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 |
|---|---|---|---|---|
math | For any set $A=\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}$ of four distinct positive integers with sum $s_{A}=a_{1}+a_{2}+a_{3}+a_{4}$, let $p_{A}$ denote the number of pairs $(i, j)$ with $1 \leq i<j \leq 4$ for which $a_{i}+a_{j}$ divides $s_{A}$. Among all sets of four distinct positive integers, determine those sets... | \{d, 5d, 7d, 11d\} \text{ and } \{d, 11d, 19d, 29d\} | 134 | 43 |
math | Example 8 Find the range of the function $y=x+\sqrt{x^{2}-3 x+2}$.
(2001 National High School Mathematics Competition) | [1,\frac{3}{2})\cup[2,+\infty) | 35 | 18 |
math | [ Measurement of segment lengths and angle measures. Adjacent angles.]
On a straight line, points $A, B$, and $C$ are given. It is known that $A B=5$, and segment $A C$ is 1 unit longer than $B C$. Find $A C$ and $B C$.
# | BC=2,AC=3 | 69 | 7 |
math | 1. Determine the digits $x, y, z$ such that the equation
$$
\frac{x+y}{z}=\overline{z, y x}
$$
holds, where $\overline{z, y x}$ denotes the number composed of $z$ units, $y$ tenths, and $x$ hundredths. | 0,5,2 | 72 | 5 |
math | $12.3 \lim _{x \rightarrow-1} \frac{\sqrt{2 x+3}-1}{\sqrt{5+x}-2}$.
$12.3 \lim _{x \rightarrow-1} \frac{\sqrt{2 x+3}-1}{\sqrt{5+x}-2}$.
The above text is already in a mathematical expression format, so the translation is the same as the original text. If you need an explanation or solution for the limit, please let ... | 4 | 111 | 1 |
math | Extend 1 Let non-negative real numbers $a, b, c, x, y, z$ satisfy $a+b+c=x+y+z=1$.
Find the minimum value of $\left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right)$. | -\frac{1}{4} | 64 | 7 |
math | Example 1 Let $x, y, z > 0$, and $x+y+z=1$. Find
$$
f(x, y, z)=\sum \frac{x(2 y-z)}{1+x+3 y}
$$
the maximum value. | \frac{1}{7} | 56 | 7 |
math | 10. $a, b, m, n$ satisfy: $a m m^{2001}+b n^{2001}=3 ; a m^{20002}+b n^{2002}=7 ; a m^{2003}+$ $b n^{2003}=24 ; a m^{21004}+b m^{2004}=102$. Then the value of $m^{2}(n-1)$ is $\qquad$ . | 6 | 117 | 1 |
math | 314. Find the derivative of the function $y=\sin \left(x^{3}-3 x^{2}\right)$. | (3x^{2}-6x)\cos(x^{3}-3x^{2}) | 28 | 19 |
math | 2. The maximum value of the function $f(x)=7 \sin x+\sin 2 x$ is $\qquad$ . | \frac{15 \sqrt{15}}{8} | 28 | 14 |
math | Let's determine the maximum possible value of the expression $x^{2} y-y^{2} x$, where $x$ and $y$ independently run through the interval $[0,1]$. $(\mathbf{H})$ | \frac{1}{4} | 50 | 7 |
math | 6. Given that $n, k$ are positive integers, $n>k$. Given real numbers $a_{1}, a_{2}, \cdots, a_{n} \in(k-1, k)$. Let positive real numbers $x_{1}, x_{2}$, $\cdots, x_{n}$ satisfy that for any $k$-element subset $I$ of $\{1,2, \cdots, n\}$, we have $\sum_{i \in I} x_{i} \leqslant \sum_{i \in I} a_{i}$. Find the maximum ... | a_{1}a_{2}\cdotsa_{n} | 150 | 14 |
math | Find the smallest positive value taken by $a^3 + b^3 + c^3 - 3abc$ for positive integers $a$, $b$, $c$ .
