 | 3 |
math | Compute the only element of the set \[\{1, 2, 3, 4, \ldots\} \cap \left\{\frac{404}{r^2-4} \;\bigg| \; r \in \mathbb{Q} \backslash \{-2, 2\}\right\}.\]
[i]Proposed by Michael Tang[/i] | 2500 | 86 | 4 |
math | M4. What are the solutions of the simultaneous equations:
$$
\begin{aligned}
3 x^{2}+x y-2 y^{2} & =-5 \\
x^{2}+2 x y+y^{2} & =1 ?
\end{aligned}
$$ | \frac{3}{5},-\frac{8}{5};-\frac{3}{5},\frac{8}{5} | 60 | 28 |
math | Rita the painter rolls a fair $6\text{-sided die}$that has $3$ red sides, $2$ yellow sides, and $1$ blue side. Rita rolls the die twice and mixes the colors that the die rolled. What is the probability that she has mixed the color purple? | \frac{1}{6} | 63 | 7 |
math | Problem 7.6. The distance between cities A and B is an integer number of kilometers. Along the road between the cities, there is a signpost every kilometer: on one side is the distance to city A, and on the other side is the distance to city B. Slava walked from city A to city B. During his journey, Slava calculated th... | 39 | 120 | 2 |
math | 9. (16 points) Given the function
$$
f(x)=\ln (a x+b)+x^{2} \quad (a \neq 0).
$$
(1) If the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$ is $y=x$, find the values of $a$ and $b$;
(2) If $f(x) \leqslant x^{2}+x$ always holds, find the maximum value of $a b$. | \frac{\mathrm{e}}{2} | 114 | 10 |
math | Let $ M(n )\equal{}\{\minus{}1,\minus{}2,\ldots,\minus{}n\}$. For every non-empty subset of $ M(n )$ we consider the product of its elements. How big is the sum over all these products? | -1 | 55 | 2 |
math | Let $\mathbb{R}^{+}$ be the set of positive real numbers. Determine all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that, for all positive real numbers $x$ and $y$,
$$ f(x+f(x y))+y=f(x) f(y)+1 $$
(Ukraine) Answer: $f(x)=x+1$. | f(x) = x + 1 | 90 | 8 |
math | A capricious mathematician writes a book with pages numbered from $2$ to $400$. The pages are to be read in the following order. Take the last unread page ($400$), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the ... | 397 | 125 | 3 |
math | Find the law that forms the sequence 425, 470, 535, 594, 716, $802, \ldots$ and give its next two terms. | 870,983 | 48 | 7 |
math | Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$.
[i]2021 CCA Math Bonanza Lightning Round #3.4[/i] | 164 | 66 | 3 |
math | 2. If the equation with respect to $x$
$$
x^{2}+2(m+3) x+m^{2}+3=0
$$
has two real roots $x_{1}$ and $x_{2}$, then the minimum value of $\left|x_{1}-1\right|+\left|x_{2}-1\right|$ is $\qquad$. | 6 | 81 | 1 |
math | $1 \cdot 27$ Find the smallest prime $p$ that cannot be expressed as $\left|3^{a}-2^{b}\right|$, where $a$ and $b$ are non-negative integers. | 41 | 47 | 2 |
math | 11. In the Cartesian coordinate system $x O y$, for the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, the right focus is $F(c, 0)$. If there exists a line $l$ passing through point $F$ intersecting the ellipse at points $A, B$, such that $O A \perp O B$. Find the range of the eccentricity $e=\frac{c}{a}$ ... | [\frac{\sqrt{5}-1}{2},1) | 115 | 13 |
math | ## Problem Statement
Calculate approximately using the differential.
