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 | Ati has $ 7$ pots of flower, ordered in $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$. She wants to rearrange the position of those pots to $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$ such that for every positive integer $ n<7$, $ B_1,B_2,\dots,B_n$ is not the permutation of $ P_1,P_2,\dots,P_7$. In how many ways can Ati do this? | 3447 | 124 | 4 |
math | 8、Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{n}=n(1 \leq n \leq 5), a_{n+1}=a_{1} \cdot a_{2} \cdot \ldots \cdot a_{n}-1(n \geq 5)$, then $S_{m}=a_{1} \cdot a_{2} \cdot \ldots \cdot a_{m}-a_{1}^{2}-a_{2}^{2}-\ldots-a_{m}^{2}$ takes the maximum value as $\qquad$
Given the sequence $\{a_n\}$ satisfies $a_n ... | 65 | 247 | 2 |
math | Call a $3$-digit number geometric if it has $3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers. | 840 | 43 | 3 |
math | 4. In $\triangle A B C$, $A B=A C$, the internal angle bisectors of $\angle C A B$ and $\angle A B C$ intersect the sides $B C$ and $C A$ at points $D$ and $E$, respectively. Let $K$ be the incenter of $\triangle A D C$. If $\angle B E K=45^{\circ}$, find all possible values of $\angle C A B$. (Belgium provided) | 60^{\circ} \text{ and } 90^{\circ} | 102 | 18 |
math | What is the geometric place of the intersection points of the perpendicular tangents drawn to the circle $x^{2}+y^{2}=32$? | x^{2}+y^{2}=64 | 32 | 11 |
math | 7.3. What angle do the clock hands form at 12:20? | 110 | 19 | 3 |
math | 24. $\int x^{4} d x$. | \frac{1}{5}x^{5}+C | 12 | 13 |
math | 2. Find all real-coefficient polynomials $P(x)$ such that for all real numbers $a, b, c$ satisfying $a b+b c+c a=0$, we have
$$
P(a-b)+P(b-c)+P(c-a)=2 P(a+b+c) .
$$ | P(x) = \alpha x^4 + \beta x^2 | 62 | 15 |
math | Find the number of digit of $\sum_{n=0}^{99} 3^n$.
You may use $\log_{10} 3=0.4771$.
2012 Tokyo Institute of Technology entrance exam, problem 2-A | 48 | 57 | 2 |
math | Example 5 Let $x, y \in (0, +\infty)$, and $\frac{19}{x} + \frac{98}{y} = 1$. Find the minimum value of $x + y$.
(1998, Hunan Province High School Mathematics Competition) | 117+14\sqrt{38} | 66 | 12 |
math | 4. Prince George has 100 coins, some of which may be counterfeit (possibly all or none). George can show the expert between 10 and 20 coins, and the expert will tell him how many of them are counterfeit. The problem is that the only expert in the area is Baron Münchhausen, and he exaggerates: the result given by the ba... | 110<120 | 149 | 7 |
math | Let $E(n)$ denote the largest integer $k$ such that $5^k$ divides $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$ Calculate
$$\lim_{n\to \infty} \frac{E(n)}{n^2 }.$$ | \frac{1}{8} | 75 | 7 |
math | \section*{Problem 5 - 111245}
Determine all three-digit natural numbers \(x\), written in the decimal positional system, for which the following holds:
If one appends the digit sequence of the number \(x+1\) to the right of the digit sequence of the number \(x\), one obtains the digit sequence of a six-digit square n... | 328,183,715,528 | 81 | 15 |
math | Let $0<c<1$ be a given real number. Determine the least constant $K$ such that the following holds: For all positive real $M$ that is greater than $1$, there exists a strictly increasing sequence $x_0, x_1, \ldots, x_n$ (of arbitrary length) such that $x_0=1, x_n\geq M$ and
\[\sum_{i=0}^{n-1}\frac{\left(x_{i+1}-x_i\ri... | K = c^{-c}(1-c)^{-(1-c)} | 160 | 15 |
math | Task 3. (15 points) Laboratory engineer Sergei received an object for research consisting of about 200 monoliths (a container designed for 200 monoliths, which was almost completely filled). Each monolith has a specific name (sandy loam or clayey loam) and genesis (marine or lake-glacial deposits). The relative frequen... | 77 | 159 | 2 |
math | \section*{Problem 16 - V01016}
For a circle with radius \(r\), four larger concentric circles are to be drawn successively so that each resulting annulus has the same area as the original circle.
