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 | 8. Given $I$ is the incenter of $\triangle A B C$, and $5 \overrightarrow{I A}=4(\overrightarrow{B I}+\overrightarrow{C I})$. Let $R, r$ be the circumradius and inradius of $\triangle A B C$, respectively. If $r=15$, then $R=$ $\qquad$ . | 32 | 81 | 2 |
math | 189. a) $\log _{2} 16$
b) $\log _{6} 36$
c) $\log _{8} 1$. | 4,2,0 | 39 | 5 |
math | 4. Find all positive integers $n$ such that $n(n+2)(n+4)$ has at most 15 positive divisors.
(2010, Slovenia National Team Selection Exam) | n=1,2,3,4,5,7,9 | 43 | 15 |
math | 12. Chris planned a $210 \mathrm{~km}$ bike ride. However, he rode $5 \mathrm{~km} / \mathrm{h}$ faster than he planned and finished his ride 1 hour earlier than he planned. His average speed for the ride was $x \mathrm{~km} / \mathrm{h}$. What is the value of $x$ ? | 35 | 84 | 2 |
math | ## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$f(x)=\left\{\begin{array}{c}\tan\left(2^{x^{2} \cos (1 /(8 x))}-1+x\right), x \neq 0 ; \\ 0, x=0\end{array}\right.$ | 1 | 80 | 1 |
math | 11. If the equation about $x$
$$
x^{2}-\left(a^{2}+b^{2}-6 b\right) x+a^{2}+b^{2}+2 a-4 b+1=0
$$
has two real roots $x_{1}, x_{2}$ satisfying $x_{1} \leqslant 0 \leqslant x_{2} \leqslant 1$, then the sum of the minimum and maximum values of $a^{2}+b^{2}+4 a+4$ is $\qquad$ | 9 \frac{1}{2}+4 \sqrt{5} | 129 | 15 |
math | Example 2 Factorize $f(x, y, z)=x^{3}(y-z)+y^{3}(z-x)+z^{3}(x-y)$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
Example 2 Factorize $f(x, y, z)=x^{3}(y-z)+y^{3}(z-x)+z^{3}(x-y)$. | f(x,y,z)=-(x-y)(y-z)(z-x)(x+y+z) | 97 | 19 |
math | 16. Find the number of ways to arrange the letters $\mathrm{A}, \mathrm{A}, \mathrm{B}, \mathrm{B}, \mathrm{C}, \mathrm{C}, \mathrm{D}$ and $\mathrm{E}$ in a line, such that there are no consecutive identical letters. | 2220 | 66 | 4 |
math | 2nd Irish 1989 Problem B2 Each of n people has a unique piece of information. They wish to share the information. A person may pass another person a message containing all the pieces of information that he has. What is the smallest number of messages that must be passed so that each person ends up with all n pieces of ... | 2(n-1) | 129 | 5 |
math | 3. A square and a regular hexagon are drawn with the same side length. If the area of the square is $\sqrt{3}$, what is the area of the hexagon? | \frac{9}{2} | 39 | 7 |
math | 6. There are three types of stocks, the sum of the shares of the first two equals the number of shares of the third; the total value of the second type of stock is four times that of the first type; the total value of the first and second types of stocks equals the total value of the third type of stock; the price per ... | 12.5\leqslantf\leqslant15 | 141 | 17 |
math | Pat wrote on the board the example:
$$
589+544+80=2013 .
$$
Mat wanted to correct the example so that both sides would actually be equal, and he searched for an unknown number which he then added to the first addend on the left side, subtracted from the second addend, and multiplied the third addend by. After perform... | 11 | 105 | 2 |
math | 7-5. In a row, there are 1000 toy bears. The bears can be of three colors: white, brown, and black. Among any three consecutive bears, there is a toy of each color. Iskander is trying to guess the colors of the bears. He made five guesses:
- The 2nd bear from the left is white;
- The 20th bear from the left is brown;
... | 20 | 167 | 2 |
math | 3. How many natural numbers less than 2016 are divisible by 2 or 3, but not by 5? | 1075 | 28 | 4 |
math | Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$ with $AC \perp BD$. Let $P=AC \cap BD$ and $W,X,Y,Z$ be the projections of $P$ on the lines $AB, BC, CD, DA$ respectively. Let $E,F,G,H$ be the mid-points of sides $AB, BC, CD, DA$ respectively.