$$
y=\frac{1}{\sqrt{2 x^{2}+x+1}}, x=1.016
$$ | 0.495 | 40 | 5 |
math | 【Question 3】There is a basket of apples. The first time, take out half of all plus 2 more. The second time, take out half of the remaining minus 3. There are still 24 left in the basket. The basket originally had $\qquad$ apples. | 88 | 61 | 2 |
math | Károly, László and Mihály went hunting for deer, foxes, and hares over three days. Each of them shot at least one of each type of game every day, and in total they shot 86 animals. On the first day, they shot 12 foxes and 14 deer, and on the second day, they shot a total of 44 animals. László shot an even number of eac... | 1 | 158 | 1 |
math | 1st Balkan 1984 Problem 4 Given positive reals a, b, c find all real solutions (x, y, z) to the equations ax + by = (x - y) 2 , by + cz = (y - z) 2 , cz + ax = (z - x) 2 . Solution | (x,y,z)=(0,0,0),(,0,0),(0,b,0),(0,,0) | 72 | 24 |
math | 29. When from the numbers from 1 to 333 Tanya excluded all numbers divisible by 3 but not divisible by 7, and all numbers divisible by 7 but not divisible by 3, she ended up with 215 numbers. Did she solve the problem correctly? | 205 | 62 | 3 |
math | In trapezoid $ABCD$, the sides $AD$ and $BC$ are parallel, and $AB = BC = BD$. The height $BK$ intersects the diagonal $AC$ at $M$. Find $\angle CDM$. | \angle CDM = 90^\circ | 50 | 11 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{3}+1}-\sqrt{n-1}}{\sqrt[3]{n^{3}+1}-\sqrt{n-1}}$ | \infty | 59 | 3 |
math | ## Zadatak B-4.7.
Odredite sve prirodne brojeve $x$ koji su rješenje nejednadžbe
$$
\log _{x}^{4} 2017+6 \cdot \log _{x}^{2} 2017>4 \cdot \log _{x}^{3} 2017+4 \cdot \log _{x} 2017
$$
| x\in{2,3,4,5,6,\ldots,44} | 107 | 20 |
math | 4・142 Solve the system of equations
$$\left\{\begin{array}{l}
x_{1} \cdot x_{2} \cdot x_{3}=x_{1}+x_{2}+x_{3}, \\
x_{2} \cdot x_{3} \cdot x_{4}=x_{2}+x_{3}+x_{4}, \\
x_{3} \cdot x_{4} \cdot x_{5}=x_{3}+x_{4}+x_{5}, \\
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\
x_{1985} \cdot x_{1986} \cdot... | x=0, \pm \sqrt{3} | 282 | 11 |
math | What is the maximum value that the area of the projection of a regular tetrahedron with an edge of 1 can take?
# | 0.50 | 28 | 4 |
math | For example, $9 N_{0}$ is the set of all non-negative integers, $f(n)$ is a function such that $f: N_{0} \rightarrow N_{0}$, and for each $n \in N_{0}, f[f(n)]+f(n)=2 n+3$. Find $f(1993)$.
(1994 Bulgarian Olympiad Problem) | 1994 | 85 | 4 |
math | Example 10 Let $a, b, c$ be positive real numbers, find the minimum value of $\frac{a+3 c}{a+2 b+c}+\frac{4 b}{a+b+2 c}-\frac{8 c}{a+b+3 c}$. (3rd China Girls Mathematical Olympiad) | -17 + 12\sqrt{2} | 70 | 12 |
math | Paulinho was studying the Greatest Common Divisor (GCD) at school and decided to practice at home. He called the ages of three people living with him $a, b$, and $c$. Then, he performed some operations with their prime factors and obtained the greatest common divisors of the 3 pairs of numbers. A few days later, he for... | =15,b=60,=20 | 184 | 11 |
math | 1. If real numbers $x, y$ satisfy $4 x^{2}+y^{2}=1$, then the minimum value of $\frac{4 x y}{2 x+y-1}$ is . $\qquad$ | 1-\sqrt{2} | 48 | 6 |
math | 7. Let $a, b>0$, satisfy: the equation $\sqrt{|x|}+\sqrt{|x+a|}=b$ has exactly three distinct real solutions $x_{1}, x_{2}, x_{3}$, and $x_{1}<x_{2}<x_{3}=b$, then the value of $a+b$ is $\qquad$ . | 144 | 79 | 3 |
math | 8.164 Suppose there are 3 piles of stones, and A moves 1 stone from one pile to another each time. A can receive payment from B for each move, the amount of which is equal to the difference between the number of stones in the pile where the stone is placed and the number of stones left in the pile where the stone is ta... | 0 | 133 | 1 |
math | Example 4 If $x \neq 0$, find the maximum value of
$$
\frac{\sqrt{1+x^{2}+x^{4}}-\sqrt{1+x^{4}}}{x}
$$
(1992, National Junior High School Mathematics League) | \sqrt{3}-\sqrt{2} | 61 | 10 |
math | 5. Find all surjective functions $f: \mathbf{N}_{+} \rightarrow \mathbf{N}_{+}$ such that for any $m, n \in \mathbf{N}_{+}$ and any prime $p$, $f(m+n)$ is divisible by $p$ if and only if $f(m) + f(n)$ is divisible by $p$. | f(n)=n | 81 | 4 |
math | Divide 283 into two (positive) parts such that one part is divisible by 13 and the other by 17.
