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 | Example 4. A tangent line is drawn through the point $\mathrm{P}(1,-1)$ to the parabola $\mathrm{y}^{2}-2 \mathrm{x}$ $-2 y+3=0$. Find the equation of the tangent line. | x+4 y+3=0 \text{ and } x=1 | 56 | 16 |
math | 10. Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. How many positive integers less than 2005 can be expressed in the form $\lfloor x\lfloor x\rfloor\rfloor$ for some positive real $x$ ? | 990 | 64 | 3 |
math | 7.3. On an island, there live 100 people, some of whom always lie, while the rest always tell the truth. Each resident of the island worships one of three gods: the Sun God, the Moon God, and the Earth God. Each resident was asked three questions:
(1) Do you worship the Sun God?
(2) Do you worship the Moon God?
(3) Do ... | 30 | 141 | 2 |
math | 3. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{n+1}=a_{n}^{2}-2\left(n \in \mathbf{N}_{+}\right) \text {, }
$$
and $a_{1}=a, a_{2012}=b(a 、 b>2)$.
Then $a_{1} a_{2} \cdots a_{2011}=$
(express in terms of $a, b$). | \sqrt{\frac{b^{2}-4}{a^{2}-4}} | 112 | 17 |
math |
N3
Find the integer solutions of the equation
$$
x^{2}=y^{2}\left(x+y^{4}+2 y^{2}\right)
$$
| (x,y)=(0,0),(12,-2),(12,2),(-8,-2),(-8,2) | 38 | 27 |
math | Given a square with side length $a$, we draw semicircles over each side such that they intersect at the center of the square. What is the area of the resulting four-leaf clover shape enclosed by the arcs? | ^{2}(\frac{\pi}{2}-1) | 46 | 12 |
math | Example. A die is thrown. Let $m$ be the number of points that come up. Then $m$ shots are fired at a target with the probability of hitting the target in a single shot being $p$.
1. Find the probability that the target will be hit.
2. Given that the target is hit, find the probability that $m=6$. | \sum_{=1}^{6}\frac{1}{6}(1-(1-p)^{}) | 76 | 21 |
math | Example 6 Suppose a circle satisfies: (1) the length of the chord intercepted on the $y$-axis is 2; (2) it is divided into two arcs by the $x$-axis, with the ratio of their arc lengths being $3: 1$. Among all circles satisfying (1) and (2), find the equation of the circle whose center is closest to the line $l: x-2y=0$... | (x-1)^{2}+(y-1)^{2}=2or(x+1)^{2}+(y+1)^{2}=2 | 93 | 33 |
math | Example 6 In the second quadrant of the complex plane, the equation
$$
z^{6}+6 z+10=0
$$
has how many complex roots? | 2 | 38 | 1 |
math | Let's determine the positive integer values of $x$ and $y$ that satisfy the equation
$$
(x-6)(y-6)=18
$$
Based on this problem, determine the right-angled triangles whose sides can be expressed as integers and for which the measure of three times the perimeter is equal to twice the area. | (9,12,15),(8,15,17),(7,24,25) | 70 | 25 |
math | Problem 80. Suppose that \(a_{1}, a_{2}, \ldots, a_{n}\) are non-negative real numbers which add up to \(n\). Find the minimum of the expression
\[S=a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}+a_{1} a_{2} \ldots a_{n}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}\right)\] | \min \left(2 n, \frac{n^{2}}{n-2}, \frac{n^{2}}{n-1}+\left(\frac{n}{n-1}\right)^{n-1}\right) | 123 | 49 |
math | Problem 11.3. A natural number $n$ is called interesting if $2 n$ is a perfect square, and $15 n$ is a perfect cube. Find the smallest interesting number. | 1800 | 43 | 4 |
