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 | 15. A tangent line is drawn through a point $A(1,1)$ on the parabola $y=x^{2}$, intersecting the $x$-axis at point $D$ and the $y$-axis at point $B$. Point $C$ is on the parabola, and point $E$ is on the line segment $A C$, satisfying $\frac{A E}{E C}=\lambda_{1}$. Point $F$ is on the line segment $B C$, satisfying $\frac{B F}{F C}=\lambda_{2}$, and $\lambda_{1}+\lambda_{2}=1$. The line segment $C D$ intersects $E F$ at point $P$. When point $C$ moves along the parabola, find the equation of the trajectory of point $P$. | 3x^{2}-2x+\frac{1}{3}(x\neq\frac{2}{3}) | 178 | 25 |
math | Jirka collects signatures of famous athletes and singers. He has a special notebook for this purpose and has decided that signatures will be on the front side of each page only. He numbered all these pages as $1,3,5,7,9, \ldots$, to know if any page gets lost. He wrote a total of 125 digits.
- How many pages did his notebook have?
- How many ones did he write in total? | 60 | 94 | 2 |
math | 9. (14 points) In $\triangle ABC$, it is known that the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively, the circumradius of $\triangle ABC$ is $R=\sqrt{3}$, and it satisfies
$\tan B + \tan C = \frac{2 \sin A}{\cos C}$.
Find: (1) $\angle B$, $b$;
(2) the maximum area of $\triangle ABC$. | \frac{9 \sqrt{3}}{4} | 112 | 12 |
math | 9. (16 points) Given real numbers $x, y$ satisfy $2^{x}+2^{y}=4^{x}+4^{y}$, try to find the range of values for $U=8^{x}+8^{y}$. | (1,2] | 57 | 5 |
math | Let be a real number $ a\in \left[ 2+\sqrt 2,4 \right] . $ Find $ \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| . $ | 1 | 64 | 1 |
math | A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then $p$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m$.
[i]Proposed by Ahaan Rungta[/i] | 1333 | 114 | 4 |
math | Find the least positive integer $N$ which is both a multiple of 19 and whose digits add to 23. | 779 | 26 | 3 |
math | Example 7 A shipping company has a ship leaving Harvard for New York every noon, and at the same time every day, a ship also leaves New York for Harvard. It takes seven days and seven nights for the ships to complete their journeys in both directions, and they all sail on the same route. How many ships of the same company will the ship leaving Harvard at noon today encounter on its way to New York? | 15 | 84 | 2 |
math | 10. Let the line pass through the intersection of the two lines $3 x+2 y-5=0, 2 x+3 y-5=0$, and have a y-intercept of -5, then the equation of this line is $\qquad$ | 6x-y-5=0 | 57 | 7 |
math | Find the set of all quadruplets $(x,y, z,w)$ of non-zero real numbers which satisfy
$$1 +\frac{1}{x}+\frac{2(x + 1)}{xy}+\frac{3(x + 1)(y + 2)}{xyz}+\frac{4(x + 1)(y + 2)(z + 3)}{xyzw}= 0$$ | (x, y, z, w) | 88 | 10 |
math | Find the natural number $A$ such that there are $A$ integer solutions to $x+y\geq A$ where $0\leq x \leq 6$ and $0\leq y \leq 7$.
[i]Proposed by David Tang[/i] | 10 | 62 | 2 |
math | A regular pentagon can have the line segments forming its boundary extended to lines, giving an arrangement of lines that intersect at ten points. How many ways are there to choose five points of these ten so that no three of the points are collinear? | 12 | 50 | 2 |
math | Liesl has a bucket. Henri drills a hole in the bottom of the bucket. Before the hole was drilled, Tap A could fill the bucket in 16 minutes, tap B could fill the bucket in 12 minutes, and tap C could fill the bucket in 8 minutes. A full bucket will completely drain out through the hole in 6 minutes. Liesl starts with the empty bucket with the hole in the bottom and turns on all three taps at the same time. How many minutes will it take until the instant when the bucket is completely full? | \frac{48}{5} | 115 | 8 |
math | 17. (12 points) Given the function $f(x)$ for any real numbers $x, y$, it satisfies $f(x+y)=f(x)+f(y)-3$, and when $x>0$, $f(x)<3$.