a) Express the radii of the four additional circles \(r_{1}, r_{2}, r_{3}, r_{4}\) in terms of the original radius \(r\) i... | r=2\mathrm{~},r_{1}=2\sqrt{2}\approx2.8\mathrm{~},r_{2}=2\sqrt{3}\approx3.5\mathrm{~},r_{3}=2\sqrt{4}=4\mathrm{~},r_{4}=2\sqrt{5}\approx4.5\mathrm{~} | 112 | 81 |
math | 24. When $0<x<\frac{\pi}{2}$, the function $y=\tan 3 x \cdot \cot ^{3} x$ cannot take values within the open interval $(a, b)$. Find the value of $a+b$.
| 34 | 57 | 2 |
math | In triangle $ABC$, $AB = 52$, $BC = 34$ and $CA = 50$. We split $BC$ into $n$ equal segments by placing $n-1$ new points. Among these points are the feet of the altitude, median and angle bisector from $A$. What is the smallest possible value of $n$? | 102 | 78 | 3 |
math | One writes, initially, the numbers $1,2,3,\dots,10$ in a board. An operation is to delete the numbers $a, b$ and write the number $a+b+\frac{ab}{f(a,b)}$, where $f(a, b)$ is the sum of all numbers in the board excluding $a$ and $b$, one will make this until remain two numbers $x, y$ with $x\geq y$. Find the maximum val... | 1320 | 105 | 4 |
math | 9. A. 2 eighth-grade students and $m$ ninth-grade students participate in a single round-robin chess tournament, where each participant plays against every other participant exactly once. The scoring rule is: the winner of each match gets 3 points, the loser gets 0 points, and in the case of a draw, both players get 1 ... | 8 | 120 | 1 |
math | Find all integers $n \leq 3$ such that there is a set $S_n$ formed by $n$ points of the plane that satisfy the following two conditions:
Any three points are not collinear.
No point is found inside the circle whose diameter has ends at any two points of $S_n$.
[b]NOTE: [/b] The points on the circumference are ... | n = 1, 2, 3 | 89 | 11 |
math | 1. Find all polynomials satisfying $(x-1) \cdot p(x+1)-(x+2) \cdot P(x) \equiv 0, x \in \mathbf{R}$. | p(x)=(x^3-x) | 43 | 8 |
math | 16. 7 (US MO 16) In the plane, there are three circles $C_{i}(i=1,2,3)$, where the diameter of $C_{1}$ is $A B=1$; $C_{2}$ is concentric with $C_{1}$, has a diameter of $k$, and satisfies $1<k<3$; $C_{3}$ has $A$ as its center and $2 k$ as its diameter ($k$ is a constant). Consider all line segments $X Y$, one end $X$ ... | 1 | 173 | 1 |
math | Let $x$ be a strictly positive real number such that $x+\frac{1}{x}=\sqrt{2020}$. What is the value of $x^{2}+\frac{1}{x^{2}}$? | 2018 | 51 | 4 |
math | 10.3. Given a trapezoid $A B C D$ and a point $M$ on the lateral side $A B$, such that $D M \perp A B$. It turns out that $M C=C D$. Find the length of the upper base $B C$, if $A D=d$. | \frac{}{2} | 69 | 6 |
math | 4. Find all integers n for which the fraction
$$
\frac{n^{3}+2010}{n^{2}+2010}
$$
is equal to an integer. | 0,1,-2010 | 43 | 8 |
math | Eva thought of two natural numbers. She first correctly added the numbers, then subtracted them. In both cases, she got a two-digit result. The product of the two-digit numbers thus created was 645.
