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 | 4. (4 points) When moving in the air, a ball is acted upon by a resistance force proportional to the square of its velocity. Immediately before the volleyball player's hit, the ball was flying horizontally at a speed of $V_{1}$. After the hit, the ball flew vertically upwards at a speed of $V_{2}$ with an acceleration of $a_{2}$. Determine the acceleration of the ball immediately before the hit.
Possible solution. Let's write the equation connecting the ball's acceleration and its velocity in horizontal motion:
$$
a_{1}=\frac{F_{1}}{m}=\frac{k V_{1}^{2}}{m}
$$
Here $F_{1}$ is the resistance force, $m$ is the mass of the ball, and $k$ is the proportionality coefficient. After the hit, the ball flew upwards, and the forces of gravity and resistance acted on it, both directed downwards. Its acceleration in this case will be
$$
a_{2}=\frac{k V_{2}^{2}+m g}{m}=\frac{\frac{m a_{1}}{V_{1}^{2}} V_{2}^{2}+m g}{m}=a_{1}\left(\frac{V_{2}}{V_{1}}\right)^{2}+g
$$
For the acceleration $a_{1}$, we get
$$
a_{1}=\left(\frac{V_{1}}{V_{2}}\right)^{2}\left(a_{2}-g\right)
$$ | a_{1}=(\frac{V_{1}}{V_{2}})^{2}(a_{2}-) | 334 | 26 |
math | Let $a$ be a natural number. What are the possible values of $a^{5}$ modulo 11? | -1,0,1 | 25 | 6 |
math | 21. $\sqrt{x^{\lg \sqrt{x}}}=10$. | x_{1}=100;\quadx_{2}=0.01 | 17 | 17 |
math | Find all integer solutions of the equation:
$$
x^{2}+y^{2}=3 z^{2}
$$ | (x,y,z)=(0,0,0) | 25 | 10 |
math | Let $a,b,c$ be distinct real numbers such that $a+b+c>0$. Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$. Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value.
[i]Mircea Becheanu[/i] | 6 | 92 | 1 |
math | 111. Reduce the equations of the lines to normal form:
1) $2 x-3 y-10=0$
2) $3 x+4 y=0$ | \frac{2}{\sqrt{13}}x-\frac{3}{\sqrt{13}}y-\frac{10}{\sqrt{13}}=0\frac{3}{5}x+\frac{4}{5}0 | 39 | 54 |
math | A1. At a party, if each kid took one apple from the fruit bucket then 7 apples would still remain in the bucket. However, if each kid had an appetite for two apples, the supply would be 16 apples short. How many kids were at the party? | 23 | 58 | 2 |
math | ## Task 4 - 100524
A group of young mathematicians went on an excursion. Each participant paid 1.50 marks for travel expenses. When paying for the collective ticket, a remainder of 1.10 marks was left.
If each participant had paid 1.40 marks, there would have been a shortage of 1.10 marks to cover the cost of the collective ticket.
Determine the number of participants in this excursion! How much money did each of these participants receive back when the excess amount paid was evenly distributed among them? | 22 | 121 | 2 |
math | 3・17 (1) Simplify $\frac{1-a^{2}}{(1+a x)^{2}-(a+x)^{2}}$;
(2) When $x=0.44$, find the value of $\sqrt{1-x-x^{2}+x^{3}}$. | 0.672 | 65 | 5 |
math | 17. (3 points) Given that $a$, $b$, and $c$ are three distinct prime numbers, if $a + b \times c = 37$, then the maximum value of $a + b - c$ is | 32 | 51 | 2 |
math | Problem 2. For what least $n$ do there exist $n$ numbers from the interval $(-1 ; 1)$ such that their sum is 0 and the sum of their squares is 30? | 32 | 45 | 2 |
math | 8. Let $n$ be a positive integer. For positive real numbers $a_{1}, a_{2}, \cdots, a_{n}$, we have $a_{1}+a_{2}+\cdots+a_{n}=1$. Let $A$ be the minimum of the following $n$ numbers $\frac{a_{1}}{1+a_{1}}, \frac{a_{2}}{1+a_{1}+a_{2}}, \cdots, \frac{a_{n}}{1+a_{1}+a_{2}+\cdots+a_{n}}$. Find the maximum value of $A$ when $a_{1}$, $a_{2}, \cdots, a_{n}$ are variables. (2004 Japan Mathematical Olympiad Problem) | 1-\frac{1}{\sqrt[n]{2}} | 171 | 12 |
math | ## 225. Math Puzzle $2 / 84$
In the quartz clock, an oscillator generates a vibration with a frequency of 32768 Hz (Hertz), which is converted by a 15-stage frequency divider so that one pulse is generated every second to drive the display mechanism.