Find all $a$, $b$, $c$ which give the smallest value | 4 | 53 | 1 |
math | 7.5. The children went to the forest to pick mushrooms. If Anya gives half of her mushrooms to Vitya, all the children will have the same number of mushrooms, and if instead Anya gives all her mushrooms to Sasha, Sasha will have as many mushrooms as all the others combined. How many children went to pick mushrooms | 6 | 69 | 1 |
math | 3. (5 points) Around a rectangular square that is 500 meters long and 300 meters wide, a pot of flowers is placed every 2.5 meters. Now, it is to be changed to placing a pot of flowers every 2 meters, and the flower pots at the four corners of the square will remain unchanged. Then, the number of additional pots of flo... | 160,160 | 109 | 7 |
math | 20. The variable $a$ can take the values: $-32,-30,-28, \ldots, 2$, $4,6,8$, and the variable $b: -17,-15,-13, \ldots, 11,13$. How many different values can the expression $a+b$ take? Find the product of the largest and smallest values of the expression $a+b$.
76 | -1029 | 97 | 5 |
math | 8. In square $A B C D$, $A B=2, E$ is the midpoint of $A B$, $F$ is a point on side $B C$, let $B F=x$, to make $\triangle A E D, \triangle D C F$ fold up (so that $A, C$ coincide at point $A^{\prime}$) to form a tetrahedron $A^{\prime}-E F D$, then the range of values for $x$ is . $\qquad$ | 0<x<\frac{4}{3} | 110 | 10 |
math | Task 4.
Every day after lunch, 7 half-eaten pieces of white bread are left on the desks of the second-grade students. If these pieces are put together, they make up half a loaf of bread. How many loaves of bread will the second-grade students save in 20 days if they do not leave these pieces? How much money will the s... | 350 | 132 | 3 |
math | ## Task 1 - V00901
The sum of two numbers is 20, and the sum of their squares is 202. Solve the problem mathematically. | 911 | 41 | 3 |
math | How many different lists $a, b, c, d$ of distinct odd positive integers with $a<b<c<d$ have the property that $a+b+c+d=24$ ? | 5 | 39 | 1 |
math | Amy and Bob choose numbers from $0,1,2,\cdots,81$ in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all $82$ numbers are chosen, let $A$ be the sum of all the numbers Amy chooses, and let $B$ be the sum of all the numbers Bob chooses. Durin... | \gcd(A, B) = 41 | 145 | 11 |
math | Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$.
Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized. | a = \sqrt{3}, b = 0 | 66 | 12 |
math | 20. (5 points) Person A and Person B start from points $A$ and $B$ respectively at the same time. If they walk in the same direction, A catches up with B in 30 minutes; if they walk towards each other, they meet in 6 minutes. It is known that B walks 50 meters per minute, then the distance between $A$ and $B$ is $\qqua... | 750 | 92 | 3 |
math | 5. For a line segment $A B$ of fixed length 3, whose endpoints move on the parabola $y^{2}=x$, the midpoint of segment $A B$ is $M$. Try to find the shortest distance from point $M$ to the $y$-axis. | \frac{5}{4} | 62 | 7 |
math | At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different? | 5 | 34 | 1 |
math | Given $AB = 2a$ is a line segment with midpoint $C$. We draw a line $XY$ through $C$ that makes a $60^{\circ}$ angle with $CB$. Choose a point $P$ on $XY$ such that the ratio of the lateral areas of the cones formed by the rotation of $PA$ and $PB$ around $XY$ is equal to a given value $m$. For which values of $m$ is t... | \frac{\sqrt{3}}{3}\leqq\leqq\sqrt{3} | 101 | 20 |
math | How many integers $n$ are there subject to the constraint that $1 \leq n \leq 2020$ and $n^n$ is a perfect square? | 1032 | 39 | 4 |
math | On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number. | 8 | 83 | 1 |
math | 2. In the tetrahedron $A B C D$ with all edges congruent, let $D O \perp(A B C), O \in(A B C)$. The point $M$ is the projection of point $O$ onto the edge $[D B]$, and $M C=2 \sqrt{7} \mathrm{~cm}$. Calculate the value of the sine of the angle between the line $M C$ and the plane (BOD).
Testarea Naţională, 2007 | \frac{3\sqrt{7}}{14} | 114 | 13 |
math | I1.4 If $\log _{2} a+\log _{2} b \geq \gamma$, determine the smallest positive value $\delta$ for $a+b$. | 16 | 38 | 2 |
math | 129. A body moves in a straight line with acceleration $a=6 t-4$. At $t=0$, the initial path $s_{0}=0$, initial velocity $v_{0}=4$. Find the velocity and the distance traveled as functions of time. | v=3^{2}-4,=^{3}-2^{2}+4 | 58 | 18 |
math | 9. Let $y=f(x)$ be an odd function on $(-\infty,+\infty)$, $f(x+2)=-f(x)$, and when $-1 \leqslant x \leqslant 1$, $f(x)=x^{3}$.
