task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
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math | ## Task 36/69
We are looking for a natural, four-digit number with the following properties:
1. The sum of the thousands place and the hundreds place is equal to the number that results when the two middle digits are removed from the sought number.
2. This sum is less than double the tens place.
3. Exactly one of the... | 1970 | 81 | 4 |
math | What values can the expression $x+y+z$ take if $\sin x=\cos y, \sin y=\cos z, \sin z=\cos x, 0 \leq x, y, z \leq$ $\pi / 2 ?$ | \frac{3\pi}{4} | 54 | 9 |
math | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{2}-y^{2}\right)=x f(x)-y f(y)
$$
for all pairs of real numbers $x$ and $y$. | f(x)=,\in\mathbb{R} | 62 | 11 |
math | 13.045. A piece of copper and zinc alloy weighing 36 kg contains $45 \%$ copper. What mass of copper needs to be added to this piece so that the resulting new alloy contains $60 \%$ copper? | 13.5 | 52 | 4 |
math | 5. Calculate $f(2)$, if $25 f\left(\frac{x}{1580}\right)+(3-\sqrt{34}) f\left(\frac{1580}{x}\right)=2017 x$. Round the answer to the nearest integer. | 265572 | 63 | 6 |
math | a) Find all integers $n \in \mathbb{N}^{*}$ that have an odd number of divisors (positive).
b) Let $n \in \mathbb{N}^{*}$. We denote $d$ as the number of divisors of $n$, and $D$ as the product of its divisors. Show that $n^{d}=D^{2}$.
c) Suppose the prime factorization of $n$ is written as $p_{1}^{\alpha_{1}} \ldots ... | n^{}=D^{2} | 144 | 8 |
math | 3. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that
$$
n+f(m) \text { divides } f(n)+n f(m)
$$
for all $m, n \in \mathbb{N}$. (We denote the set of natural numbers by $\mathbb{N}$.)
(Albania) | f(x)\equivx^{2}f(x)\equiv1 | 81 | 13 |
math | 9. (16 points) Let the function $f(x)=|\lg (x+1)|$, and real numbers $a, b (a<b)$ satisfy
$$
f(a)=f\left(-\frac{b+1}{b+2}\right), f(10 a+6 b+21)=4 \lg 2 \text {. }
$$
Find the values of $a, b$. | a=-\frac{2}{5}, b=-\frac{1}{3} | 89 | 18 |
math | 5. The clock shows 00:00, at which the hour and minute hands of the clock coincide. Counting this coincidence as number 0, determine how much time (in minutes) will pass before they coincide for the 23rd time. Round your answer to the nearest hundredth. | 1505.45 | 63 | 7 |
math | Problem 7. (8 points)
The earned salary of the citizen was 23,000 rubles per month from January to June inclusive, and 25,000 rubles from July to December. In August, the citizen, participating in a poetry competition, won a prize and was awarded an e-book worth 10,000 rubles. What amount of personal income tax needs ... | 39540 | 121 | 5 |
math | $11 \cdot 28$ in which base, $4 \cdot 13=100$?
(Kyiv Mathematical Olympiad, 1953) | 6 | 39 | 1 |
math | 8. (5 points) Mr. Wang, who teaches mathematics, is preparing to visit a friend. Before leaving, Mr. Wang calls his friend, whose phone number is 27433619. After Mr. Wang finishes the call, he realizes that this phone number is exactly the product of 4 consecutive prime numbers. What is the sum of these 4 prime numbers... | 290 | 85 | 3 |
math | Let $x$ be in the interval $\left(0, \frac{\pi}{2}\right)$ such that $\sin x - \cos x = \frac12$ . Then $\sin^3 x + \cos^3 x = \frac{m\sqrt{p}}{n}$ , where $m, n$, and $p$ are relatively prime positive integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$. | 28 | 102 | 2 |
math | 1. Two trains, each containing 15 identical cars, were moving towards each other at constant speeds. Exactly 28 seconds after the first cars of the trains met, passenger Sasha, sitting in the third car, passed passenger Valera from the oncoming train, and another 32 seconds later, the last cars of these trains had comp... | 12 | 93 | 2 |
math | 13.050. The worker's wages for October and November were in the ratio of $3 / 2: 4 / 3$, and for November and December, they were in the ratio of $2: 8 / 3$. In December, he received 450 rubles more than in October, and for exceeding the quarterly plan, the worker was awarded a bonus of $20 \%$ of his three-month earni... | 1494 | 101 | 4 |
math | Let $\mathrm{n}$ be a strictly positive integer. How many polynomials $\mathrm{P}$ with coefficients in $\{0,1,2,3\}$ are there such that $P(2)=n$?