(a) Prove that $E,F,G,H,W,X,Y,Z$ are concyclic.
(b) If $R... | \frac{\sqrt{2R^2 - d^2}}{2} | 154 | 17 |
math | $\underline{\text { Gopovanov A.S. }}$The sum of the digits in the decimal representation of a natural number $n$ is 100, and the sum of the digits of the number $44 n$ is 800. What is the sum of the digits of the number $3 n$? | 300 | 71 | 3 |
math | Find all pairs $ (m$, $ n)$ of integers which satisfy the equation
\[ (m \plus{} n)^4 \equal{} m^2n^2 \plus{} m^2 \plus{} n^2 \plus{} 6mn.\] | (0, 0) | 55 | 7 |
math | 9.7. Find all triples of prime numbers $p, q, r$ such that the fourth power of any of them, decreased by 1, is divisible by the product of the other two.
(V. Senderov) | 2,3,5 | 47 | 5 |
math | 21. A triangle whose angles are $A, B, C$ satisfies the following conditions
$$
\frac{\sin A+\sin B+\sin C}{\cos A+\cos B+\cos C}=\frac{12}{7},
$$
and
$$
\sin A \sin B \sin C=\frac{12}{25} .
$$
Given that $\sin C$ takes on three possible values $s_{1}, s_{2}$ and $s_{3}$, find the value of $100 s_{1} s_{2} s_{3}$ - | 48 | 127 | 2 |
math | In a warehouse, the inventory is stored in packages weighing no more than 1 ton each. We have a 1-ton and a 4-ton truck. What is the maximum load that we can definitely deliver in one trip? | 4 | 46 | 1 |
math | 13. Find the smallest positive number $a$ such that there exists a positive number $b$, for which the inequality
$$
\sqrt{1-x}+\sqrt{1+x} \leqslant 2-b x^{a}
$$
holds for all $x \in [0,1]$.
For the value of $a$ found, determine the largest positive number $b$ that satisfies the above inequality. | a=2, b=\frac{1}{4} | 90 | 12 |
math | Let's say a positive integer $ n$ is [i]atresvido[/i] if the set of its divisors (including 1 and $ n$) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have. | 16 | 71 | 2 |
math | Given a sequence of positive integers $\left\{a_{n}\right\}$ satisfying: $a_{2}, a_{3}$ are prime numbers, and for any positive integers $m, n(m<n)$, we have $a_{m+n}=a_{m}+a_{n}+31, \frac{3 n-1}{3 m-1}<\frac{a_{n}}{a_{m}}<\frac{5 n-2}{5 m-2}$. Find the general term formula of $\left\{a_{n}\right\}$. | a_{n}=90n-31 | 124 | 10 |
math | (10) Divide a line segment of length $a$ into three segments. The probability that these three segments can form a triangle is $\qquad$ | \frac{1}{4} | 32 | 7 |
math | ## Task 2 - 020612
Of the 296 minutes previously set, 96 minutes were saved by the workers of VEB Druck- und Prägemaschinen Berlin during a work process in the production allocation. This amounts to 2.40 DM per machine produced.
a) How large is the savings if 60 stamping machines are produced?
b) As a result of the ... | 144\mathrm{DM} | 121 | 8 |
math | 13.250. A brigade of lumberjacks was supposed to prepare $216 \mathrm{~m}^{3}$ of wood over several days according to the plan. For the first three days, the brigade met the daily planned quota, and then each day they prepared 8 m $^{3}$ more than planned, so by the day before the deadline, they had prepared 232 m $^{3... | 24\mathrm{~}^{3} | 113 | 10 |
math | $$
\begin{gathered}
\sqrt{2+\sqrt{3}}, \sqrt{2+\sqrt{2+\sqrt{3}}}, \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \\
\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}
\end{gathered}
$$
Multiply the following nested radicals together:
$$
\begin{gathered}
\sqrt{2+\sqrt{3}}, \sqrt{2+\sqrt{2+\sqrt{3}}}, \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \\
\sq... | 1 | 158 | 1 |
math | Is there an $n$ for which the number $n!$ ends with 1971 zeros? Is there an $n$ for which the number of trailing zeros in $n!$ is 1972? (By $n!$ we mean the product of natural numbers from $n$ down to 1.) | 1971 | 70 | 4 |
math | 14. $f(x)$ is a function defined on $(0,+\infty)$ that is increasing, and $f\left(\frac{x}{y}\right)=f(x)-f(y)$.