(1) Find the analytical expression of $f(x)$ when $x \in[1,5]$;
(2) If $A=\{x \mid f(x)>a, x \in \mathbf{R}\}$, and $A \neq \varnothing$, find the range of real number $a$. | <1 | 132 | 2 |
math | 12. (12 points) Nine cards are labeled with the numbers $2,3,4,5,6,7,8,9,10$ (they cannot be read upside down). Four people, A, B, C, and D, each draw two of these cards.
A says: "The two numbers I got are coprime, because they are consecutive"
B says: "The two numbers I got are not coprime, and they are not multiples ... | 7 | 169 | 1 |
math | The roots of $x^{2}+b x+c=0$ are the squares of the roots of $x^{2}-5 x+2=0$. What is the value of $\frac{c}{b}$ ? | -\frac{4}{21} | 47 | 8 |
math | 20. (2002 Shanghai Spring College Entrance Examination) Triangular prism $O A B-O A_{1} B_{1}$, plane $O B B_{1} O_{1} \perp O A B, \angle O_{1} O B=$ $60^{\circ}, \angle A O B=90^{\circ}$, and $O B=O O_{1}=2, O A=\sqrt{3}$. Find:
(1) The size of the dihedral angle $O_{1}-A B-O$;
(2) The distance $d$ between the skew l... | \sqrt{3} | 155 | 5 |
math | 5. The edges of the tetrahedron $ABCD$ have lengths 7, 13, 18, 27, 36, and 41 (in some order). If $AB$ has a length of 41, determine the length of the edge $CD$.
## Fourth grade - B category | 13 | 72 | 2 |
math | Find all positive integers $n$ for which $(x^n+y^n+z^n)/2$ is a perfect square whenever $x$, $y$, and $z$ are integers such that $x+y+z=0$. | n = 1, 4 | 46 | 7 |
math | ## Task Condition
Find the derivative.
$$
y=\ln \left(2 x-3+\sqrt{4 x^{2}-12 x+10}\right)-\sqrt{4 x^{2}-12 x+10} \cdot \operatorname{arctg}(2 x-3)
$$ | -\frac{\operatorname{arctg}(2x-3)}{\sqrt{4x^{2}-12x+10}} | 68 | 30 |
math | Let $ a,\ b$ be postive real numbers. For a real number $ t$, denote by $d(t)$ the distance between the origin and the line $ (ae^t)x \plus{} (be^{ \minus{} t})y \equal{} 1$.
Let $ a,\ b$ vary with $ ab \equal{} 1$, find the minimum value of $ \int_0^1 \frac {1}{d(t)^2}\ dt$. | e - \frac{1}{e} | 98 | 9 |
math | 3. In the arithmetic progression $\left(a_{n}\right) a_{1000}=150, d=0.5$.
Calculate: $99 \cdot 100 \cdot\left(\frac{1}{a_{1580} \cdot a_{1581}}+\frac{1}{a_{1581} \cdot a_{1582}}+\ldots+\frac{1}{a_{2019} \cdot a_{2020}}\right)$. | 15 | 117 | 2 |
math | 1. Let $A$ and $B$ be two moving points on the ellipse $\frac{x^{2}}{2}+y^{2}=1$, and $O$ be the origin. Also, $\overrightarrow{O A} \cdot \overrightarrow{O B}=0$. Let point $P$ be on $AB$, and $O P \perp A B$. Find the value of $|O P|$. | \frac{\sqrt{6}}{3} | 91 | 10 |
math | 9.2. In the basket, there are oranges and bananas. If you add as many oranges as there are currently bananas (in pieces), then the percentage of oranges
will be twice as much as it would be if you added as many bananas as there are currently oranges. What is the current percentage of oranges in the basket? | 50 | 67 | 2 |
math | Example 6 If $x=\frac{\sqrt{5}-1}{2}$, then $x^{4}+x^{2}+2 x-$
$$
1=
$$ | 3-\sqrt{5} | 39 | 6 |
math | Find all positive integers $n$ and $p$ if $p$ is prime and \[ n^8 - p^5 = n^2+p^2 . \]
[i]Adrian Stoica[/i] | (n, p) = (2, 3) | 47 | 13 |
math | ## Task B-4.4.
