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 | 154. Find $\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{3 x}$. | e^6 | 33 | 3 |
math | Question 102: Let the complex number $z$ satisfy $z+\frac{1}{z} \in[1,2]$, then the minimum value of the real part of $z$ is $\qquad$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | \frac{1}{2} | 73 | 7 |
math | B2. Find all real numbers $x$ that satisfy the equation
$$
\log _{\sin x}\left(\frac{1}{2} \sin 2 x\right)=2
$$ | \frac{\pi}{4}+2k\pi | 43 | 12 |
math | 6. (10 points) In three consecutive small ponds $A$, $B$, and $C$, several goldfish are placed. If 12 goldfish swim from pond $A$ to pond $C$, then the number of goldfish in pond $C$ will be twice that in pond $A$. If 5 goldfish swim from pond $B$ to pond $A$, then the number of goldfish in ponds $A$ and $B$ will be eq... | 40 | 154 | 2 |
math | 4. Quadrilateral $ABCD$ is inscribed in a circle, $BC=CD=4$, $AC$ and $BD$ intersect at $E$, $AE=6$, and the lengths of $BE$ and $DE$ are both integers. Then the length of $BD$ is $\qquad$ | 7 | 66 | 1 |
math | 1. Find the minimum value of the expression $\frac{25 x^{2} \sin ^{2} x+16}{x \sin x}$ for $0<x<\pi$. | 40 | 42 | 2 |
math | Let's determine the minimum possible value of the expression
$$
4+x^{2} y^{4}+x^{4} y^{2}-3 x^{2} y^{2}
$$
where $(x, y)$ runs through the set of all real number pairs. | 3 | 58 | 1 |
math | ## 246. Math Puzzle $11 / 85$
A hollow round ceiling support in a large bakery is loaded with 28 tons. The outer diameter of the support is 145 millimeters, and the clear width, i.e., the inner diameter, is 115 millimeters.
What is the load per square centimeter of the cross-sectional area? | 457\mathrm{~}/\mathrm{}^{2} | 81 | 14 |
math | 6. [5] Determine the value of $\lim _{n \rightarrow \infty} \sum_{k=0}^{n}\binom{n}{k}^{-1}$. | 2 | 40 | 1 |
math | In equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has area $36\pi$, then find the perimeter of the triangle.
[i]Proposed by Isabella Grabski
[/i] | 36 | 60 | 2 |
math | Problem 8.2. The arithmetic mean of three two-digit natural numbers $x, y, z$ is 60. What is the maximum value that the expression $\frac{x+y}{z}$ can take? | 17 | 45 | 2 |
math | $\left.\begin{array}{l}\left.\text { [ } \begin{array}{l}\text { Rebus } \\ \text { [Divisibility of numbers. General properties }\end{array}\right] \\ \text { [ Decimal system } \\ \text { [ Case enumeration }\end{array}\right]$
Which digits can stand in place of the letters in the example $A B \cdot C=D E$, if diffe... | 13\cdot6=78 | 110 | 8 |
math |
10.3. Laura has 2010 lamps connected with 2010 buttons in front of her. For each button, she wants to know the corresponding lamp. In order to do this, she observes which lamps are lit when Richard presses a selection of buttons. (Not pressing anything is also a possible selection.) Richard always presses the buttons ... | 11 | 163 | 2 |
math | Let $M$ be a set and $A,B,C$ given subsets of $M$. Find a necessary and sufficient condition for the existence of a set $X\subset M$ for which $(X\cup A)\backslash(X\cap B)=C$. Describe all such sets. | X_0 \cup P \cup Q | 59 | 10 |
math | 5. Each football is sewn from pieces of leather in the shape of regular pentagons and regular hexagons. The ball has a total of 32 leather pieces. Each piece of leather in the shape of a pentagon is connected along its sides only to pieces of leather in the shape of hexagons. Each piece of leather in the shape of a hex... | 20 | 120 | 2 |