(1) Is $f(x)$ a monotonic function on the set of real numbers $\mathbf{R}$? Explain your reasoning;
(2) If $f(6)=-9$, find $f\left(\left(\frac{1}{2}\right)^{2010}\right)$. | 3-\left(\frac{1}{2}\right)^{2009} | 119 | 18 |
math | 4. The sequence $\left(a_{n}\right)_{n=1}^{\infty}$ is defined as
$$
a_{n}=\sin ^{2} \pi \sqrt{n^{2}+n}
$$
Determine
$$
\lim _{n \rightarrow \infty} a_{n}
$$ | 1 | 72 | 1 |
math | Find all rational numbers $k$ such that $0 \le k \le \frac{1}{2}$ and $\cos k \pi$ is rational. | k = 0, \frac{1}{2}, \frac{1}{3} | 33 | 20 |
math | Example 12 Let $n$ be a fixed integer, $n \geqslant 2$.
(1) Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c \cdot\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$
holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$;
(2) For this constant $c$, determine the necessary and sufficient conditions for equality to hold.
(IMO - 40 Problem) | \frac{1}{8} | 169 | 7 |
math | 7. [5] Find $p$ so that $\lim _{x \rightarrow \infty} x^{p}(\sqrt[3]{x+1}+\sqrt[3]{x-1}-2 \sqrt[3]{x})$ is some non-zero real number. | \frac{5}{3} | 60 | 7 |
math | Given two positive integers $m,n$, we say that a function $f : [0,m] \to \mathbb{R}$ is $(m,n)$-[i]slippery[/i] if it has the following properties:
i) $f$ is continuous;
ii) $f(0) = 0$, $f(m) = n$;
iii) If $t_1, t_2\in [0,m]$ with $t_1 < t_2$ are such that $t_2-t_1\in \mathbb{Z}$ and $f(t_2)-f(t_1)\in\mathbb{Z}$, then $t_2-t_1 \in \{0,m\}$.
Find all the possible values for $m, n$ such that there is a function $f$ that is $(m,n)$-slippery. | \gcd(m,n) = 1 | 189 | 9 |
math | Let $ 2^{1110} \equiv n \bmod{1111} $ with $ 0 \leq n < 1111 $. Compute $ n $. | 1024 | 42 | 4 |
math | 10. (3 points) On a highway, there are three points $A$, $O$, and $B$. $O$ is between $A$ and $B$, and the distance between $A$ and $O$ is 1360 meters. Two people, Jia and Yi, start from points $A$ and $O$ respectively at the same time and head towards point $B$. After 10 minutes, Jia and Yi are at the same distance from point $O$: After 40 minutes, Jia and Yi meet at point $B$. What is the distance between points $O$ and $B$ in meters? | 2040 | 138 | 4 |
math | For the vertices of an $n$-sided polygon, we wrote different real numbers such that any written number is the product of the numbers written at the two adjacent vertices.
Determine the value of $n$! | 6 | 45 | 1 |
math | 7. Let $A B C D$ be a tetrahedron such that edges $A B, A C$, and $A D$ are mutually perpendicular. Let the areas of triangles $A B C, A C D$, and $A D B$ be denoted by $x, y$, and $z$, respectively. In terms of $x, y$, and $z$, find the area of triangle $B C D$. | \sqrt{x^{2}+y^{2}+z^{2}} | 91 | 16 |
math | The product of three natural numbers is 600. If we decreased one of the factors by 10, the product would decrease by 400. If we increased that one factor by 5 instead, the product would double. Which three natural numbers have this property?
