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 6. Find the inverse function of the function $y=\sin x, x \in[-\pi$, $-\frac{\pi}{2}$]. | y=-\pi-\arcsin x, x \in[-1,0] | 33 | 18 |
math | 10. (5 points) For $n$ consecutive natural numbers starting from 1, if one of the numbers is removed, the average of the remaining numbers is $\frac{152}{7}$, then the removed number is $\qquad$ | 34 | 53 | 2 |
math | 12. There is a well, at the bottom of which there is a spring that continuously gushes out water, with the same amount of water gushing out every minute. If 4 water pumps are used to pump out the water, it takes 40 minutes to finish; if 5 water pumps are used, it takes 30 minutes to finish. Now, it is required to pump ... | 6 | 104 | 1 |
math | 3. Anita and Boris are shooting at a target with a ball, each 50 times. One part of the target is painted yellow, and the other part is blue. For each hit on the target, a certain number of points is awarded, and if the target is missed, no points are awarded. Anita hit the yellow part 36 times and missed the target 2 ... | x=8,y=6 | 157 | 6 |
math | Example 1. The roots of the equation
$$
x^{3}+a x^{2}+b x+c=0
$$
form a geometric progression. What necessary and sufficient condition must the coefficients of the equation satisfy? | ^{3}=b^{3} | 49 | 7 |
math | 2A. Determine all three-digit numbers with distinct digits that are divisible by any two-digit number obtained by removing one of its digits without changing the order of the remaining digits. | 120,150,240,360,480 | 35 | 19 |
math | H1. No digit of the positive integer $N$ is prime. However, all the single-digit primes divide $N$ exactly.
What is the smallest such integer $N$ ? | 840 | 38 | 3 |
math | 15. $A B C D$ is a convex quadrilateral and $E, F$ are the mid-points of $B C$ and $C D$ respectively. The line segments $A E, A F$ and $E F$ divide $A B C D$ into four triangles, whose areas are four consecutive integers. Find the greatest possible area of $\triangle A B D$.
(2 marks) $A B C D$ 是凸四邊形, $E 、 F$ 分別是 $B C... | 6 | 182 | 1 |
math | Two convex polygons have a total of 33 sides and 243 diagonals. Find the number of diagonals in the polygon with the greater number of sides. | 189 | 35 | 3 |
math | Determine the real numbers $x$, $y$, $z > 0$ for which
$xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$ | x = y = z = \sqrt{2} | 67 | 12 |
math | Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and
\[a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}\]
for each $n \in \mathbb{N}$. | a_i \in \{a, b\} \text{ where } ab = 2006 \text{ and } a_n = a_{n+2} \text{ for all } n | 75 | 44 |
math | 5. (7 points) There is a sequence of numbers: $1,1,2,3,5,8,13,21, \cdots$, starting from the third number, each number is the sum of the two preceding ones. How many of the first 2007 numbers are even? | 669 | 67 | 3 |
math | Let's determine the numbers $a, b, c$ such that
$$
x^{3}-a x^{2}+b x-c=(x-a)(x-b)(x-c)
$$
is an identity. | =b==0or=b=-1,=1 | 46 | 10 |
math | 1. The range of the function $f(x)=\sin x+\cos x+\tan x+$ $\arcsin x+\arccos x+\arctan x$ is $\qquad$ . | [-\sin 1+\cos 1-\tan 1+\frac{\pi}{4}, \sin 1+\cos 1+\tan 1+\frac{3 \pi}{4}] | 43 | 41 |
math | ## Zadatak B-3.3.
Riješite jednadžbu $x^{2}-x-\sin (\pi x)=-1.25$.
