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 | 3. Let $f(x)=x^{2}+a x+b$ have two real roots in the interval $[0,1]$. Then the range of $a^{2}-2 b$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | [0,2] | 72 | 5 |
math | 4. Find all natural numbers $n$ such that $n-4$, $2n+2$, and $4n+1$ are perfect cubes. | 31 | 33 | 2 |
math | Find the value of $\alpha+2 \beta$ angle, if $\operatorname{tg} \alpha=\frac{1}{7}, \operatorname{tg} \beta=\frac{1}{3}$? | \alpha+2\beta=45 | 45 | 9 |
math | 10. Determine all integers $n$ for which $2 n^{3}-3 n^{2}+n+3$ is a prime number. | n=0n=1 | 32 | 6 |
math | Consider the parabola $C: y^{2}=4 x$ and the straight line $L: y=x+2$. Let $P$ be a variable point on $L$. Draw the two tangents from $P$ to $C$ and let $Q_{1}$ and $Q_{2}$ denote the two points of contact on $C$. Let $Q$ be the mid-point of the line segment joining $Q_{1}$ and $Q_{2}$. Find the locus of $Q$ as $P$ mov... | (y - 1)^2 = 2 \left( x - \frac{3}{2} \right) | 116 | 26 |
math | 5. A positive integer $N$ and $N^{2}$ end with the same sequence of digits $\overline{a b c d}$, where $a-$ is a non-zero digit. Find $\overline{a b c}$. | 937 | 51 | 3 |
math | \section*{Exercise 2 - 330932}
For each positive integer \(n\), imagine another number \(n^{\prime}\) formed according to the following rule: From the digit representation of \(n\) in the decimal system, the first digit is removed and appended to the end of the last digit.
Then let \(n^{\prime}\) be the number with t... | 10^{21}+\frac{10^{21}-7}{69} | 115 | 20 |
math | 11. Given the function defined on the set of non-zero natural numbers
$$
f(n)=\left\{\begin{array}{ll}
n+2, & n \leqslant 2005 ; \\
f(f(n-4)), & n>2005 .
\end{array}\right.
$$
then when $n \leqslant 2005$, $n-f(n)=$ $\qquad$
when $2005<n \leqslant 2007$, $n-f(n)=$
$\qquad$ | -2,0 | 126 | 4 |
math | 2. Find a natural number $N$, such that it is divisible by 5 and 49, and including 1 and $N$, it has a total of 10 divisors. | 5 \times 7^{4} | 41 | 8 |
math | 3.106.
$$
\frac{1+\cos (4 \alpha-2 \pi)+\cos \left(4 \alpha-\frac{\pi}{2}\right)}{1+\cos (4 \alpha+\pi)+\cos \left(4 \alpha+\frac{3}{2} \pi\right)}
$$ | \operatorname{ctg}2\alpha | 72 | 10 |
math | 12. (5 points) Teacher Wang drives from home to location $A$. On the way there, the first $\frac{1}{2}$ of the journey is traveled at a speed of 50 kilometers per hour, and the remaining distance is traveled at a speed increased by $20 \%$; On the return trip, the first $\frac{1}{3}$ of the journey is traveled at a spe... | 330 | 143 | 3 |
math | Upon cutting a certain rectangle in half, you obtain two rectangles that are scaled down versions of the original. What is the ratio of the longer side length to the shorter side length? | \sqrt{2} | 36 | 5 |
math | Determine all pairs $(x, y)$ of integers that satisfy the equation
$$
\sqrt[3]{7 x^{2}-13 x y+7 y^{2}}=|x-y|+1
$$ | (x_1, y_1) = (m^3 + 2m^2 - m - 1, m^3 + m^2 - 2m - 1), (x_2, y_2) = (-m^3 - m^2 + 2m + 1, -m^3 - 2m^2 + m + 1) | 47 | 82 |
math | In triangle $A B C$, the angle bisectors $A M$ and $C N$ are drawn. It is known that $A C=6, A N=2, C M=3$. Find $M N$.