(L. Hozová)
Hint. From each declarative sentence in the problem, one factor can be directly determined. | 5,8,15 | 83 | 6 |
math | (1) For $ 0 < x < 1$, prove that $ (\sqrt {2} \minus{} 1)x \plus{} 1 < \sqrt {x \plus{} 1} < \sqrt {2}.$
(2) Find $ \lim_{a\rightarrow 1 \minus{} 0} \frac {\int_a^1 x\sqrt {1 \minus{} x^2}\ dx}{(1 \minus{} a)^{\frac 32}}$. | \frac{2\sqrt{2}}{3} | 105 | 12 |
math | Task B-4.3. Determine $a \in \mathbb{C}$ so that the number $z_{0}=-\sqrt{3}+i$ is a root of the polynomial $P(z)=z^{15}-a$. From the remaining roots of the polynomial $P$, determine the one with the smallest argument. | w_{0}=2(\cos\frac{\pi}{30}+i\sin\frac{\pi}{30}) | 71 | 27 |
math | 13. Given the function
$$
f(x)=\sin ^{2} \omega x+\sqrt{3} \sin \omega x \cdot \sin \left(\omega x+\frac{\pi}{2}\right)
$$
has the smallest positive period of $\frac{\pi}{2}$, where $\omega>0$. Find the maximum and minimum values of $f(x)$ on $\left[\frac{\pi}{8}, \frac{\pi}{4}\right]$. | 1 \leqslant f(x) \leqslant \frac{3}{2} | 102 | 21 |
math | 11.010. In a cube with an edge length of $a$, the center of the upper face is connected to the vertices of the base. Find the total surface area of the resulting pyramid. | ^{2}(1+\sqrt{5}) | 43 | 9 |
math | XXIV OM - I - Problem 8
Find a polynomial with integer coefficients of the lowest possible degree, for which $ \sqrt{2} + \sqrt{3} $ is a root. | x^4-10x^2+1 | 41 | 11 |
math | 2. Let $A B C$ be a right triangle with $\angle B C A=90^{\circ}$. A circle with diameter $A C$ intersects the hypotenuse $A B$ at $K$. If $B K: A K=1: 3$, find the measure of the angle $\angle B A C$. | 30 | 72 | 2 |
math | Task 5. Determine all pairs $(m, n)$ of positive integers for which
$$
(m+n)^{3} \mid 2 n\left(3 m^{2}+n^{2}\right)+8 .
$$ | (,n)=(1,1)(,n)=(n+2,n)forn\geq1 | 49 | 21 |
math | 11.082. The height of the cone and its slant height are 4 and 5 cm, respectively. Find the volume of a hemisphere inscribed in the cone, with its base lying on the base of the cone. | \frac{1152}{125}\pi\mathrm{}^{3} | 50 | 19 |
math | Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of [i]balanced[/i] tokens is even or odd. | \text{Odd} | 80 | 6 |
math | 1. In some cells of a $1 \times 2100$ strip, one chip is placed. In each of the empty cells, a number is written that is equal to the absolute difference between the number of chips to the left and to the right of this cell. It is known that all the written numbers are distinct and non-zero. What is the minimum number of chips that can be placed in the cells? | 1400 | 87 | 4 |
math | Example 4 Find the largest constant $c$ such that for all real numbers $x, y$ satisfying $x>0, y>0, x^{2}+y^{2}=1$, we always have $x^{6}+y^{6} \geqslant c x y$.
| \frac{1}{2} | 65 | 7 |
math | 8.4. There are two types of five-digit numbers as follows:
(1) The sum of the digits is 36, and it is an even number;
(2) The sum of the digits is 38, and it is an odd number.