## - Other examples - | a_{n}=[\frac{n}{2}]+1 | 51 | 13 |
math | 4. How many numbers divisible by 4 and less than 1000 do not contain any of the digits $6,7,8,9$ or 0. | 31 | 37 | 2 |
math | Find all functions from $\mathbb{R}-\{0,1\}$ to $\mathbb{R}$ such that
$$
f(x)+f\left(\frac{1}{1-x}\right)=x
$$ | f(x)=\frac{1}{2}(x+1-\frac{1}{x}-\frac{1}{1-x}) | 48 | 28 |
math | In how many ways can two kings be placed on an $8 \times 8$ chessboard so that they do not attack each other, that is, the squares occupied by the two kings do not share a common corner? | 1806 | 46 | 4 |
math | ## 3. Solve the equation in the set of real numbers
$$
x\lfloor x\rfloor-5 x+7=0 \text {, }
$$
where $\lfloor x\rfloor$ denotes the floor function of $x$, i.e., the greatest integer $k$ such that $k \leqq x$. (For example, $\lfloor\sqrt{2}\rfloor=1 \quad \text{and} \quad \lfloor-3.1\rfloor=-4$). | x_{1}=\frac{7}{4},x_{2}=\frac{7}{3},x_{3}=\frac{7}{2} | 112 | 33 |
math | 7. Given the function $f(x)=x^{2}+2 x+a \ln x$, for any $t \geqslant 1$ it always holds that $f(2 t-1) \geqslant 2 f(t)-3$. Then the range of the real number $a$ is . $\qquad$ | \leqslant2 | 73 | 6 |
math | 3.070. $\cos \left(\frac{\pi}{6}-\frac{\alpha}{2}\right) \sin \left(\frac{\pi}{3}-\frac{\alpha}{2}\right) \sin \frac{\alpha}{2}$. | \frac{1}{4}\sin\frac{3}{2}\alpha | 56 | 16 |
math | Problem 10.3. On the board, natural numbers $a, b, c, d$ are written. It is known that among the six sums
$$
a+b, \quad b+c, c+d, d+a, a+c, b+d
$$
three are equal to 23, and the other three are equal to 34.
(a) (1 point) What is the value of $a+b+c+d$?
(b) (3 points) What is the smallest of the numbers $a, b, c, d$... | 57 | 118 | 2 |
math | Let $ABCD$ be a quadrilateral and let $O$ be the point of intersection of diagonals $AC$ and $BD$. Knowing that the area of triangle $AOB$ is equal to $ 1$, the area of triangle $BOC$ is equal to $2$, and the area of triangle $COD$ is equal to $4$, calculate the area of triangle $AOD$ and prove that $ABCD$ is a trapezo... | 2 | 97 | 3 |
math | Cyclic pentagon $ ABCDE$ has side lengths $ AB\equal{}BC\equal{}5$, $ CD\equal{}DE\equal{}12$, and $ AE \equal{} 14$. Determine the radius of its circumcircle. | 13 | 52 | 2 |
math | Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The in finite series
$$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)}$$
has the value $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 451 | 107 | 3 |
math | Which polynomial $p(x)$ satisfies the equation
$$
(x-16) p(2 x)=16(x-1) p(x)
$$
? (H) | p(x)=(x-2)(x-4)(x-8)(x-16) | 37 | 20 |
math | 16. Given that $\vec{a}$ and $\vec{b}$ are non-zero vectors, and $\vec{a}+3 \vec{b}$ is perpendicular to $7 \vec{a}-5 \vec{b}$, $\vec{a}-4 \vec{b}$ is perpendicular to $7 \vec{a}-2 \vec{b}$, find the angle between $\vec{a}$ and $\vec{b}$. | 60 | 95 | 2 |
math | XLVI OM - III - Problem 1
Determine the number of subsets of the set $ \{1,2, \ldots , 2n\} $ in which the equation $ x + y = 2n+1 $ has no solutions. | 3^n | 55 | 2 |
math | In the language of the mumbo-jumbo tribe, there are only 4 letters: $a, u, o$ and $e$. The letter $e$ is special: if we insert $e$ into a word - once or multiple times, at the beginning, end, or somewhere in between - the meaning of the word does not change. Additionally, the letter groups aаaaaaa, ииuиuии, ооооооo in ... | 343 | 172 | 3 |
math | ## 15. Puzzle
Martin once told Martina: “I am three times as old as you were when I was as old as you are now.”