Which numbers did Eva think of?
(E. Novotná)
Hint. Every natural number has a finite number of divisors. | 29,14 | 73 | 5 |
math | Example 1. Calculate the definite integral $\int_{1}^{4} x^{2} d x$. | 21 | 23 | 2 |
math | 1. Inside square $A B C D$, a point $E$ is chosen so that triangle $D E C$ is equilateral. Find the measure of $\angle A E B$. | 150 | 39 | 3 |
math | 2. For any point $A(x, y)$ in the plane region $D$:
$$
\left\{\begin{array}{l}
x+y \leqslant 1, \\
2 x-y \geqslant-1, \\
x-2 y \leqslant 1
\end{array}\right.
$$
and a fixed point $B(a, b)$, both satisfy $\overrightarrow{O A} \cdot \overrightarrow{O B} \leqslant 1$. Then the maximum value of $a+b$ is $\qquad$ | 2 | 126 | 1 |
math | Question 1 Find the minimum value of the function $y=2 \sqrt{(x-1)^{2}+4}+$ $\sqrt{(x-8)^{2}+9}$. | 5 \sqrt{5} | 42 | 6 |
math | 1. A 2019-digit number written on the board is such that any number formed by any two adjacent digits (in the order they follow) is divisible by 13. Find the last digit of this number, given that the first digit is 6. | 2 | 56 | 1 |
math | 8. In $\triangle A B C$ and $\triangle A E F$, $B$ is the midpoint of $E F$, $A B=E F=1, B C=6, C A=\sqrt{33}$, if $\overrightarrow{A B} \cdot \overrightarrow{A E}+\overrightarrow{A C} \cdot \overrightarrow{A F}=2$, then the cosine value of the angle between $\overrightarrow{E F}$ and $\overrightarrow{B C}$ is $\qquad$... | \frac{2}{3} | 113 | 7 |
math | 43. Given that $a, b, c$ are positive integers satisfying
$$
a+b+c=\operatorname{gcd}(a, b)+\operatorname{gcd}(b, c)+\operatorname{gcd}(c, a)+120 \text {, }
$$
determine the maximum possible value of $a$. | 240 | 72 | 3 |
math | 13. (25 points) Given the function
$$
f(x)=4 \cos x \cdot \sin \left(x+\frac{7 \pi}{6}\right)+a
$$
has a maximum value of 2. Find:
(1) the value of $a$ and the smallest positive period of $f(x)$;
(2) the intervals where $f(x)$ is monotonically decreasing. | 1,[-\frac{\pi}{3}+k\pi,\frac{\pi}{6}+k\pi](k\in{Z}) | 89 | 32 |
math | 10. (20 points) Let \( a, b, c \in (0,1], \lambda \) be a real number such that
\[
\frac{\sqrt{3}}{\sqrt{a+b+c}} \geqslant 1+\lambda(1-a)(1-b)(1-c) .