By how many seconds does the display error increase per day if the oscillator is incorrectly set and generates $32769 \mathrm{~Hz}$? | 2.64 | 97 | 4 |
math | 5. For several identical books, 104 rubles were paid. The price of one book is a natural number. How much does one book cost if more than 10 but less than 60 were bought? | 2,4,8 | 47 | 5 |
math | 8. Let $A, B, C, D$ be four points in space that are not coplanar. With a probability of $\frac{1}{2}$, connect an edge between each pair of points, and whether any two pairs of points are connected is independent. Then the probability that points $A$ and $B$ can be connected by (a single edge or a sequence of edges forming) a spatial polyline is $\qquad$ | \frac{3}{4} | 92 | 7 |
math | ## Task A-1.8. (20 points)
Determine all natural numbers $n$ for which $n^{2}-440$ is a perfect square. | 111,57,27,21 | 37 | 12 |
math | In rectangle $ ABCD$, $ AB\equal{}100$. Let $ E$ be the midpoint of $ \overline{AD}$. Given that line $ AC$ and line $ BE$ are perpendicular, find the greatest integer less than $ AD$. | 141 | 54 | 3 |
math | Four, $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns,
\begin{tabular}{llllll}
$a_{11}$ & $a_{12}$ & $a_{13}$ & $a_{14}$ & $\cdots$ & $a_{1 n}$ \\
$a_{21}$ & $a_{22}$ & $a_{23}$ & $a_{24}$ & $\cdots$ & $a_{2 n}$ \\
$a_{31}$ & $a_{32}$ & $a_{33}$ & $a_{34}$ & $\cdots$ & $a_{3 n}$ \\
$a_{41}$ & $a_{42}$ & $a_{43}$ & $a_{41}$ & $\cdots$ & $a_{4 n}$ \\
$\cdots \cdots$ & & & & & \\
$a_{n 1}$ & $a_{n 2}$ & $a_{n 3}$ & $a_{n 4}$ & $\cdots$ & $a_{n n}$
\end{tabular}
where the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, find
$$
a_{11}+a_{22}+a_{33}+a_{44}+\cdots+a_{n n}
$$ | 2-\frac{1}{2^{n-1}}-\frac{n}{2^{n}} | 359 | 20 |
math | Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible. | M | 57 | 2 |
math | 2. Metka stands $60 \mathrm{~m}$ east and $80 \mathrm{~m}$ south of the point where Tine is standing. Both are equally distant from a linden tree in the city park, which is directly east of the point where Tine is. At the same time, each sets off directly towards the linden tree from their respective positions. How many meters will each of them walk until they meet under the linden tree? | 83\frac{1}{3}\mathrm{~} | 97 | 13 |
math | 8.3. On the island of knights and liars, each resident was asked about each of the others: is he a knight or a liar. In total, 42 answers of "knight" and 48 answers of "liar" were received. What is the maximum number of knights that could have been on the island? Justify your answer. (It is known that knights always tell the truth, while liars always lie.) | 6 | 93 | 1 |
math | 4. Given the three sides of a triangle $a, b, c$ are integers, and $a+b+c=11$. Then when the product $abc$ takes the minimum value, the area of the triangle is $\qquad$ . | \frac{3 \sqrt{11}}{4} | 51 | 13 |
math | 2. According to analysts' forecasts, next year the number of the economically active population (employed and unemployed) in a certain city will increase by $4 \%$, while the number of unemployed will decrease by $9 \%$. What percentage of the economically active population next year will be unemployed, if this year they made up $5.6 \%$? | 4.9 | 71 | 3 |
math | Example 7 Let the set $M=\{1,2, \cdots, 1000\}$, and for any non-empty subset $X$ of $M$, let $a_{x}$ denote the sum of the largest and smallest numbers in $X$. Then, the arithmetic mean of all such $a_{x}$ is $\qquad$
(1991, National High School Mathematics Competition) | 1001 | 88 | 4 |
math | 10. (10 points) A rectangular chessboard composed of unit squares of size $m \times n$ (where $m, n$ are positive integers not exceeding 10), has a chess piece placed in the unit square at the bottom-left corner. Two players, A and B, take turns moving the piece. The rules are: move up any number of squares, or move right any number of squares, but you cannot move out of the board or not move at all. The player who cannot make a move loses (i.e., the first player to move the piece to the top-right corner wins). How many pairs of positive integers $(m, n)$ allow the first player A to have a winning strategy? $\qquad$ | 90 | 153 | 2 |
math | 12.3. Express the area of triangle $ABC$ in terms of the length of side $BC$ and the measures of angles $B$ and $C$. | ^{2}\sin\beta\sin\gamma/2\sin(\beta+\gamma) | 35 | 19 |
math | Four, (50 points) Find all integers $a$, such that there exist distinct positive integers $x, y$, satisfying $(a x y+1) \mid\left(a x^{2}+1\right)^{2}$.