And in response, Martina noted: “When I am as old as you are now, together we will be 77 years old.”
How old are Martin and Martina? | Martinis33old,Martinais22old | 70 | 12 |
math | Let be given a triangle $ABC$ and points $D,M,N$ belong to $BC,AB,AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. If $S_{\vartriangle BMD} = 9$ cm $^2, S_{\vartriangle DNC} = 25$ cm$^2$, compute $S_{\vartriangle AMN}$?
| 15 \, \text{cm}^2 | 94 | 11 |
math | Example 2 Let $k \geqslant 9$. Solve the equation
$$
x^{3}+2 k x^{2}+k^{2} x+9 k+27=0 \text {. }
$$ | x=\frac{(3-k) \pm \sqrt{(k-9)(k+3)}}{2} \text{ or } x=k+3 | 50 | 32 |
math | Let $a \circ b=a+b-a b$. Find all triples $(x, y, z)$ of integers such that $(x \circ y) \circ z+(y \circ z) \circ x+(z \circ x) \circ y=0$. | (0,0,0),(0,2,2),(2,0,2),(2,2,0) | 55 | 25 |
math | For which positive integer values of $n$ will $2^{n}+1$ be a power of a positive integer greater than one? | 3 | 29 | 1 |
math | 11.1. Arkady, Boris, and Vasili decided to run the same distance, consisting of several laps. Arkady ran each lap 2 minutes faster than Boris, and Boris ran 3 minutes faster than Vasili, and all of them ran at a constant speed. When Arkady finished the distance, Boris had one lap left to run, and Vasili had two laps le... | 6 | 102 | 1 |
math | In a set of 100 numbers. If one number is removed, the median of the remaining numbers will be 78. If another number is removed, the median of the remaining numbers will be 66. Find the median of the entire set.
# | 72 | 55 | 2 |
math | 14. From the 20 numbers 11, 12, 13, 14, ... 30, at least ( ) numbers must be taken to ensure that there are two numbers among the taken numbers whose sum is a multiple of ten. | 11 | 57 | 2 |
math | 2. On the first day, Masha collected $25 \%$ fewer berries than Vanya, and on the second day, $20 \%$ more than Vanya. Over the two days, Masha collected $10 \%$ more berries than Vanya. What is the smallest number of berries they could have collected together? | 189 | 69 | 3 |
math | ## Aufgabe 1 - 281241
Man ermittle alle reellen Lösungen $(x, y, z)$ des Gleichungssystems
$$
\begin{aligned}
x^{2}+y^{2}+z^{2} & =1 \\
x+2 y+3 z & =\sqrt{14}
\end{aligned}
$$
| (x,y,z)=(\frac{\sqrt{14}}{14},\frac{2\sqrt{14}}{14},\frac{3\sqrt{14}}{14}) | 83 | 44 |
math | $12$ friends play a tennis tournament, where each plays only one game with any of the other eleven. Winner gets one points. Loser getos zero points, and there is no draw. Final points of the participants are $B_1, B_2, ..., B_{12}$. Find the largest possible value of the sum $\Sigma_3=B_1^3+B_2^3+ ... + B_{12}^3$ . | 4356 | 97 | 4 |
math | 7.2. Students in the seventh grade send each other New Year's stickers on Telegram. It is known that exactly 26 people received at least one sticker, exactly 25 - at least two stickers, ..., exactly 1 - at least 26 stickers. How many stickers did the students in this class receive in total, if it is known that no one r... | 351 | 97 | 3 |