| \frac{1}{2} | 37 | 7 |
math | 6.4. Find all solutions of the inequality
$$
\cos ^{2018} x+\sin ^{-2019} x \leqslant \sin ^{2018} x+\cos ^{-2019} x
$$
belonging to the interval $\left[-\frac{\pi}{3} ; \frac{5 \pi}{3}\right]$. | [-\frac{\pi}{3};0)\cup[\frac{\pi}{4};\frac{\pi}{2})\cup(\pi;\frac{5\pi}{4}]\cup(\frac{3\pi}{2};\frac{5\pi}{3}] | 90 | 58 |
math | 9-6-1. Six pirates - a captain and five members of his crew - are sitting around a campfire facing the center. They need to divide a treasure: 180 gold coins. The captain proposes a way to divide the treasure (i.e., how many coins each pirate should receive: each pirate will receive a non-negative integer number of coi... | 59 | 155 | 2 |
math | 13.298. Two athletes are running on the same closed track of a stadium. The speed of each is constant, but the first one takes 10 seconds less to run the entire track than the second one. If they start running from a common starting point in the same direction, they will meet again after 720 seconds. What fraction of t... | \frac{1}{80} | 87 | 8 |
math | 1. (8 points) Define a new operation $a \preccurlyeq b$ such that: $a \preccurlyeq b=b \times 10+a \times 2$, then 2011 approx $130=$ | 5322 | 56 | 4 |
math | Let $P(x)=x^3+ax^2+bx+c$ be a polynomial where $a,b,c$ are integers and $c$ is odd. Let $p_{i}$ be the value of $P(x)$ at $x=i$. Given that $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$, find the value of $p_{2}+2p_{1}-3p_{0}.$ | 18 | 113 | 2 |
math | 1. On January 1, 2013, a little boy was given a bag of chocolate candies, containing 300 candies. Each day, the little boy ate one candy. On Sundays, Karlson would fly over, and the little boy would treat him to a couple of candies. How many candies did Karlson eat? (January 1, 2013, was a Tuesday).
2. Petya can swap ... | 66 | 122 | 2 |
math | 9. [6] Newton and Leibniz are playing a game with a coin that comes up heads with probability $p$. They take turns flipping the coin until one of them wins with Newton going first. Newton wins if he flips a heads and Leibniz wins if he flips a tails. Given that Newton and Leibniz each win the game half of the time, wha... | \frac{3-\sqrt{5}}{2} | 87 | 12 |
math | Let $a_{1}, \ldots, a_{n}$ be distinct and $b_{1}, \ldots, b_{n}$ be real numbers.
What are the polynomials such that $P\left(a_{i}\right)=b_{i}$ for all $i$? | P_{0}+R\prod_{i}(X-a_{i}) | 61 | 16 |
math | Let $S$ be a subset of $\{0,1,2,\dots ,9\}$. Suppose there is a positive integer $N$ such that for any integer $n>N$, one can find positive integers $a,b$ so that $n=a+b$ and all the digits in the decimal representations of $a,b$ (expressed without leading zeros) are in $S$. Find the smallest possible value of $|S|$.
... | 5 | 202 | 1 |
math | 4.4. A smooth sphere with a radius of 1 cm was dipped in blue paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a blue trail. During its m... | 38.25 | 134 | 5 |
math | 7. (10 points) The teacher is doing calculation practice with Jiajia, Fangfang, and Mingming. The teacher first gives each of them a number, and then asks them to each pick 3 cards with numbers on them. Jiajia picks 3, 6, 7, Fangfang picks 4, 5, 6, and Mingming picks 4, 5, 8. The teacher then asks them to each multiply... | 7 | 157 | 1 |
math | 1. (17 points) Three cyclists, Dima, Misha, and Petya, started a 1 km race simultaneously. At the moment Misha finished, Dima still had to cover one tenth of the distance, and at the moment Dima finished, Petya still had to cover one tenth of the distance. How far apart (in meters) were Petya and Misha when Misha finis... | 190 | 97 | 3 |
math | 6. There are 900 three-digit numbers (100, $101, \cdots, 999$). If these three-digit numbers are printed on cards, with one number per card, some cards, when flipped, still show a three-digit number, such as 198, which when flipped becomes 861 (1 is still considered 1 when flipped); some cards do not, such as 531, whic... | 34 | 135 | 2 |
math | 4. Considering $\boldsymbol{x}$ and $\boldsymbol{y}$ as integers, solve the system of equations (11 points):
$$
\left\{\begin{array}{l}
8^{x^{2}-2 x y+1}=(z+4) 5^{|y|-1} \\
\sin \frac{3 \pi z}{2}=-1
\end{array}\right.
$$ | (-1;-1;-3),(1;1;-3) | 89 | 13 |
math | 1. Solve the inequality $\frac{\sqrt{\frac{x}{\gamma}+(\alpha+2)}-\frac{x}{\gamma}-\alpha}{x^{2}+a x+b} \geqslant 0$.