(1) Find the value of $f(1)$; (2) If $f(6)=1$, solve the inequality $f(x+3)-f\left(\frac{1}{x}\right)<2$. | 0<x<\frac{-3+3\sqrt{17}}{2} | 90 | 18 |
math | \section*{Task 1 - 171021}
For four circles \(k_{1}, k_{2}, k_{3}, k_{4}\), it is required that they have the following two properties (1), (2):
(1) The diameter of \(k_{4}\) is \(1 \mathrm{~cm}\) larger than the diameter of \(k_{3}\), whose diameter is \(1 \mathrm{~cm}\) larger than that of \(k_{2}\), and whose diam... | \sqrt{2} | 185 | 5 |
math | 9.3. The number 2019 is represented as the sum of different odd natural numbers. What is the maximum possible number of addends? | 43 | 32 | 2 |
math | Example 11 For all real numbers $p$ satisfying $0 \leqslant p \leqslant 4$, the inequality $x^{2}+p x>4 x+p-3$ always holds. Try to find the range of values for $x$. | x>3orx<-1 | 59 | 7 |
math | [ [Volume helps solve the problem]
Does a triangular pyramid exist whose heights are 1, 2, 3, and 6?
# | No | 30 | 1 |
math | Find all polynomials $P(x)$ which have the properties:
1) $P(x)$ is not a constant polynomial and is a mononic polynomial.
2) $P(x)$ has all real roots and no duplicate roots.
3) If $P(a)=0$ then $P(a|a|)=0$
[i](nooonui)[/i] | P(x) = x(x-1)(x+1) | 74 | 14 |
math | Problem 1. Consider $A$, the set of natural numbers with exactly 2019 natural divisors, and for each $n \in A$, we denote
$$
S_{n}=\frac{1}{d_{1}+\sqrt{n}}+\frac{1}{d_{2}+\sqrt{n}}+\ldots+\frac{1}{d_{2019}+\sqrt{n}}
$$
where $d_{1}, d_{2}, \ldots, d_{2019}$ are the natural divisors of $n$.
Determine the maximum valu... | \frac{673}{2^{337}} | 142 | 13 |
math | ## Task B-4.5.
The ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{16}=1$ and the parabola $y^{2}=2 p x$ intersect at the point $T(4 \sqrt{3}, 2)$. What is the area of the triangle $T F_{E} F_{P}$, if $F_{E}$ is the focus of the ellipse located on the positive part of the $x$-axis, and $F_{P}$ is the focus of the parabola? | \frac{47\sqrt{3}}{12} | 123 | 14 |
math | Let $(a_n)_{n\ge 1}$ be a sequence such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n + 2}$, for any $n\ge 1$. Show that the sequence $(x_n)_{n\ge 1}$ given by $x_n = \log_{a_n} a_{n + 1}$ for $n\ge 1$ is convergent and compute its limit. | 1 | 105 | 1 |
math | Task 2. Find all prime numbers $p$ and natural numbers $n$ for which $\frac{1}{p}=\frac{n}{2010}$? | p=2,n=1005;p=3,n=670;p=5,n=402;p=67,n=30 | 36 | 33 |
math | 12. Two people take turns rolling dice, each rolling two
at a time. The first one to get a sum greater than 6 on the two dice wins, otherwise the turn passes to the other person. The probability that the first person to roll wins is $\qquad$. | \frac{12}{17} | 58 | 9 |
math | Let's determine the value of $c$ such that the ratio of the roots of the equation
$$
5 x^{2}-2 x+c=0
$$
is $-\frac{3}{5}$. | -3 | 45 | 2 |
math | 9. Put 12 small balls marked with numbers $1, 2, 3, \cdots \cdots, 12$ into a paper box. Three people, A, B, and C, each take 4 balls from the box. It is known that the sum of the numbers on the balls they each took is equal. A has two balls marked with numbers 6 and 11, B has two balls marked with numbers $4$ and $8$,... | 3,10,12 | 126 | 7 |
math | The number sequence $1,4,5,7,12,15,16,18,23, \ldots$ was written based on a certain rule. Continue the number sequence until it exceeds the number 50.