Try to determine: which type of number is more? Explain your reasoning. | There\ are\ more\ even\ numbers\ whose\ sum\ of\ digits\ equals\ 36. | 70 | 24 |
math | 5. A company's working hours are from 8:30 AM to 5:30 PM. During this period, the hour and minute hands of the clock overlap times. | 9 | 38 | 1 |
math | Two candidates participated in an election with $p+q$ voters. Candidate $A$ received $p$ votes and candidate $B$ received $q$ votes, with $p>q$. During the counting, only one vote is recorded at a time on a board. Let $r$ be the probability that the number associated with candidate $A$ on the board is always greater than the number associated with candidate $B$ throughout the entire counting process.
a) Determine the value of $r$ if $p=3$ and $q=2$.
b) Determine the value of $r$ if $p=1010$ and $q=1009$. | \frac{1}{2019} | 142 | 10 |
math | Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible value for $\tan \theta$ can be expressed in the form $\dfrac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
[i]Proposed by James Lin[/i] | 503 | 179 | 3 |
math | For any finite set $X$, let $| X |$ denote the number of elements in $X$. Define
\[S_n = \sum | A \cap B | ,\]
where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\left\{ 1 , 2 , 3, \cdots , n \right\}$ with $|A| = |B|$.
For example, $S_2 = 4$ because the sum is taken over the pairs of subsets
\[(A, B) \in \left\{ (\emptyset, \emptyset) , ( \{1\} , \{1\} ), ( \{1\} , \{2\} ) , ( \{2\} , \{1\} ) , ( \{2\} , \{2\} ) , ( \{1 , 2\} , \{1 , 2\} ) \right\} ,\]
giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4$.
Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by
1000. | 245 | 311 | 3 |
math | 1A. Determine the second term of the arithmetic progression if the sum of the first 10 terms is 300, and the first, second, and fifth terms, in that order, form a geometric progression. | 9 | 46 | 1 |
math | 8. The sequence $\left\{a_{n}\right\}$ satisfies $a_{n+1}=(-1)^{n} n-a_{n}, n=1,2,3, \cdots$, and $a_{10}=a_{1}$, then the maximum value of $a_{n} a_{n+1}$ is $\qquad$ . | \frac{33}{4} | 80 | 8 |
math | Example 10. What set of points in the complex plane $z$ is defined by the condition
$$
\operatorname{Im} z^{2}>2 ?
$$ | xy>1 | 37 | 3 |
math | $$
\begin{array}{ll}
1^{\circ} . & \frac{\sin 3 \alpha+\sin \alpha}{\cos 3 \alpha-\cos \alpha}+\operatorname{ctg} \alpha=? \\
2^{\circ} . & \frac{\sin 3 \alpha-\sin \alpha}{\cos 3 \alpha+\cos \alpha}-\operatorname{tg} \alpha=?
\end{array}
$$
| 0 | 99 | 1 |
math | 7.2. What is the smallest number of digits that can be appended to the right of the number 2013 so that the resulting number is divisible by all natural numbers less than 10? | 2013480 | 43 | 7 |
math | Example 16 Find the product $T_{n}=\left(\cos \frac{A}{2}+\cos \frac{B}{2}\right) \cdot\left(\cos \frac{A}{2^{2}}+\cos \frac{B}{2^{2}}\right) \cdots \cdots\left(\cos \frac{A}{2^{n}}+\cos \frac{B}{2^{n}}\right)$. | \frac{\sin\frac{A+B}{2}\sin\frac{A-B}{2}}{2^{n}\sin\frac{A+B}{2^{n+1}}\sin\frac{A-B}{2^{n+1}}} | 97 | 53 |
math | \section*{Problem 3 - 071033}
Inge says to her sister Monika:
"We performed calculations on a square pyramid in math class yesterday and obtained the same numerical values for the volume and the surface area.
I still remember that all the numerical values were natural numbers, but I can't recall what they were."
"Which numerical values did you mean when you said 'all numerical values'?"
"I meant the numerical values of the side length of the base, the height, the volume, and the surface area of the pyramid."
"Were these quantities labeled with matching units, for example, were the lengths in cm, the surface area in \(\mathrm{cm}^{2}\), and the volume in \(\mathrm{cm}^{3}\)?"
"Yes, that's how it was."