In your answer, specify the number equal to the number of integer roots of this inequality. If there are no integer roots, or if there are infinitely many roots, enter the digit 0 in the answer sheet.
... | 7 | 115 | 1 |
math | 3. Find all pairs of numbers (a, b) for which the equality $(\mathrm{a}+\mathrm{b}-1)^{2}=\mathrm{a}^{2}+\mathrm{b}^{2}-1$ holds. | (1,),(,1) | 52 | 7 |
math | 7. The sum of the reciprocals of all positive integers $n$ that make $\left[\frac{n^{2}}{5}\right]$ a prime number is . $\qquad$ | \frac{37}{60} | 40 | 9 |
math | Robert was born in the year $n^{2}$. On his birthday in the year $(n+1)^{2}$, he will be 89 years old. In what year was he born? | 1936 | 43 | 4 |
math | 3. If the orthocenter of $\triangle O A B$ is exactly the focus of the parabola $y^{2}=4 x$, where $O$ is the origin, and points $A$ and $B$ are on the parabola, then the area $S$ of $\triangle O A B$ is $\qquad$ . | 10\sqrt{5} | 74 | 7 |
math | 3. It is known that for some positive coprime numbers $m$ and $n$, the numbers $m+2024 n$ and $n+2024 m$ have a common prime divisor $d>7$. Find the smallest possible value of the number $d$ under these conditions. | 17 | 66 | 2 |
math |
4. Find the sum of all 3-digit natural numbers which contain at least one odd digit and at least one even digit.
| 370775 | 27 | 6 |
math | [Trigonometric Systems of Equations and Inequalities]
Find all solutions to the system of equations
$$
\left\{\begin{array}{l}
\sin (x+y)=0 \\
\sin (x-y)=0
\end{array}\right.
$$
satisfying the conditions $0 \leq x \leq \pi, ; ; 0 \leq y \leq \pi$. | (0,0),(0,\pi),(\pi/2,\pi/2),(\pi,0),(\pi,\pi) | 90 | 28 |
math | 9. (12 points) Four people, A, B, C, and D, have a total of 251 stamps. It is known that A has 2 more stamps than twice the number of B's stamps, 6 more stamps than three times the number of C's stamps, and 16 fewer stamps than four times the number of D's stamps. Therefore, D has $\qquad$ stamps. | 34 | 88 | 2 |
math | 4. If $f(\sin x+1)=\cos ^{2} x+\sin x-3$, then $f(x)=$ | -x^{2}+3x-4,x\in[0,2] | 30 | 17 |
math | 2. Given the line $L: x+y-9=0$ and the circle $M: 2 x^{2}+2 y^{2}-8 x-8 y-1=0$, point $A$ is on line $L$, $B, C$ are two points on circle $M$, in $\triangle A B C$, $\angle B A C=45^{\circ}, A B$ passes through the center of circle $M$, then the range of the x-coordinate of point $A$ is $\qquad$ | 3\leqslantx\leqslant6 | 114 | 13 |
math | 9.5. The cells of a chessboard are painted in 3 colors - white, gray, and black - in such a way that adjacent cells, sharing a side, must differ in color, but a sharp change in color (i.e., a white cell adjacent to a black cell) is prohibited. Find the number of such colorings of the chessboard (colorings that coincide... | 2^{33} | 98 | 5 |
math | ## Problem 4
Let $A B C D$ be a rectangle, with side lengths $A B=4 \mathrm{~cm}$ and $A D=3 \mathrm{~cm}$, and $\{O\}=A C \cap B D$.
Determine the length of the vector $\vec{v}=\overrightarrow{A B}+\overrightarrow{A O}+\overrightarrow{A D}$.
## Note
- Effective working time 3 hours.
- All problems are mandatory.
-... | \frac{15}{2}\mathrm{~} | 157 | 12 |
math | Example 12 Let $x, y, z$ be real numbers greater than -1. Find the minimum value of
$$\frac{1+x^{2}}{1+y+z^{2}}+\frac{1+y^{2}}{1+z+x^{2}}+\frac{1+z^{2}}{1+x+y^{2}}$$ | 2 | 73 | 1 |
math | \section*{Problem 3 - 021133}
In how many different ways can the number 99 be expressed as the sum of three distinct prime numbers?
