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 |
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math | Let $n\geq 2$ be an integer and let $f$ be a $4n$-variable polynomial with real coefficients. Assume that, for any $2n$ points $(x_1,y_1),\dots,(x_{2n},y_{2n})$ in the Cartesian plane, $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$ if and only if the points form the vertices of a regular $2n$-gon in some order, or are all equal.
Determine the smallest possible degree of $f$.
(Note, for example, that the degree of the polynomial $$g(x,y)=4x^3y^4+yx+x-2$$ is $7$ because $7=3+4$.)
[i]Ankan Bhattacharya[/i] | 2n | 181 | 4 |
math | 1. Given $f(x)=\frac{10}{x+1}-\frac{\sqrt{x}}{3}$. Then the number of elements in the set $M=\left\{n \in \mathbf{Z} \mid f\left(n^{2}-1\right) \geqslant 0\right\}$ is $\qquad$. | 6 | 79 | 1 |
math | A regular 1976-sided polygon has the midpoints of all its sides and diagonals marked. Of the points thus obtained, what is the maximum number that can lie on a circle? | 1976 | 40 | 4 |
math | 1st APMO 1989 Problem 5 f is a strictly increasing real-valued function on the reals. It has inverse f -1 . Find all possible f such that f(x) + f -1 (x) = 2x for all x. Solution | f(x)=x+b | 59 | 5 |
math | 3. The number of triples of positive integers $(x, y, z)$ that satisfy $x y z=3^{2010}$ and $x \leqslant y \leqslant z<x+y$ is $\qquad$.
| 336 | 54 | 3 |
math | Solve the equation $1 / a+1 / b+1 / c=1$ in integers.
# | (3,3,3),(2,3,6),(2,4,4),(1,,-) | 23 | 23 |
math | Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$ | (1, 2, 3) | 46 | 10 |
math | 308. The scale division value of the amperemeter is 0.1 A. The readings of the amperemeter are rounded to the nearest whole division. Find the probability that an error exceeding $0.02 \mathrm{~A}$ will be made during the reading. | 0.6 | 62 | 3 |
math | 10.122. The larger base of the trapezoid has a length of 24 cm (Fig. 10.113). Find the length of its smaller base, given that the distance between the midpoints of the diagonals of the trapezoid is 4 cm. | 16 | 66 | 2 |
math | 2. The system of equations
$$
\left\{\begin{array}{l}
x^{2}-y^{2}+x-2 y-6=0, \\
2 x y+2 x+y-4=0
\end{array}\right.
$$
has the real solutions $(x, y)$ as $\qquad$ . | (2,0),(-3,-2) | 74 | 10 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(x+f(y))-f(x)=(x+f(y))^{4}-x^{4}
$$
for all $x, y \in \mathbb{R}$. | f(x)=0f(x)=x^{4}+kforanyrealconstantk | 61 | 18 |
math | $n$ is a positive integer. Let $a(n)$ be the smallest number for which $n\mid a(n)!$
Find all solutions of:$$\frac{a(n)}{n}=\frac{2}{3}$$ | n = 9 | 49 | 4 |
math | United States 1998
We color a $98 \times 98$ checkerboard in a chessboard pattern. A move consists of selecting a rectangle made up of small squares and inverting their colors. What is the minimum number of moves required to make the checkerboard monochrome? | 98 | 62 | 2 |
math | ## problem statement
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4_{\text {and }}}$ its height dropped from vertex $A_{4 \text { to the face }} A_{1} A_{2} A_{3}$.