Grandpa Ante, looking for a way to entertain his grandchildren Iva and Mato, found three cards. On the first card, one side has the number 1, and the other side has the number 4. On the second card, one side has the number 2, and the other side has the number 4. On the third card, one side has the numbe... | 10434 | 202 | 5 |
math | 3. Given the circle $C: x^{2}+y^{2}=24$, the line $l: \frac{x}{12}+\frac{y}{8}=1$, and point $P$ on $l$, the ray $O P$ intersects the circle at point $R$. Point $Q$ is on $O P$ and satisfies: $|O Q| \cdot|O P|=|O R|^{2}$. When point $P$ moves along $l$, find the equation of the trajectory of point $Q$ and describe what... | (x-1)^{2}+(y-\frac{3}{2})^{2}=\frac{13}{4} | 127 | 27 |
math | 3. Find the number of four-digit numbers in which all digits are different, the first digit is divisible by 2, and the sum of the first and last digits is divisible by 3. | 672 | 40 | 3 |
math | 8,9
Two circles touch each other internally. It is known that two radii of the larger circle, the angle between which is $60^{\circ}$, touch the smaller circle. Find the ratio of the radii of the circles. | \frac{1}{3} | 52 | 7 |
math | 14. (15 points) The teacher gives fruits to students, preparing two types of fruits. The number of oranges is 3 more than 3 times the number of apples. If each student gets 2 apples, there will be 6 apples left; if each student gets 7 oranges, the last student can only get 1 orange. Find the number of students.
| 27 | 78 | 2 |
math | Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer. | n = 3, 20, 119 | 35 | 14 |
math | 8. There is an unlimited number of test tubes of three types - A, B, and C. Each test tube contains one gram of a solution of the same substance. Test tubes of type A contain a $10\%$ solution of this substance, type B $-20\%$ solution, and type C $-90\%$ solution. Sequentially, one after another, the contents of the t... | 73 | 161 | 2 |
math | 3. From two pieces of alloy weighing $6 \mathrm{~kg}$ and $3 \mathrm{~kg}$ with different percentages of copper, pieces of the same weight are cut off. Each of the cut pieces is then fused with the remainder of the other piece. After this fusion, the percentage of copper in both alloys is equal. How much do the cut pie... | 2 | 78 | 1 |
math | 4. What number can the first addend be? Justify your answer.
$$
\begin{array}{r}
\mathrm{XX} 4 \\
+\quad 3 \mathrm{X} \\
\hline \mathrm{XXXX}
\end{array}
$$ | 964,974,984,994 | 57 | 15 |
math | 11. Let $f(x)=x^{2}+a x+b(a, b \in \mathbf{R}), A=\{x \mid f(x)=x, x \in \mathbf{R}\}$, $B=\{x \mid f(f(x))=x, x \in \mathbf{R}\}$. If $A=\{-1,3\}$, then $B=$ $\qquad$ | {-1,\sqrt{3},-\sqrt{3},3} | 93 | 14 |
math | 11. (20 points) Given non-zero complex numbers $x, y$ satisfy $y^{2}\left(x^{2}-x y+y^{2}\right)+x^{3}(x-y)=0$.