math | 4. Given a tetrahedron $S-ABC$ with the base being an isosceles right triangle with hypotenuse $AB$, $SA=SB=SC=2$, $AB=2$. Suppose points $S, A, B, C$ all lie on a sphere with center $O$. Then the distance from point $O$ to the plane $ABC$ is $\qquad$. | \frac{\sqrt{3}}{3} | 85 | 10 |
math | 4. (5 points) Hongfu Supermarket purchased a batch of salt. In the first month, 40% of the salt was sold, and in the second month, another 420 bags were sold. At this point, the ratio of the salt sold to the remaining salt is 3:1. How many bags of salt did Hongfu Supermarket purchase? | 1200 | 78 | 4 |
math | 12. [7] $\triangle P N R$ has side lengths $P N=20, N R=18$, and $P R=19$. Consider a point $A$ on $P N . \triangle N R A$ is rotated about $R$ to $\triangle N^{\prime} R A^{\prime}$ so that $R, N^{\prime}$, and $P$ lie on the same line and $A A^{\prime}$ is perpendicular to $P R$. Find $\frac{P A}{A N}$. | \frac{19}{18} | 119 | 9 |
math | Three distinct integers are chosen uniformly at random from the set
$$\{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\}.$$
Compute the probability that their arithmetic mean is an integer. | \frac{7}{20} | 88 | 8 |
math | Example 33 (2002-2003 Hungarian Mathematical Olympiad) Let $n$ be an integer greater than 2, and $a_{n}$ be the largest $n$-digit number that is neither the sum of two perfect squares nor the difference of two perfect squares.
(1) Find $a_{n}$ (expressed as a function of $n$);
(2) Find the smallest value of $n$ such th... | 66 | 115 | 2 |
math | 8. Find all positive integers $a$ such that for any positive integer $n \geqslant 5$, we have $\left(2^{n}-n^{2}\right) \mid\left(a^{n}-n^{a}\right)$. | 2, 4 | 55 | 4 |
math | Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and
\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$.
Prove that they are converges and find their limits. | \lim_{n \to \infty} x_n = 0 | 127 | 16 |
math | 4. [6 points] Solve the inequality $2 x^{4}+x^{2}-2 x-3 x^{2}|x-1|+1 \geqslant 0$. | (-\infty;-\frac{1+\sqrt{5}}{2}]\cup[-1;\frac{1}{2}]\cup[\frac{\sqrt{5}-1}{2};+\infty) | 42 | 45 |
math | Example 15 Find all non-zero real-coefficient polynomials $P(x)$ satisfying $P(x) \cdot P\left(2 x^{2}\right) \equiv P\left(2 x^{3}+x\right)(x \in \mathbf{R})$.
(IMO - 21 Shortlist) | P(x)=(x^2+1)^k,k\in{Z}^+ | 71 | 18 |
math | ## Task 1 - 250921
Determine all triples $(a, b, c)$ of natural numbers $a, b, c$ for which $a \leq b \leq c$ and $a \cdot b \cdot c=19 \cdot 85$ hold! | (1,1,1615),(1,5,323),(1,17,95),(1,19,85),(5,17,19) | 67 | 42 |
math | 5th Mexico 1991 Problem A1 Find the sum of all positive irreducible fractions less than 1 whose denominator is 1991. | 900 | 33 | 3 |
math | 8. Several buses (more than three) at the beginning of the workday sequentially leave from one point to another at constant and equal speeds. Upon arrival at the final point, each of them, without delay, turns around and heads back in the opposite direction. All buses make the same number of trips back and forth, and t... | 52or40 | 140 | 5 |
math | 2. Given $p+q=96$, and the quadratic equation $x^{2}+p x$ $+q=0$ has integer roots. Then its largest root is | 98 | 39 | 2 |
math | 9.2. Each of the 10 people is either a knight, who always tells the truth, or a liar, who always lies. Each of them thought of some integer. Then the first said: “My number is greater than 1”, the second said: “My number is greater than 2”, \ldots, the tenth said: “My number is greater than 10”. After that, all ten, sp... | 8 | 163 | 1 |
math | Example 3 (1) Find the domain of the function $y=\sqrt{\frac{1}{2}-\log _{2}(3-x)}$;