$A_{1}(4 ;-1 ; 3)$
$A_{2}(-2 ; 1 ; 0)$
$A_{3}(0 ;-5 ; 1)$
$A_{4}(3 ; 2 ;-6)$ | \frac{17}{\sqrt{5}} | 122 | 11 |
math | Given $x \geqslant 0, y \geqslant 0$, and $x^{2}+y^{2}=1$, then the maximum value of $x(x+y)$ is
保留源文本的换行和格式，直接输出翻译结果。
Given $x \geqslant 0, y \geqslant 0$, and $x^{2}+y^{2}=1$, then the maximum value of $x(x+y)$ is | \frac{\sqrt{2}+1}{2} | 103 | 12 |
math | # Problem 9.4
Solve the equation in natural numbers $\mathrm{n}$ and $\mathrm{m}$
$(n+1)!(m+1)!=(n+m)!$
Number of points 7 | (2,4),(4,2) | 45 | 9 |
math | Example 3 Given the circle $x^{2}+y^{2}=r^{2}$ passes through the two foci $F_{1}(-c, 0)$, $F_{2}(c, 0)$ of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, and the two curves have four intersection points, one of which is $P$. If the area of $\triangle F_{1} P F_{2}$ is 26, and the length of the major axis of the ellipse is 15, find the value of $a+b+c$.
(2000 "Hope Cup" Competition Question) | 13+\sqrt{26} | 155 | 8 |
math | 6. (3 points) On a $2 \times 4$ chessboard, there are knights, who always tell the truth, and liars, who always lie. Each of them said: "Among my neighbors, there are exactly three liars." How many liars are on the board?
Neighbors are considered to be people on cells that share a common side. | 6 | 76 | 1 |
math | 14. Can $1971^{26}+1972^{27}+1973^{28}$ be divisible by 3? | 2(\bmod3) | 37 | 6 |
math | 4. Given the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0), l$ is an asymptote of the hyperbola $C$, and $F_{1}, F_{2}$ are the left and right foci of the hyperbola $C$. If the symmetric point of $F_{1}$ with respect to the line $l$ lies on the circle $(x-c)^{2}+y^{2}=c^{2}$ (where $c$ is the semi-focal distance of the hyperbola), then the eccentricity of the hyperbola $C$ is $\qquad$ | 2 | 154 | 1 |
math | Let $p,q,r$ be distinct prime numbers and let
\[A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} \]
Find the least $n\in\mathbb{N}$ such that for any $B\subset A$ where $|B|=n$, has elements $x$ and $y$ such that $x$ divides $y$.
[i]Ioan Tomescu[/i] | 28 | 100 | 4 |
math | 2. A 13-digit display shows the number 1201201201201. Robots C3PO and R2D2 take turns rearranging its digits. In one move, they can swap two adjacent digits, but it is forbidden to swap digits on positions that have already been swapped by either robot. Additionally, a zero cannot be placed in the first position. The player who cannot make a move loses. Who will win with correct play if C3PO starts? | C3PO | 104 | 3 |
math | Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of
\[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\]
Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$ | n | 122 | 2 |
math | In the parallelogram $ABCD$, the angle bisector of $\angle ABC$ intersects the side $AD$ at point $P$. We know that $PD=5$, $BP=6$, $CP=6$. What is the length of side $AB$? | 4or9 | 57 | 3 |
math | Let $ABCD$ be a trapezoid with $AB\parallel DC$. Let $M$ be the midpoint of $CD$. If $AD\perp CD, AC\perp BM,$ and $BC\perp BD$, find $\frac{AB}{CD}$.
[i]Proposed by Nathan Ramesh | \frac{\sqrt{2}}{2} | 69 | 10 |
math | 1. [5] A polynomial $P$ with integer coefficients is called tricky if it has 4 as a root.
A polynomial is called teeny if it has degree at most 1 and integer coefficients between -7 and 7 , inclusive.
How many nonzero tricky teeny polynomials are there? | 2 | 62 | 1 |
math | 10. Disks (from 9th grade. 3 points). At a familiar factory, metal disks with a diameter of 1 m are cut out. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, there is a measurement error, and therefore the standard deviation of the radius is 10 mm. Engineer Sidorov believes that a stack of 100 disks will on average weigh 10000 kg. By how much is Engineer Sidorov mistaken? | 4 | 115 | 1 |
math | A $0 \leq t \leq \pi$ real parameter for which values does the equation $\sin (x+t)=1-\sin x$ have no solution? | \frac{2\pi}{3}<\leq\pi | 36 | 14 |
math | 4. (2000 National High School Mathematics Competition) Let $S_{n}=1+2+3+\cdots+n, n \in \mathbf{N}^{\cdot}$, find the maximum value of $f(n)=\frac{S_{n}}{(n+32) S_{n+1}}$.