(For the latter, see the article "Integer Solutions of Linear Equations" in the current issue.) | 130153 | 51 | 6 |
math | 18. A positive integer is said to be 'good' if each digit is 1 or 2 and there is neither four consecutive 1 's nor three consecutive 2 's. Let $a_{n}$ denote the number of $n$-digit positive integers that are 'good'. Find the value of $\frac{a_{10}-a_{8}-a_{5}}{a_{7}+a_{6}}$.
(2 marks)
若某正整數的每個數字均為 1 或 2 , 且當中既沒有四個連續的 ... | 2 | 203 | 1 |
math | 6. Inside a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, there is a small ball that is tangent to the diagonal segment $A C_{1}$. Then the maximum radius of the ball is $\qquad$ | \frac{4-\sqrt{6}}{5} | 63 | 12 |
math | Let $ABC$ be a triangle. Find the point $D$ on its side $AC$ and the point $E$ on its side $AB$ such that the area of triangle $ADE$ equals to the area of the quadrilateral $DEBC$, and the segment $DE$ has minimum possible length. | AD = AE = \sqrt{\frac{1}{2} AB \cdot AC} | 64 | 19 |
math | 2.5 A can run a circular track in 40 seconds, B runs in the opposite direction, and meets A every 15 seconds, how many seconds does it take for B to run the track? | 24 | 44 | 2 |
math | 1. (8 points) The calculation result of the expression $(11 \times 24-23 \times 9) \div 3+3$ is | 22 | 36 | 2 |
math | 8. If for any $\alpha, \beta$ satisfying $\alpha \pm \beta \neq k \cdot 360^{\circ}$, we have
$$
\begin{array}{l}
\frac{\sin \left(\alpha+30^{\circ}\right)+\sin \left(\beta-30^{\circ}\right)}{\cos \alpha-\cos \beta} \\
=m \cot \frac{\beta-\alpha}{2}+n,
\end{array}
$$
then the array $(m, n)=$ $\qquad$ | (m, n)=\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) | 122 | 25 |
math | H1. The five-digit integer ' $a 679 b$ ' is a multiple of 72 . What are the values of $a$ and $b$ ? | =3,b=2 | 38 | 5 |
math | Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exists $k$ nonzero vectors $v_1,v_2,…,v_k$ in $\mathbb{R}^n$ such that for every pair $i,j$ of indices with $|i-j|>1$ the vectors $v_i$ and $v_j$ are orthogonal.
[i]Proposed by Alexey Balitskiy, Moscow Institute of Physics and Techno... | \left\lceil \frac{k}{2} \right\rceil | 108 | 15 |
math | The Garfield Super Winners play $100$ games of foosball, in which teams score a non-negative integer number of points and the team with more points after ten minutes wins (if both teams have the same number of points, it is a draw). Suppose that the Garfield Super Winners score an average of $7$ points per game but all... | 81 | 134 | 2 |
math | 5.22 Let \( r \) and \( s \) be positive integers. Derive a formula for the number of ordered quadruples of positive integers \((a, b, c, d)\) that satisfy the following condition:
\[
3^{r} \cdot 7^{s} = [a, b, c] = [a, b, d] = [a, c, d] = [b, c, d] \text{.}
\]
The notation \([x, y, z]\) denotes the least common multi... | (1+4r+6r^2)(1+4s+6s^2) | 141 | 21 |
math | 389. Determine the force of water pressure on the gate wall, which is $20 \mathrm{~m}$ long and $5 \mathrm{m}$ high (assuming the gate is completely filled with water). | 2,45\cdot10^{6} | 46 | 11 |
math | Example 1 Let $P(x)$ be a polynomial of degree $2n$, such that
$$
\begin{array}{l}
P(0)=P(2)=\cdots=P(2 n)=0, \\
P(1)=P(3)=\cdots=P(2 n-1)=2, \\
P(2 n+1)=-6 .