(2) Given that the domain of $f(x)$ is $[0,1]$, find the domain of the function $y=f\left(\log _{\frac{1}{2}}(3-x)\right)$. | [2,\frac{5}{2}] | 82 | 9 |
math | 1. Can the first 8 natural numbers be arranged in a circle so that each number is divisible by the difference of its neighbors? | Yes | 27 | 1 |
math | 10. Given
$$
S_{n}=|n-1|+2|n-2|+\cdots+10|n-10| \text {, }
$$
where, $n \in \mathbf{Z}_{+}$. Then the minimum value of $S_{n}$ is $\qquad$ | 112 | 72 | 3 |
math | 2. (5 points) After converting three simplest improper fractions with the same numerator into mixed numbers, they are $\mathrm{a} \frac{2}{3}, b \frac{3}{4}, c \frac{3}{5}$, where $a, b, c$ are natural numbers not exceeding 10, then $(2 a+b) \div c=$ $\qquad$ . | 4.75 | 83 | 4 |
math | ## Problem Statement
Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=2(2 \cos t-\cos 2 t) \\
y=2(2 \sin t-\sin 2 t)
\end{array}\right. \\
& 0 \leq t \leq \frac{\pi}{3}
\end{aligned}
$$ | 8(2-\sqrt{3}) | 98 | 8 |
math | An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence ... | a + b - \gcd(a, b) | 132 | 11 |
math | 1. Given the equation $\left|x^{2}-2 a x+b\right|=8$ has exactly three real roots, and they are the side lengths of a right triangle. Find the value of $a+b$.
(Bulgaria) | 264 | 50 | 3 |
math | 2. Given the complex number $z=\frac{-1+i \sqrt{3}}{2}$. Calculate the product:
$$
\left(z+\frac{1}{z}\right)\left(z^{2}+\frac{1}{z^{2}}\right) \ldots\left(z^{2012}+\frac{1}{z^{2012}}\right)
$$ | 2^{670} | 85 | 6 |
math | Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC =60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.
[i]Proposed by Kaan Dokmeci[/i] | 408 | 89 | 3 |
math | 11. The number of real solutions to the equation $\left(x^{2006}+1\right)\left(1+x^{2}+x^{4}+\cdots+x^{2004}\right)=2006 x^{2005}$ is | 1 | 61 | 1 |
math | Find all prime numbers $p$ and all strictly positive integers $x, y$ such that $\frac{1}{x}-\frac{1}{y}=\frac{1}{p}$. | (p-1,p(p-1)) | 41 | 8 |
math | 10.325. The base of the triangle is 20 cm, the medians of the lateral sides are 18 and 24 cm. Find the area of the triangle. | 288\mathrm{~}^{2} | 42 | 11 |
math | 1. Person A writes a four-digit number $A$ on the blackboard, and Person B adds a digit before and after this four-digit number to form a six-digit number $B$. If the quotient of $B$ divided by $A$ is 21 and the remainder is 0, find the values of $A$ and $B$. | A=9091,B=190911 | 73 | 14 |
math | 2. A person will turn as many years old in 2008 as the sum of the digits of their birth year. In which year was this person born? | 1985or2003 | 35 | 9 |
math | 5. (20 points) A voltmeter connected to the terminals of a current source with an EMF of $12 \mathrm{~V}$ shows a voltage $U=9 \mathrm{~V}$. Another identical voltmeter is connected to the terminals of the source. Determine the readings of the voltmeters. (The internal resistance of the source is non-zero, the resistan... | 7.2\mathrm{~V} | 91 | 9 |
math | Example 13. Map the unit circle $|z|<1$ onto the unit circle $|\boldsymbol{w}|<1$. | e^{i\alpha}\frac{z-z_{0}}{1-z\bar{z}_{0}} | 30 | 23 |
math | 7. (3 points) There are 10 cards each of the numbers “3”, “4”, and “5”. If 8 cards are randomly selected, and their sum is 31, then the maximum number of cards that can be “3” is $\qquad$. | 4 | 59 | 1 |
math | A $300 \mathrm{~m}$ high cliff is where two water droplets fall freely one after the other. The first has already fallen $\frac{1}{1000} \mathrm{~mm}$ when the second begins its fall.