| \geqslant-1 | 51 | 7 |
math | What is the remainder, in base $10$, when $24_7 + 364_7 + 43_7 + 12_7 + 3_7 + 1_7$ is divided by $6$?
| 3 | 54 | 1 |
math | In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ? | 2 | 113 | 1 |
math | For each positive integer $n$, let $g(n)$ be the sum of the digits when $n$ is written in binary. For how many positive integers $n$, where $1\leq n\leq 2007$, is $g(n)\geq 3$? | 1941 | 63 | 4 |
math | 9. (3 points) A natural number that can only be divided by 1 and itself is called a prime number, such as: $2,3,5,7$, etc. Then, the prime number greater than 40 and less than 50 is $\qquad$, and the largest prime number less than 100 is $\qquad$. | 41,43,47,97 | 76 | 11 |
math | 12.119. A sphere is inscribed in a cone. The radius of the circle where the cone and the sphere touch is $r$. Find the volume of the cone if the angle between the height and the slant height of the cone is $\alpha$. | \frac{\pir^{3}\operatorname{ctg}^{3}(\frac{\pi}{4}-\frac{\alpha}{2})}{3\cos^{2}\alpha\sin\alpha} | 56 | 43 |
math | 4. In a certain store, 2400 kilograms of flour were ordered. They planned to repackage it into 5 kg bags and sell each bag for 14 kn. When the goods arrived, they noticed that 300 kg of flour had been damaged during transport. By how much should the price of the 5 kg packaging be increased so that the planned profit remains the same despite the reduced amount of flour? | 2 | 89 | 1 |
math | Let $ABC$ be a triangle with $AB=10$, $AC=11$, and circumradius $6$. Points $D$ and $E$ are located on the circumcircle of $\triangle ABC$ such that $\triangle ADE$ is equilateral. Line segments $\overline{DE}$ and $\overline{BC}$ intersect at $X$. Find $\tfrac{BX}{XC}$. | \frac{8}{13} | 89 | 8 |
math | Find all positive integers $x,y,z$ with $z$ odd, which satisfy the equation:
$$2018^x=100^y + 1918^z$$ | x = y = z = 1 | 42 | 9 |
math | (IMO 2016, problem 5)
We write on the board the equality:
$$
(x-1)(x-2) \ldots(x-2016)=(x-1)(x-2) \ldots(x-2016)
$$
We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimum number of factors that must be erased to achieve this?
## - Solutions - | 2016 | 107 | 4 |
math | ## Task 34/74
Choose a natural number $n$ with at least two digits, whose decimal representation contains no zero. By arbitrarily swapping the digits in it, a second natural number $n^{\prime}$ is obtained.
If one digit is erased from the difference $n-n^{\prime}$, the erased digit can be determined from the sum of the remaining digits. How is this possible? | 9m-S | 85 | 3 |
math | Evaluate the sum
$$
\frac{1}{1+\tan 1^{\circ}}+\frac{1}{1+\tan 2^{\circ}}+\frac{1}{1+\tan 3^{\circ}}+\cdots+\frac{1}{1+\tan 89^{\circ}}
$$
(The tangent $(\tan )$ of an angle $\alpha$ is the ratio $B C / A C$ in a right triangle $A B C$ with $\angle C=90^{\circ}$ and $\angle A=\alpha$, and its value does not depend on the triangle used.) | \frac{89}{2} | 129 | 8 |
math | 8.309. $\operatorname{tg}\left(t^{2}-t\right) \operatorname{ctg} 2=1$.