math | 5. Each of two ants, Fat and Thin, needs to transport 150 g of cargo from point $A$ (where they are currently located) to point $B$, the distance between which is 15 meters. The Fat ant walks at a speed of 3 m/min but can carry 5 g of cargo, while the Thin ant walks at a speed of 5 m/min but can carry only 3 g of cargo... | The\Fat\one\will\finish\2\\earlier | 124 | 14 |
math | How many ways are there to write $91$ as the sum of at least $2$ consecutive positive integers? | 3 | 24 | 1 |
math | A positive integer $n$ is called[i] bad [/i]if it cannot be expressed as the product of two distinct positive integers greater than $1$. Find the number of bad positive integers less than $100. $
[i]Proposed by Michael Ren[/i] | 30 | 57 | 2 |
math | 148 The function $f(x)=\left|x^{2}-a\right|$ has a maximum value $M(a)$ in the interval $-1 \leqslant x \leqslant 1$. The minimum value of $M(a)$ is
$\qquad$ | \frac{1}{2} | 61 | 7 |
math | 2. On a line, four points $A, B, C$, $D$ are chosen, such that $A B=1, B C=2, C D=4$. What can $A D$ be? List all possibilities. | 1,3,5,7 | 51 | 7 |
math | 20. In the performance of a group gymnastics, it is required that when the formation changes to 10 rows, 15 rows, 18 rows, and 24 rows, the formation can always form a rectangle. How many people are needed at minimum for the group gymnastics performance? | 360 | 64 | 3 |
math | Show that the expression $(a + b + 1) (a + b - 1) (a - b + 1) (- a + b + 1)$, where $a =\sqrt{1 + x^2}$, $b =\sqrt{1 + y^2}$ and $x + y = 1$ is constant ¸and be calculated that constant value.
| 4 | 83 | 1 |
math | Example 15 Simplify $m=\frac{a^{2}(x-b)(x-c)}{(a-b)(a-c)}$ $+\frac{b^{2}(x-c)(x-a)}{(b-c)(b-a)}+\frac{c^{2}(x-a)(x-b)}{(c-a)(c-b)}$. | m=x^{2} | 71 | 5 |
math | 12. The Zmey Gorynych has 2000 heads. A legendary hero can cut off 33, 21, 17, or 1 head with one strike of his sword, but in return, the Zmey grows 48, 0, 14, or 349 heads respectively. If all heads are cut off, no new ones grow. Can the hero defeat the Zmey? How should he act? | 17 | 101 | 2 |
math | Three. (20 points) Given that $a$ and $b$ are integers, and satisfy $\sqrt{9-8 \sin 50^{\circ}}=a+b \csc 50^{\circ}$. Find all possible values of $a$ and $b$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | a=3, b=-1 | 88 | 7 |
math | For what $a$ and $b$ does the polynomial $P(x)=(a+b) x^{5}+a b x^{2}+1$ divide by $x^{2}-3 x+2$?
# | {-1,\frac{31}{28}} | 47 | 11 |
math | a)Find a matrix $A\in \mathcal{M}_3(\mathbb{C})$ such that $A^2\neq O_3$ and $A^3=O_3$.
b)Let $n,p\in\{2,3\}$. Prove that if there is bijective function $f:\mathcal{M}_n(\mathbb{C})\rightarrow \mathcal{M}_p(\mathbb{C})$ such that $f(XY)=f(X)f(Y),\ \forall X,Y\in \mathcal{M}_n(\mathbb{C})$, then $n=p$.