How far apart will the two droplets be at the moment when the first one reaches the base of the cliff? (The result should be calculated... | 34.6\mathrm{~} | 111 | 9 |
math | 1. [20] Let $A B C$ be an equilateral triangle with side length 2 that is inscribed in a circle $\omega$. A chord of $\omega$ passes through the midpoints of sides $A B$ and $A C$. Compute the length of this chord. | \sqrt{5} | 61 | 5 |
math | 79. Without performing the operations, represent the product as a polynomial in standard form
$$
(x-1)(x+3)(x+5)
$$ | (x-1)(x+3)(x+5)=x^{3}+7x^{2}+7x-15 | 33 | 28 |
math | 1. Given $f(x)=x^{2}-2 x$, set $A=\{x \mid f(f(x))=0\}$, then the sum of all elements in set $A$ is $\qquad$ . | 4 | 48 | 1 |
math | 9. The military district canteen needs 1000 pounds of rice and 200 pounds of millet for dinner. The quartermaster goes to the rice store and finds that the store is having a promotion. “Rice is 1 yuan per pound, and for every 10 pounds purchased, 1 pound of millet is given as a gift (no gift for less than 10 pounds); m... | 1168 | 151 | 4 |
math | G10.1 If $A$ is the area of a square inscribed in a circle of diameter 10 , find $A$.
G10.2 If $a+\frac{1}{a}=2$, and $S=a^{3}+\frac{1}{a^{3}}$, find $S$.
G10.3 An $n$-sided convex polygon has 14 diagonals. Find $n$.
G10.4 If $d$ is the distance between the 2 points $(2,3)$ and $(-1,7)$, find $d$. | 2 | 130 | 1 |
math | 12. (20 points) Given the ellipse $\frac{x^{2}}{6}+\frac{y^{2}}{2}=1$ with its right focus at $F$, the line $y=k(x-2)(k \neq 0)$ passing through $F$ intersects the ellipse at points $P$ and $Q$. If the midpoint of $PQ$ is $N$, and $O$ is the origin, the line $ON$ intersects the line $x=3$ at point $M$, find
(1) the siz... | \sqrt{3} | 143 | 5 |
math | $14 \cdot 25$ Find the smallest natural number $n$ such that the equation $\left[\frac{10^{n}}{x}\right]=1989$ has an integer solution $x$.
(23rd All-Soviet Union Mathematical Olympiad, 1989) | 7 | 68 | 1 |
math | For any real number $t$, let $\lfloor t \rfloor$ denote the largest integer $\le t$. Suppose that $N$ is the greatest integer such that $$\left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4$$Find the sum of digits of $N$. | 24 | 88 | 2 |
math | The twelve students in an olympiad class went out to play soccer every day after their math class, forming two teams of 6 players each and playing against each other. Each day they formed two different teams from those formed on previous days. By the end of the year, they found that every group of 5 students had played... | 132 | 85 | 3 |
math | Find all polynomials $P$ with real coefficients such that $P\left(X^{2}\right)=P(X) P(X-1)$. | (1+X+X^{2})^{n} | 31 | 12 |
math | Let $a,b,c\in \mathbb{R}$ with $a^2+b^2+c^2=1$ and $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$. Then for each solution $(x,y,z)$ of the system of equations:
\[
\begin{cases}
x-\lambda y=a,\\
y-\lambda z=b,\\
z-\lambda x=c.
\end{cases}
\]
we have $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$.
[i]Radu Golog... | \frac{1}{(1 - \lambda)^2} | 142 | 14 |
math | Solve the equation in natural numbers: $x^{3}+y^{3}+1=3 x y$.