# | \frac{\sqrt{145}}{5} | 50 | 12 |
math | Meena owns a bottle cap collection. While on a vacation, she finds a large number of bottle caps, increasing her collection size by $40\%$. Later on her same vacation, she decides that she does not like some of the bottle caps, so she gives away $20\%$ of her current collection. Suppose that Meena owns $21$ more bottle... | 175 | 99 | 3 |
math | 3. If a, b, c are distinct non-negative real numbers, what is the smallest possible number of distinct numbers among the numbers $a+b, b+c, c+a, a^{2}+b^{2}, b^{2}+c^{2}, c^{2}+a^{2}$? (Patrik Bak) | 4 | 71 | 1 |
math | 9.083. $\frac{1}{3^{x}+5}<\frac{1}{3^{x+1}-1}$. | x\in(-1;1) | 32 | 8 |
math | 7. Given $a_{1}, a_{2}, \cdots, a_{n}$ are pairwise distinct real numbers, find the minimum value of the function defined by $f(x)=\sum_{i=1}^{n}\left|x-a_{i}\right|$. Here, $x$ is a real number. (1969 Polish Mathematical Olympiad) | -a_{1} - a_{2} - \cdots - a_{m} + a_{m+1} + \cdots + a_{n} | 78 | 34 |
math | We distribute the integers $1, 2, \ldots, 8$ into two sets $A$ and $B$, then we denote $P_{A}$ as the product of all elements in $A$ and $P_{B}$ as the product of all elements in $B$.
What are the minimum and maximum values that the sum $P_{A} + P_{B}$ can take?
Note: if a set $E$ is empty, we consider the product of i... | 40321 \text{ and } 402 | 110 | 14 |
math | The two-variable function $f(x, y)$ defined on natural numbers satisfies the following conditions:
$$
f(x, x)=x
$$
$$
f(x, y)=f(y, x)
$$
$$
(x+y) \cdot f(x, y)=f(x, x+y) \cdot y
$$
What is $f(980,1980)$? | 97020 | 83 | 5 |
math | Example 3 Given that $a, b, c$ are positive real numbers, and $a+b+c=12, ab+bc$ $+ca=45$, try to find the maximum value of $abc$.
保留了原文的换行和格式。 | 54 | 57 | 2 |
math | 2. Find the smallest positive real number $k$ such that for any 4 distinct real numbers $a, b, c, d$ not less than $k$, there exists a permutation $p, q, r, s$ of $a, b, c, d$ such that the equation $\left(x^{2}+p x+q\right)\left(x^{2}+r x+s\right)=0$ has 4 distinct real roots. (Feng Zhigang) | 4 | 104 | 1 |
math | 8.3. In a cinema, five friends took seats numbered 1 to 5 (the leftmost seat is number 1). During the movie, Anya left to get popcorn. When she returned, she found that Varya had moved one seat to the right, Galia had moved three seats to the left, and Diana and Elia had swapped places, leaving the edge seat for Anya. ... | 3 | 94 | 1 |
math | 1. Andrei, Boris, and Valentin participated in a 1 km race. (We assume that each of them ran at a constant speed). Andrei was 100 m ahead of Boris at the finish line. And Boris was 50 m ahead of Valentin at the finish line. What was the distance between Andrei and Valentin at the moment Andrei finished? | 145\mathrm{} | 80 | 6 |
math | Example 2 Solve the system of simultaneous equations
$$
\left\{\begin{array}{l}
x+y+z=3, \\
x^{2}+y^{2}+z^{2}=3, \\
x^{5}+y^{5}+z^{5}=3 .
\end{array}\right.
$$
Find all real or complex roots.
(2nd USA Mathematical Olympiad) | 1 | 86 | 1 |
math | Example 10. Find the domain of the function
$$
y=\arcsin \frac{2 x-1}{3}
$$ | [-1,2] | 30 | 5 |
math | Find the gcd of $n^{17}-n$ for $n$ an integer. | 510 | 19 | 3 |
math | ## Task 2 - 280732
In a factory for the production of alcoholic essences, a remaining stock of $300 \mathrm{~kg}$ of 32% alcohol is to be converted into a new stock of 40% alcohol by adding 90% alcohol.