(Two cases are considered the same if the same addends appear, merely in a different order.) | 21 | 58 | 2 |
math | Find all functions $f: \mathbf{Q}_{+} \rightarrow \mathbf{Q}_{+}$, such that
$$
f(x)+f(y)+2 x y f(x y)=\frac{f(x y)}{f(x+y)},
$$
where $\mathbf{Q}_{+}$ denotes the set of positive rational numbers.
(Li Shenghong) | f(x)=\frac{1}{x^{2}} | 81 | 12 |
math | 1. Calculate: $7.625-6 \frac{1}{3}+5.75-1 \frac{3}{8}=$ | 5\frac{2}{3} | 34 | 8 |
math | 9.4. Vasya cut out a triangle from cardboard and numbered its vertices with the digits 1, 2, and 3. It turned out that if Vasya's triangle is rotated 15 times clockwise around its vertex numbered 1 by the angle at this vertex, the triangle will return to its original position. If Vasya's triangle is rotated 6 times clo... | 5 | 166 | 1 |
math | 10. What is the smallest 3-digit positive integer $N$ such that $2^{N}+1$ is a multiple of 5 ? | 102 | 32 | 3 |
math | 20. Calculate: $1101_{(2)} \times 101_{(2)}=$ $\qquad$ (2). | 1000001_{(2)} | 32 | 11 |
math | In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$,$\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$. Then find the angle $\angle{DBA}$ | 45^\circ | 61 | 4 |
math | B4. In the square of numbers below, positive numbers are listed. The product of the numbers in each row, each column, and each of the two diagonals is always the same.
What number is in the position of $H$?
| $\frac{1}{2}$ | 32 | $A$ | $B$ |
| :---: | :---: | :---: | :---: |
| $C$ | 2 | 8 | 2 |
| 4 | 1 | $D$ | $E$ |
... | \frac{1}{4} | 134 | 7 |
math | 7.3. There are three automatic coin exchange machines. Among them, the first machine can only exchange 1 coin for 2 other coins; the second machine can only exchange 1 coin for 4 other coins; the third machine can exchange 1 coin for 10 other coins. A person made a total of 12 exchanges, turning 1 coin into 81 coins. T... | A=B=2, C=8 | 101 | 8 |
math | \section*{Problem 3B - 251043B}
Given are real numbers \(a_{1}, a_{2}, a_{3}, a_{4}\).
Determine for each possible case of these \(a_{1}, \ldots, a_{4}\) all triples \(\left(b_{1}, b_{2}, b_{3}\right)\) of real numbers (or prove, if applicable, that no such triples exist) for which the system of equations
\[
\begin{... | (b_{1},b_{2},b_{3})=(0,0,0) | 237 | 19 |
math | 2. The integer solution to the equation $(\lg x)^{\lg (\lg x)}=10000$ is $x=$ | 10^{100} | 30 | 7 |
math | 13. The game "Unicum". Petya and 9 other people are playing such a game: each throws a die. A player wins a prize if they roll a number of points that no one else has rolled.
a) (6th - 11th grades, 1 point) What is the probability that Petya will win a prize?
b) (8th - 11th grades, 3 points) What is the probability t... | )(\frac{5}{6})^{9}\approx0.194;b)\approx0.919 | 103 | 25 |
math | In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. The probability t... | 25 | 126 | 2 |
math | ## Task 1 - 020621
During the group flight of Soviet cosmonauts Nikolayev and Popovich, the spacecraft Vostok III and Vostok IV orbited the Earth once in about 88 minutes (approximately $41000 \mathrm{~km}$).
a) What distance did each spacecraft cover in one hour?
b) What distance did it cover in each second?
The r... | 28000\mathrm{~},7.8\mathrm{~} | 98 | 18 |
math | 3. (43rd American High School Mathematics Examination) Let $S$ be a subset of the set $\{1,2, \cdots, 50\}$ with the following property: the sum of any two distinct elements of $S$ cannot be divisible by 7. What is the maximum number of elements that $S$ can have? | 23 | 74 | 2 |
math | # Problem 4. (3 points)
In triangle $A B C$, the median $B M$ is drawn, in triangle $A B M$ - the median $B N$, in triangle $B N C$ - the median $N K$. It turns out that $N K \perp B M$. Find $A B: A C$.