[i]Ion Savu[/i] | n = p | 150 | 4 |
math | 27. At a fun fair, coupons can be used to purchsse food. Each coupon is worth $\$ 5, \$ 8$ or $\$ 12$. For example, for a $\$ 15$ purchase you can use three coupons of $\$ 5$, or use one coupon of $\$ 5$ and one coupon of $\$ 8$ and pay $\$ 2$ by cash. Suppose the prices in the fun fair are all whole dollars. What is t... | 19 | 114 | 2 |
math | Example 3. The distribution function of a random variable $X$ is given by
$$
F(x)= \begin{cases}0 & \text { if } x \leq 0 \\ \frac{x^{2}}{1+x^{2}} & \text { if } x>0\end{cases}
$$
Find its probability density function. | p(x)=\begin{cases}0&\text{for}x\leq0\\\frac{2x}{(1+x^{2})^{2}}&\text{for}x>0\end{cases} | 76 | 49 |
math | 26. Find the function $f$, defined on the set of all real numbers, if it is known that $f^{2}(x+y)=f^{2}(x)+f^{2}(y)$ for any $x$ and $y$. | f(x)=0 | 52 | 4 |
math | 2. (8 points) 2010 consecutive natural numbers are arranged in ascending order. The average of the numbers in the odd positions is 2345. Then the average of the numbers in the even positions is $\qquad$ . | 2346 | 52 | 4 |
math | 10. (40 points) Given positive integers $a$, $b$, and $c$ to 甲, 乙, and 丙 respectively, each person only knows their own number. They are told that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$, and each is asked the following two questions:
(1) Do you know the value of $a+b+c$?
(2) Do you know the values of $a$, $b$, and $c$... | 11 | 166 | 2 |
math | 9. (16 points) Let the real number $t \in [0, \pi]$. If the equation $\cos (x+t)=1-\cos x$ has a solution for $x$, find the range of values for $t$.
| \in[0,\frac{2\pi}{3}] | 53 | 13 |
math | 22. Suppose that $x_{1}, x_{2}$ and $x_{3}$ are the three roots of $(11-x)^{3}+(13-x)^{3}=(24-2 x)^{3}$. Find the value of $x_{1}+x_{2}+x_{3}$. | 36 | 72 | 2 |
math | Find all integers $n$ such that $2^{n}+3$ is a perfect square. The same question with $2^{n}+1$. | 3 | 33 | 1 |
math | 1. Let the real number $x>0$, and $x^{2}+\frac{1}{x^{2}}=7$. Then $x^{5}+\frac{1}{x^{5}}$ | 123 | 45 | 3 |
math | 9. Given the function $f(x)=a x^{3}+b x^{2}+c x+d(a \neq 0)$, when $0 \leqslant x \leqslant 1$, $\left|f^{\prime}(x)\right| \leqslant 1$, try to find the maximum value of $a$. | \frac{8}{3} | 79 | 7 |
math | 12 Given $\frac{1-|\cos x|}{1+|\cos x|}=\sin x$, then $x=$ $\qquad$ . | k\pi,2k\pi+\frac{\pi}{2}(k\in{Z}) | 33 | 21 |
math | Determine all real functions $f(x)$ that are defined and continuous on the interval $(-1, 1)$ and that satisfy the functional equation
\[f(x+y)=\frac{f(x)+f(y)}{1-f(x) f(y)} \qquad (x, y, x + y \in (-1, 1)).\] | f(x) = \tan(ax) | 74 | 8 |
math | 6. B. Given that the lengths of the two legs are integers $a$ and $b$ $(b<2011)$. Then the number of right triangles with the hypotenuse length $b+1$ is | 31 | 48 | 2 |
math | 4. (15 points) Two heaters are connected sequentially to the same DC power source. The water in the pot boiled after $t_{1}=3$ minutes from the first heater. The same water, taken at the same initial temperature, boiled after $t_{2}=6$ minutes from the second heater. How long would it take for the water to boil if the ... | 2 | 92 | 1 |
math | [ Law of Cosines
The center of the circle inscribed in a right triangle is at distances $\sqrt{5}$ and $\sqrt{10}$ from the ends of the hypotenuse. Find the legs.
# | 3\times4 | 46 | 4 |
math | 13. (6 points) Squirrels $A$, $B$, and $C$ have a total of some pine nuts. Squirrel $A$ originally had 26 pine nuts, from which it took out 10 pine nuts and divided them equally between $B$ and $C$. Then, Squirrel $B$ took out 18 of its own pine nuts and divided them equally between $A$ and $C$. Finally, Squirrel $C$ d... | 86 | 148 | 2 |
math | Problem 9.6. Find all pairs of natural prime numbers $p$, $q$, that satisfy the equation
$$
3 p^{4}+5 q^{4}+15=13 p^{2} q^{2}
$$ | (2,3) | 52 | 5 |
math | A set $S$ is called [i]neighbouring [/i] if it has the following two properties:
a) $S$ has exactly four elements
b) for every element $x$ of $S$, at least one of the numbers $x - 1$ or $x+1$ belongs to $S$.