# | (1,1) | 27 | 5 |
math | Ilya Muromets meets the three-headed Zmei Gorynych. And the battle begins. Every minute Ilya cuts off one of Zmei's heads. With a probability of $1 / 4$, two new heads grow in place of the severed one, with a probability of $1 / 3$ - only one new head, and with a probability of $5 / 12$ - no heads at all. The Zmei is c... | 1 | 118 | 1 |
math | 6.358 Solve the equations $x^{3}-7 x^{2}+12 x-10=0$ and $x^{3}-10 x^{2}-2 x+20=0$, given that one of the roots of the first equation is half of one of the roots of the second equation. | 5,x_{1}=10,x_{2}=-\sqrt{2},x_{3}=\sqrt{2} | 70 | 26 |
math | \section*{Problem 4 - 071014}
On May 30, 1967, Mr. \(X\) notices that he uses each digit from 0 to 9 exactly once when he writes his date of birth in the format just used for date specifications and adds his age in years. Additionally, he observes that the number of his life years is a prime number.
When was Mr. \(X\... | 26.05.1894,73 | 100 | 13 |
math | 24. Calculate the sum $\left(x+\frac{1}{x}\right)^{2}+\left(x^{2}+\frac{1}{x^{2}}\right)^{2}+\ldots+\left(x^{n}+\frac{1}{x^{n}}\right)^{2}$. | 2n+\frac{1-x^{2n}}{1-x^{2}}\cdot\frac{x^{2n+2}+1}{x^{2n}} | 67 | 36 |
math | Five. (20 points) Given that the side lengths of $\triangle A B C$ are $a, b, c$, and they satisfy
$$
a b c=2(a-1)(b-1)(c-1) .
$$
Does there exist a $\triangle A B C$ with all side lengths being integers? If so, find the three side lengths; if not, explain the reason. | 3,7,8or4,5,6 | 86 | 11 |
math | 11.3. Solve the system of equations $\left\{\begin{array}{l}\sin x \cos y=\sin z \\ \cos x \sin y=\cos z\end{array}\right.$, if the numbers $x, y$ and $z$ lie in the interval $\left[0 ; \frac{\pi}{2}\right]$ | (\frac{\pi}{2};0;\frac{\pi}{2}),(0;\frac{\pi}{2};0) | 76 | 26 |
math | 1. Let the sum of the digits of the natural number $x$ be $S(x)$, then the solution set of the equation $x+S(x)+S(S(x))+S(S(S(x)))=2016$ is | 1980 | 49 | 4 |
math |
Problem 8.2. Given a $\triangle A B C$. Let $M$ be the midpoint of $A B$, $\angle C A B=15^{\circ}$ and $\angle A B C=30^{\circ}$.
a) Find $\angle A C M$.
b) Prove that $C M=\frac{A B \cdot B C}{2 A C}$.
Chavdar Lozanov
| MC=\frac{AB\cdotBC}{2AC} | 93 | 12 |
math | Tony has an old sticky toy spider that very slowly "crawls" down a wall after being stuck to the wall. In fact, left untouched, the toy spider crawls down at a rate of one inch for every two hours it's left stuck to the wall. One morning, at around $9$ o' clock, Tony sticks the spider to the wall in the living room t... | 8 | 156 | 1 |
math | G1.1 Given that when 81849,106392 and 124374 are divided by an integer $n$, the remainders are equal. If $a$ is the maximum value of $n$, find $a$. | 243 | 58 | 3 |
math | Into how many regions do $n$ great circles, no three of which meet at a point, divide a sphere? | f(n) = n^2 - n + 2 | 24 | 13 |
math | A sequence $a_0,a_1,a_2,\cdots,a_n,\cdots$ satisfies that $a_0=3$, and $(3-a_{n-1})(6+a_n)=18$, then the value of $\sum_{i=0}^{n}\frac{1}{a_i}$ is________. | \frac{2^{n+2} - n - 3}{3} | 70 | 17 |
math | 28. In response to a question about his age, the grandfather answered: “The number expressing my age in years is a two-digit number equal to the sum of the number of its tens and the square of its units.” How old is the grandfather? | 89 | 52 | 2 |
math | [Methods for solving problems with parameters]
[Investigation of a quadratic trinomial]
For what positive value of $p$ do the equations $3 x^{2}-4 p x+9=0$ and $x^{2}-2 p x+5=0$ have a common root? | 3 | 62 | 1 |
math | 4. Solve the equation $\sqrt{2+\cos 2 x-\sqrt{3} \tan x}=\sin x-\sqrt{3} \cos x$. Find all its roots that satisfy the condition $|x-3|<1$.