Determine the amount of 90% alcohol needed to achieve this! | 48 | 81 | 2 |
math | In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have... | 11 | 63 | 2 |
math | Task 1. The fractions $\frac{\overline{3 a 5 b}}{36}$ and $\frac{\overline{4 c 7 d}}{45}$ are natural numbers, where $a, b, c, d$ are digits. Arrange all such numbers in ascending order. | \frac{4275}{45}<\frac{3456}{36}<\frac{4770}{45}<\frac{3852}{36} | 64 | 44 |
math | Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $, we have $ a_{pk+1}=pa_k-3a_p+13 $.Determine all possible values of $ a_{2013} $. | 2016 | 71 | 4 |
math | 一、Fill in the Blanks (8 questions in total, 8 points each, 64 points in total)
1. $2017^{\ln \ln 2017}-(\ln 2017)^{\ln 2017}=$ $\qquad$ | 0 | 65 | 1 |
math | 13.378 In a piece of alloy weighing 6 kg, copper is contained. In another piece of alloy weighing 8 kg, copper is contained in a different percentage than in the first piece. From the first piece, a certain part was separated, and from the second - a part twice as heavy as from the first. Each of the separated parts wa... | 2.4 | 116 | 3 |
math | 【Example 7】10 students line up and walk, requiring that 4 female students be adjacent, and neither walk at the very front nor at the very back. | 5!\cdot6! | 35 | 5 |
math | 4. Given point $P$ inside $\triangle A B C$, and satisfying $\overrightarrow{A P}=\frac{1}{3} \overrightarrow{A B}+\frac{1}{4} \overrightarrow{A C}$, let the areas of $\triangle P B C, \triangle P C A, \triangle P A B$ be $S_{1}, S_{2}, S_{3}$ respectively, then $S_{1}: S_{2}: S_{3}=$ $\qquad$. | 5:4:3 | 109 | 5 |
math | 936. What is the greatest rate at which the function $u(M)=\frac{10}{x^{2}+y^{2}+z^{2}+1}$ can increase as the point $M(x, y, z)$ passes through the point $M_{0}(-1 ; 2 ;-2)$? In what direction should the point $M$ move as it passes through the point $M_{1}(2 ; 0 ; 1)$, so that the function $u(M)$ decreases at the grea... | \frac{3}{5} | 115 | 7 |
math | 6.144. $\left(x^{2}+2 x\right)^{2}-(x+1)^{2}=55$.
6.144. $\left(x^{2}+2 x\right)^{2}-(x+1)^{2}=55$. | x_{1}=-4,x_{2}=2 | 65 | 11 |
math | 1. A four-digit number, $n$, is written as ' $A B C D$ ' where $A, B, C$ and $D$ are all different odd digits. It is divisible by each of $A, B, C$ and $D$. Find all the possible numbers for $n$. | 1395,1935,3195,3915,9135,9315 | 65 | 29 |
math | 5. Given real numbers $x, y$ satisfy $x+y=1$. Then, the maximum value of $\left(x^{3}+1\right)\left(y^{3}+1\right)$ is $\qquad$ . | 4 | 50 | 1 |
math | Problem 1. A train is moving at a speed of $4 \mathrm{~m} / \mathrm{s}$. A bird is flying at a speed of $12 \mathrm{~m} / \mathrm{s}$. In 60 seconds, the bird flew from the end to the beginning of the train and back. How long is the train? | 320 | 76 | 3 |
math | IMO 1998 Problem B3 Consider all functions f from the set of all positive integers into itself satisfying f(t 2 f(s)) = s f(t) 2 for all s and t. Determine the least possible value of f(1998). | 120 | 56 | 3 |
math | 208. Random variables $X$ and $Y$ are independent. Find the variance of the random variable $Z=3X+2Y$, given that $D(X)=5, D(Y)=6$. | 69 | 45 | 2 |
math | 10.157. Through points $R$ and $E$, belonging to sides $A B$ and $A D$ of parallelogram $A B C D$, and such that $A R=(2 / 3) A B, A E=(1 / 3) A D$, a line is drawn. Find the ratio of the area of the parallelogram to the area of the resulting triangle. | 9 | 88 | 1 |
math | 2. The pages of materials for conducting a math club session are numbered. The last page is numbered 128. How many digits were needed to number all the pages? | 276 | 36 | 3 |
math | (3) In the coordinate plane, there are two regions $M$ and $N$. $M$ is defined by
\[
\left\{\begin{array}{l}
y \geqslant 0, \\
y \leqslant x, \\
y \leqslant 2-x,
\end{array} \quad N\right.