Continue the number sequence until it exceeds the number 50. | 51 | 65 | 2 |
math | Find all real solutions to $ x^3 \minus{} 3x^2 \minus{} 8x \plus{} 40 \minus{} 8\sqrt[4]{4x \plus{} 4} \equal{} 0$ | x = 3 | 53 | 5 |
math | In 1988, the Chinese Junior High School Mathematics League had the following problem: If natural numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ satisfy $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=x_{1} x_{2} x_{3} x_{4} x_{5}$, what is the maximum value of $x_{5}$? | 5 | 102 | 1 |
math | 11. Given that $a$ and $b$ are real numbers, satisfying:
$$
\sqrt[3]{a}-\sqrt[3]{b}=12, \quad a b=\left(\frac{a+b+8}{6}\right)^{3} \text {. }
$$
Then $a-b=$ $\qquad$ (Proposed by Thailand) | 468 | 79 | 3 |
math | Example 2. For the direction vector of the line
$$
\left\{\begin{array}{l}
2 x-3 y-3 z+4=0 \\
x+2 y+z-5=0
\end{array}\right.
$$
find the direction cosines. | \cos\alpha=\frac{3}{\sqrt{83}},\quad\cos\beta=-\frac{5}{\sqrt{83}},\quad\cos\gamma=\frac{7}{\sqrt{83}} | 62 | 50 |
math | 2. Find the smallest possible value of the function
$$
f(x)=|x+1|+|x+2|+\ldots+|x+100|
$$
$(25$ points. $)$ | 2500 | 48 | 4 |
math | Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? | 156 | 40 | 3 |
math | 4.2. The line $p: y=-x+1$ and two points $\mathrm{A}(-1,5)$ and $\mathrm{B}(0,4)$ are given. Determine the point $P$ on the line $p$ for which the sum of the lengths of the segments $|A P|+|P B|$ is minimal. | (-2,3) | 76 | 5 |
math | 4. Alloy $A$ of two metals with a mass of 6 kg, in which the first metal is twice as much as the second, placed in a container with water, creates a pressure force on the bottom of $30 \mathrm{N}$. Alloy $B$ of the same metals with a mass of 3 kg, in which the first metal is five times less than the second, placed in a... | 40 | 135 | 2 |
math | Positive integers $a_0<a_1<\dots<a_n$, are to be chosen so that $a_j-a_i$ is not a prime for any $i,j$ with $0 \le i <j \le n$. For each $n \ge 1$, determine the smallest possible value of $a_n$. | 4n + 1 | 68 | 7 |
math | 10. (USA 6) ${ }^{\mathrm{IMO4}}$ Find the largest number obtainable as the product of positive integers whose sum is 1976 . | 2\cdot3^{658} | 39 | 9 |
math | 4. Let $a, b, c$ be the sides opposite to the interior angles $A, B, C$ of $\triangle A B C$, respectively, and the area $S=\frac{1}{2} c^{2}$. If $a b=\sqrt{2}$, then the maximum value of $a^{2}+b^{2}+c^{2}$ is . $\qquad$ | 4 | 87 | 1 |
math | # Task 5. (20 points)
Find all values of the parameter $a$ for which the roots $x_{1}$ and $x_{2}$ of the equation
$$
2 x^{2}-2016(x-2016+a)-1=a^{2}
$$
satisfy the double inequality $x_{1}<a<x_{2}$.
# | \in(2015,2017) | 82 | 13 |
math | 5. Draw the tangent line to the curve $y=3x-x^{3}$ through the point $A(2,-2)$. Then the equation of the tangent line is $\qquad$ . | y=-2 \text{ or } 9x+y-16=0 | 42 | 17 |
math | Determine the type of body obtained by rotating a square around its diagonal.
## Answer
Two equal cones with a common base.
Find the area of the section of a sphere with radius 3 by a plane that is 2 units away from its center.