8.309. $\tan\left(t^{2}-t\right) \cot 2=1$. | t_{1,2}=\frac{1\\sqrt{9+4\pik}}{2},wherek=0;1;2;\ldots | 59 | 34 |
math | Let $n$ be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set $S$ containing $m$ integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing $i\in\{1,2,\dots, n\}$ that has not been chosen and setting the value of $v_i$ to either $0$ or $1$. At the end of the game, when all of $v_1,v_2,\dots,v_n$ have been set, the expression $$E=v_1\cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1}$$ is calculated. Determine the minimum $m$ such that Alice can always ensure that $E\in S$ regardless of how Bob plays. | m = 2^{\frac{n}{2}} | 187 | 12 |
math | 5. Let $a, b \in \mathbf{R}$, the equation $\left(x^{2}-a x+1\right)\left(x^{2}-b x+1\right)=0$ has 4 real roots that form a geometric sequence with common ratio $q$. If $q \in\left[\frac{1}{3}, 2\right]$, then the range of $a b$ is $\qquad$. | [4,\frac{112}{9}] | 94 | 11 |
math | 12. The five digits $1,1,2,2,3$ can form $\qquad$ four-digit numbers. | 30 | 27 | 2 |
math | Let $S$ be a subset of $\{1,2, \ldots, 9\}$, such that the sums formed by adding each unordered pair of distinct numbers from $S$ are all different. For example, the subset $\{1,2,3,5\}$ has this property, but $\{1,2,3,4,5\}$ does not, since the pairs $\{1,4\}$ and $\{2,3\}$ have the same sum, namely 5.
What is the maximum number of elements that $S$ can contain? | 5 | 122 | 1 |
math | 2. The product of five numbers is not equal to zero. Each of these numbers was decreased by one, yet their product remained unchanged. Provide an example of such numbers. | -\frac{1}{15} | 35 | 8 |
math | $\left[\begin{array}{l}\text { Arithmetic. Mental calculation, etc. } \\ {[\underline{\text { Rebus }}]}\end{array}\right.$
Authors: Galnierein G.A., Grieorenko D.:
2002 is a palindrome year, meaning it reads the same backward as forward. The previous palindrome year was 11 years earlier (1991). What is the maximum number of consecutive non-palindrome years that can occur (between 1000 and 9999 years)? | 109 | 118 | 3 |
math | 10.12 A natural number minus 45 is a perfect square, and this natural number plus 44 is also a perfect square. Try to find this natural number.
(China Beijing Junior High School Grade 3 Mathematics Competition, 1981) | 1981 | 56 | 4 |
math | 7.50 Given that $n$ planes divide a cube into 300 parts, find the minimum value of $n$.
| 13 | 29 | 2 |
math | 5. In an arbitrary triangular pyramid $A B C D$, a section is made by a plane intersecting the edges $A B, D C$, and $D B$ at points $M, N, P$ respectively. Point $M$ divides edge $A B$ in the ratio $A M: M B=2: 3$. Point $N$ divides edge $D C$ in the ratio $D N: N C=3: 5$. Point $P$ divides edge $D B$ in the ratio $D P: P B=4$. Find the ratio $A Q: Q C$. | AQ:QC=1:10 | 129 | 8 |
math | Find all integers $n$ such that $20n+2$ divides $2003n+2002$.