\end{array}
$$
Determine $n$ and $P(x)$. | P(x)=-2 x^{2}+4 x | 97 | 12 |
math | 2. Arrange the order of operations in the expression
$$
1891-(1600: a+8040: a) \times c .
$$
and calculate its value when $a=40$ and $c=4$. Show how the expression can be modified without changing its numerical value. | 927 | 68 | 3 |
math | One, (40 points) Find the smallest integer $c$, such that there exists a positive integer sequence $\left\{a_{n}\right\}(n \geqslant 1)$ satisfying:
$$
a_{1}+a_{2}+\cdots+a_{n+1}<c a_{n}
$$
for all $n \geqslant 1$. | 4 | 83 | 1 |
math | Four, (50 points) On a roadside, there are $n$ parking spaces, and $n$ drivers each driving a car. Each driver parks their car in front of their favorite parking space. If that space is already occupied, they park in the nearest available space down the road. If that space and all the spaces below it are occupied, they... | (n+1)^{n-1} | 161 | 9 |
math | Let $ABCD$ be a tetrahedron. Let $a$ be the length of $AB$ and let $S$ be the area of the projection of the tetrahedron onto a plane perpendicular to $AB$. Determine the volume of the tetrahedron in terms of $a$ and $S$. | \frac{1}{3} S a | 67 | 9 |
math | ## Task 5 - 030715
How many zeros does the product of all natural numbers from 1 to 40 end with? (Justification!) | 9 | 37 | 1 |
math | 4.41 Given that $n$ is a positive integer, determine the number of solutions in ordered pairs of positive integers $(x, y)$ for the equation $\frac{x y}{x+y}=n$.
(21st Putnam Mathematical Competition, 1960) | (2\alpha_{1}+1)(2\alpha_{2}+1)\cdots(2\alpha_{k}+1) | 59 | 31 |
math | The $ 52$ cards in a deck are numbered $ 1, 2, \ldots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let... | 263 | 180 | 5 |
math | 2. $36 S$ is a subset of $\{1,2, \cdots, 1989\}$, and the difference between any two numbers in $S$ cannot be 4 or 7. How many elements can $S$ have at most? | 905 | 59 | 3 |
math | ## Task A-4.2.
In one row, the numbers $1,2, \ldots, 2016$ are written in sequence. In each subsequent row, the sums of two adjacent numbers are written in sequence. For example, in the second row, the numbers $3,5, \ldots$, 4031 are written. In the last row, there is only one number. What is that number? | 2^{2014}\cdot2017 | 93 | 12 |
math | 7. For a real number $x$, $[x]$ denotes the greatest integer not exceeding the real number $x$. For some integer $k$, there are exactly 2008 positive integers $n_{1}, n_{2}, \cdots, n_{2008}$, satisfying
$$
k=\left[\sqrt[3]{n_{1}}\right]=\left[\sqrt[3]{n_{2}}\right]=\cdots=\left[\sqrt[3]{n_{2008}}\right],
$$
and $k \m... | 668 | 151 | 3 |
math | The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the remainder when the smallest possible sum $m+n$ is divided by 1000. | 371 | 77 | 3 |
math | 13. (6 points) There are 6 numbers arranged in a row, their average is 27, it is known that the average of the first 4 numbers is 23, the average of the last 3 numbers is 34, the 4th number is $\qquad$ .
| 32 | 65 | 2 |
math | An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.