\]
Find the maximum value of \(\lambda\). | \frac{64}{27} | 81 | 9 |
math | 18. Calculate $1^{2}-2^{2}+3^{2}-4^{2}+\cdots+2005^{2}-$ $2006^{2}=$ $\qquad$ . | -2013021 | 48 | 8 |
math | In a condominium, 29 families live, each of them has either 1 cat or 3 cats or 5 cats. The number of families that have only 1 cat is the same as the number of families that have 5 cats. How many cats are there in this condominium? | 87 | 60 | 2 |
math | Let's write the following sum in the form of a monomial expression:『
$$
\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\ldots+\frac{n+2}{n!+(n+1)!+(n+2)!}
$$[^0]
[^0]: ${ }^{1}$ The $n!$ symbol abbreviates the product 1. 2. 3... $(n-1) n$. | S_{n}=\frac{1}{2}-\frac{1}{(n+2)!} | 108 | 22 |
math | 2. Arrange all positive integers that are coprime with 105 in ascending order, and find the 1000th term of this sequence. | 2186 | 34 | 4 |
math | One container of paint is exactly enough to cover the inside of an old rectangle which is three times as long as it is wide. If we make a new rectangle by shortening the old rectangle by $18$ feet and widening it by $8$ feet as shown below, one container of paint is also exactly enough to cover the inside of the new r... | 172 | 259 | 3 |
math | 11. Given an integer array consisting of 121 integers, where each number is between 1 and 1000 (repetition is allowed), this array has a unique mode (i.e., the integer that appears most frequently). Let $D$ be the difference between this mode and the arithmetic mean of the array. When $D$ reaches its maximum value, wha... | 947 | 87 | 3 |
math | Example 3 Let $\forall x, y \in \mathbf{R}$, if $f(1)=0, \alpha(y)$ is a constant function, try to find the continuous solutions of the functional inequality
$$
f(x y) \geqslant \alpha(y) f(x)+f(y)
$$ | f(y)=f^{\}(1)\int_{1}^{y}\frac{\alpha(y)}{y}\mathrm{~}y | 68 | 29 |
math | 12. Let the function $f(x)$ be a differentiable function defined on the interval $(-\infty, 0)$, with its derivative being $f^{\prime}(x)$, and $2 f(x) + x f^{\prime}(x) > x^{2}$. Then
$$
(x+2017)^{2} f(x+2017)-f(-1)>0
$$
The solution set is $\qquad$ . | (-\infty,-2018) | 102 | 10 |
math | ## Task B-3.1.
Determine all solutions of the equation $\left|\cos ^{2} x-2 \sin x\right|=2$. | \frac{\pi}{2}+k\pi,k\in\mathbb{Z} | 34 | 20 |
math | In a class at school, all students are the same age, except for seven who are 1 year younger and two who are 2 years older.
The sum of the ages of all the students in this class is 330. How many students are in this class? | 37 | 57 | 2 |
math | Consider a rectangle $ABCD$ with $BC = 2 \cdot AB$. Let $\omega$ be the circle that touches the sides $AB$, $BC$, and $AD$. A tangent drawn from point $C$ to the circle $\omega$ intersects the segment $AD$ at point $K$. Determine the ratio $\frac{AK}{KD}$.
[i]Proposed by Giorgi Arabidze, Georgia[/i] | \frac{1}{2} | 90 | 8 |
math | 4. In the set of prime numbers, solve the equation
$$
p^{2}+p q+q^{2}=r^{2}
$$ | (p,q,r)\in{(3,5,7),(5,3,7)} | 32 | 18 |
math | The $\emph{Stooge sort}$ is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of size $n$, we swap the first and last elements if they are out of order; we then (if $n\ge3$) Stooge sort the first $\lceil\tfrac{2n}3\rceil$ elements, then the last $\lceil\tfrac{2n}3\rceil$, the... | 243 | 146 | 3 |
math | 12.49 At what points are the tangents to the curve $y=\frac{x^{3}}{3}-x^{2}-x+1$ parallel to the line $y=2 x-1$? | (3;-2)(-1;\frac{2}{3}) | 47 | 14 |
math | 12. (3 points) A rectangular photo frame is 40 cm long and 32 cm wide. A photo that is 32 cm long and 28 cm wide is placed inside. The area of the part of the frame not covered by the photo is $\qquad$ square centimeters. | 384 | 65 | 3 |
math | Example 5 Find the maximum constant $k$, such that $\frac{k a b c}{a+b+c} \leqslant(a+b)^{2}+(a+b+4 c)^{2}$ holds for all positive real numbers $a, b, c$.