\]
is a region that varies with $t$, determined by the inequality $t \leqslant x \leqslant t+1$, where the range of $t$ is $0 \leq... | -^{2}++\frac{1}{2} | 157 | 12 |
math | 3. Determine all pairs $(a, b)$ of real numbers for which the quadratic trinomials $P(x)=$ $=x^{2}+a x+b$ and $Q(x)=x^{2}+b x+a$ have the following property: each of the equations
$$
a P(x)+b Q(x)=0 \quad \text{and} \quad a Q(x)+b P(x)=0
$$
is a quadratic equation with a double root.
(Jaroslav Švrček) | (4,4) | 112 | 5 |
math | Example 8 Find all positive integers $n$ greater than 1 such that $2^{2}+3^{2}+\cdots+n^{2}$ is a $k$-th power of a prime $p$, where $k$ is a positive integer.
(2004 Polish Mathematical Olympiad) | 2,3,4,7 | 66 | 7 |
math | 1. Let the general term formula of the sequence $\left\{a_{n}\right\}$ be $a_{n}=3^{3-2 n}$, then the sum of the first $n$ terms of the sequence $S_{n}=$ | \frac{27}{8}[1-(\frac{1}{9})^{n}] | 54 | 20 |
math | 285. Solve the equation:
$$
2 x^{5}+5 x^{4}-13 x^{3}-13 x^{2}+5 x+2=0
$$ | -1,2,1/2,-2\\sqrt{3} | 42 | 15 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{6 ;-1 ; 7\}$
$p=\{1 ;-2 ; 0\}$
$q=\{-1 ; 1 ; 3\}$
$r=\{1 ; 0 ; 4\}$ | -p-3q+4r | 73 | 7 |
math | Caden, Zoe, Noah, and Sophia shared a pizza. Caden ate 20 percent of the pizza. Zoe ate 50 percent more of the pizza than Caden ate. Noah ate 50 percent more of the pizza than Zoe ate, and Sophia ate the rest of the pizza. Find the percentage of the pizza that Sophia ate. | 5\% | 72 | 3 |
math | 11.4 The eccentric has $N$ unit squares, from which he managed to form a rectangle with sides differing by 9. The eccentric did not stop there and formed another rectangle from the same $N$ squares, but this time with sides differing by 6. Find $N$. | 112 | 60 | 3 |
math | 1. Kolya, Seryozha, and Vanya regularly go to the cinema: Kolya goes there every 4th day, Seryozha - every 5th day, and Vanya - every 6th day. Today all the boys were at the cinema. When will all three meet at the cinema again? | 60 | 71 | 2 |
math | 31. How many days did the gardener work?
- I want to give you a task, - said the Griffin. - Usually it is solved using algebra, but if you use my method, you will do wonderfully without it!
Once the King hired one of the spade suit gardeners for twenty-six days to do some work in the garden. The King set the conditio... | 22 | 413 | 2 |
math | A car is called flex if it can be fueled with gasoline or ethanol. Consider that the prices of ethanol and gasoline are, respectively, R$ 1.59 and R$ 2.49 per liter.
(a) Suppose a flex car can travel $12.3 \mathrm{~km}$ per liter of gasoline, which we denote as $12.3 \mathrm{~km} / \mathrm{l}$. What must be the $\math... | y>7.85,7.22\mathrm{~}/\mathrm{},3.61\mathrm{~}/\mathrm{},y>0.64x | 297 | 38 |
math | 161. Find the greatest integer $x$ such that the number
$$
4^{27}+4^{1000}+4^{x}
$$
is a perfect square. | 1972 | 43 | 4 |
math | 10 If the vector $\vec{a}+3 \vec{b}$ is perpendicular to the vector $7 \vec{a}-5 \vec{b}$, and the vector $\vec{a}-4 \vec{b}$ is perpendicular to the vector $7 \vec{a}-2 \vec{b}$, then the angle between vector $\vec{a}$ and $\vec{b}$ is $\qquad$. | \frac{\pi}{3} | 90 | 7 |
math | Example 2 Solve the equation $3 x^{3}-[x]=3$
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
However, it seems the translation request was for the provided content itself. Here is the translation:
Example 2 Solve the equation $3 ... | \sqrt[3]{\frac{4}{3}} | 78 | 12 |
math | 10. For $i=2,3, \cdots, k$, the remainder when the positive integer $n$ is divided by $i$ is $i-1$. If the smallest value of $n$, $n_{0}$, satisfies $2000<n_{0}<3000$, then the smallest value of the positive integer $k$ is | 9 | 79 | 1 |
math | Let $P(x)=x^{2020}+x+2$, which has $2020$ distinct roots. Let $Q(x)$ be the monic polynomial of degree $\binom{2020}{2}$ whose roots are the pairwise products of the roots of $P(x)$. Let $\alpha$ satisfy $P(\alpha)=4$. Compute the sum of all possible values of $Q(\alpha^2)^2$.