From these statements, Monika can reconstruct the problem. How can this be done? | =12,=8,V=A=384 | 186 | 12 |
math | Example 4 In the positive term sequence $\left\{a_{n}\right\}$, $a_{1}=10, a_{n+1}$ $=10 \sqrt{a_{n}}$. Find the general formula for this sequence. | a_{n}=10^{2-\left(\frac{1}{2}\right)^{n-1}} | 54 | 24 |
math | 7. For $n \in \mathbf{Z}_{+}, n \geqslant 2$, let
$$
S_{n}=\sum_{k=1}^{n} \frac{k}{1+k^{2}+k^{4}}, T_{n}=\prod_{k=2}^{n} \frac{k^{3}-1}{k^{3}+1} \text {. }
$$
Then $S_{n} T_{n}=$ . $\qquad$ | \frac{1}{3} | 107 | 7 |
math | Compute
\[
\left(\frac{4-\log_{36} 4 - \log_6 {18}}{\log_4 3} \right) \cdot \left( \log_8 {27} + \log_2 9 \right).
\]
[i]2020 CCA Math Bonanza Team Round #4[/i] | 12 | 83 | 2 |
math | 7.2. Does there exist a number of the form $1000 \cdots 001$ that can be divided by a number of the form $111 \cdots 11$? | 11 | 47 | 2 |
math | Example 4. The probability of hitting the target with a single shot is $p=0.8$. Find the probability of five hits in six shots. | 0.3934 | 32 | 6 |
math | 1.1.3 $\star \star$ A 4-element real number set $S$ has the sum of the elements of all its subsets equal to 2008 (here the sum of elements of the empty set is considered to be 0). Find the sum of all elements of $S$. | 251 | 64 | 3 |
math | Problem 7.2. For natural numbers $m$ and $n$, it is known that if their sum is multiplied by the absolute value of their difference, the result is 2021. What values can the numbers $m$ and $n$ take? (7 points)
# | =1011,n=1010=45,n=2 | 61 | 17 |
math | 6. Draw a normal line (a normal line is a line passing through the point of tangency and perpendicular to the tangent line) at point $A$ on the parabola $y=x^{2}$, intersecting the parabola at another point $B$, and $O$ is the origin. When the area of $\triangle O A B$ is minimized, the y-coordinate of point $A$ is $\qquad$ (Liu Kaifeng) | \frac{-3+\sqrt{33}}{24} | 97 | 14 |
math | Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$?
[i]Proposed by Giacomo Rizzo[/i] | 444 | 58 | 3 |
math | Let $a, b, c, d, e \in \mathbb{R}^{+}$ such that $a+b+c+d+e=8$ and $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16$. What is the maximum value of $e$? (American Olympiads, 1992) | \frac{16}{5} | 85 | 8 |
math | 7.3. A young artist had one jar of blue and one jar of yellow paint, each enough to cover 38 dm ${ }^{2}$ of area. Using all of this paint, he painted a picture: a blue sky, green grass, and a yellow sun. He obtained the green color by mixing two parts of yellow paint and one part of blue. What area on his picture is painted with each color, if the area of the grass in the picture is 6 dm $^{2}$ more than the area of the sky | Blue=27\,^2,\,Green=33\,^2,\,Yellow=16\,^2 | 111 | 28 |
math | All of the digits of a seven-digit positive integer are either $7$ or $8.$ If this integer is divisible by $9,$ what is the sum of its digits? | 54 | 36 | 2 |
math | 12.246. Find the angle between the generatrix of the cone and the plane of the base, if the lateral surface of the cone is equal to the sum of the areas of the base and the axial section. | 2\operatorname{arcctg}\pi | 47 | 10 |
math | 2. Find the smallest possible value of $\left|2015 m^{5}-2014 n^{4}\right|$, given that $m, n$ are natural numbers. | 0 | 41 | 1 |
math | Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that:
$ (i)$ $ a\plus{}c\equal{}d;$
$ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$
$ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$.