# | 5\pi | 53 | 3 |
math | 10.3. Given a trapezoid $A B C D$ and a point $M$ on the lateral side $A B$, such that $D M \perp A B$. It turns out that $M C=C D$. Find the length of the upper base $B C$, if $A D=d$. | \frac{}{2} | 69 | 6 |
math | 34. What is the smallest possible integer value of $n$ such that the following statement is always true?
In any group of $2 n-10$ persons, there are always at least 10 persons who have the same birthdays.
(For this question, you may assume that there are exactly 365 different possible birthdays.) | 1648 | 71 | 4 |
math | Example 5 Find all positive integer arrays $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$, such that
$$\left\{\begin{array}{l}
a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}, \\
a_{1}+a_{2}+\cdots+a_{n}=26, \\
a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=62, \\
a_{1}^{3}+a_{2}^{3}+\cdots+a_{n}^{3}=164 .
\end{array}\right.$... | (1,1,1,1,1,2,2,2,3,3,3,3,3) \text{ and } (1,2,2,2,2,2,2,2,2,2,3,4) | 159 | 57 |
math | 1. Let $x=\frac{1-\sqrt{5}}{2}$, then $\sqrt{\frac{1}{x}-\frac{1}{x^{3}}}-$ $\sqrt[3]{x^{4}-x^{2}}=$ $\qquad$ | \sqrt{5} | 57 | 5 |
math | 1. The arithmetic sequence $\left\{a_{n}\right\}$ satisfies $a_{2021}=a_{20}+a_{21}=1$, then the value of $a_{1}$ is $\qquad$ | \frac{1981}{4001} | 52 | 13 |
math | 14. Given the sets $A=\left\{(x, y) \left\lvert\, \frac{y-3}{x-2}=a+1\right.\right\}, B=\left\{(x, y) \mid\left(a^{2}-1\right) x+\right.$ $(a-1) y=15\}$. If $A \cap B=\varnothing$, find the value of the real number $a$. | 1,-1,\frac{5}{2},-4 | 100 | 12 |
math | 1. Determine all pairs of real numbers $(x, y)$, satisfying $x=3 x^{2} y-y^{3}, y=x^{3}-3 x y^{2}$. | (x,y)=(0,0),(\frac{\sqrt{2+\sqrt{2}}}{2},\frac{\sqrt{2-\sqrt{2}}}{2}),(-\frac{\sqrt{2-\sqrt{2}}}{2},\frac{\sqrt{2+\sqrt{2}}}{2}),(-\frac{\sqrt{2+\sqrt{2}}}{2},-\frac{\sqrt{2-} | 40 | 87 |
math | Determine all pairs $(x, y)$ of integers that satisfy the equation
$$
\sqrt[3]{7 x^{2}-13 x y+7 y^{2}}=|x-y|+1
$$ | (x_1, y_1) = (m^3 + 2m^2 - m - 1, m^3 + m^2 - 2m - 1), (x_2, y_2) = (-m^3 - m^2 + 2m + 1, -m^3 - 2m^2 + m + 1) | 47 | 82 |
math | 5. Solve the equation $a^{b}+a+b=b^{a}$ in prime numbers. | =5,b=2 | 21 | 5 |
math | $$
\text { 1. Given } \sqrt[3]{a}+a=\sqrt{6}, b^{3}+b=\sqrt{6} \text {. Then } a+b
$$ | a+b=\sqrt{6} | 44 | 7 |
math | Exercise 1. N.B. In this exercise, and only this one, an answer without justification is requested.
Let $m>n>p$ be three (positive integer) prime numbers such that $m+n+p=74$ and $m-n-p=44$. Determine $m, n$, and $p$.
(A prime number is an integer strictly greater than one, and whose only divisors are one and itself.... | =59,n=13,p=2 | 88 | 10 |
math | 26. Find all integer solutions $(x, y, z)$ that satisfy the following system of equations:
$$\left\{\begin{array}{l}
x+1=8 y^{2}, \\
x^{2}+1=2 z^{2} .