Find the value of $\sum_{m=0}^{29} \sum_{n=0}^{29} x^{18 m n} y^{-18 m n}$. | 180 | 88 | 3 |
math | Example 4.3. Investigate the convergence of the series $\sum_{n=0}^{\infty} a^{n}$ (infinite geometric progression) $a \in R$. | S_{n}=\frac{1}{1-}if||<1 | 41 | 16 |
math | 2. For a natural number ending not in zero, one of its digits (not the most significant) was erased. As a result, the number decreased by 9 times. How many numbers exist for which this is possible? | 28 | 46 | 2 |
math | 2.301. $\frac{\sqrt{5-2 \sqrt{6}} \cdot(5+2 \sqrt{6})(49-20 \sqrt{6})}{\sqrt{27}-3 \sqrt{18}+3 \sqrt{12}-\sqrt{8}}=1$. | 1 | 70 | 1 |
math | A rook has traversed the chessboard, visiting each square at least once. What is the minimum number of turns it could have made
# | 14 | 30 | 2 |
math | 55. Let $x, y, z$ be positive numbers, and $(x+y+z)^{3}=32 x y z$, find the maximum and minimum values of $\frac{x^{4}+y^{4}+z^{4}}{(x+y+z)^{4}}$. (2004 Vietnam Mathematical Olympiad) | \frac{383-165 \sqrt{5}}{256}, \frac{9}{128} | 72 | 29 |
math | Let's determine the remainders when these numbers
$$
65^{6 n}, \quad 65^{6 n+1}, \quad 65^{6 n+2}, \quad 65^{6 n+3}
$$
are divided by 9. | 1,2,4,8 | 59 | 7 |
math | Find the greatest value $M$ that the expression $7 x+10 y+z$ can take when $x, y, z$ are real numbers satisfying $x^{2}+2 x+\frac{1}{5} y^{2}+7 z^{2}=6$. In which cases is this value achieved? Go to the puzzle at the cell corresponding to $M$.
Go to the puzzle at the cell corresponding to $M$. | 55 | 93 | 2 |
math | Define the [i]hotel elevator cubic [/i]as the unique cubic polynomial $P$ for which $P(11) = 11$, $P(12) = 12$, $P(13) = 14$, $P(14) = 15$. What is $P(15)$?
[i]Proposed by Evan Chen[/i] | 13 | 84 | 2 |
math | 6. Let the set $M=\{1,2, \cdots, 2020\}$. For any non-empty subset $X$ of $M$, let $\alpha_{X}$ denote the sum of the largest and smallest numbers in $X$. Then the arithmetic mean of all such $\alpha_{X}$ is $\qquad$ . | 2021 | 74 | 4 |
math | 4. Let the two branches of the hyperbola $xy=1$ be $C_{1}$ and $C_{2}$, and let the three vertices of the equilateral triangle $PQR$ lie on this hyperbola.
(1) Prove that $P$, $Q$, and $R$ cannot all lie on the same branch of the hyperbola;
(2) Suppose $P(-1,-1)$ is on $C_{2}$, and $Q$, $R$ are on $C_{1}$. Find the coo... | (2+\sqrt{3}, 2-\sqrt{3}),(2-\sqrt{3}, 2+\sqrt{3}) | 138 | 28 |
math | 4. If the positive integer $a$ makes the maximum value of the function $f(x)=x+\sqrt{13-2 a x}$ also a positive integer, then this maximum value is equal to . $\qquad$ | 7 | 48 | 1 |
math | 3. Find all integers $x$ for which both numbers $(x-3)^{2}-2,(x-7)^{2}+1$ are prime numbers. | {1,5,6,8} | 36 | 9 |
math | Problem 11.6. The polynomial $P(x)$ has all coefficients as non-negative integers. It is known that $P(1)=4$ and $P(5)=152$. What is $P(11) ?$ | 1454 | 52 | 4 |
math | Let $P(x),\ Q(x)$ be polynomials such that :
\[\int_0^2 \{P(x)\}^2dx=14,\ \int_0^2 P(x)dx=4,\ \int_0^2 \{Q(x)\}^2dx=26,\ \int_0^2 Q(x)dx=2.\]
Find the maximum and the minimum value of $\int_0^2 P(x)Q(x)dx$. | [-8, 16] | 103 | 7 |
math | Example 3. Calculate the area of the part of the surface of the paraboloid of revolution $2z = x^2 + y^2$, enclosed within the cylinder $x^2 + y^2 = R^2$. | \frac{2\pi}{3}(\sqrt{(1+R^{2})^{3}}-1) | 49 | 24 |
math | 4.39 Given: $\operatorname{ctg} \alpha=\frac{3}{4}, \operatorname{ctg} \beta=\frac{1}{7}, 0<\alpha<\frac{\pi}{2}, 0<\beta<\frac{\pi}{2}$. Find $\alpha+\beta$. | \frac{3\pi}{4} | 71 | 9 |
math | 2. In $\triangle A B C$, it is known that $A B=4, A C=3, P$ is a point on the perpendicular bisector of side $B C$, then $\overrightarrow{B C} \cdot \overrightarrow{A P}=$ | -\frac{7}{2} | 58 | 7 |
math |
Problem N1. Find all positive integers $n$ such that $n 2^{n+1}+1$ is a perfect square.