math | We call a permutation $ \left(a_1, a_2, ..., a_n\right)$ of $ \left(1, 2, ..., n\right)$ [i]quadratic[/i] if there exists at least a perfect square among the numbers $ a_1$, $ a_1 \plus{} a_2$, $ ...$, $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n$. Find all natural numbers $ n$ such that all permutations in $ S_n$ are qua... | n = 1, 8, 49, \ldots | 145 | 16 |
math | 12.8 $f(x)=\sqrt{x^{2}+3}+\frac{2 x}{x+1} ; f^{\prime}(1)=?$ | 1 | 36 | 1 |
math | 1.4.7 ** Summation
$$
S=[\lg 2]+[\lg 3]+\cdots+[\lg 2008]+\left[\lg \frac{1}{2}\right]+\left[\lg \frac{1}{3}\right]+\cdots+\left[\lg \frac{1}{2008}\right]
$$
Find the value of the sum. Here $[x]$ denotes the greatest integer not exceeding $x$. | -2004 | 100 | 5 |
math | 5. Given the vector $\boldsymbol{a}=(\cos \theta, \sin \theta)$, vector $\boldsymbol{b}=(\sqrt{3},-1)$, then the maximum value of $|2 \boldsymbol{a}-\boldsymbol{b}|$ is | 4 | 62 | 1 |
math | 5. The power in the kingdom of gnomes was seized by giants. The giants decided to get rid of the gnomes and told them the following: "Tomorrow we will line you up so that each of you will see those who stand after and not see those who stand before (i.e., the 1st sees everyone, the last sees no one). We will put either... | 1 | 177 | 1 |
math | Four, (50 points) Let $n \in \mathbf{N}$ and $n \geqslant 2$, $n$ distinct sets $A_{1}, A_{2}, \cdots, A_{n}$ satisfy $\left|A_{i}\right|=n(i=1$, $2, \cdots, n)$, and for any $k(2 \leqslant k \leqslant n-1)$ of the sets $A_{1}, A_{2}, \cdots, A_{n}$, we have
$$
\left|A_{i_{1}} \cap A_{i_{2}} \cap \cdots \cap A_{i_{k}}\... | S_{\text{}}=n+1,\,S_{\text{max}}=2n-1 | 206 | 24 |
math | ## Task 1 - 220921
Determine all natural numbers $n$ that satisfy the following conditions (1) and (2):
(1) $n-9$ is a prime number.
(2) $n^{2}-1$ is divisible by 10. | 11 | 63 | 2 |
math | Problem 9.1. Find all values of $a$ such that the equation
$$
\sqrt{\left(4 a^{2}-4 a-1\right) x^{2}-2 a x+1}=1-a x-x^{2}
$$
has exactly two solutions.
Sava Grozdev, Svetlozar Doychev | =\frac{1}{3},=\frac{1}{2},\in(\frac{5}{6},\frac{3}{2})\backslash{1} | 75 | 36 |
math | 4. 209 There are two small piles of bricks. If 100 bricks are taken from the first pile and placed in the second pile, then the second pile will be twice as large as the first pile. If a certain number of bricks are taken from the second pile and placed in the first pile, then the first pile will be six times the size ... | 170 | 110 | 3 |
math | 17. (GBR) A sequence of integers $a_{1}, a_{2}, a_{3}, \ldots$ is defined as follows: $a_{1}=1$, and for $n \geq 1, a_{n+1}$ is the smallest integer greater than $a_{n}$ such that $a_{i}+a_{j} \neq 3 a_{k}$ for any $i, j, k$ in $\{1,2, \ldots, n+1\}$, not necessarily distinct. Determine $a_{1998}$. | 4494 | 128 | 4 |
math | Given a number $\overline{abcd}$, where $a$, $b$, $c$, and $d$, represent the digits of $\overline{abcd}$, find the minimum value of
\[\frac{\overline{abcd}}{a+b+c+d}\]
where $a$, $b$, $c$, and $d$ are distinct
[hide=Answer]$\overline{abcd}=1089$, minimum value of $\dfrac{\overline{abcd}}{a+b+c+d}=60.5$[/hide] | 60.5 | 116 | 4 |
math | 2. Given $x, y \in \mathbf{R}, 2 x^{2}+3 y^{2} \leqslant 12$, then the maximum value of $|x+2 y|$ is $\qquad$ | \sqrt{22} | 53 | 6 |
math | 5. Let point $P$ be on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\left(a>b>0, c=\sqrt{a^{2}-b^{2}}\right)$, the equation of line $l$ is $x=-\frac{a^{2}}{c}$, and the coordinates of point $F$ are $(-c, 0)$. Draw $P Q \perp l$ at point $Q$. If points $P, Q, F$ form an isosceles triangle, then the eccentricity... | \frac{\sqrt{2}}{2} | 141 | 10 |