The original text has been translated into English while preserving the original line breaks and format. | 0 \text{ or } -42 | 46 | 9 |
math | 3. The class mini bookshelf has a total of 12 science popularization books. According to statistics, each member of the math group has borrowed exactly two of them, and each science popularization book has been borrowed by exactly 3 members of the math group. How many people are there in this math group? | 18 | 65 | 2 |
math | Example 1. Find the domain of convergence of the series
$$
\sum_{n=1}^{\infty} \frac{n^{3}}{\left(n^{2}+\sqrt{n}+1\right)^{x+1}}
$$ | (1,\infty) | 54 | 6 |
math | 5. Each pair of numbers $A$ and $B$ is assigned a number $A * B$. Find $2021 * 1999$, if it is known that for any three numbers $A, B, C$ the identities are satisfied: $A * A=0$ and $A *(B * C)=(A * B)+C$. | 22 | 77 | 2 |
math | 1. $[\mathbf{5}]$ Let $x$ and $y$ be complex numbers such that $x+y=\sqrt{20}$ and $x^{2}+y^{2}=15$. Compute $|x-y|$. | \sqrt{10} | 53 | 6 |
math | Problem 4. In triangle $A B C$ with the ratio of sides $A B: A C=7: 2$, the bisector of angle $B A C$ intersects side $B C$ at point $L$. Find the length of segment $A L$, if the length of the vector $2 \cdot \overrightarrow{A B}+7 \cdot \overrightarrow{A C}$ is 2016. | 224 | 93 | 3 |
math | The set $\{1,2, \ldots, 100\}$ has a subset $H$ with the property that the tenfold of any element in $H$ is not in $H$. What is the maximum number of elements $H$ can have? | 91 | 57 | 2 |
math | II. (40 points) Let $p$ be a prime number, and the sequence $\left\{a_{n}\right\}(n \geqslant 0)$ satisfies $a_{0}=0, a_{1}=1$, and for any non-negative integer $n$, $a_{n+2}=2 a_{n+1}-p a_{n}$. If -1 is a term in the sequence $\left\{a_{n}\right\}$, find all possible values of $p$.
保留了原文的换行和格式。 | 5 | 121 | 1 |
math | 1. Given the length of the major axis of an ellipse $\left|A_{1} A_{2}\right|=6$, and the focal distance $\left|F_{1} F_{2}\right|$ $=4 \sqrt{2}$, a line is drawn through the left focus $F_{1}$ intersecting the ellipse at points $M$ and $N$. Let $\angle F_{2} F_{1} M=\alpha(0 \leqslant \alpha<\pi)$. When $|M N|$ equals the length of the minor axis of the ellipse, $\alpha=$ $\qquad$ . | \alpha=\frac{\pi}{6} \text{ or } \frac{5 \pi}{6} | 132 | 23 |
math | ## 9. Distribution of Euros
Ana, Janko, and Tara have certain amounts of euros and want to redistribute them among themselves. First, Ana gives Janko and Tara a portion of her money so that after this, both Janko and Tara have twice as much money as they had before. Then, Janko gives Ana and Tara a portion of his money so that after this, both Ana and Tara have twice as much money as they had before. Finally, Tara gives Ana and Janko a portion of her money so that after this, both Ana and Janko have twice as much money as they had before. If Tara has 73 euros at the beginning and at the end, how many euros do Ana, Janko, and Tara have in total?
Result: $\quad 511$ | 511 | 165 | 3 |
math | Let $K$ be a point on the angle bisector, such that $\angle BKL=\angle KBL=30^\circ$. The lines $AB$ and $CK$ intersect in point $M$ and lines $AC$ and $BK$ intersect in point $N$. Determine $\angle AMN$. | 90^\circ | 66 | 4 |
math | Let's assume that the birth of a girl and a boy is equally likely. It is known that a certain family has two children.
a) What is the probability that one is a boy and one is a girl?
b) It is additionally known that one of the children is a boy. What is the probability now that there is one boy and one girl in the family?
c) It is additionally known that the boy was born on a Monday. What is the probability now that there is one boy and one girl in the family? | \frac{1}{2};\frac{2}{3};\frac{14}{27} | 108 | 23 |
math | 5. Find all subsequences $\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ of the sequence $\{1,2, \cdots, n\}$, such that for $1 \leqslant i \leqslant n$, the property $i+1 \mid 2\left(a_{1}+a_{2}+\cdots+a_{i}\right)$ holds. | (1,2,\cdots,n)or(2,1,3,4,\cdots,n) | 96 | 23 |
math | 10.3. In a bag, there are chips of two colors, blue and green. The probability of drawing two chips of the same color is equal to the probability of drawing two chips of different colors (drawing one chip after another without replacement). It is known that there are more blue chips than green chips, and the total number of chips is more than 2 but less than 50. How many blue chips can there be in the bag? Find all answers. | 3,6,10,15,21,28 | 97 | 15 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{1-2+3-4+\ldots+(2 n-1)-2 n}{\sqrt[3]{n^{3}+2 n+2}}
$$ | -1 | 63 | 2 |
math | 907. Find the correlation function of a stationary random function, given its spectral density $\delta_{x}(\omega)=D \alpha / \pi\left(\alpha^{2}+\omega^{2}\right)$. | k_{x}(\tau)=De^{-\alpha|\tau|} | 47 | 15 |
math | 5.22. Let
$$
f(x)=\left\{\begin{array}{l}
e^{3 x}, \text { if } x<0, \\
a+5 x, \text { if } x \geqslant 0
\end{array}\right.