Determine $ n$. | 2002 | 93 | 4 |
math | Find all functions $f: \mathbf{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that the number $x f(x)+f^{2}(y)+2 x f(y)$ is a perfect square for all positive integers $x, y$. | f(x)=x | 59 | 4 |
math | Find all triplets $(a, b, c)$ of strictly positive real numbers such that
\[
\left\{\begin{array}{l}
a \sqrt{b}-c=a \\
b \sqrt{c}-a=b \\
c \sqrt{a}-b=c
\end{array}\right.
\] | (4,4,4) | 67 | 7 |
math | 4 [
On a sphere of radius 11, points $A, A 1, B, B 1, C$ and $C 1$ are located. Lines $A A 1, B B 1$ and $C C 1$ are pairwise perpendicular and intersect at point $M$, which is at a distance of $\sqrt{59}$ from the center of the sphere. Find $A A 1$, given that $B B 1=18$, and point $M$ divides the segment $C C 1$ in the ratio $(8+\sqrt{2}):(8-\sqrt{2})$.
# | 20 | 136 | 2 |
math | 2. In the set of natural numbers, solve the equation
$$
x^{5-x}=(6-x)^{1-x}
$$ | 1,5,9 | 29 | 5 |
math | 2. When a five-digit number is divided by 100, we get a quotient $k$ and a remainder $o$. For how many five-digit numbers is the sum $k+o$ divisible by 11? | 8181 | 48 | 4 |
math | ## Task A-4.6.
Each term of the sequence $\left(a_{n}\right)_{n \in \mathbb{N}}$ of positive real numbers, starting from the second, is equal to the arithmetic mean of the geometric and arithmetic means of its two neighboring terms.
If $a_{1}=\frac{1}{505}$ and $a_{505}=505$, determine $a_{1010}$. | 2020 | 97 | 4 |
math | 9. (16 points) Let $a \geqslant 0$, for any $m, x (0 \leqslant m \leqslant a, 0 \leqslant x \leqslant \pi)$, we have
$$
|\sin x - \sin (x+m)| \leqslant 1 \text{.}
$$
Find the maximum value of $a$. | \frac{\pi}{3} | 93 | 7 |
math | [ Special cases of parallelepipeds (other). ] [ Skew lines, angle between them ]
On the diagonals $A B 1$ and $B C 1$ of the faces of the parallelepiped $A B C D A 1 B 1 C 1 D 1$, points $M$ and $N$ are taken, such that the segments $M N$ and $A 1 C$ are parallel. Find the ratio of these segments. | 1:3 | 100 | 3 |
math | 3. There are $n$ ellipses centered at the origin and symmetric with respect to the coordinate axes
the directrices of which are all $x=1$. If the eccentricity of the $k$-th $(k=1,2, \cdots, n)$ ellipse is $e_{k}=2^{-k}$, then the sum of the major axes of these $n$ ellipses is $\qquad$ . | 2-2^{1-n} | 92 | 7 |
math | 7.1. Six consecutive numbers were written on the board. When one of them was erased and the remaining ones were added up, the result was 10085. Which number could have been erased? List all possible options. | 2020or2014 | 49 | 9 |
math | Three. (20 points) Given a function $f(x)$ defined on $\mathbf{R}$ that satisfies $f\left(f(x)-x^{2}+x\right)=f(x)-x^{2}+x$.
(1) When $f(2)=3$, find $f(1)$; when $f(0)=a$, find $f(a)$.
(2) If there is exactly one real number $x_{0}$ such that $f\left(x_{0}\right)=x_{0}$, find the analytical expression of the function $f(x)$. | f(x)=x^{2}-x+1 | 127 | 10 |
math | Determine all positive integers $n$ such that there exists a polynomial $P$ of degree $n$ with integer coefficients and $n$ distinct integers $k_{1}, k_{2}, \ldots, k_{n}$ satisfying $P\left(k_{i}\right)=n$ for all integers $1 \leqslant i \leqslant n$ and $P(0)=0$. | 1,2,3,4 | 87 | 7 |
math | 455. The amount of electricity flowing through a conductor, starting from the moment of time $t=0$, is given by the formula $Q=$ $=3 t^{2}-3 t+4$. Find the current strength at the end of the 6th second. | 33 | 58 | 2 |
math | How many $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$? (The leading digit of $N$ cannot be zero.)