Answer: $\frac{1}{2}$ | \frac{1}{2} | 84 | 7 |
math | Let 5 be a point inside an acute triangle $\triangle ABC$ such that $\angle ADB = \angle ACB + 90^{\circ}$, and $AC \cdot BD = AD \cdot BC$. Find the value of $\frac{AB \cdot CD}{AC \cdot BD}$. | \sqrt{2} | 64 | 5 |
math | ## Task B-4.6.
For what value of the real number $x$ is the sum of the third and fifth term in the expansion of $\left(\sqrt{2^{x}}+\frac{1}{\sqrt{2^{x-1}}}\right)^{m}$ less than or equal to 135 if the sum of the binomial coefficients of the last three terms is 22? | x\in[-1,2] | 87 | 8 |
math | Let $m$ be a fixed positive integer. The infinite sequence $\left\{a_{n}\right\}_{n \geq 1}$ is defined in the following way: $a_{1}$ is a positive integer, and for every integer $n \geq 1$ we have
$$
a_{n+1}= \begin{cases}a_{n}^{2}+2^{m} & \text { if } a_{n}<2^{m} \\ a_{n} / 2 & \text { if } a_{n} \geq 2^{m}\end{case... | =2,a_{1}=2^{\ell}for\ell\geq1 | 213 | 18 |
math | 3. Given a sequence of numbers $a_{1}, a_{2}, \cdots, a_{2006}$, where $a_{1}$ $=1$, and the sum of each pair of consecutive terms is 3. Then $a_{1}-a_{2}+$ $a_{3}-a_{4}+\cdots+a_{2003}-a_{2004}+a_{2005}=$ $\qquad$ | -1001 | 101 | 5 |
math | 9. (6 points) Three mice found a pile of peanuts and agreed to come and share them equally the next day. The next day, the first mouse arrived the earliest. He found that the peanuts could not be divided equally, so he ate one, and the remaining could be divided into 3 equal parts, and he took one part. The second and ... | 25 | 121 | 2 |
math | Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique)
What's the volume of $A\cup B$? | \frac{1}{2} | 43 | 7 |
math | In a line, there are 2020 white stones and one black stone. A possible operation is as follows: choose a black stone that is not at the edge and change the color of its neighboring stones. Find all possible initial positions for the black stone such that it is possible to color all the stones black with a finite number... | 1011 | 71 | 4 |
math | 1. [5 points] The altitudes $C F$ and $A E$ of an acute triangle $A B C$ intersect at point $H$. Points $M$ and $N$ are the midpoints of segments $A H$ and $C H$ respectively. It is known that $F M=1, E N=4$, and $F M \| E N$. Find $\angle A B C$, the area of triangle $A B C$, and the radius of the circumscribed circle... | \angleABC=60,S_{\triangleABC}=18\sqrt{3},R=2\sqrt{7} | 147 | 27 |
math | Determine all possible values of integer $k$ for which there exist positive integers $a$ and $b$ such that $\dfrac{b+1}{a} + \dfrac{a+1}{b} = k$. | 3 \text{ and } 4 | 49 | 8 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{(3-4 n)^{2}}{(n-3)^{3}-(n+3)^{3}}$ | -\frac{8}{9} | 51 | 7 |
math | Example 2 If $x$, $y$, $z$ are all real numbers, and $x^{2}+y^{2}+z^{2}$ $=1$, then the maximum value of $\sqrt{2} x y+y z$ is $\qquad$ | \frac{\sqrt{3}}{2} | 58 | 10 |
math | 8.5. One hundred and one numbers are written in a circle. It is known that among any five consecutive numbers, there are at least two positive numbers. What is the minimum number of positive numbers that can be among these 101 written numbers? | 41 | 53 | 2 |
math | ## Task 3 - 331213
In a beauty contest for poodles, Asta, Benno, Cäsar, and Dolly face a jury of four members. Each jury member votes for one of the dogs by raising a card with the initial letter of the dog's name. As a rule for evaluating this voting result, it was stipulated: If two dogs clearly receive more votes t... | 172 | 205 | 3 |