math | 4. If the acute angle $\alpha$ satisfies
$$
\frac{1}{\sqrt{\tan \frac{\alpha}{2}}}=\sqrt{2 \sqrt{3}} \cdot \sqrt{\tan 10^{\circ}}+\sqrt{\tan \frac{\alpha}{2}},
$$
then $\alpha=$ $\qquad$ | 50^{\circ} | 72 | 6 |
math | Exercise 1. Calculate
$$
\frac{1 \times 2 \times 4+2 \times 4 \times 8+3 \times 6 \times 12+4 \times 8 \times 16}{1 \times 3 \times 9+2 \times 6 \times 18+3 \times 9 \times 27+4 \times 12 \times 36}
$$
Only a numerical answer is expected here. The answer should be given as an irreducible fraction (i.e., in the form $... | \frac{8}{27} | 147 | 8 |
math | 9. For a regular hexagon $A B C D E F$, the diagonals $A C$ and $C E$ are internally divided by points $M$ and $N$ in the following ratios: $\frac{A M}{A C}=\frac{C M}{C E}=r$. If points $B, M, N$ are collinear, find the ratio $r$. | \frac{\sqrt{3}}{3} | 83 | 10 |
math | 4. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Then the set
$$
\{[x]+[2 x]+[3 x] \mid x \in \mathbf{R}\} \cap\{1,2, \cdots, 100\}
$$
has $\qquad$ elements. | 67 | 77 | 2 |
math | 5. In $\triangle A B C$, $A B=B C>A C, A H$ and $A M$ are the altitude and median from vertex $A$ to side $B C$, respectively, and $\frac{S_{\triangle A M H}}{S_{\triangle A B C}}=\frac{3}{8}$. Determine the value of $\cos \angle B A C$. | \frac{1}{4} | 84 | 7 |
math | Exercise 9. Alexie and Baptiste each own a building. Each floor of Alexie's building has 3 bathrooms and 2 bedrooms. Baptiste, on the other hand, has 4 bathrooms and 3 bedrooms per floor. There are a total of 25 bathrooms and 18 bedrooms. Find the number of floors in Alexie's and Baptiste's buildings.
Only a numerical... | =3,b=4 | 87 | 5 |
math | II. (40 points) Find the real solution of the equation
$$
\sqrt[3]{x(3+\sqrt{8 x-3})-1}+\sqrt[3]{x(3-\sqrt{8 x-3})-1}=1
$$ | x\geqslant\frac{3}{8} | 58 | 13 |
math | 4.024. The first term of an arithmetic progression is 429, and its difference is -22. How many terms of this progression need to be taken so that their sum is equal to 3069? | 9or31 | 50 | 4 |
math | How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?
[i]2019 CCA Math Bonanza Individual Round #5[/i] | 40 | 42 | 2 |
math | Problem 2. Solve the system of equations $(n>2)$ :
$$
\begin{aligned}
& \sqrt{x_{1}}+\sqrt{x_{2}+\cdots+x_{n}}=\sqrt{x_{2}}+\sqrt{x_{3}+\cdots+x_{n}+x_{1}}=\cdots \\
& \cdots=\sqrt{x_{n}}+\sqrt{x_{1}+\cdots+x_{n-1}} ; \quad x_{1}-x_{2}=1
\end{aligned}
$$ | x_{1}=1,x_{2}=0,x_{3}=\cdots=x_{n}=0 | 114 | 22 |
math | 1. Find the real number $m$ such that the equation
$$
\left(x^{2}-2 m x-4\left(m^{2}+1\right)\right)\left(x^{2}-4 x-2 m\left(m^{2}+1\right)\right)=0
$$
has exactly three distinct real roots. | 3 | 74 | 1 |
math | One. (20 points) Determine for which positive integers $a$, the equation
$$
5 x^{2}-4(a+3) x+a^{2}-29=0
$$
has positive integer solutions? And find all positive integer solutions of the equation. | x=14, a=29 \text { or } 27; x=13, a=22 \text { or } 30; x=5, a=2 \text { or } 18; x=7, a=6 \text { or } 22; x=2, a=11; x=10, a=13 \text { or } 27 | 57 | 94 |
math | Sylvia has a bag of 10 coins. Nine are fair coins, but the tenth has tails on both sides. Sylvia draws a coin at random from the bag and flips it without looking. If the coin comes up tails, what is the probability that the coin she drew was the 2-tailed coin? | \frac{2}{11} | 65 | 8 |
math | ## Task B-4.3.