[i]Based on a proposal by Yannick Yao[/i] | 256 | 74 | 3 |
math | A two-digit number $A$ is a supernumber if it is possible to find two numbers $B$ and $C$, both also two-digit numbers, such that:
- $A = B + C$;
- the sum of the digits of $A$ = (sum of the digits of $B$) + (sum of the digits of $C$).
For example, 35 is a supernumber. Two different ways to show this are $35 = 11 + 24$ and $35 = 21 + 14$, because $3 + 5 = (1 + 1) + (2 + 4)$ and $3 + 5 = (2 + 1) + (1 + 4)$. The only way to show that 21 is a supernumber is $21 = 10 + 11$.
a) Show in two different ways that 22 is a supernumber and in three different ways that 25 is a supernumber.
b) In how many different ways is it possible to show that 49 is a supernumber?
c) How many supernumbers exist? | 80 | 246 | 2 |
math | 11.042. Determine the volume of a regular truncated square pyramid if its diagonal is 18 cm, and the lengths of the sides of the bases are 14 and 10 cm. | 872\mathrm{~}^{3} | 44 | 11 |
math | 4.48. Find the remainder when the number
$$
10^{10}+10^{\left(10^{2}\right)}+10^{\left(10^{3}\right)}+\ldots+10^{\left(10^{10}\right)}
$$
is divided by 7. | 5 | 74 | 1 |
math | Let $k>1, n>2018$ be positive integers, and let $n$ be odd. The nonzero rational numbers $x_{1}$, $x_{2}, \ldots, x_{n}$ are not all equal and satisfy
$$
x_{1}+\frac{k}{x_{2}}=x_{2}+\frac{k}{x_{3}}=x_{3}+\frac{k}{x_{4}}=\cdots=x_{n-1}+\frac{k}{x_{n}}=x_{n}+\frac{k}{x_{1}}
$$
Find:
a) the product $x_{1} x_{2} \ldots x_{n}$ as a function of $k$ and $n$
b) the least value of $k$, such that there exist $n, x_{1}, x_{2}, \ldots, x_{n}$ satisfying the given conditions. | k=4 | 197 | 3 |
math | Let's calculate the value of the expression under a) and simplify the expression under b) as much as possible;
a) $\left(1-\cos 15^{\circ}\right)\left(1+\sin 75^{\circ}\right)+\cos 75^{\circ} \cos 15^{\circ} \operatorname{cotg} 15^{\circ}$
b) $\sin \left(45^{\circ}-\alpha\right)-\cos \left(30^{\circ}+\alpha\right)+\sin ^{2} 30^{\circ}-\cos \left(45^{\circ}+\alpha\right)+$ $+\sin \left(60^{\circ}-\alpha\right)+\sin ^{2} 60^{\circ}$. | 1 | 182 | 1 |
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 12 Given $a, b, c, d \in \mathbf{R}^{+}$, try to find the minimum value of $f(a, b, c, d)=\frac{a}{b+c+d}+\frac{b}{c+d+a}+\frac{c}{a+b+d}+\frac{d}{a+b+c}$. | \frac{4}{3} | 79 | 7 |
math | 13.25 Given 4 coins, one of which may be counterfeit, each genuine coin weighs 10 grams, and the counterfeit coin weighs 9 grams. Now there is a balance scale with one pan that can measure the total weight of the objects on the pan. To identify whether each coin is genuine or counterfeit, what is the minimum number of weighings required? | 3 | 77 | 1 |
math | 4. Let $z$ be a complex number. If the equation $\left|z^{2}\right|-\left|z^{2}-9\right|=7$ represents a conic section, then the eccentricity $e=$
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | \frac{3\sqrt{2}}{4} | 75 | 12 |
math | 9 If for any real number $x$, there is $f(x)=\log _{a}\left(2+\mathrm{e}^{x-1}\right) \leqslant-1$, then the range of real number $a$ is $\qquad$ . (where $e$ is an irrational number, $e=2.71828 \cdots$ ) | [\frac{1}{2},1) | 84 | 9 |
math | 6. How many different right-angled triangles exist, one of the legs of which is equal to $\sqrt{2016}$, and the other leg and the hypotenuse are expressed as natural numbers?