math | Task 1. The frog has a jump of $50 \mathrm{~cm}$, the rabbit $75 \mathrm{~cm}$, and the kangaroo $90 \mathrm{~cm}$. What is the smallest distance at which all will make an integer number of jumps. How many jumps will each of the animals then make? | 450 | 73 | 3 |
math | 7. (15 points) Insert 2 " $\div$ " and 2 "+" between the 9 "1"s below to make the calculation result an integer. The smallest integer is $\qquad$
\begin{tabular}{|lllllllllll|}
\hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & $=$ & $?$ \\
\hline
\end{tabular} | 3 | 103 | 1 |
math | Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and
\[a^3 + b^3 = p^n\mbox{.}\]
[i](2nd Benelux Mathematical Olympiad 2010, Problem 4)[/i] | (2^k, 2^k, 2, 3k + 1), (3^k, 2 \cdot 3^k, 3, 3k + 2), (2 \cdot 3^k, 3^k, 3, 3k + 2) | 68 | 68 |
math | 1469. Calculate $\sqrt{1.004}$ with an accuracy of 0.0001. | 1.002 | 27 | 5 |
math | 61st Putnam 2000 Problem A1 k is a positive constant. The sequence x i of positive reals has sum k. What are the possible values for the sum of x i 2 ? Solution | any\value\in\the\open\interval\(0,k^2) | 46 | 17 |
math | Triangle $ABC$ is a triangle with side lengths $13$, $14$, and $15$. A point $Q$ is chosen uniformly at random in the interior of $\triangle{ABC}$. Choose a random ray (of uniformly random direction) with endpoint $Q$ and let it intersect the perimeter of $\triangle{ABC}$ at $P$. What is the expected value of $QP^2$?
... | \frac{84}{\pi} | 105 | 9 |
math | 18. Given the quadratic function $f(x)=4 x^{2}-4 a x+\left(a^{2}-2 a+2\right)$ has a minimum value of 2 on $0 \leqslant x \leqslant 1$, find the value of $a$.
| 0 | 64 | 1 |
math | Let $N$ be a positive integer. Two persons play the following game. The first player writes a list of positive integers not greater than 25 , not necessarily different, such that their sum is at least 200 . The second player wins if he can select some of these numbers so that their sum $S$ satisfies the condition $200-... | 11 | 119 | 2 |
math | 7. In the Cartesian coordinate system, the ellipse $\Omega: \frac{x^{2}}{4}+y^{2}=1$, $P$ is a moving point on $\Omega$, $A, B$ are two fixed points, where the coordinates of $B$ are $(0,3)$. If the minimum area of $\triangle P A B$ is 1 and the maximum area is 5, then the length of line segment $A B$ is $\qquad$. | \sqrt{7} | 101 | 5 |
math | 151. $\int \sqrt{x+1} d x$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
151. $\int \sqrt{x+1} d x$. | \frac{2}{3}(x+1)\sqrt{x+1}+C | 55 | 18 |
math | ## Task 4 - 090614
A working group received a sum of exactly $240 \mathrm{M}$ as an award for very good performance.
If this money had been distributed evenly among all members of the working group, each member would have received an integer amount (in Marks). However, the members decided to spend the $240 \mathrm{M}... | 15 | 154 | 2 |
math | 10. (20 points) Find the number of all positive integer solutions $(x, y, z)$ to the equation $\arctan \frac{1}{x}+\arctan \frac{1}{y}+\arctan \frac{1}{z}=\frac{\pi}{4}$. | 15 | 67 | 2 |
math | 1. $2019^{\ln \ln 2019}-(\ln 2019)^{\ln 2019}$ $=$ . $\qquad$
Fill in the blank (8 points per question, total 64 points)
1. $2019^{\ln \ln 2019}-(\ln 2019)^{\ln 2019}$ $=$ . $\qquad$ | 0 | 100 | 1 |
math | 3. Let the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>1, b>1)$ have a focal distance of $2 c$, and let the line $l$ pass through the points $(a, 0)$ and $(0, b)$. If the distance from the point $(1, 0)$ to the line $l$ plus the distance from the point $(-1,0)$ to the line $l$ is $S \geqslant \frac{4 c}{5}$.... | [\frac{\sqrt{5}}{2},\sqrt{5}] | 142 | 15 |