A sequence of numbers is defined by $a_{n}=n^{4}-360 n^{2}+400$. Calculate the sum of all terms of this sequence that are prime numbers. | 802 | 50 | 3 |
math | 10. Determine the maximum value attained by
$$
\frac{x^{4}-x^{2}}{x^{6}+2 x^{3}-1}
$$
over real numbers $x>1$. | \frac{1}{6} | 45 | 7 |
math | 4. Determine the set of points $z$ in the complex plane for which there exists a real number $c$ such that $z=\frac{c-i}{2 c-i}$. | {x+iy\lvert\,(x-\frac{3}{4})^{2}+y^{2}=\frac{1}{16}.,(x,y)\neq(\frac{1}{2},0)} | 39 | 48 |
math | 2. Oleg and Sergey take turns writing down one digit from left to right until a nine-digit number is formed. At the same time, they cannot write down digits that have already been written. Oleg starts (and finishes). Oleg wins if the resulting number is divisible by 4; otherwise, Sergey wins. Who will win with correct ... | Sergey | 86 | 3 |
math | 4. On the board, the numbers $2,3,4, \ldots ., n+1$ are written. Then the products of every two of them are written, followed by the products of every three of them, and the process continues until the product of all the numbers 2,3,4,...., $n+1$ is written. Calculate the sum of the reciprocals of all the numbers writt... | \frac{n}{2} | 93 | 6 |
math | We recall that $n!=1 \times 2 \times 3 \times \cdots \times n$.
How many zeros are there at the end of 100! ? and at the end of 1000! ? | 249 | 52 | 3 |
math | Example 1 Let $p$ be a prime number, and the number of distinct positive divisors of $p^{2}+71$ does not exceed 10. Find the value of $p$. | 2or3 | 44 | 3 |
math | 3. A three-digit number $x y z$ (where $x$, $y$, and $z$ are distinct), rearranging its digits to form the largest and smallest possible three-digit numbers. If the difference between the largest and smallest three-digit numbers is equal to the original three-digit number, then this three-digit number is $\qquad$ | 495 | 71 | 3 |
math | 2. Let $0 \leqslant x \leqslant 2$, the minimum value of the function $y=4^{x-\frac{1}{2}}-3 \cdot 2^{x}+5$ is
保留源文本的换行和格式，翻译结果如下：
2. Let $0 \leqslant x \leqslant 2$, the function $y=4^{x-\frac{1}{2}}-3 \cdot 2^{x}+5$'s minimum value is | \frac{1}{2} | 116 | 7 |
math | 2. Every 20 minutes for a week, an exact amount of liters of water (always the same amount) is transferred from a tank with 25,000 liters to another initially empty reservoir. From this second reservoir, at regular intervals of time, 1 liter is extracted first, then 2 liters, then 3 liters, and so on. At the end of the... | 2016 | 160 | 4 |
math | 13. In a drawer, there are red and blue socks, no more than 1991 in total. If two socks are drawn without replacement, the probability that they are the same color is $\frac{1}{2}$. How many red socks can there be at most in this case? | 990 | 63 | 3 |
math | 358. Given: $f(x)=\operatorname{arcctg} \frac{x}{2}-\operatorname{arctg} \sqrt{x}$. Find $f^{\prime}(1)$. | -\frac{13}{20} | 47 | 9 |
math | 23. Let $x$, $y$, and $a$ be real numbers. And satisfy $x+y=x^{3}+y^{3}=$ $x^{5}+y^{5}=a$. Find all possible values of $a$.
(Greece for the 43rd IMO selection exam) | -2, -1, 0, 1, 2 | 66 | 14 |
math | . How many pairs $(m, n)$ of positive integers with $m<n$ fulfill the equation
$$
\frac{3}{2008}=\frac{1}{m}+\frac{1}{n} ?
$$
## Answer: 5. | 5 | 56 | 1 |
math | For each positive integer $ k$, find the smallest number $ n_{k}$ for which there exist real $ n_{k}\times n_{k}$ matrices $ A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold:
(1) $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$,
(2) $ A_{i}A_{j}= A_{j}A_{i}$ for all $ 1 \le i, j \le k$, a... | n_k = 2^k | 149 | 8 |
math | 17. (6 points) Gymnasts are arranged in a square formation with the same number of people in each row and column. The outermost ring of each square formation has 16 people. If four such square formations can be combined to form a larger square formation, then the outermost ring of the larger square formation has $\qqua... | 36 | 74 | 2 |
math | 4. In $\triangle A B C$, $a+c=2 b$. Then $\tan \frac{A}{2} \cdot \tan \frac{C}{2}=$ $\qquad$ | \frac{1}{3} | 42 | 7 |
math | Example 5 Given that $p$ and $q$ are both prime numbers, and that the quadratic equation in $x$
$$
x^{2}-(8 p-10 q) x+5 p q=0
$$
has at least one integer root. Find all pairs of prime numbers $(p, q)$.