[i]Proposed by Milan Haiman.[/i] | 2020 \times 2^{2019} | 109 | 14 |
math | 13.138. Two workers, the second of whom started working 1.5 days later than the first, independently wallpapered several rooms in 7 days, counting from the moment the first worker started. If this work had been assigned to each separately, the first would have needed 3 days more than the second to complete it. How many... | 14 | 86 | 2 |
math | 7.277. $\left\{\begin{array}{l}x^{y}=2, \\ (2 x)^{y^{2}}=64(x>0)\end{array}\right.$ | (\frac{1}{\sqrt[3]{2}};-3),(\sqrt{2};2) | 45 | 22 |
math | For all $x \in\left(0, \frac{\pi}{2}\right)$, the largest positive integer $n$ that satisfies the inequality $\sin ^{n} x+\cos ^{n} x>\frac{1}{2}$ is | 3 | 54 | 1 |
math | 3. The diagonals of a rhombus differ by $14 \mathrm{~cm}$. If we extend the longer diagonal by $2 \mathrm{~cm}$, and the shorter one by $8 \mathrm{~cm}$, the area of the rhombus will increase by $144 \mathrm{~cm}^{2}$. What are the lengths of the diagonals of the original rhombus? | 30 | 91 | 2 |
math | Consider the polynomial $P(n) = n^3 -n^2 -5n+ 2$. Determine all integers $n$ for which $P(n)^2$ is a square of a prime.
[hide="Remark."]I'm not sure if the statement of this problem is correct, because if $P(n)^2$ be a square of a prime, then $P(n)$ should be that prime, and I don't think the problem means that.[/hide... | n = -1, -3, 0, 3, 1 | 99 | 17 |
math | 12.043. Through the vertex $A$ of an isosceles acute triangle $ABC$ and the center of the circumscribed circle around this triangle, a line is drawn, intersecting the side $BC$ at point $D$. Find the length of $AD$, if $AB=BC=b$ and $\angle ABC=\alpha$. | \frac{b\sin\alpha}{\sin\frac{3\alpha}{2}} | 75 | 20 |
math | ## Task B-3.1.
Simplify the numerical expression $\frac{6-\log _{14} 2401}{4-\log _{14} 49}$ and write it in the form of a logarithm $\log _{28} n$, where $n$ is a natural number. | \log_{28}56 | 70 | 8 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{1}{4}} \frac{\sqrt[3]{\frac{x}{16}}-\frac{1}{4}}{\sqrt{\frac{1}{4}+x}-\sqrt{2 x}}$ | -\frac{2}{3}\sqrt{\frac{1}{2}} | 65 | 15 |
math | II Let $a, d \geqslant 0, b, c>0$ and $b+c \geqslant a+d$. Find the minimum value of $\frac{b}{c+d}+\frac{c}{a+b}$. | \sqrt{2}-\frac{1}{2} | 54 | 12 |
math | 8. Two individuals, A and B, independently toss a fair coin. A tosses the coin 10 times, while B tosses it 11 times. The probability that the number of times the coin lands heads up for B is greater than that for A is $\qquad$ (Provided by Gan Zhiguo) | \frac{1}{2} | 69 | 7 |
math | Exercise 5. A number has been written on the board. At each step, we add to it the largest of its digits (for example, if 142 is written, the next number will be 146). What is the maximum number of odd numbers that can be written consecutively by proceeding in this way? | 5 | 69 | 1 |
math | 1. In the field of real numbers, solve the system of equations
$$
\begin{aligned}
x^{2}-x y+y^{2} & =7, \\
x^{2} y+x y^{2} & =-2 .