| \frac{1}{50} | 73 | 8 |
math | 4・82 Solve the equation $\log _{3} x+\log _{x} 3-2 \log _{3} x \log _{x} 3=\frac{1}{2}$. | x=\sqrt{3}, x=9 | 47 | 9 |
math | Example 7. Solve the equation:
a) $\log _{x^{2}-1}\left(x^{3}+6\right)=\log _{x^{2}-1}\left(4 x^{2}-x\right)$;
b) $\log _{x^{3}+x}\left(x^{2}-4\right)=\log _{4 x^{2}-6}\left(x^{2}-4\right)$. | x_2=2,x_3=3 | 95 | 10 |
math | (British MO 1998)
Solve for $x, y, z \in \mathbb{R}^{+}$ real positive numbers the following system of equations:
$$
\begin{aligned}
& x y + y z + z x = 12 \\
& x y z = 2 + x + y + z
\end{aligned}
$$ | 2 | 80 | 1 |
math | II. (50 points)
Given that $x, y, z$ are all positive real numbers, find the minimum value of $\iota=\frac{(x+y-z)^{2}}{(x+y)^{2}+z^{2}}+\frac{(z+x-y)^{2}}{(z+x)^{2}+y^{2}}+\frac{(y+z-x)^{2}}{(y+z)^{2}+x^{2}}$. | \frac{3}{5} | 97 | 7 |
math | [ Decimal numeral system ]
$[$ Division with remainder $]$
Find all such two-digit numbers that when multiplied by some integer, the resulting number has a second-to-last digit of 5. | Allnotdivisible20 | 39 | 6 |
math | Every cell of a $3\times 3$ board is coloured either by red or blue. Find the number of all colorings in which there are no $2\times 2$ squares in which all cells are red. | 417 | 47 | 3 |
math | The nine delegates to the Economic Cooperation Conference include $2$ officials from Mexico, $3$ officials from Canada, and $4$ officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 139 | 97 | 3 |
math | 10. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\frac{1}{2}$, the upper and lower endpoints of the ellipse's minor axis are $A, B$ respectively. A circle centered at $A$ with the semi-major axis length of the ellipse as its radius intersects the ellipse at points $C, D$. The y-coordinate of the midpoint of $C D$ is $6-3 \sqrt{3}$.
(1) Find the equation of the ellipse;
(2) A line $l$ passes through the right focus $F$ of the ellipse and is not perpendicular to the $x$-axis. $l$ intersects the ellipse at points $M, N$. Let the point $N$'s reflection over the $x$-axis be $N^{\prime}$. Determine whether the line $M N^{\prime}$ passes through a fixed point. If it does, find this fixed point; otherwise, explain the reason. | \frac{x^{2}}{4}+\frac{y^{2}}{3}=1P(4,0) | 229 | 26 |
math | What is the value of $a+b+c+d$, if
$$
\begin{gathered}
6 a+2 b=3848 \\
6 c+3 d=4410 \\
a+3 b+2 d=3080
\end{gathered}
$$ | 1986 | 63 | 4 |
math | 12. Write the consecutive odd numbers from 1 to 103 as a single large number: \( A = \) 13579111315171921….9799101103.