(2005, National Junior High School Mathematics Competition) | (7,3),(11,3) | 82 | 10 |
math | 4. Evaluate the sum
$$
\frac{1}{2\lfloor\sqrt{1}\rfloor+1}+\frac{1}{2\lfloor\sqrt{2}\rfloor+1}+\frac{1}{2\lfloor\sqrt{3}\rfloor+1}+\cdots+\frac{1}{2\lfloor\sqrt{100}\rfloor+1} .
$$ | \frac{190}{21} | 91 | 10 |
math | ## 139. Math Puzzle $12 / 76$
Ralf and Marion had set their watches, real "vintages," at the beginning of a longer hike. At the destination, Marion's watch shows 13:46 and Ralf's 14:13.
What time is it really, if Marion's watch gains 2 minutes daily and Ralf's old alarm clock loses 4 minutes daily? | 13:55 | 92 | 5 |
math | 4. If $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ satisfy the following system of equations
$$
\cdot\left\{\begin{array}{c}
2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=6, \\
x_{1}+2 x_{2}+x_{3}+x_{4}+x_{5}=12, \\
x_{1}+x_{2}+2 x_{3}+x_{4}+x_{5}=24, \\
x_{1}+x_{2}+x_{3}+2 x_{4}+x_{5}=48, \\
x_{1}+x_{2}+x_{3}+x_{4}+2 x_{5}=96
\end{array}\ri... | 181 | 212 | 3 |
math | A [i]Benelux n-square[/i] (with $n\geq 2$) is an $n\times n$ grid consisting of $n^2$ cells, each of them containing a positive integer, satisfying the following conditions:
$\bullet$ the $n^2$ positive integers are pairwise distinct.
$\bullet$ if for each row and each column we compute the greatest common divisor of... | n = 2 | 208 | 5 |
math | 19. (MEX 1) Let $f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $f(f(m)+f(n))=m+n$ for all positive integers $n, m$. Find all possible values for $f(1988)$. | 1988 | 71 | 4 |
math | Example 3. Find the integral $\int \frac{x}{x^{3}+1} d x$. | -\frac{1}{3}\ln|x+1|+\frac{1}{6}\ln(x^{2}-x+1)+\frac{1}{\sqrt{3}}\operatorname{arctg}\frac{2x-1}{\sqrt{3}}+C | 23 | 60 |
math | 15. Given that $a$, $b$, and $c$ are distinct integers. Then
$$
4\left(a^{2}+b^{2}+c^{2}\right)-(a+b+c)^{2}
$$
the minimum value is $\qquad$ | 8 | 59 | 1 |
math | 5. (15 points) A massive vertical plate is fixed on a car moving at a speed of $5 \mathrm{M} / \mathrm{c}$. A ball is flying towards it at a speed of $6 \mathrm{m} / \mathrm{s}$ relative to the Earth. Determine the speed of the ball relative to the Earth after a perfectly elastic normal collision. | 16\mathrm{~}/\mathrm{} | 79 | 10 |
math | Example 1. Let $n$ be an integer, calculate the following expression:
$$
\begin{array}{l}
{\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{2^{2}}\right]+\left[\frac{n+2^{2}}{2^{3}}\right]} \\
+\cdots .
\end{array}
$$
where the symbol $[x]$ denotes the greatest integer not exceeding $x$.
(IMO-10)
Analysis: In the expressio... | n | 250 | 1 |
math | 10. (40 points) Let $P(n)$ denote the product of the digits of a natural number $n$. For what largest natural number $k$ does there exist a natural number $n>10$ such that
$$
P(n)<P(2 n)<\ldots<P(k n) ?
$$ | 9 | 68 | 1 |
math | The teacher wrote two numbers on the board one under the other and called Klára to add them. Klára correctly wrote the result under the given numbers. The teacher erased the topmost number, and the remaining two numbers formed a new addition problem. This time, Lukáš correctly wrote the result under the numbers. The te... | 22or26 | 136 | 5 |
math | Condition of the problem
Find the point of intersection of the line and the plane.