\end{aligned}
$$
(J. Földes) | {x,y}={-1,2},{x,y}={1+\sqrt{2},1-\sqrt{2}},{x,y}={\frac{-9+\sqrt{57}}{6},\frac{-9-\sqrt{57}}{6}} | 66 | 56 |
math | 4. Let the function $y=f(x)$ be defined on $\mathbf{R}$, and have an inverse function $f^{-1}(x)$. It is known that the inverse function of $y=f(x+1)-2$ is $y=f^{-1}(2 x+1)$, and $f(1)=4$. If $n \in \mathbf{N}_{+}$, then $f(n)=$ $\qquad$ | 3+\left(\frac{1}{2}\right)^{n-1} | 95 | 17 |
math | 8. The number of all integer solutions $(x, y, z)$ for the equation $x y z=2009$ is $\qquad$ groups. | 72 | 35 | 2 |
math | Two prime numbers differ by a hundred. When their decimal representations are written one after the other, we get another prime number. Which are these numbers? | p=3,103,r=1033 | 30 | 13 |
math | 9.103. For what values of $m$ are the roots of the equation $4 x^{2}-(3 m+1) x-m-2=0$ contained in the interval between -1 and 2? | \in(-\frac{3}{2};\frac{12}{7}) | 49 | 18 |
math | 1. The range of the function $f(x)=\frac{\sin 2 x}{2+\sin x+\cos x}$ is | [2\sqrt{3}-4,1+\frac{\sqrt{2}}{2}] | 28 | 20 |
math | $\mathbf{N 4 2}$ (37-4, Russia) Let positive integers $a, b$ be such that $15a + 16b$ and $16a - 15b$ are both squares of positive integers. Find the smallest value that the smaller of these two squares can take. | 231361 | 71 | 6 |
math | In the "Magical Six" TV contest, players multiply their prize if they roll a six with one of the two dice or if the sum of the numbers rolled is 6. What is the probability that a player will multiply their prize? | \frac{4}{9} | 49 | 7 |
math | 15. (3 points) There is a three-digit number $A$, after adding a decimal point before one of its digits, the resulting number is $B$. If $A-B=478.8$, then $A=$ . $\qquad$ | 532 | 54 | 3 |
math | (IMO 2016, problem 5)
We write on the board the equality:
$$
(x-1)(x-2) \ldots(x-2016)=(x-1)(x-2) \ldots(x-2016)
$$
We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimum number of factors that must be erased to achieve this?
## - Solutions ... | 2016 | 107 | 4 |
math | 6. A reel of film needs to be rewound from one spool to another. The diameters of the empty spools are the same and equal to $a$. Find the time required for rewinding if the length of the film is $L$, the thickness of the film is small and equal to $S$, and the receiving spool rotates at a constant angular velocity $\o... | \frac{\pi}{S\omega}(\sqrt{^{2}+\frac{4SL}{\pi}}-) | 79 | 25 |
math | 1. Given the product of four real numbers is 1, and the sum of any one of the numbers with the product of the other three is 1000. Then the sum of these four numbers is | 2000 | 44 | 4 |
math | 11. (12 points) $0 \sim 9$ can form two five-digit numbers $A$ and $B$. If the sum $A+B$ is a six-digit number with the last five digits being the same, then the number of different values of $A \times B$ is $\qquad$.