Then the number \( \mathrm{a} \) has \_\_\_\_ digits, and the remainder when \( \mathrm{a} \) is divided by 9 is \_\_\_\_ _\. | 101;4 | 101 | 5 |
math | 2 Find all pairs of positive integers $(a, b)$, such that
$$
\frac{a^{2}}{2 a b^{2}-b^{3}+1}
$$
is an integer. | (2,1),(,2),(8^{4}-,2) | 45 | 15 |
math | Example 10 Given $\left(x^{2}-x+1\right)^{6}=a_{12} x^{12}+$
$$
\begin{array}{l}
a_{11} x^{11}+\cdots+a_{2} x^{2}+a_{1} x+a_{0} \text {. Then } a_{12}+a_{10} \\
+\cdots+a_{2}+a_{0}=\ldots
\end{array}
$$
(Fifth National Partial Provinces Junior High School Mathematics Correspondence Competition) | 365 | 126 | 3 |
math | Let $ \{a_n\}_{n=1}^\infty $ be an arithmetic progression with $ a_1 > 0 $ and $ 5\cdot a_{13} = 6\cdot a_{19} $ . What is the smallest integer $ n$ such that $ a_n<0 $? | 50 | 69 | 2 |
math | 68. From Pickleminster to Quickville. Trains $A$ and $B$ depart from Pickleminster to Quickville at the same time as trains $C$ and $D$ depart from Quickville to Pickleminster. Train $A$ meets train $C$ 120 miles from Pickleminster and train $D$ 140 miles from Pickleminster. Train $B$ meets train $C$ 126 miles from Quickville, and train $D$ at the halfway point between Pickleminster and Quickville. What is the distance from Pickleminster to Quickville? All trains travel at constant speeds, not too different from the usual. | 210 | 144 | 3 |
math | Example. Find the flux of the vector field
$$
\vec{a}=x \vec{i}+(y+z) \vec{j}+(z-y) \vec{k}
$$
through the part of the surface
$$
x^{2}+y^{2}+z^{2}=9
$$
cut by the plane $z=0 \quad(z \geq 0)$ (the normal is external to the closed surface formed by these surfaces). | 54\pi | 98 | 4 |
math | Evokimov
A natural number is written on the board. If you erase the last digit (in the units place), the remaining non-zero number will be divisible by 20, and if you erase the first digit, it will be divisible by 21. What is the smallest number that can be written on the board if its second digit is not equal to 0?
# | 1609 | 80 | 4 |
math | Example 2 The sum of the 6 edge lengths of the tetrahedron $P-ABC$ is $l$, and $\angle APB=\angle BPC=\angle CPA=90^{\circ}$. Then the maximum volume of the tetrahedron is $\qquad$
(5th United States of America Mathematical Olympiad) | \frac{1}{6}\left(\frac{l}{3+3 \sqrt{2}}\right)^{3} | 72 | 26 |
math | Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 157 | 132 | 3 |
math | Solve the system of equations for positive real numbers:
$$\frac{1}{xy}=\frac{x}{z}+ 1,\frac{1}{yz} = \frac{y}{x} + 1, \frac{1}{zx} =\frac{z}{y}+ 1$$ | x = y = z = \frac{1}{\sqrt{2}} | 66 | 16 |
math | ## Task 1 - 150611
The People's Republic of Poland delivered a total of 132,000 tons of coal and 24,000 tons of coke from December 6, 1974 (first delivery day) to December 18, 1974 (last delivery day) by waterway to the capital of the GDR. The delivery was made on barges with a capacity of 600 tons each.
How many of these barge loads arrived in Berlin on average each day during the specified period (with Sundays counted as delivery days)? | 20 | 131 | 2 |
math | # Problem 7. (4 points)
$O A B C$ is a rectangle on the Cartesian plane, with sides parallel to the coordinate axes. Point $O$ is the origin, and point $B$ has coordinates $(11; 8)$. Inside the rectangle, a point $X$ with integer coordinates is taken. What is the smallest value that the area of triangle $O B X$ can take?
Answer: $\frac{1}{2}$ | \frac{1}{2} | 96 | 7 |
math | 1. The price of one pencil is a whole number of eurocents. The total price of 9 pencils is greater than 11, but less than 12 euros, while the total price of 13 pencils is greater than 15, but less than 16 euros. How much does one pencil cost? | 123 | 68 | 3 |
math | 1373. Integrate the equation
$$
x d y=(x+y) d x
$$
and find the particular solution that satisfies the initial conditions $y=2$ when $x=-1$. | x\ln|x|-2x | 45 | 7 |
math | Twelve students participated in a theater festival consisting of $n$ different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of $n$. | n = 4 | 47 | 5 |
math | 13.435 The desired number is greater than 400 and less than 500. Find it, if the sum of its digits is 9 and it is equal to 47/36 of the number represented by the same digits but written in reverse order. | 423 | 61 | 3 |
math | 14. In 1993, American mathematician F. Smarandache proposed many number theory problems, attracting the attention of scholars both at home and abroad. One of these is the famous Smarandache function. The Smarandache function of a positive integer \( n \) is defined as:
\[
S(n)=\min \left\{m \left| m \in \mathbb{N}^{*}, n \mid m!\right\} .