| 100 | 58 | 3 |
math | 1. Given the set $A=\left\{x \mid \log _{a}(a x-1)>1\right\}$. If $3 \in A$, then the range of values for $a$ is | (\frac{1}{3},\frac{1}{2})\cup(1,+\infty) | 48 | 23 |
math | 10. In $\triangle A B C$, $A B=\sqrt{2}, A C=\sqrt{3}$, $\angle B A C=30^{\circ}$, and $P$ is any point in the plane of $\triangle A B C$. Then the minimum value of $\mu=\overrightarrow{P A} \cdot \overrightarrow{P B}+\overrightarrow{P B} \cdot \overrightarrow{P C}+\overrightarrow{P C} \cdot \overrightarrow{P A}$ is . ... | \frac{\sqrt{2}}{2}-\frac{5}{3} | 117 | 17 |
math | Let 7 be the first term and the common difference of an arithmetic sequence both be non-negative integers, the number of terms is no less than 3, and the sum of all terms is $97^{2}$. How many such sequences are there? | 4 | 53 | 1 |
math | 373. A discrete random variable $X$ is given by the distribution law:
$$
\begin{array}{cccc}
X & 1 & 3 & 5 \\
p & 0.4 & 0.1 & 0.5
\end{array}
$$
Find the distribution law of the random variable $Y=3X$. | \begin{pmatrix}Y&3&9&15\\p&0.4&0.1&0.5\end{pmatrix} | 76 | 34 |
math | 4. A number, its fractional part, integer part, and itself form a geometric sequence, then the number is
保留源文本的换行和格式，所以翻译结果如下：
4. A number, its fractional part, integer part, and itself form a geometric sequence, then the number is | \frac{1+\sqrt{5}}{2} | 61 | 12 |
math | Let $n{}$ and $m$ be positive integers, $n>m>1$. Let $n{}$ divided by $m$ have partial quotient $q$ and remainder $r$ (so that $n = qm + r$, where $r\in\{0,1,...,m-1\}$). Let $n-1$ divided by $m$ have partial quotient $q^{'}$ and remainder $r^{'}$.
a) It appears that $q+q^{'} =r +r^{'} = 99$. Find all possible values o... | n = 5000 | 157 | 8 |
math | 16. $[7]$ Let $\mathbb{R}$ be the set of real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that for all real numbers $x$ and $y$, we have
$$
f\left(x^{2}\right)+f\left(y^{2}\right)=f(x+y)^{2}-2 x y \text {. }
$$
Let $S=\sum_{n=-2019}^{2019} f(n)$. Determine the number of possible values of $S... | 2039191 | 127 | 7 |
math | 18. (15 points) Let $\left\{a_{n}\right\}$ be a sequence of positive numbers, and its first $n$ terms sum $S_{n}$ satisfies $S_{n}=\frac{1}{4}\left(a_{n}-1\right)\left(a_{n}+3\right)$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Let $b_{n}=\frac{1}{S_{n}}$, try to find the first $n$ te... | \frac{3}{4}-\frac{2 n+3}{2(n+1)(n+2)} | 143 | 24 |
math | 321. Spheres and a cube. Once, during transportation, it was required to pack a sphere with a diameter of 30 cm into a cubic box with a side of 32 cm. To prevent the sphere from moving during transportation, 8 identical small spheres had to be placed in the corners of the box. What is the diameter of such a small spher... | 63-31\sqrt{3}\approx9.308 | 78 | 16 |
math | ## Subject IV
Determine all pairs of natural numbers $\overline{a b}$ and $\overline{x y z}$, with $x<y<z$, such that
$$
\overline{a b} \cdot\left(x^{2}+y^{2}+z^{2}\right)=1 \cdot 2 \cdot 19 \cdot 53
$$
Note:
Working time: 2 hours
All subjects are mandatory
Each subject is graded from 0 to 7
No points are given ... | (19;349),(38;146),(53;235) | 158 | 22 |
math | Determine all pairs $(x, y)$ of real numbers such that: $|x+y|=3$ and $x y=-10$. | (5,-2),(-2,5),(2,-5),(-5,2) | 30 | 19 |
math | 2- 107 Let $n \geqslant 5$ be a natural number, and $a_{1}, a_{2}, \cdots, a_{n}$ be $n$ different natural numbers with the following property: for any two different non-empty subsets $A$ and $B$ of the set
$$S=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$$
the sum of the numbers in $A$ and the sum of the numbers in $B$... | 2-\frac{1}{2^{n-1}} | 162 | 12 |
math | 9. (14 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 $a$. | 2 | 108 | 1 |
math | 79. $\int\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2} d x$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
79. $\int\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2} d x$. | x-\cosx+C | 88 | 5 |
math | 19. (6b, 9-11) A die is rolled six times. Find the expected value of the number of different faces that appear.