\end{array}\right.$$ | (x, y, z)=(-1,0, \pm 1),(7, \pm 1, \pm 5) | 63 | 28 |
math | Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which
\[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\]
is a rational number. | n = 1, 3 | 79 | 7 |
math | 2. Let the arithmetic sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$, if $a_{1}=2022, S_{20}=22$, then the common difference $d$ is $\qquad$ . | -\frac{20209}{95} | 81 | 12 |
math | 8. (10 points) In $\triangle A B C$, $B D=D E=E C$, $C F: A C=1: 3$. If the area of $\triangle A D H$ is 24 square centimeters more than the area of $\triangle H E F$, find the area of triangle $A B C$ in square centimeters? | 108 | 77 | 3 |
math | ## Task 4
A company has two cars of the type "Wartburg". One car drove 600 km in a week, and the other drove 900 km.
How many liters of gasoline did each car need, if the second car, which drove 900 km, used 27 liters more than the first? | 54 | 73 | 2 |
math | Higher Secondary P2
Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$. | 0 | 74 | 1 |
math | 18. School A and School B each send out $\mathbf{5}$ students to participate in a long-distance running competition, with the rule being: The $\mathbf{K}$-th student to reach the finish line scores $\mathbf{K}$ points (no students arrive at the finish line simultaneously), and the school with the lower total score wins... | 13 | 88 | 2 |
math | 5. From the smallest six-digit number divisible by 2 and 3, subtract the smallest five-digit number divisible by 5 and 9. What is the resulting difference? Round that difference to the nearest hundred. What number did you get?
## Tasks worth 10 points: | 90000 | 58 | 5 |
math | 4. Given that $\odot O_{1}$ and $\odot O_{2}$ are externally tangent, their radii are $112$ and $63$, respectively. The segment $A B$ is intercepted by their two external common tangents on their internal common tangent. Then, the length of $A B$ is $\qquad$ . | 168 | 75 | 3 |
math | 7. Let the 10 complex roots of the equation $x^{10}+(13 x-1)^{10}=0$ be $x_{1}, x_{2}, \cdots, x_{10}$. Then
$$
\frac{1}{x_{1} \overline{x_{1}}}+\frac{1}{x_{2} \overline{x_{2}}}+\cdots+\frac{1}{x_{5} \overline{x_{5}}}=
$$
$\qquad$ | 850 | 113 | 3 |
math | Problem 5. (Option 2).
Find the form of all quadratic trinomials $f(x)=a x^{2}+b x+c$, where $a, b, c$ are given constants, $\quad a \neq 0$, such that for all values of $x$ the condition $f(3.8 x-1)=f(-3.8 x)$ is satisfied. | f(x)=^{2}++ | 85 | 7 |
math | 5. In the tournament, 15 volleyball teams are playing, and each team plays against all other teams only once. Since there are no draws in volleyball, there is a winner in each match. A team is considered to have performed well if it loses no more than two matches. Find the maximum possible number of teams that performe... | 5 | 69 | 1 |
math | 8. A staircase has a total of 12 steps. When a person climbs the stairs, they sometimes take one step at a time, and sometimes two steps at a time. The number of ways this person can climb the stairs is $\qquad$. | 233 | 52 | 3 |
math | 5. Yesterday at the market, with one hundred tugriks you could buy 9 gingerbreads and 7 pastries (and even get some change), but today this amount is no longer enough. However, with the same one hundred tugriks today, you can buy two gingerbreads and 11 pastries (also with some change), but yesterday this amount would ... | 5 | 126 | 1 |
math | 28.2.11 ** Let $M$ be a set of $n$ points in the plane, satisfying:
(1) $M$ contains 7 points which are the 7 vertices of a convex heptagon;
(2) For any 5 points in $M$, if these 5 points are the 5 vertices of a convex pentagon, then this convex pentagon contains at least one point from $M$ inside it.
Find the minimum ... | 11 | 102 | 2 |
math | 18. (1994 Bulgarian Mathematical Olympiad) $n$ is a positive integer, $A$ is a set of subsets of the set $\{1,2, \cdots, n\}$, such that no element of $A$ contains another element of $A$. Find the maximum number of elements in $A$. | C_{n}^{[\frac{n}{2}]} | 71 | 12 |
math | 8・3 Let the sequence $x_{1}, x_{2}, x_{3}, \cdots$ satisfy
$$
\begin{array}{ll}
& 3 x_{n}-x_{n-1}=n, n=2,3, \cdots \\
\text { and } \quad\left|x_{1}\right|<1971 .