$$
For which choice of the number $a$ will the function $f(x)$ be continuous? | 1 | 84 | 1 |
math | 53rd Putnam 1992 Problem A6 Four points are chosen independently and at random on the surface of a sphere (using the uniform distribution). What is the probability that the center of the sphere lies inside the resulting tetrahedron? Solution | \frac{1}{8} | 53 | 7 |
math | 2. Given $\boldsymbol{a}=\left(\cos \frac{2}{3} \pi, \sin \frac{2}{3} \pi\right), \overrightarrow{O A}=\boldsymbol{a}-$ $\boldsymbol{b}, \overrightarrow{O B}=\boldsymbol{a}+\boldsymbol{b}$, if $\triangle O A B$ is an isosceles right triangle with $O$ as the right-angle vertex, then the area of $\triangle O A B$ is $\qquad$ | 1 | 117 | 1 |
math | A certain number leaves a remainder of 1 when divided by 3, leaves a remainder of 2 when divided by 4, leaves a remainder of 3 when divided by 5, and leaves a remainder of 4 when divided by 6. What is the smallest positive integer that satisfies these properties? | 58 | 63 | 2 |
math | Oh no! While playing Mario Party, Theo has landed inside the Bowser Zone. If his next roll is between $1$ and $5$ inclusive, Bowser will shoot his ``Zero Flame" that sets a player's coin and star counts to zero. Fortunately, Theo has a double dice block, which lets him roll two fair $10$-sided dice labeled $1$-$10$ and take the sum of the rolls as his "roll". If he uses his double dice block, what is the probability he escapes the Bowser zone without losing his coins and stars?
[i]Proposed by Connor Gordon[/i] | \frac{9}{10} | 131 | 8 |
math | 11.2. Find the greatest and the least value of the function $y=(\arcsin x) \cdot(\arccos x)$. | Themaximumvalue=\frac{\pi^2}{16}(atx=\frac{\sqrt{2}}{2}),theminimumvalue=-\frac{\pi^2}{2}(atx=-1). | 33 | 45 |
math | ## Task $3 / 83$
For which natural numbers $n$ does the area $A_{2 n}$ of the regular $2 n$-gon equal twice the area $A_{n}$ of the regular $n$-gon with the same circumradius? | 3 | 57 | 1 |
math | 20. Let $m=76^{2006}-76$. Find the remainder when $m$ is divided by 100 . | 0 | 33 | 1 |
math | 11.1. When composing the options for the district mathematics olympiad for grades $7, 8, 9, 10, 11$, the jury aims to ensure that in the option for each grade, there are exactly 7 problems, of which exactly 4 do not appear in any other option. What is the maximum number of problems that can be included in the olympiad? | 27 | 85 | 2 |
math | (12) The complex number $z$ satisfies $|z|(3 z+2 \mathrm{i})=2(\mathrm{i} z-6)$, then $|z|=$ | 2 | 40 | 1 |
math | A group of boys and girls waits at a bus stop. In the first bus that passes, only 15 girls board, and 2 boys remain for each girl at the bus stop. In the second bus that passes, only 45 boys board, and 5 girls remain for each boy at the bus stop. Determine the number of boys and girls that were at the stop before the first bus arrived. | M=40,H=50 | 84 | 8 |
math | 1. Let $x, y \in \mathbf{R}$, if $2 x, 1, y-1$ form an arithmetic sequence, and $y+3,|x+1|+|x-1|, \cos (\arccos x)$ form a geometric sequence, then the value of $(x+1)(y+1)$ is $\qquad$. | 4 | 82 | 1 |
math | 2. If $n$ is a positive integer, $a(n)$ is the smallest positive integer such that $(a(n))!$ is divisible by $n$. Find all positive integers $n$ such that $\frac{a(n)}{n}=\frac{2}{3}$. (2003 German Mathematical Olympiad) | 9 | 70 | 1 |