| n=0n=3 | 31 | 6 |
math | A A trapezoid is divided into seven strips of equal width as shown. What fraction of the trapezoid's area is shaded? Explain why your answer is correct. | \frac{4}{7} | 37 | 7 |
math | 19 The sequence of positive integers $\left\{a_{n}\right\}$ satisfies: for any positive integers $m, n$, if $m \mid n, m<n$, then $a_{m} \mid a_{n}$, and $a_{m}<a_{n}$. Find the minimum possible value of $a_{2000}$. | 128 | 78 | 3 |
math | Find all functions $f: \mathbb N \cup \{0\} \to \mathbb N\cup \{0\}$ such that $f(1)>0$ and
\[f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2 \quad \forall m,n \in \mathbb N\cup \{0\}.\] | f(n) = n | 84 | 6 |
math | 10. If $a^{3}+b^{3}+c^{3}=3 a b c=6$ and $a^{2}+b^{2}+c^{2}=8$, find the value of $\frac{a b}{a+b}+\frac{b c}{b+c}+\frac{c a}{c+a}$. | -8 | 76 | 2 |
math | You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. (That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$.) At any step, you take an... | 695 | 181 | 3 |
math | The 17th question: Find all positive integers $n$ such that $\left(n^{2}+11 n-4\right) \cdot n!+33 \times 13^{n}+4$ is a perfect square. | 1,2 | 54 | 3 |
math | Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac mn,$ where $m$ and $n$ are relatively prime posi... | 77 | 106 | 2 |
math | Four, in an election, there are 12 candidates, and each member of the electoral committee casts 6 votes. It is known that any two members' votes have at most 2 candidates in common. Find the maximum number of members in the committee.
(Proposed by the Problem Committee) | 4 | 61 | 1 |
math | Problem 1. Let $a, b, c$ be numbers different from zero, such that $b(c+a)$ is the arithmetic mean of the numbers $a(b+c)$ and $c(a+b)$. If $b=\frac{2019}{2020}$, calculate the arithmetic mean of the numbers $\frac{1}{a}, \frac{1}{b}$ and $\frac{1}{c}$. | \frac{2020}{2019} | 91 | 13 |
math | 4. For all complex numbers $z$ satisfying $z \neq \mathrm{i}$, we have $F(z) = \frac{z-\mathrm{i}}{z+\mathrm{i}}$. For all positive integers $n$, $z_{n}=F\left(z_{n-1}\right)$. If $z_{0}=2016+\mathrm{i}$, then $z_{2016}=$ $\qquad$ . | 2016+\mathrm{i} | 96 | 8 |
math | 2. Polynomials $P(x)$ and $Q(x)$ of equal degree are called similar if one can be obtained from the other by permuting the coefficients (for example, the polynomials $2 x^{3}+x+7$ and $x^{3}+2 x^{2}+7 x$ are similar). For what largest $k$ is it true that for any similar polynomials $P(x), Q(x)$, the number $P(2009)-Q(2... | 2008 | 118 | 4 |
math | 11. (This question is worth 20 points) In the plane rectangular coordinate system $x O y$, let $A B$ be a chord of the parabola $y^{2}=4 x$ passing through the point $F(1,0)$, and the circumcircle of $\triangle A O B$ intersects the parabola at point $P$ (different from points $O, A, B$). If $P F$ bisects $\angle A P B... | \sqrt{13}-1 | 115 | 7 |
math | II. (50 points) Let $x_{i} \in \mathbf{R}^{+}(i=1,2, \cdots, n)$, and $x_{1} x_{2} \cdots x_{n}=1, n$ be a given positive integer. Try to find the smallest positive number $\lambda$, such that the inequality $\frac{1}{\sqrt{1+2 x_{1}}}+\frac{1}{\sqrt{1+2 x_{2}}}+\cdots+\frac{1}{\sqrt{1+2 x_{n}}} \leqslant \lambda$ alwa... | \lambda={\begin{pmatrix}\frac{\sqrt{3}n}{3}(n=1,2),\\n-1(n\geqslant3)0\end{pmatrix}.} | 137 | 45 |
math | Which are those positive integers $n$ for which $n^{3}+1$ and $n^{2}-1$ are both divisible by 101? | k\cdot101-1, | 35 | 9 |
math | 4. Problem: Let $p$ be a prime and let $f(x)=a x^{2}+b x+c$ be a quadratic polynomial with integer coefficients such that $0<a, b, c \leqslant p$. Suppose $f(x)$ is divisible by $p$ whenever $x$ is a positive integer. Find all possible values of $a+b+c$. | +b+=3pforallp+b+=4whenp=2 | 80 | 13 |
math | 2. The pond has a square shape. On the first frosty day, the part of the pond within 10 meters of the nearest shore froze. On the second day, the part within 20 meters froze, on the third day, the part within 30 meters, and so on. On the first day, the area of open water decreased by $19 \%$. How long will it take for ... | 10 | 92 | 2 |
math | 9. Given $\sin \alpha+\sin (\alpha+\beta)+\cos (\alpha+\beta)=\sqrt{3}, \beta \in\left[\frac{\pi}{4}, \pi\right]$, find the value of $\beta$.