\]
For example: \( S(2)=2, S(3)=3, S(6)=3 \).
(1) Find the values of \( S(16) \) and \( S(2016) \);
(2) If \( S(n)=7 \), find the maximum value of the positive integer \( n \);
(3) Prove: There exist infinitely many composite numbers \( n \) such that \( S(n)=p \). Here, \( p \) is the largest prime factor of \( n \). | 8 | 217 | 1 |
math | 12. (6 points) In a parking lot, there are a total of 24 vehicles, among which cars have 4 wheels, and motorcycles have 3 wheels. These vehicles have a total of 86 wheels. How many three-wheeled motorcycles are there? $\qquad$ vehicles. | 10 | 64 | 2 |
math | 3. The function
$$
f(x)=\left|\sin x+\frac{1}{2} \sin 2 x\right|(x \in \mathbf{R})
$$
has the range | \left[0, \frac{3 \sqrt{3}}{4}\right] | 44 | 19 |
math | Find the number of divisors of $2^{9} \cdot 3^{14}$. | 150 | 21 | 3 |
math | Seven, (25 points) Let $n$ be a positive integer, $a=[\sqrt{n}]$ (where $[x]$ denotes the greatest integer not exceeding $x$). Find the maximum value of $n$ that satisfies the following conditions:
(1) $n$ is not a perfect square;
(2) $a^{3} \mid n^{2}$.
(Zhang Tongjun
Zhu Yachun, problem contributor) | 24 | 95 | 2 |
math | 9. (12 points) Shuaishuai memorized words for 7 days. Starting from the 2nd day, he memorized 1 more word each day than the previous day, and the sum of the number of words memorized in the first 4 days is equal to the sum of the number of words memorized in the last 3 days. How many words did Shuaishuai memorize in total over these 7 days? $\qquad$ | 84 | 98 | 2 |
math | 75. There are two identical urns. The first contains 2 black and 3 white balls, the second - 2 black and 1 white ball. First, an urn is chosen at random, and then one ball is drawn at random from it. What is the probability that a white ball will be selected? | \frac{7}{15} | 66 | 8 |
math | 3. A journey is divided into three segments: uphill, flat, and downhill, with the length ratios of these segments being $1: 2: 3$. The time ratios spent on each segment by a person are $4: 5: 6$. It is known that the person's speed uphill is 3 kilometers per hour, and the total length of the journey is 50 kilometers. How long did it take this person to complete the entire journey? | 10\frac{5}{12} | 96 | 10 |
math | 【Question 23】
How many consecutive "0"s are at the end of the product $5 \times 10 \times 15 \times 20 \times \cdots \times 2010 \times 2015$? | 398 | 58 | 3 |
math | 8. The real value range of the function $f(x)=\sqrt{\cos ^{2} x-\frac{3}{4}}+\sin x$ is | [-\frac{1}{2},\frac{\sqrt{2}}{2}] | 34 | 18 |
math | 1. From the odd natural numbers, we form the sets
$$
A_{1}=\{1\}, \quad A_{2}=\{3,5\}, \quad A_{3}=\{7,9,11\}, \quad A_{4}=\{13,15,17,19\}, \ldots
$$
Calculate the sum of the numbers in the set $A_{n}$. | n^3 | 95 | 3 |
math | Three. (Full marks 20 points) Given the equation $x^{2}+m x-m+1=0$ (where $m$ is an integer) has two distinct positive integer roots. Find the value of $m$.
---
The translation maintains the original format and line breaks as requested. | -5 | 64 | 2 |
math | Example 1. Select 4 people from 6 boys and 4 girls to participate in an extracurricular interest group. How many ways are there to select them?