ANSWER: 12. | 12 | 51 | 2 |
math | 15. (6 points) There are 4 different digits that can form a total of 18 different four-digit numbers, which are arranged in ascending order. The first number is a perfect square, and the second-to-last four-digit number is also a perfect square. What is the sum of these two numbers? $\qquad$ | 10890 | 69 | 5 |
math | A crew of rowers travels $3.5 \mathrm{~km}$ downstream and the same distance upstream in a total of 1 hour and 40 minutes in a river. The speed of the river current is $2 \mathrm{~km}$ per hour. How far would the rowing team travel in 1 hour in still water? | 5 | 72 | 1 |
math | 5. Given that $A$ took an integer number of hours to travel from location A to location B, and the number of kilometers $A$ walks per hour is the same as the number of hours he took to travel from location A to location B. $B$ walks 2 kilometers per hour from location A to location B, and rests for 1 hour after every 4 kilometers, and it took him a total of 11 hours. Then the distance between location A and location B is $\qquad$ kilometers. | 16 | 108 | 2 |
math | Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$. | \frac{1}{4} | 59 | 7 |
math | 2. Let $s(n)$ denote the sum of the digits (in decimal notation) of a natural number $n$. Find all natural $n$ for which $n+s(n)=2011$. | 1991 | 43 | 4 |
math | Let $m, n, p \in \mathbf{Z}_{+} (m < n < p)$. There are $p$ lamps arranged in a circle, and initially, the pointer points to one of the lamps. Each operation can either rotate the pointer $m$ positions clockwise and then change the on/off state of the lamp it reaches, or rotate the pointer $n$ positions clockwise and then change the on/off state of the lamp it reaches. Find the necessary and sufficient conditions that $m, n, p$ should satisfy so that any of the $2^{p}$ possible on/off states of the lamps can be achieved through these two operations. | (n-m, p) \leq 2 | 136 | 10 |
math | What is the largest positive integer $n$ for which there is a unique integer $k$ such that
$$
\frac{8}{15}<\frac{n}{n+k}<\frac{7}{13} ?
$$ | 112 | 49 | 3 |
math | 10,11
Construct a rational parametrization of the circle $x^{2}+y^{2}=1$ by drawing lines through the point $(1,0)$. | (\frac{^{2}-1}{^{2}+1},\frac{-2}{^{2}+1}) | 39 | 25 |
math | 6.50. $\lim _{x \rightarrow 1} \frac{x^{4}-1}{\ln x}$. | 4 | 28 | 1 |
math | Let $p$ be an odd prime number. For positive integer $k$ satisfying $1\le k\le p-1$, the number of divisors of $kp+1$ between $k$ and $p$ exclusive is $a_k$. Find the value of $a_1+a_2+\ldots + a_{p-1}$. | p-2 | 75 | 3 |
math | Let $M=\{1,2,3, \cdots, 1995\}, A$ be a subset of $M$ and satisfy the condition: when $x \in A$, $15x \notin A$. Then the maximum number of elements in $A$ is $\qquad$ . | 1870 | 67 | 4 |
math | Task 1. Calculate:
$$
20130+2 \cdot(480 \cdot 4 \cdot 14+30 \cdot 44 \cdot 16)-(5 \cdot 80 \cdot 43+19 \cdot 400 \cdot 3) \cdot 2
$$ | 36130 | 75 | 5 |
math | 25. In one book, the following 100 statements were written:
1) “In this book, there is exactly one false statement.”
2) “In this book, there are exactly two false statements ...”
3) “In this book, there are exactly one hundred false statements.”
Which of these statements is true? | 99 | 68 | 2 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.