Find the number of all [i]neighbouring [/i] subsets of the set $\{1,2,... ,n\}$. | \binom{n-2}{2} | 100 | 9 |
math | $3+$ [ The transfer helps solve the task_ ]
On the side AB of the square ABCD, an equilateral triangle AKB was constructed (outside). Find the radius of the circle circumscribed around triangle CKD, if $\mathrm{AB}=1$.
# | 1 | 57 | 1 |
math | 1. find all natural numbers $a, b$ and $n$ so that the following equation holds:
$$
a!+b!=2^{n}
$$
## Solution: | (1,1,1),(2,2,2),(2,3,3),(3,2,3) | 38 | 25 |
math | 11. Given a periodic sequence $\left\{x_{n}\right\}$ satisfying $x_{n}=\left|x_{n-1}-x_{n-2}\right|(n \geqslant 3)$. If $x_{1}=1, x_{2}=a \geqslant 0$. When the smallest positive period of the sequence is minimal, then the sum of the first 2008 terms of the sequence is $\qquad$. | 1339 | 102 | 4 |
math | Two lines passing through point $M$, lying outside a circle with center $O$, touch the circle at points $A$ and $B$. The segment $OM$ is bisected by the circle. In what ratio is the segment $OM$ divided by the line $AB$? | 1:3 | 58 | 3 |
math | Example 7 Find the value: $\cos \frac{\pi}{13}+\cos \frac{3 \pi}{13}+\cos \frac{9 \pi}{13}$. | \frac{1+\sqrt{13}}{4} | 42 | 13 |
math | 4. Calculate: $\sqrt{9+8 \cos 20^{\circ}}-\sec 20^{\circ}=$ | 3 | 29 | 1 |
math | For example, a bug crawls along a triangular iron ring. At each vertex, it has an equal chance of crawling to one of the other two vertices. Find the probability that it returns to the starting point after 10 crawls.
untranslated text remains the same:
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。
However, since the instructi... | \frac{171}{2^{9}} | 149 | 11 |
math | 3. For the polynomial $p(x)=2x+1$, find a polynomial $q(x)$ of the first degree such that $p^{2}(q(x))=q\left(p^{2}(x)\right)$ for any $x$. | q(x)=x | 52 | 4 |
math | 5. Given $P$ as a moving point on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, draw the perpendiculars from the foci $F_{1}$ and $F_{2}$ to the external angle bisector of $\angle F_{1} P F_{2}$. Then the locus of the feet of these perpendiculars is $\qquad$ | x^{2}+y^{2}=^{2} | 92 | 12 |
math | 4. Find two numbers whose sum is 2017, and the sum of the numbers written with the same digits but in reverse order is 8947. | 1408,609 | 36 | 8 |
math | 6. Let the circumcenter of the tetrahedron $S-ABC$ be $O$, the midpoints of $SB$ and $AC$ be $N$ and $M$ respectively, and the midpoint of segment $MN$ be $P$. Given that $SA^2 + SB^2 + SC^2 = AB^2 + BC^2 + AC^2$, if $SP = 3\sqrt{7}$ and $OP = \sqrt{21}$, then the radius of the sphere $O$ is $\qquad$ | 2\sqrt{21} | 117 | 7 |
math | 1021. Find the Maclaurin series for the function $\arcsin x$, starting from its representation as an integral: $\arcsin x=\int_{0}^{x} \frac{d t}{\sqrt{1-t^{2}}}$. | \begin{aligned}\arcsinx&=x+\frac{1}{2}\frac{x^{3}}{3}+\frac{1\cdot3}{2\cdot4}\frac{x^{5}}{5}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\frac{x^{7}}{7}+\ldots+\\&+\frac{1} | 57 | 85 |
math | 20th BMO 1984 Problem 1 In the triangle ABC, ∠C = 90 o . Find all points D such that AD·BC = AC·BD = AB·CD/√2. | (x,y)=(\frac{}{+b},\frac{}{+b})or(\frac{}{-b},\frac{}{-b}) | 48 | 30 |
math | 16th APMC 1993 Problem 1 Find all positive integers m, n such that 2 m - 3 n = 7. | =4,n=2 | 33 | 5 |
math | B1. Let $(1+\sqrt{2})^{5}=a+b \sqrt{2}$, where $a$ and $b$ are positive integers. Determine the value of $a+b$. | 70 | 42 | 2 |
math | Example 2. Find $\lim _{x \rightarrow 0} \frac{\sin x e^{2 x}-x}{5 x^{2}+x^{3}}$. | \frac{2}{5} | 38 | 7 |
math | Example 6 If the parabola $y=x^{2}+m x+2$ intersects the line segment $MN$ (including points $M$ and $N$) connecting points $M(0,1)$ and $N(2,3)$ at two distinct points, find the range of values for $m$. | -\frac{3}{2} \leqslant m < -1 | 69 | 16 |
math | Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$th day of January. What is $m$?