$\frac{x+3}{1}=\frac{y-2}{-5}=\frac{z+2}{3}$
$5 x-y+4 z+3=0$ | (-2,-3,1) | 57 | 7 |
math | 1. If points $A$ and $B$ are on the same side of the $x$-axis, you can first find the symmetric point $B^{\prime}\left(x_{2}, -y_{2}\right)$ (or $A^{\prime}\left(x_{1}, -y_{1}\right)$) of point $B$ (or $A$) with respect to the $x$-axis, then calculate $\left|A B^{\prime}\right|$. The value of $\left|A B^{\prime}\right|... | \sqrt{26} | 167 | 6 |
math | Example 5. Solve the equation $y^{\prime \prime}+\left(y^{\prime}\right)^{2}=2 e^{-y}$. | e^{y}+\tilde{C}_{1}=(x+C_{2})^{2} | 33 | 21 |
math | (1) In an arithmetic sequence $\left\{a_{n}\right\}$ with a common difference of $d$ and all terms being positive integers, if $a_{1}=1949$, $a_{n}=2009$, then the minimum value of $n+d$ is $\qquad$ . | 17 | 69 | 2 |
math | 6.24 In a geometric sequence with a common ratio greater than 1, what is the maximum number of terms that are integers between 100 and 1000.
(4th Canadian Mathematics Competition, 1972) | 6 | 52 | 1 |
math | Problem 1. Sasho thought of a number and multiplied it by 7 and by 16. He added the obtained products and got the number 230. Which number did Sasho think of? | 10 | 46 | 2 |
math | 1. (5 points) Calculate: $(3 \div 2) \times(4 \div 3) \times(5 \div 4) \times \cdots \times(2012 \div 2011) \times(2013 \div 2012)$ $=$ . $\qquad$ | \frac{2013}{2} | 75 | 10 |
math | For which values of $k$ does the polynomial $X^{2017}-X^{2016}+X^{2}+k X+1$ have a rational root? | k=-2ork=0 | 41 | 6 |
math | Example. Given a positive integer $n \geqslant 2, x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R}^{+}$ and $x_{1}+x_{2}+\cdots+x_{n}=\pi$, find the minimum value of $\left(\sin x_{1}+\frac{1}{\sin x_{1}}\right)\left(\sin x_{2}+\frac{1}{\sin x_{2}}\right) \cdots\left(\sin x_{n}+\frac{1}{\sin x_{n}}\right)$. | (\sin\frac{\pi}{n}+\frac{1}{\sin\frac{\pi}{n}})^{n} | 138 | 27 |
math | 19. (6 points) Use small cubes with edge length $m$ to form a large cube with edge length 12. Now, paint the surface (6 faces) of the large cube red, where the number of small cubes with only one red face is equal to the number of small cubes with only two red faces. Then $m=$ $\qquad$ . | 3 | 77 | 1 |
math | Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive integ... | 25 | 87 | 2 |
math | 4.20. The length of the hypotenuse of an isosceles right triangle is 40. A circle with a radius of 9 touches the hypotenuse at its midpoint. Find the length of the segment cut off by this circle on one of the legs. | \sqrt{82} | 59 | 6 |
math | The equation
$$ (x-1)(x-2) \cdots(x-2016)=(x-1)(x-2) \cdots(x-2016) $$
is written on the board. One tries to erase some linear factors from both sides so that each side still has at least one factor, and the resulting equation has no real roots. Find the least number of linear factors one needs to erase to achieve ... | 2016 | 102 | 4 |
math | . Let $n$ be an integer with $n \geq 3$. Consider all dissections of a convex $n$-gon into triangles by $n-3$ non-intersecting diagonals, and all colourings of the triangles with black and white so that triangles with a common side are always of a different colour. Find the least possible number of black triangles. | \lfloor\frac{n-1}{3}\rfloor | 78 | 13 |
math | 27.6. Several points are marked on a circle, $A$ is one of them. Which are there more of: convex polygons with vertices at these points that contain point $A$ or those that do not contain it? | (n-1)(n-2)/2 | 48 | 9 |
math | 5. A right regular equilateral triangular prism is given, whose base has an area of $6.25 \sqrt{3} \mathrm{~cm}^{2}$.
a) Calculate the surface area of the given prism.