(1) At least one boy and one girl participate;
(2) At most 3 boys participate. | 195 | 58 | 3 |
math | Let $M$ be the intersection of two diagonal, $AC$ and $BD$, of a rhombus $ABCD$, where angle $A<90^\circ$. Construct $O$ on segment $MC$ so that $OB<OC$ and let $t=\frac{MA}{MO}$, provided that $O \neq M$. Construct a circle that has $O$ as centre and goes through $B$ and $D$. Let the intersections between the circle and $AB$ be $B$ and $X$. Let the intersections between the circle and $BC$ be $B$ and $Y$. Let the intersections of $AC$ with $DX$ and $DY$ be $P$ and $Q$, respectively. Express $\frac{OQ}{OP}$ in terms of $t$. | \frac{t + 1}{t - 1} | 170 | 13 |
math | 3. Let $\mathbb{R}^{\star}$ be the set of all real numbers, except 1 . Find all functions $f: \mathbb{R}^{\star} \rightarrow \mathbb{R}$ that satisfy the functional equation
$$
x+f(x)+2 f\left(\frac{x+2009}{x-1}\right)=2010 .
$$ | f(x)=\frac{x^{2}+2007x-6028}{3(x-1)} | 86 | 26 |
math | Let $\mathbb{N}=\{1,2,\ldots\}$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that for all $m,n\in \mathbb{N}$ the number $f^2(m)+f(n)$ is a divisor of $(m^2+n)^2$. | f(m) = m | 76 | 6 |
math | 3 Let $n$ be a given positive integer, find the smallest positive integer $u_{n}$, satisfying: for any positive integer $d$, the number of integers divisible by $d$ in any $u_{n}$ consecutive odd numbers is not less than the number of integers divisible by $d$ in the odd numbers $1,3,5, \cdots, 2 n-1$. (Provided by Yonggao Chen) | u_{n}=2n-1 | 93 | 8 |
math | 4. (2003 Turkey Mathematical Olympiad) Find the triple positive integer solutions $(x, m, n)$ of the equation $x^{m}=2^{2 n+1}+2^{n}+1$. | (2^{2 n+1}+2^{n}+1,1, n),(23,2,4) | 48 | 27 |
math | 14. Given $a, b, c \in \mathbf{R}_{+}$, satisfying $a b c(a+b+c)=1$.
(1) Find the minimum value of $S=(a+c)(b+c)$;
(2) When $S$ takes the minimum value, find the maximum value of $c$.
保留了源文本的换行和格式。 | \sqrt{2}-1 | 82 | 6 |
math | 8.373. $\operatorname{tg} x+\operatorname{tg} \alpha+1=\operatorname{tg} x \operatorname{tg} \alpha$.
8.373. $\tan x+\tan \alpha+1=\tan x \tan \alpha$. | -\alpha+\frac{\pi}{4}(4k-1)for\alpha\neq\frac{\pi}{4},k\inZ | 63 | 31 |
math | 2.2.13 * Given the function $f(x)=4 \pi \arcsin x-[\arccos (-x)]^{2}$ with the maximum value $M$ and the minimum value $m$. Then $M-m=$ $\qquad$ | 3\pi^{2} | 56 | 6 |
math | A number $N$ is in base 10, $503$ in base $b$ and $305$ in base $b+2$ find product of digits of $N$ | 64 | 43 | 2 |
math | Among all fractions (whose numerator and denominator are positive integers) strictly between $\tfrac{6}{17}$ and $\tfrac{9}{25}$, which one has the smallest denominator? | \frac{5}{14} | 41 | 8 |
math | 8. In the tetrahedron $S-ABC$, $SA=4$, $SB \geqslant 7$, $SC \geqslant 9$, $AB=5$, $BC \leqslant 6$, $AC \leqslant 8$. Then the maximum volume of the tetrahedron is $\qquad$. | 8 \sqrt{6} | 78 | 6 |
math | $\mathbb{R}^2$-tic-tac-toe is a game where two players take turns putting red and blue points anywhere on the $xy$ plane. The red player moves first. The first player to get $3$ of their points in a line without any of their opponent's points in between wins. What is the least number of moves in which Red can guarantee a win? (We count each time that Red places a point as a move, including when Red places its winning point.) | 4 | 108 | 1 |
math | Let $s(a)$ denote the sum of digits of a given positive integer $a$. The sequence $a_{1}, a_{2}, \ldots a_{n}, \ldots$ of positive integers is such that $a_{n+1}=a_{n}+s\left(a_{n}\right)$ for each positive integer $n$. Find the greatest possible $n$ for which it is possible to have $a_{n}=2008$.