# | \frac{3\pi}{4},\pi,\frac{5\pi}{4} | 54 | 20 |
math | ## Task 5 - 040935
At a puzzle afternoon, the best young mathematician in the class is given the task of guessing a certain real number. His classmates take turns naming properties of this number:
Klaus: "The number is divisible by 4 without a remainder."
Inge: "The number is the radius of a circle whose circumferen... | \frac{1}{\pi} | 191 | 8 |
math | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
$f(f(x)f(1-x))=f(x)$ and $f(f(x))=1-f(x)$,
for all real $x$. | f(x) = \frac{1}{2} | 55 | 11 |
math | 8. In an acute triangle $\triangle A B C$, the lengths of the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively. If
$$
\frac{b^{2}}{a c} \geqslant \frac{\cos ^{2} B}{\cos A \cdot \cos C},
$$
then the range of $\angle B$ is $\qquad$ . | \left[\frac{\pi}{3}, \frac{\pi}{2}\right) | 96 | 18 |
math | Let's determine $k$ so that the expression
$$
x^{2}+2 x+k
$$
is greater than 10 for all values of $x$. | k>11 | 37 | 4 |
math | 1. From point $A$ of a circular track, a car and a motorcycle started simultaneously and in the same direction. The car drove two laps without stopping in one direction. At the moment when the car caught up with the motorcyclist, the motorcyclist turned around and increased his speed by 16 km/h, and after $3 / 8$ hours... | 21 | 134 | 2 |
math | Putnam 1993 Problem B1 What is the smallest integer n > 0 such that for any integer m in the range 1, 2, 3, ... , 1992 we can always find an integral multiple of 1/n in the open interval (m/1993, (m + 1)/1994)? Solution | 3987 | 79 | 4 |
math | What is the maximum number of parts that $n$ circles can divide a plane into?
# | n(n-1)+2 | 19 | 6 |
math | 12.8. Determine the continuous functions $f:\left[\frac{1}{e^{2}} ; e^{2}\right] \rightarrow \mathbb{R}$, for which
$$
\int_{-2}^{2} \frac{1}{\sqrt{1+e^{x}}} f\left(e^{-x}\right) d x-\int_{-2}^{2} f^{2}\left(e^{x}\right) d x=\frac{1}{2}
$$ | f()=\frac{1}{2}\cdot\sqrt{\frac{}{1+}},\forall\in[\frac{1}{e^{2}};e^{2}] | 107 | 37 |
math | 11. Given a plane that makes an angle of $\alpha$ with all 12 edges of a cube, then $\sin \alpha=$ $\qquad$ | \frac{\sqrt{3}}{3} | 34 | 10 |
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 | Example 2. Derive the equations of the tangent and normal to the curve $x^{2}+3 y^{2}=4$, drawn at the point with ordinate $y=1$ and negative abscissa. | ():\frac{1}{3}x+\frac{4}{3};\quad(n):-3x-2 | 47 | 24 |
math | 2・104 Let $n \geqslant 2, x_{1}, x_{2}, \cdots, x_{n}$ be real numbers, and
$$
\sum_{i=1}^{n} x_{i}^{2}+\sum_{i=1}^{n-1} x_{i} x_{i+1}=1 .
$$
For each fixed $k(k \in N, 1 \leqslant k \leqslant n)$, find the maximum value of $\left|x_{k}\right|$. | \max|x_{k}|=\sqrt{\frac{2k(n-k+1)}{n+1}} | 123 | 23 |
math | Example 3 If a positive integer has eight positive divisors, and the sum of these eight positive divisors is 3240, then this positive integer is called a "good number". For example, 2006 is a good number, because the sum of its divisors 1, $2,17,34,59,118,1003,2006$ is 3240. Find the smallest good number. ${ }^{[3]}$
(... | 1614 | 120 | 4 |
math | 10. Given that $a$ and $b$ are real numbers, the system of inequalities about $x$
$$
\left\{\begin{array}{l}
20 x+a>0, \\
15 x-b \leqslant 0
\end{array}\right.
$$
has only the integer solutions $2, 3, 4$. Then the maximum value of $ab$ is $\qquad$ | -1200 | 92 | 5 |
math | Find all triples of natural numbers $a, b, c$ such that $a<b<c$ and the value of the fraction $\frac{1}{7}(44-a b c)$ is a natural number.