\end{array}
$$
Try to find $x_{1971}$, accurate to 0.000001. | 985.250000 | 112 | 10 |
math | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-4.5,4.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 90 | 57 | 2 |
math | The Sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by $a_{0}=1, a_{1}=-4$ and $a_{n+2}=-4a_{n+1}-7a_{n}$ , for $n \geqslant 0$. Find the number of positive integer divisors of $a^2_{50}-a_{49}a_{51}$. | 51 | 97 | 2 |
math | # 5. The factory paints cubes in 6 colors (each face in its own color, the set of colors is fixed). How many varieties of cubes can be produced? | 30 | 36 | 2 |
math | 9.5 What is the largest number of non-overlapping groups into which all integers from 1 to 20 can be divided so that the sum of the numbers in each group is a perfect square? | 11 | 42 | 2 |
math | 8.030. $2+\tan x \cot \frac{x}{2}+\cot x \tan \frac{x}{2}=0$. | \frac{2\pi}{3}(3k\1),k\inZ | 32 | 18 |
math | ## Task 24/71
Determine the greatest common divisor of $2^{n-1}-1$ and $2^{n+1}-1$ for $n=2 ; 3 ; \ldots$ ! | If\n\is\odd,\the\greatest\common\divisor\is\3;\if\n\is\even,\the\greatest\common\divisor\is\1 | 49 | 39 |
math | a,b,c are positives with 21ab+2bc+8ca \leq 12 . Find the least possoble value of the expresion 1/a + 2/b + 3/c.
Thanks | \frac{15}{2} | 52 | 8 |
math | 9.23 How many four-digit numbers, composed of the digits $0,1,2,3,4,5$, contain the digit 3 (the digits in the numbers do not repeat)? | 204 | 42 | 3 |
math | Let $M$ and $N$ be the midpoints of sides $C D$ and $D E$ of a regular hexagon $A B C D E F$. Find the angle between the lines $A M$ and $B N$.
# | 60 | 53 | 2 |
math | 5. Let $Q(x)=a_{2023} x^{2023}+a_{2022} x^{2022}+\cdots+a_{1} x+a_{0}$ be a polynomial with integer coefficients. For every odd prime number $p$, we define the polynomial $Q_{p}(x)=a_{2023}^{p-2} x^{2023}+a_{2022}^{p-2} x^{2022}+\cdots+a_{1}^{p-2} x+a_{0}^{p-2}$. It is known that for infinitely many odd prime numbers $... | \frac{2023^{2024}-1}{2022} | 202 | 20 |
math | 11. B. Let real numbers $a, b$ satisfy
$$
3 a^{2}-10 a b+8 b^{2}+5 a-10 b=0 \text {. }
$$
Find the minimum value of $u=9 a^{2}+72 b+2$. | -34 | 67 | 3 |
math | 2. Let $P$ be any point inside a regular tetrahedron $ABCD$ with edge length $\sqrt{2}$, and let the distances from point $P$ to the four faces be $d_{1}, d_{2}, d_{3}, d_{4}$. Then the minimum value of $d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$ is | \frac{1}{3} | 99 | 7 |
math | Solve the following equation:
$$
\sqrt{0.12^{0.12 x}+5}+\sqrt{0.12^{0.12 x}+6}=5
$$ | 1.079 | 45 | 5 |
math | In the May 1989 issue of this journal, there was a problem from the Zhejiang Province High School Mathematics Competition as follows:
Arrange the positive rational numbers: $\frac{1}{1}, \frac{2}{1}, \frac{1}{2}, \frac{3}{1}, \frac{2}{2}, \frac{1}{3}, \frac{4}{1}, \frac{3}{2}, \frac{2}{3}, \frac{1}{4}, \cdots$, then t... | 7749965 | 135 | 7 |
math | Expression: $1^{2015}+2^{2015}+3^{2015}+\cdots+2013^{2015}+2014^{2015}$, the unit digit of the calculation result is | 5 | 60 | 1 |
math | 51. If in a certain six-digit number the leftmost digit 7 is moved to the end of the number, the resulting number is 5 times smaller than the original. Find the original number.
18 | 714285 | 44 | 6 |
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