| \beta=\frac{\pi}{4} | 53 | 9 |
math | Suggestion: Count the even and odd numbers separately.
Tiago writes down all four-digit numbers with non-zero distinct digits that have the same parity. What is the probability that, if we choose one of these numbers, it will be even? | \frac{1}{6} | 49 | 7 |
math | 1. Daria Dmitrievna is preparing a test on number theory. She promised to give each student as many problems as the number of addends they create in the numerical example
$$
a_{1}+a_{2}+\ldots+a_{n}=2021
$$
where all numbers $a_{i}$ are natural numbers, greater than 10, and are palindromes (do not change if their dig... | 3 | 140 | 1 |
math | Given a triangle $ABC$ with the base $a$ and the sum of the other two sides $b+c$. Determine this triangle when the median and the angle bisector from $A$ form the largest possible angle with each other. | =\frac{\sqrt{2}+2(b+)}{4};b=\frac{-\sqrt{2}+2(b+)}{4} | 48 | 32 |
math | 8. If $x>0, y>0$, and $2 \lg (x-2 y)$ $=\lg x+\lg y$, then $\mathrm{x}: \mathrm{y}$ is:
(A) 43
(B) Is
(C) 1 or 43
(D) $\frac{1}{4}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | A | 97 | 1 |
math | Determine all positive integer solutions $x, y$ of the equation $x^{2}-2 \cdot y!=2021$. | (45,2) | 29 | 6 |
math | \section*{Task 2 - 131222}
Each of the 41 students in a class had to participate in exactly three track and field running events.
Each of these students had to start once on lanes 1, 2, and 3.
Student A claims that due to these rules alone, there must be at least seven students in the class whose sequence of startin... | 7 | 114 | 1 |
math | 12.4. Let the matrix $A=\left(\begin{array}{cc}1 & -2 \\ -2 & 1\end{array}\right)$. Determine $A^{2021}$. | (\begin{pmatrix}\frac{1}{2}(3^{2021}-1)&-\frac{1}{2}(3^{2021}+1)\\-\frac{1}{2}(3^{2021}+1)&\frac{1}{2}(3^{2021}-1)\end{pmatrix} | 47 | 77 |
math | 12. If:
(1) $a, b, c, d$ all belong to $\{1,2,3,4\}$;
(2) $a \neq b, b \neq c, c \neq d, d \neq a$;
(3) $a$ is the smallest value among $a, b, c, d$.
Then, the number of different four-digit numbers $\overline{a b c d}$ that can be formed is $\qquad$. | 28 | 109 | 2 |
math | 2. Given $a<b<0$, and $\frac{a}{b}+\frac{b}{a}=6$. Then $\left(\frac{a+b}{a-b}\right)^{3}=$ $\qquad$ | 2 \sqrt{2} | 48 | 6 |
math | A line $g$ is given in a plane. $n$ distinct points are chosen arbitrarily from $g$ and are named as $A_1, A_2, \ldots, A_n$. For each pair of points $A_i,A_j$, a semicircle is drawn with $A_i$ and $A_j$ as its endpoints. All semicircles lie on the same side of $g$. Determine the maximum number of points (which are not... | \binom{n}{4} | 116 | 8 |
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