(Bird's) | [1,2,8],[1,2,15],[1,3,10],[1,5,6],[2,3,5] | 48 | 33 |
math | Question 4 Let $a, b, c \in \mathbf{C}$, satisfying
$$
\left\{\begin{array}{l}
\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=9, \\
\frac{a^{2}}{b+c}+\frac{b^{2}}{c+a}+\frac{c^{2}}{a+b}=32, \\
\frac{a^{3}}{b+c}+\frac{b^{3}}{c+a}+\frac{c^{3}}{a+b}=122 .
\end{array}\right.
$$
Find the value of $a b c$.
[4]
(2017, Carnegie Me... | 13 | 171 | 2 |
math | 2. (10 points) Divide 120 boys and 140 girls into several groups, requiring that the number of boys in each group is the same, and the number of girls in each group is also the same, then the maximum number of groups that can be formed is $\qquad$ groups. | 20 | 66 | 2 |
math | 11. The maximum value of the function $y=2 x+\sqrt{1-2 x}$ is | \frac{5}{4} | 23 | 7 |
math | Solve the following equation:
$$
3 \sin ^{2} x-4 \cos ^{2} x=\frac{\sin 2 x}{2} .
$$ | x_{1}=135\k\cdot180,\quadx_{2}=537^{\}\k\cdot180(k=0,1,2\ldots) | 37 | 43 |
math | 3. Evaluate the sum
$$
\sum_{n=1}^{6237}\left\lfloor\log _{2}(n)\right\rfloor
$$
where $|x|$ denotes the greatest integer less than or equal to $x$. | 66666 | 57 | 5 |
math | 30. [15] The function $f: \mathbb{Z}^{2} \rightarrow \mathbb{Z}$ satisfies
- $f(x, 0)=f(0, y)=0$, and
- $f(x, y)=f(x-1, y)+f(x, y-1)+x+y$
for all nonnegative integers $x$ and $y$. Find $f(6,12)$. | 77500 | 95 | 5 |
math | 1.2. Inside the rectangle $A B C D$, whose sides are $A B=C D=15$ and $B C=A D=10$, there is a point $P$ such that $A P=9, B P=12$. Find $C P$. | 10 | 61 | 2 |
math | Example 6 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 values for $a$. | a > 3 | 93 | 4 |
math | Isabella has a sheet of paper in the shape of a right triangle with sides of length 3, 4, and 5. She cuts the paper into two pieces along the altitude to the hypotenuse, and randomly picks one of the two pieces to discard. She then repeats the process with the other piece (since it is also in the shape of a right trian... | 64 | 172 | 2 |
math | 6. 27 The function $f(k)$ is defined on $N$, taking values in $N$, and is a strictly increasing function (if for any $x_{1}, x_{2} \in A$, when $x_{1}<x_{2}$, we have $f\left(x_{1}\right)<f\left(x_{2}\right)$, then $f(x)$ is called a strictly increasing function on $A$), and satisfies the condition $f(f(k))=3 k$. Try t... | 197 | 129 | 3 |
math | 12. Given $\triangle A B C$, where $\overrightarrow{A B}=\vec{a}, \overrightarrow{A C}=\vec{b}$, express the area of $\triangle A B C$ using vector operations of $\vec{a}$ and $\vec{b}$, i.e., $S_{\triangle A B C}=$ $\qquad$ | \frac{1}{2}\sqrt{(|\vec{}|\cdot|\vec{b}|)^{2}-(\vec{}\cdot\vec{b})^{2}} | 79 | 37 |
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