Let $s(a)$ denote the sum of digits of a given positive integer $a$. The sequence $a_{1}, a_{2}, \ldots a_{n}, \ldots$ of positive integers is such that $a_{n+1}=a_{n}+s\left(a_{n}\right)$ for each positive integer $n$. Find the greatest possible $n$ for which it is possible to have $a_{n}=2008$. | 6 | 197 | 1 |
math | 18. There are 2012 students standing in a row, numbered from left to right as $1, 2, \cdots \cdots 2012$. In the first round, they report numbers from left to right as “1, 2”, and those who report 2 stay; from the second round onwards, each time the remaining students report numbers from left to right as “1, 2, 3”, and those who report 3 stay, until only one student remains. What is the number of the last remaining student? | 1458 | 118 | 4 |
math | ## Task 6 - V00506
$1 ; 4 ; 5 ; 7 ; 12 ; 15 ; 16 ; 18 ; 23 ;$ etc.
This sequence of numbers is constructed according to a certain rule. Continue this sequence of numbers beyond 50! | 1;4;5;7;12;15;16;18;23;26;27;29;34;37;38;40;45;48;49;51;56;\ldots | 69 | 61 |
math | 10.148. Find the area of an isosceles trapezoid if its height is equal to $h$, and the lateral side is seen from the center of the circumscribed circle at an angle of $60^{\circ}$. | ^{2}\sqrt{3} | 56 | 7 |
math | 1. (6 points) Calculate: $\frac{(2015-201.5-20.15)}{2.015}=$ | 890 | 36 | 3 |
math | 7. Given $4 \sin \theta \cos \theta-5 \sin \theta-5 \cos \theta-1=0, \sin ^{3} \theta+\cos ^{3} \theta=$ | -\frac{11}{16} | 47 | 9 |
math | 6.289. $x^{\frac{4}{5}}-7 x^{-\frac{2}{5}}+6 x^{-1}=0$. | x_{1}=1,x_{2}=2\sqrt[3]{4},x_{3}=-3\sqrt[3]{9} | 35 | 30 |
math | There are balance scales without weights and 3 identical-looking coins, one of which is counterfeit: it is lighter than the genuine ones (the genuine coins weigh the same). How many weighings are needed to determine the counterfeit coin? Solve the same problem in the cases where there are 4 coins and 9 coins. | 1;2;2 | 64 | 5 |
math | 46th Putnam 1985 Problem A1 How many triples (A, B, C) are there of sets with union A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A ∩ B ∩ C = ∅ ? Solution | 6^{10} | 81 | 5 |
math | 13. In $\triangle A B C$, $\angle A, \angle B, \angle C$ are opposite to sides $a, b, c$ respectively. Let
$$
\begin{array}{l}
f(x)=\boldsymbol{m} \cdot \boldsymbol{n}, \boldsymbol{m}=(2 \cos x, 1), \\
\boldsymbol{n}=(\cos x, \sqrt{3} \sin 2 x), \\
f(A)=2, b=1, S_{\triangle A B C}=\frac{\sqrt{3}}{2} . \\
\text { Then } \frac{b+c}{\sin B+\sin C}=
\end{array}
$$ | 2 | 154 | 1 |
math | The first term of a number sequence is 2, the second term is 3, and the subsequent terms are formed such that each term is 1 less than the product of its two neighbors. What is the sum of the first 1095 terms of the sequence? | 1971 | 57 | 4 |
math | 5. Given the inequality $\sqrt{2}(2 a+3) \cos \left(\theta-\frac{\pi}{4}\right)+\frac{6}{\sin \theta+\cos \theta}-2 \sin 2 \theta<3 a+6$, for $\theta \in\left[0, \frac{\pi}{2}\right]$ to always hold. Find the range of $\theta$. (1st China Southeast Mathematical Olympiad) | a>3 | 96 | 3 |
math | 6. (5 points) 2015 minus its $\frac{1}{2}$, then minus the remaining $\frac{1}{3}$, then minus the remaining $\frac{1}{4}, \cdots$, and finally minus the remaining $\frac{1}{2015}$, the final number obtained is $\qquad$ . | 1 | 74 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}} \frac{e^{\sin 2 x}-e^{\tan 2 x}}{\ln \left(\frac{2 x}{\pi}\right)}$ | -2\pi | 60 | 4 |
math | 6. A class arranges for some students to participate in activities over a week (6 days), with several people arranged each day. However, among any three days, there must be at least one student who participates in all three days. Among any four days, there should not be a student who participates in all four days. How many students are needed at minimum, and provide a specific arrangement plan for the activities according to the conditions.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 20 | 110 | 2 |
math | 5 Mathematical Induction
Mathematical induction can be used to solve function problems on the set of natural numbers.