[i]Based on a proposal by Neelabh Deka[/i] | 23 | 87 | 2 |
math | 9. [7] A set of points is convex if the points are the vertices of a convex polygon (that is, a non-selfintersecting polygon with all angles less than or equal to $180^{\circ}$ ). Let $S$ be the set of points $(x, y)$ such that $x$ and $y$ are integers and $1 \leq x, y \leq 26$. Find the number of ways to choose a conv... | 4958 | 111 | 4 |
math | 14.3.17 $\star$ Let $n>0$, find the greatest common divisor of $\mathrm{C}_{2 n}^{1}, \mathrm{C}_{2 n}^{3}, \cdots, \mathrm{C}_{2 n}^{2 n-1}$. | 2^{k+1} | 63 | 6 |
math | 3.297. $\sin ^{2}(2 \alpha-\beta)-\sin ^{2} 2 \alpha-\sin ^{2} \beta$. | -2\sin2\alpha\sin\beta\cos(2\alpha-\beta) | 37 | 20 |
math | 3.376. Knowing that $\operatorname{tg} \frac{\alpha}{2}=m$, find $\frac{1-2 \sin ^{2} \frac{\alpha}{2}}{1+\sin \alpha}$. | \frac{1-}{1+} | 51 | 9 |
math | Calculate the sum of matrix commutators $[A, [B, C]] + [B, [C, A]] + [C, [A, B]]$, where $[A, B] = AB-BA$ | 0 | 48 | 1 |
math | Problem 7. Which number is greater: $\frac{1}{99}$ or
$$
\frac{1}{9903}+\frac{1}{9903+200}+\frac{1}{9903+200+202}+\ldots+\frac{1}{9903+200+202+\ldots+2018} ?
$$ | \frac{1}{99} | 94 | 8 |
math | Example 7.5 Let $\left(D^{\prime}, \circ\right)$ be the vertex permutation group derived from the improper rotation group of a regular hexagon. Find the cycle index of $\left(D^{\prime}, \circ\right)$. | \frac{1}{12}(x_{1}^{6}+4x_{2}^{3}+2x_{3}^{2}+3x_{1}^{2}x_{2}^{2}+2x_{6}) | 53 | 55 |
math | 1. (16 points) Given the equation in $x$
$$
x^{2}-\left(m^{2}+2 m-3\right) x+2(m+1)=0
$$
has two real roots that are opposites of each other.
(1) Find the value of the real number $m$;
(2) If the equation in $x$ $x^{2}-(k+m) x-3 m-k-5=0$ has roots that are all integers, find all real numbers $k$ that satisfy the condi... | k=4 \text{ or } k=-2 | 119 | 11 |
math | 5. (1999 National High School Competition Question) Given a positive integer $n$ does not exceed 2000, and can be expressed as the sum of no less than 60 consecutive positive integers, then the number of such $n$ is $\qquad$ . | 6 | 61 | 1 |
math | 371. A body is thrown vertically upward with a velocity that varies according to the law \( v = (29.4 - 9.8 t) \) m/s. Find the maximum height of ascent. | 44.1 | 46 | 4 |
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