math | 4. Given an isosceles triangle $\triangle A B C$ with side lengths $a$, $b$, and $c$ all being integers, and satisfying $a+b c+b+c a=24$. Then the number of such triangles is $\qquad$. | 3 | 56 | 1 |
math | One, (20 points) A research study group from Class 1, Grade 3 of Yuhong Middle School conducted a survey on students' lunch time at the school canteen. It was found that within a unit of time, the number of people buying lunch at each window and the number of people choosing to eat outside due to unwillingness to wait ... | 5 | 240 | 1 |
math | XXXVIII OM - II - Task 1
From an urn containing one ball marked with the number 1, two balls marked with the number 2, ..., $ n $ balls marked with the number $ n $, we draw two balls without replacement. We assume that drawing each ball from the urn is equally probable. Calculate the probability that both drawn balls... | \frac{4}{3(n+2)} | 78 | 10 |
math | Two positive whole numbers differ by $3$. The sum of their squares is $117$. Find the larger of the two
numbers. | 9 | 29 | 1 |
math | 2. Six people are standing in a circle, each of whom is either a knight - who always tells the truth, or a liar - who always lies. Each of them said one of two phrases: "There is a liar next to me" or "There is a liar opposite me." What is the minimum number of liars among them? Provide an example and prove that there ... | 2 | 83 | 1 |
math | 13. $A B C D$ is a convex quadrilateral in which $A C$ and $B D$ meet at $P$. Given $P A=1$, $P B=2, P C=6$ and $P D=3$. Let $O$ be the circumcentre of $\triangle P B C$. If $O A$ is perpendicular to $A D$, find the circumradius of $\triangle P B C$.
(2 marks)
$A B C D$ is a convex quadrilateral, in which $A C$ and $B ... | 3 | 190 | 1 |
math | Example 2.18. Estimate the upper and lower bounds of the value of the integral $\int_{0}^{\pi / 6} \frac{d x}{1+3 \sin ^{2} x}$. | \frac{2\pi}{21}\leqslant\int_{0}^{\pi/6}\frac{}{1+3\sin^{2}x}\leqslant\frac{\pi}{6} | 49 | 49 |
math | 13. Given the function $f(x)$ $=\sin 2 x \cdot \tan x+\cos \left(2 x-\frac{\pi}{3}\right)-1$, find the intervals where the function $f(x)$ is monotonically decreasing. | [k\pi+\frac{\pi}{3},k\pi+\frac{\pi}{2}),(k\pi+\frac{\pi}{2},k\pi+\frac{5\pi}{6}](k\in{Z}) | 55 | 50 |
math | 3. $A D 、 B E 、 C F$ are the angle bisectors of $\triangle A B C$. If $B D+B F=C D+C E=A E+A F$, then the degree measure of $\angle B A C$ is $\qquad$ . | 60^{\circ} | 58 | 6 |
math | 3. (3 points) Find the number of integers from 1 to 1000 inclusive that give the same remainder when divided by 11 and by 12. | 87 | 38 | 2 |
math | In the triangle $A B C$, let $l$ be the bisector of the external angle at $C$. The line through the midpoint $O$ of the segment $A B$ parallel to $l$ meets the line $A C$ at $E$. Determine $|C E|$, if $|A C|=7$ and $|C B|=4$. | \frac{11}{2} | 78 | 8 |
math | 7. Given $a, b, c, d \in \mathbf{N}$ and satisfy $342(a b c d+$ $a b+a d+c d+1)=379(b c d+b+d)$, let $M=a$ $\cdot 10^{3}+b \cdot 10^{2}+c \cdot 10+d$ then the value of $M$ is $\qquad$ . | 1949 | 95 | 4 |
math | ## Task 3 - 241213
Determine all real numbers $x$ for which $2 x-3,5 x-14$ and $\frac{2 x-3}{5 x-14}$ are integers. | 3 | 53 | 1 |
math | Let $ABC$ be an acute-angled triangle with circumcenter $O$ and let $M$ be a point on $AB$. The circumcircle of $AMO$ intersects $AC$ a second time on $K$ and the circumcircle of $BOM$ intersects $BC$ a second time on $N$.
Prove that $\left[MNK\right] \geq \frac{\left[ABC\right]}{4}$ and determine the equality case. | \left[MNK\right] \geq \frac{\left[ABC\right]}{4} | 99 | 22 |
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