math | For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.$ | \frac{\pi}{2} | 139 | 8 |
math | Find all integers $n \in \mathbb{N}$ such that $(n+1) \mid\left(n^{2}+1\right)$. | 01 | 34 | 2 |
math | 6・151 Let $k$ be a positive integer, find all polynomials
$$
p(x)=a_{0}+a_{1} x+\cdots \cdot+a_{n} x^{n},
$$
such that $a_{i}$ are real numbers $(i=0,1,2, \cdots, n)$, and satisfy the equation
$$
p(p(x))=[p(x)]^{k} \text {. }
$$ | p(x)=x^{k}\quad(k\geqslant1) | 98 | 16 |
math | 13.355. A car, having traveled a distance from $A$ to $B$, equal to 300 km, turned back and after 1 hour 12 minutes from leaving $B$, increased its speed by 16 km/h. As a result, it spent 48 minutes less on the return trip than on the trip from $A$ to $B$. Find the original speed of the car. | 60 | 91 | 2 |
math | 14. For the quadratic equation in $x$, $x^{2}+z_{1} x+z_{2}+m=0$, where $z_{1}, z_{2}, m$ are all complex numbers, and $z_{1}^{2}-4 z_{2}=$ $16+20 \mathrm{i}$. Let the two roots of this equation be $\alpha, \beta$, and $\alpha, \beta$ satisfy $|\alpha-\beta|=2 \sqrt{7}$. Find the maximum and minimum values of $|m|$. | ||_{\max}=\sqrt{41}+7;||_{\}=7-\sqrt{41} | 121 | 25 |
math | 123. In a regular triangular pyramid $S A B C$ ( $S$ - vertex), point $E$ is the midpoint of the apothem of face $S B C$, and points $F, L$, and $M$ lie on edges $A B, A C$, and $S C$ respectively, such that $|A L|=\frac{1}{10}|A C|$. It is known that $\boldsymbol{E F L M}$ is an isosceles trapezoid and the length of its base $\boldsymbol{E} \boldsymbol{F}$ is $\sqrt{7}$. Find the volume of the pyramid. | \frac{16}{3}\sqrt{2} | 143 | 12 |
math | Determine the set of all polynomials $P(x)$ with real coefficients such that the set $\{P(n) | n \in \mathbb{Z}\}$ contains all integers, except possibly finitely many of them. | P(x) = \frac{x + c}{k} | 48 | 13 |
math | Find all integers $a, b, c, d$ such that $$\begin{cases} ab - 2cd = 3 \\ ac + bd = 1\end{cases}$$ | (1, 3, 1, 0), (-1, -3, -1, 0), (3, 1, 0, 1), (-3, -1, 0, -1) | 41 | 49 |
math | Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $n/2$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processed. Shirley wants to create the longest possible output sequence possible with initial input at most $100$. What number should she input? | 67 | 98 | 2 |
math | ## Task 30/79
Solve the following system of Diophantine equations
$$
\begin{array}{r}
135 x_{1}+100 x_{2}-x_{3}=-4 \\
97 x_{1}+132 x_{2}-x_{4}=20 \\
7 x_{1}+193 x_{2}-x_{4}=0
\end{array}
$$
with the conditions $x_{1} ; x_{2} ; x_{3} ; x_{4} \in N, x_{2} \leq 30$ and form the triples $\left(x_{1} ; x_{2}, x_{3}\right)$ and $\left(x_{1} ; x_{2}, x_{4}\right)$. | (7,10,1949)(7,10,1979) | 176 | 21 |
math | How many unordered triples $A,B,C$ of distinct lattice points in $0\leq x,y\leq4$ have the property that $2[ABC]$ is an integer divisible by $5$?
[i]2020 CCA Math Bonanza Tiebreaker Round #3[/i] | 300 | 63 | 3 |
math | 206. Распределение орехов. Тетушка Марта купила орехов. Томми она дала один орех и четверть оставшихся, и Бесси получила один орех и четверть оставшихся, Боб тоже получил один орех и четверть оставшихся, и, наконец, Джесси получила один орех и четверть оставшихся. Оказалось, что мальчики получили на 100 орехов больше, чем девочки.