Example 6 Find all functions $f: Z_{+} \rightarrow Z_{+}$, such that for any positive integer $n$, we have
$$
f(f(f(n)))+f(f(n))+f(n)=3 n \text {. }
$$
(2008, Dutch National Team Selection Exam) | f(n)=n | 90 | 4 |
math | Example 2 Given that $f(x)$ is an $n(>0)$-degree polynomial of $x$, and for any real number $x$, it satisfies $8 f\left(x^{3}\right)-x^{6} f(2 x)-$ $2 f\left(x^{2}\right)+12=0$
(1). Find $f(x)$. | f(x)=x^{3}-2 | 81 | 8 |
math | Problem 3. Let $a, b, c, d, x, y, z, t$ be digits such that $0<a<b \leq$ $c<d$ and
$$
\overline{d c b a}=\overline{a b c d}+\overline{x y z t}
$$
Determine all possible values of the sum
$$
S=\overline{x y z t}+\overline{t z y x}
$$ | 10989or10890 | 100 | 11 |
math | There are $ n \plus{} 1$ cells in a row labeled from $ 0$ to $ n$ and $ n \plus{} 1$ cards labeled from $ 0$ to $ n$. The cards are arbitrarily placed in the cells, one per cell. The objective is to get card $ i$ into cell $ i$ for each $ i$. The allowed move is to find the smallest $ h$ such that cell $ h$ has a card with a label $ k > h$, pick up that card, slide the cards in cells $ h \plus{} 1$, $ h \plus{} 2$, ... , $ k$ one cell to the left and to place card $ k$ in cell $ k$. Show that at most $ 2^n \minus{} 1$ moves are required to get every card into the correct cell and that there is a unique starting position which requires $ 2^n \minus{} 1$ moves. [For example, if $ n \equal{} 2$ and the initial position is 210, then we get 102, then 012, a total of 2 moves.] | 2^n - 1 | 243 | 7 |
math | ## T-7
Find all positive integers $n$ for which there exist positive integers $a>b$ satisfying
$$
n=\frac{4 a b}{a-b}
$$
Answer. Any composite $n \neq 4$.
| Anycompositen\neq4 | 52 | 8 |
math | 13.372 A consignment shop has accepted for sale cameras, watches, fountain pens, and receivers for a total of 240 rubles. The sum of the prices of a receiver and a watch is 4 rubles more than the sum of the prices of a camera and a fountain pen, and the sum of the prices of a watch and a fountain pen is 24 rubles less than the sum of the prices of a camera and a receiver. The price of a fountain pen is an integer number of rubles, not exceeding 6. The number of cameras accepted is equal to the price of one camera in rubles, divided by 10; the number of watches accepted is equal to the number of receivers, as well as the number of cameras. The number of fountain pens is three times the number of cameras. How many items of the specified types were accepted by the shop? | 18 | 187 | 2 |
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