a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$;
b) Calculate the distance of $p$ from either base;
c) Determine under what conditions such points $P$ actu... | h^2 \leq ac | 99 | 8 |
math | 5. There are six boxes $B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6}$, and initially each box contains exactly 1 coin. Each time, one of the following two operations can be chosen to perform on them:
(1) Select a box $B_{j}$ $(1 \leqslant j \leqslant 5)$ that contains at least 1 coin, remove 1 coin from box $B_{j}$, and ad... | (0,0,0,0,0, A) | 288 | 13 |
math | 4. (8 points) There is a magical tree with 123 fruits on it. On the first day, 1 fruit will fall from the tree. Starting from the second day, the number of fruits that fall each day is 1 more than the previous day. However, if the number of fruits on the tree is less than the number that should fall on a certain day, t... | 17 | 120 | 2 |
math | 5. (10 points) The product of two decimal numbers, when rounded to the nearest tenth, is 27.6. It is known that both decimals have one decimal place and their units digits are both 5. What is the exact product of these two decimals? $\qquad$ . | 27.55 | 62 | 5 |
math | ## Problem 1
Let the set $G=(k, \infty) \backslash\{k+1\}, k>0$, on which the operation * is defined such that:
$\log _{a}[(x * y)-k]=\log _{a}(x-k) \cdot \log _{a}(y-k), \forall x, y \in G, a>0, a \neq 1$.
a) Determine the number of pairs of integers $(k, a)$ for which the neutral element of the operation * is $e=2... | x_1=2009,x_2=1005+\frac{1}{1004} | 196 | 26 |
math | 10,11 In the parallelepiped $A B C D A 1 B 1 C 1 D 1$, points $K$ and $M$ are taken on the lines $A C$ and $B A 1$, respectively, such that $K M \| D B 1$. Find the ratio $K M: D B 1$. | 1:3 | 77 | 3 |
math | Folklore
The volume of a bottle of kvass is 1.5 liters. The first person drank half of the bottle, the second drank a third of what was left after the first, the third drank a quarter of what remained from the previous, and so on, the fourteenth drank a fifteenth of what was left. How much kvass is left in the bottle? | 0.1 | 79 | 3 |
math | Let $A M C$ be an isosceles triangle with $M$ as the vertex and such that the angle $\widehat{A M C}$ is acute. Let $B$ be the symmetric point of $A$ with respect to $M$ and let $H$ be the foot of the altitude from vertex $C$ in the triangle $A B C$. Suppose that $A H = H M$. Calculate the values of the three angles of... | 30,90,60 | 102 | 8 |
math | How many integers exist from 1 to 1000000 that are neither perfect squares, nor perfect cubes, nor fourth powers?
# | 998910 | 31 | 6 |
math | Example 1: From a container filled with 20 liters of pure alcohol, 1 liter is poured out, then it is refilled with water. After pouring out 1 liter of the mixed solution and refilling with water again, this process continues. If after the $k$-th pour $(k \geq 1)$, a total of $x$ liters of pure alcohol have been poured ... | 1+\frac{19}{20}x(1\leqslantx<20) | 131 | 23 |
math | # Problem 7.4 (7 points)
Find all three-digit numbers that decrease by 6 times after the first digit is erased. # | 120,240,360,480 | 29 | 15 |
math | Example 4.1. Investigate the convergence of the series:
$$
\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\ldots+\frac{1}{4 n^{2}-1}+\ldots
$$ | \frac{1}{2} | 59 | 7 |
math | 4. In the number $2016 * * * * 02 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,2,4,5,7,9$ (digits can be repeated) so that the resulting 11-digit number is divisible by 15. In how many ways can this be done? | 864 | 79 | 3 |
math | 3. Solve the inequality:
$$
\frac{2|2 x-1|+2}{3}+\frac{6}{1+|2 x-1|} \leq 4-\sqrt{16 x^{4}-8 x^{2}+1}
$$ | -0.5 | 60 | 4 |