b) Determine the ratio of the volumes of the circumscribed and inscribed cylinders of this prism. Is this ratio the same for every right regular equil... | 4:1 | 84 | 3 |
math | (Following Benoît's course 2021)
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy:
$$
\forall x, y \in \mathbb{R}, \quad f(2 f(x)+f(y))=2 x+f(y)
$$ | f(x)=x | 70 | 4 |
math | Task B-1.3. If $a+b=2$ and $a^{2}+b^{2}=6$, what is $a^{-1}+b^{-1}$? | -2 | 40 | 2 |
math | Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$. | 8 | 43 | 1 |
math | 20 Let the set $M=\{1,2,3, \cdots, 50\}$, and the set $S \subseteq M$, for any $x, y \in S, x \neq y$, we have $x+y \neq 7 k(k \in \mathbf{N})$. Among all such sets $S$, the set $S_{0}$ is the one that contains the most elements, then the number of elements in $S_{0}$ is $\qquad$. | 23 | 110 | 2 |
math | For the polynomial $f$, it holds that $f\left(x^{2}+1\right)-f\left(x^{2}-1\right)=4 x^{2}+6$. Determine the polynomial $f\left(x^{2}+1\right)-f\left(x^{2}\right)$. | 2x^{2}+4 | 67 | 7 |
math | 1. Let $x, y$ and $a$ be real numbers such that $x+y=a-1$ and $x y=a^{2}-7 a+12$. For which $a$ does the expression $x^{2}+y^{2}$ attain its maximum possible value? What are $x$ and $y$ then? | 6,2,3or3,2 | 73 | 9 |
math | $8 \cdot 57$ a (finite) sequence of the same non-zero digit can be the ending of a perfect square. Find the maximum length of such a sequence, and the smallest square number whose ending is such a sequence.
(31st Putnam Mathematical Competition, 1970) | 1444 | 64 | 4 |
math | 10. (20 points) Given that the probability of forming an obtuse triangle by randomly selecting three vertices from a regular $n$-sided polygon is $\frac{93}{125}$, find all possible values of the positive integer $n$ | 376127 | 56 | 6 |
math | 2. Let $a=\frac{\sqrt{5}-1}{2}$. Then $\frac{a^{5}+a^{4}-2 a^{3}-a^{2}-a+2}{a^{3}-a}=$ $\qquad$ | -2 | 54 | 2 |
math | Example. Let $y=f(u)=\lg u, u \in R^{+}$, $u=\varphi(x)=\sin x, x \in R$. Try to discuss the domain of the composite function $y=f[\varphi(x)]$. | \{x \mid 2 k \pi < x < (2 k+1) \pi, k \in \mathbb{Z}\} | 54 | 32 |
math | II. (40 points) Given the function
$$
f(x)=3\left(\sin ^{3} x+\cos ^{3} x\right)+m(\sin x+\cos x)^{3}
$$
has a maximum value of 2 in $x \in\left[0, \frac{\pi}{2}\right]$. Find the value of the real number $m$. | m=-1 | 86 | 3 |
math | Determine the value of the natural number $a$, knowing that $4 a^{2}$ and $\frac{4}{3} \times a^{3}$ are four-digit integers. | 18 | 38 | 2 |
math | 11.056. The diagonal of a rectangular parallelepiped is 10 cm and forms an angle of $60^{\circ}$ with the base plane. The area of the base is $12 \mathrm{~cm}^{2}$. Find the lateral surface area of the parallelepiped. | 70\sqrt{3} | 67 | 7 |
math | In an urn, there are 5 white and 4 blue balls, and in another, 2 white and 8 blue. We draw one ball from each urn and, without looking at them, put them into a third, empty urn. What is the probability that drawing one ball from the third - containing two balls - urn, it will be white? | \frac{17}{45} | 73 | 9 |
math | 2. $\sin ^{2} 100^{\circ}-\sin 50^{\circ} \sin 70^{\circ}=$ | \frac{1}{4} | 35 | 7 |
math | 6. In the geometric sequence $\left\{a_{n}\right\}$, if for any positive integer $n, a_{1}+a_{2}+\cdots+a_{n}=2^{n}-1$, then $a_{1}^{3}+a_{2}^{3}+\cdots+a_{n}^{3}=$ $\qquad$ | \frac{1}{7}(8^{n}-1) | 80 | 13 |
math | 1. Let $n$ be a natural number, $a, b$ be positive real numbers, and satisfy $a+b=2$, then the minimum value of $\frac{1}{1+a^{n}}+$ $\frac{1}{1+b^{n}}$ is $\qquad$. | 1 | 61 | 1 |
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