# | \frac{6^{6}-5^{6}}{6^{5}} | 34 | 16 |
math | 6. Find all sets of numbers $x_{1}, x_{2}, \ldots, x_{n+1}$ such that $x_{1}=x_{n+1}$ and for all $k=1, \ldots, n$ the equality
$$
2 \log _{2} x_{k} \cdot \log _{2} x_{k+1}-\log _{2}^{2} x_{k}=9
$$ | x_{k}=8,k=1,\ldots,n+1,orx_{k}=\frac{1}{8},k=1,\ldots,n+1 | 99 | 36 |
math | 6. In a mathematics competition, the first round consists of 25 questions. According to the marking rules, each correct answer earns 4 points, and each wrong answer (including unanswered questions) deducts 1 point. If a score of no less than 60 points qualifies a student for the second round, then, how many questions a... | 17 | 92 | 2 |
math | \section*{Problem 1 - 031231}
Give all two-digit numbers that have the following property!
If one forms their third power and deletes all digits of this number except the last two, one obtains the original number again. | 24,25,49,51,75,76,99 | 52 | 20 |
math | Task B-4.3. Let $f_{n}(x)=x^{n+1}+x^{n}$, for $n \in \mathbb{N}, x \in \mathbb{R}$. For which real numbers $x$, will the infinite sum $f_{1}(x)+f_{2}(x)+f_{3}(x)+\ldots$ have a value in the interval $\left\langle 0, \frac{2}{3}\right]$? | x\in\langle0,\frac{1}{3}] | 105 | 13 |
math | 11. (1 mark) Find the $2002^{\text {nd }}$ positive integer that is not the difference of two square integers.
(1 分) 求第 2002 個不能寫成兩個平方整數的差的正整數。 | 8006 | 63 | 4 |
math | 2. Determine all integer values of $n$ for which $n^{2}+6 n+24$ is a perfect square. | 4,-2,-4,-10 | 29 | 8 |
math | Given $f: k \rightarrow R$, for all $x, y \in \mathbf{R}$, it satisfies
$$
f\left(x^{2}-y^{2}\right)=x f(x)-y f(y) .