math | Determine the smallest natural number written in the decimal system with the product of the digits equal to $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$. | 45578899 | 47 | 8 |
math | Xiao Wang and Xiao Li are to process the same number of the same type of parts, and they start working at the same time. It is known that Xiao Wang processes 15 per hour, and must rest for 1 hour after every 2 hours of work; Xiao Li works without interruption, processing 12 per hour. As a result, they both finish at th... | 60 | 90 | 2 |
math | \section*{Aufgabe 2 - 021242}
Für welche Zahlen \(x\) des Intervalls \(0<x<\pi\) gilt
\[
\frac{\tan 2 x}{\tan x}-\frac{2 \cot 2 x}{\cot x} \leq 1
\]
| (\frac{\pi}{4},\frac{\pi}{3}]\cup[\frac{2\pi}{3},\frac{3\pi}{4}) | 73 | 34 |
math | 8. The domain of the function $f(x)=\frac{\sin 2 x \cos x}{1-\sin x}$ is $\qquad$ , and the range is $\qquad$ . | {x\lvert\,x\neq2k\pi+\frac{\pi}{2}.,k\in{Z}},[-\frac{1}{2},4) | 42 | 38 |
math | Boats are sailing on the sea at a speed of $v$. Exactly opposite them, an airplane is flying at a height of $h$ at a speed of $V$. At what distance from the boat should the pilot drop the aid package so that it lands precisely in the boat? What would be the distance in question if the airplane is flying behind the boat... | =(V+v)\sqrt{\frac{2}{}},\=(V-v)\sqrt{\frac{2}{}},\=V\sqrt{\frac{2}{}} | 85 | 36 |
math | Someone invests $24000 \mathrm{~K}$; part of it at $4.5 \%$, the other part at $6 \%$. Their income is the same as if the entire amount earned $5 \%$ interest. How much is invested at $4.5 \%$ and how much at $6 \%$? | 16000koronaat4.5,8000koronaat6 | 71 | 20 |
math | 4. Find the value of the expression $\frac{1}{a b}+\frac{1}{b c}+\frac{1}{a c}$, if it is known that $a, b, c$ are three different real numbers satisfying the conditions $a^{3}-2020 a^{2}+1010=0, b^{3}-2020 b^{2}+1010=0, \quad c^{3}-2020 c^{2}+1020=0$. | -2 | 116 | 2 |
math | Condition of the problem
Find the derivative.
$$
y=\cos (\ln 13)-\frac{1}{44} \cdot \frac{\cos ^{2} 22 x}{\sin 44 x}
$$ | \frac{1}{4\sin^{2}22x} | 51 | 15 |
math | \section*{Problem 6 - 031116}
The sum of 100 consecutive natural numbers is 1000050.
What is the smallest and the largest of these numbers? | 995110050 | 47 | 9 |
math | 3. (6 points) Insert two numbers between 2015 and 131 so that the four numbers are arranged in descending order, and the difference between any two adjacent numbers is equal. The sum of the two inserted numbers is
保留源文本的换行和格式，所以翻译结果如下：
3. (6 points) Insert two numbers between 2015 and 131 so that the four numbers are... | 2146 | 115 | 4 |
math | 24.1 ** Select $k$ numbers from $1,2, \cdots, 1000$. If among the selected numbers there are always three that can form the sides of a triangle, find the minimum value of $k$. | 16 | 52 | 2 |
math | Find the least positive integer \(M\) for which there exist a positive integer \(n\) and polynomials \(P_1(x)\), \(P_2(x)\), \(\ldots\), \(P_n(x)\) with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\]
[i]Proposed by Karthik Vedula[/i] | 6 | 92 | 1 |
math | Example 12 (2008 National High School Joint Competition Hubei Province Preliminary Test Question) Let the sequence $\{f(n)\}$ satisfy: $f(1)=1$, $f(2)=2, \frac{f(n+2)}{f(n)}=\frac{f^{2}(n+1)+1}{f^{2}(n)+1} \quad(n \geqslant 1)$.
(1) Find the recurrence relation between $f(n+1)$ and $f(n)$, i.e., $f(n+1)=g[f(n)]$;
(2) P... | 63<f(2008)<78 | 157 | 11 |
math | [ Coordinate method in space ] [
The edge of the cube $E F G H E 1 F 1 G 1 H 1$ is 2. Points $A$ and $B$ are taken on the edges $E H$ and $H H 1$, respectively, such that $\frac{E A}{A H}=2, \frac{B H}{B H 1}=\frac{1}{2}$. A plane is drawn through points $A, B$, and $G 1$. Find the distance from point $E$ to this plan... | 2\sqrt{\frac{2}{11}} | 122 | 11 |
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