| 384 | 68 | 3 |
math | A team of four students goes to LMT, and each student brings a lunch. However, on the bus, the students’ lunches get mixed up, and during lunch time, each student chooses a random lunch to eat (no two students may eat the same lunch). What is the probability that each student chooses his or her own lunch correctly? | \frac{1}{24} | 69 | 8 |
math | 6-11 The function $f(x, y)$ satisfies for all non-negative integers $x, y$:
(1) $f(0, y)=y+1$;
(2) $f(x+1,0)=f(x, 1)$;
(3) $f(x+1, y+1)=f[x, f(x+1, y)]$;
Determine $f(4,1981)$. | f(4,1981)=-3+2^{2^{2\cdot}}\ | 95 | 21 |
math | 2. (5 points) There are three natural numbers, their sum is 2015, the sums of each pair are $m+1, m+2011$ and $m+2012$, then $m=$ $\qquad$ | 2 | 56 | 1 |
math | 21. (1) Given that $a, b, c, d, e, f$ are real numbers, and satisfy the following two equations: $a+b+c+d+e+f=10, (a-1)^{2}+(b-1)^{2}+(c-1)^{2}+(d-1)^{2}+(e-1)^{2}+(f-1)^{2}=6$, find the maximum value of $f$. (1993 Balkan Mathematical Olympiad)
(2) Given that $a, b, c, d \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$,... | \frac{10}{3} | 244 | 8 |
math | 12.219. When a circular sector rotates about one of its extreme radii, a body is formed whose spherical surface area is equal to the area of the conical surface. Find the sine of the central angle of the circular sector. | \frac{4}{5} | 51 | 7 |
math | Example 3. Solve the equation
$$
(x+y-2) d x+(x-y+4) d y=0
$$ | x^{2}+2xy-y^{2}-4x+8C | 29 | 16 |
math | Example 1-8: There are 5 Japanese books, 7 English books, and 10 Chinese books. If two books of different languages are taken, how many possibilities are there? If two books of the same language are taken, how many schemes are there? If two books are taken regardless of the language, how many possibilities are there? | 231 | 72 | 3 |
math | How many digits are in the base $10$ representation of $3^{30}$ given $\log 3 = 0.47712$?
[i]2015 CCA Math Bonanza Lightning Round #1.4[/i] | 15 | 55 | 2 |
math | ## Task A-3.3.
Determine all natural numbers that are squares of natural numbers and in whose decimal representation two digits are different from 0, and one of these two is 3. | 36\cdot100^k,fork\in\mathbb{N}_0 | 41 | 20 |
math | Problem 1. Let the natural numbers $x$ and $y$ be such that the fractions $\frac{7 x-5 y-3}{3 x+9 y-15}$ and $\frac{1}{5}$ are equivalent. Determine the value of the expression $(16 x-17 y)^{2015}$. | 0 | 73 | 1 |
math | 1. Given the function $f(x)=\frac{x^{3}}{1+x^{3}}$. Find the value of the sum
$$
\begin{aligned}
f\left(\frac{1}{1}\right)+f\left(\frac{2}{1}\right)+\ldots+f\left(\frac{2007}{1}\right) & +f\left(\frac{1}{2}\right)+f\left(\frac{2}{2}\right)+\ldots+f\left(\frac{2007}{2}\right)+ \\
& +\ldots+f\left(\frac{1}{2007}\right)+... | \frac{2007^2}{2} | 185 | 12 |
math | 30. There are 111 lamps, and each lamp has its own switch. It is allowed to simultaneously switch 13 of them. At the initial moment, some lamps are on, and some are off.
a) Is it possible to turn off all the lamps?
b) How many switches will be required for this if all the lamps were initially on? | 9 | 76 | 1 |
math | 11. Inside a cone, there is a sphere that is tangent to the cone's side and base. The maximum ratio of the sphere's surface area to the cone's surface area is $\qquad$ . | \frac{1}{2} | 43 | 7 |
math | 14. The set $P$ consisting of certain integers has the following properties:
(1) $P$ contains both positive and negative numbers;
(2) $P$ contains both odd and even numbers;
(3) $1 \notin P$;
(4) If $x, y \in P$, then $x+y \in P$.
Determine the relationship between 0, 2 and the set $P$. | 0\inP,2\notinP | 90 | 10 |
math | In an $m \times n$ grid, each square is either filled or not filled. For each square, its [i]value[/i] is defined as $0$ if it is filled and is defined as the number of neighbouring filled cells if it is not filled. Here, two squares are neighbouring if they share a common vertex or side. Let $f(m,n)$ be the largest to... | C = 2 | 131 | 5 |
math | 9. (3 points) A natural number that can only be divided by 1 and itself is called a prime number, such as: $2,3,5,7$, etc. Then, the prime number greater than 40 and less than 50 is $\qquad$, and the largest prime number less than 100 is $\qquad$. | 41,43,47,97 | 76 | 11 |
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