$$
Find $f(x)$. | f(x)=kx | 59 | 5 |
math | Find the greatest positive integer $N$ with the following property: there exist integers $x_1, . . . , x_N$ such that $x^2_i - x_ix_j$ is not divisible by $1111$ for any $i\ne j.$
| 1000 | 59 | 6 |
math | Initial 65. Given a real-coefficient polynomial function $y=a x^{2}+b x+c$, for any $|x| \leqslant 1$, it is known that $|y| \leqslant 1$. Try to find the maximum value of $|a|+|b|+|c|$. | 3 | 74 | 1 |
math | Example 17 (2004 National College Entrance Examination, Science Question 22) Given that the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$, $S_{n}$, satisfies:
$$
S_{n}=2 a_{n}+(-1)^{n}(n \geqslant 1) \text {. }
$$
(1) Write down the first three terms of the sequence $\left\{a_{n}\right\}$, $a_{1}, a... | a_{n}=\frac{2}{3}[2^{n-2}+(-1)^{n-1}],n\in{N}^{*} | 309 | 35 |
math | 5. Let $x_{1}, x_{2}, \cdots, x_{51}$ be natural numbers, $x_{1}<x_{2}$ $<\cdots<x_{51}$, and $x_{1}+x_{2}+\cdots+x_{51}=1995$. When $x_{26}$ reaches its maximum value, the maximum value that $x_{51}$ can take is | 95 | 94 | 2 |
math | A6. Tea and a cake cost $£ 4.50$. Tea and an éclair cost $£ 4$. A cake and an éclair cost $£ 6.50$. What is the cost of tea, a cake and an éclair? | 7.50 | 56 | 4 |
math | Big candles cost 16 cents and burn for exactly 16 minutes. Small candles cost 7 cents and burn for exactly 7 minutes. The candles burn at possibly varying and unknown rates, so it is impossible to predictably modify the amount of time for which a candle will burn except by burning it down for a known amount of time. Ca... | 97 | 106 | 2 |
math | 14. (18 points) Let $x, y, z$ be positive real numbers. Find the minimum value of the function
$$
f(x, y, z)=\frac{(1+2 x)(3 y+4 x)(4 y+3 z)(2 z+1)}{x y z}
$$ | 194+112 \sqrt{3} | 69 | 12 |
math | G4.3 Given two positive integers $x$ and $y, x y-(x+y)=\operatorname{HCF}(x, y)+\operatorname{LCM}(x, y)$, where $\operatorname{HCF}(x, y)$ and $\operatorname{LCM}(x, y)$ are respectively the greatest common divisor and the least common multiple of $x$ and $y$. If $c$ is the maximum possible value of $x+y$, find $c$. | 10 | 105 | 2 |
math | 4. Given that the radius of $\odot O$ is $R$, $C, D$ are two points on the circumference of the circle on the same side of the diameter $A B$, the degree measure of $\overparen{A C}$ is $96^{\circ}$. The degree measure of $\overparen{B D}$ is $36^{\circ}$, and a moving point $P$ is on $A B$. Then the minimum value of $... | \sqrt{3} R | 111 | 6 |
math | Ten identical crates each of dimensions $3\mathrm{ft}\times 4\mathrm{ft}\times 6\mathrm{ft}$. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $\frac {m}{n}$ be the probabil... | 190 | 121 | 3 |
math | ## Zadatak B-2.6.
Riješite jednadžbu
$$
\sqrt[2015]{16+8 x+x^{2}}+\sqrt[2015]{16-x^{2}}=2 \cdot \sqrt[2015]{16-8 x+x^{2}}
$$
| x_{2}=0 | 75 | 5 |
math | Three, (50 points) Let $f(x)$ be a polynomial with integer coefficients. For any prime $p$ and integers $u, v$, if $p \mid (uv + u + v)$, then $p \mid (f(u) f(v) - 1)$. In this case, $f(x)$ is called "good". Find all good $f(x)$.
保留源文本的换行和格式，直接输出翻译结果。 | f(x)=\(x+1)^n(n\in{N}) | 97 | 15 |
math | 1. [5 points] Point $D$ lies on side $A C$ of triangle $A B C$. The circle with diameter $B D$ intersects sides $A B$ and $B C$ at points $P$ and $T$ respectively. Points $M$ and $N$ are the midpoints of segments $A D$ and $C D$ respectively. It is known that $P M \| T N$.
a) Find the angle $A B C$.