Сколько орехов тетушка Марта оставила себе | 321 | 150 | 3 |
math | 9. (This sub-question is worth 15 points) Let $a_{n}=2^{n}, n \in \mathbf{N}^{+}$, and the sequence $\left\{b_{n}\right\}$ satisfies $b_{1} a_{n}+b_{2} a_{n-1}+\cdots+b_{n} a_{1}=2^{n}-\frac{n}{2}-1$. Find the sum of the first $n$ terms of the sequence $\left\{a_{n} \cdot b_{n}\right\}$. | \frac{(n-1)\cdot2^{n}+1}{2} | 125 | 17 |
math | 7. Let $f(x)=\frac{\sin \pi x}{x^{2}}(x \in(0,1))$. Then the minimum value of $g(x)=f(x)+f(1-x)$ is . $\qquad$ | 8 | 52 | 1 |
math | 4. In triangle $A B C$, $B C=4, C A=5, A B=6$, then $\sin ^{6} \frac{A}{2}+\cos ^{6} \frac{A}{2}=$ $\qquad$ | \frac{43}{64} | 57 | 9 |
math | Rectangle $ABCD$ has side lengths $AB = 6\sqrt3$ and $BC = 8\sqrt3$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 17 | 84 | 4 |
math | 20 On the parabola $y^{2}=2 p x$, does there exist a point $M$ such that the ratio of the distance from $M$ to the vertex of the parabola and the distance from $M$ to the focus of the parabola is maximized? If the point $M$ exists, find its coordinates and the aforementioned maximum ratio; if the point $M$ does not exist, please explain the reason. | \frac{2}{\sqrt{3}} | 94 | 10 |
math | [ Volume of a parallelepiped ]
The base of an oblique prism is a parallelogram with sides 3 and 6 and an acute angle of $45^{\circ}$. The lateral edge of the prism is 4 and is inclined to the base plane at an angle of $30^{\circ}$. Find the volume of the prism. | 18\sqrt{2} | 75 | 7 |
math | [b]8.[/b] Find all integers $a>1$ for which the least (integer) solution $n$ of the congruence $a^{n} \equiv 1 \pmod{p}$ differs from 6 (p is any prime number). [b](N. 9)[/b] | a = 2 | 68 | 4 |
math | 14. Let $n$ be a positive integer, and define: $f_{n}(x)=\underbrace{f\{f[\cdots f}_{n \uparrow f}(x) \cdots]\}$, given:
$$
f(x)=\left\{\begin{array}{ll}
2(1-x), & (0 \leqslant x \leqslant 1), \\
x-1, & (1<x \leqslant 2) .
\end{array}\right.
$$
(1) Solve the inequality: $f(x) \leqslant x$;
(2) Let the set $A=\{0,1,2\}$, for any $x \in A$, prove: $f_{3}(x)=x$;
(3) Find the value of $f_{2007}\left(\frac{8}{9}\right)$;
(4) If the set $B=\left\{x \mid f_{12}(x)=x, x \in[0,2]\right\}$, prove: $B$ contains at least 8 elements. | {x\lvert\,\frac{2}{3}\leqslantx\leqslant2} | 246 | 24 |
math | 6. Let the real number $a$ and the function $f(x)$ satisfy: $f(1)=1$, and for any $x \in \mathbf{R} \backslash\{0\}$, $f\left(\frac{1}{x}\right)=a f(x)-x$. Let $A=\{x|x \in \mathbf{R} \backslash\{0\}| f,(x) \mid \leqslant 1\}$. Then the range of the function $g(x)=f(a x) f\left(\frac{1}{x}\right)(x \in A)$ is $\qquad$ | [\frac{25}{18},\frac{3}{2}] | 141 | 16 |
math | 7. Given complex numbers $z_{1}, z_{2}, z_{3}$ satisfy
$$
\begin{array}{l}
\left|z_{1}\right| \leqslant 1,\left|z_{2}\right| \leqslant 1, \\
\left|2 z_{3}-\left(z_{1}+z_{2}\right)\right| \leqslant\left|z_{1}-z_{2}\right| .
\end{array}
$$
Then the maximum value of $\left|z_{3}\right|$ is | \sqrt{2} | 127 | 5 |
math | 3. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0)=0, f(1)=1$, and $\left|f^{\prime}(x)\right| \leq 2$ for all real numbers $x$. If $a$ and $b$ are real numbers such that the set of possible values of $\int_{0}^{1} f(x) d x$ is the open interval $(a, b)$, determine $b-a$. | \frac{3}{4} | 114 | 7 |
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