b) Suppose additi... | 90;\frac{\sqrt{35}}{3} | 142 | 13 |
math | 2. In the bay, there are three pirate ships. During the night, raids occur, and on the first night, a third of the gold coins from the first ship are thrown onto the second ship. On the second night, a quarter of the gold coins from the second ship are thrown onto the third ship, and on the third night, a fifth of the ... | 405 | 120 | 3 |
math | 1. We understand a palindrome as a natural number that reads the same forwards and backwards, for example, 16 261. Find the largest four-digit palindrome whose square is also a palindrome. | 2002 | 42 | 4 |
math | 4. Given an equilateral $\triangle A B C$ with side length 1,
$$
\overrightarrow{A P}=\frac{1}{3}(\overrightarrow{A B}+\overrightarrow{A C}), \overrightarrow{A Q}=\overrightarrow{A P}+\frac{1}{2} \overrightarrow{B C} \text {. }
$$
Then the area of $\triangle A P Q$ is $\qquad$ . | \frac{\sqrt{3}}{12} | 98 | 11 |
math | 17. On the blackboard, there are $n$ consecutive positive integers starting from 1. After erasing one of these numbers, the average of the remaining numbers is $36 \frac{2}{5}$. The number that was erased is $\qquad$ . | 8 | 58 | 1 |
math | Five. (Full marks 14 points) Find the non-negative integer solutions $x, y, z$ that satisfy the equation $2^{x}+3^{y}=z^{2}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | (x, y, z) = (3, 0, 3), (0, 1, 2), (4, 2, 5) | 67 | 34 |
math | Determine all pairs $(x, y)$ of positive integers that satisfy
$$
x+y+1 \mid 2 x y \quad \text{ and } \quad x+y-1 \mid x^{2}+y^{2}-1
$$ | (x, x+1) \text{ with } x \geq 1 \text{ and } (x, x-1) \text{ with } x \geq 2 | 54 | 40 |
math | 4. In the Olympionov family, it is a tradition to especially celebrate the day when a person turns as many years old as the sum of the digits of their birth year. Kolya Olympionov had such a celebration in 2013, and Tolya Olympionov had one in 2014. Who is older and by how many years? | 4 | 79 | 1 |
math | Example 1 Given the function $f(x)=\log _{2}\left(x^{2}+1\right)(x \geqslant 0), g(x)=\sqrt{x-a}(a \in \mathbf{R})$.
(1) Try to find the inverse function $f^{-1}(x)$ of the function $f(x)$;
(2) The function $h(x)=f^{-1}(x)+g(x)$, find the domain of the function $h(x)$, and determine the monotonicity of $h(x)$;
(3) If t... | (-\infty,-4]\cup[\log_{2}5,+\infty) | 166 | 19 |
math | Problem 8. For what values of the parameter a does the equation $x^{3}+6 x^{2}+a x+8=0$ have exactly three solutions? | (-\infty;-15) | 38 | 8 |
math | 9. (15 points) Find all values of the parameter $a$ for which the equation
$$
3 x^{2}-4(3 a-2) x+a^{2}+2 a=0
$$
has roots $x_{1}$ and $x_{2}$, satisfying the condition $x_{1}<a<x_{2}$. | (-\infty;0)\cup(1,25;+\infty) | 76 | 18 |
math | 6. On the front and back of four cards, 0 and 1, 0 and 2, 3 and 4, 5 and 6 are written respectively. By placing any three of them side by side to form a three-digit number, a total of $\qquad$ different three-digit numbers can be obtained. | 124 | 69 | 3 |
math | 3-ча 1. Solve the system:
$$
\left\{\begin{aligned}
x+y+z & =a \\
x^{2}+y^{2}+z^{2} & =a^{2} \\
x^{3}+y^{3}+z^{3} & =a^{3}
\end{aligned}\right.
$$ | (0,0,),(0,,0),(,0,0) | 76 | 15 |
math | When Lisa squares her favorite $2$-digit number, she gets the same result as when she cubes the sum of the digits of her favorite $2$-digit number. What is Lisa's favorite $2$-digit number? | 27 | 48 | 2 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.