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 | 11. Arrange the following 4 numbers in ascending order: $\sin \left(\cos \frac{3 \pi}{8}\right), \cos \left(\sin \frac{3}{8} \pi\right), \cos \left(\cos \frac{3}{8} \pi\right)$. $\sin \left(\sin \frac{3}{8} \pi\right)$ $\qquad$ . | \cos(\cos\frac{3\pi}{8})>\sin(\sin\frac{3\pi}{8})>\cos(\sin\frac{3\pi}{8})>\sin(\cos\frac{3\pi}{8}) | 90 | 52 |
math | Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, suppose $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$. | 682 | 108 | 3 |
math | 1.51. The side of a regular triangle is equal to $a$. Determine the area of the part of the triangle that lies outside a circle of radius $\frac{a}{3}$, the center of which coincides with the center of the triangle. | \frac{^{2}(3\sqrt{3}-\pi)}{18} | 54 | 19 |
math | 2. Twelve students participated in a theater festival consisting of $n$ different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of $n$. | 4 | 48 | 1 |
math | 6. (5 points) The average of numbers $a, b, c, d$ is 7.1, and $2.5 \times a=b-1.2=c+4.8=0.25 \times d$, then $a \times b \times c$ $\times d=$ $\qquad$ | 49.6 | 70 | 4 |
math | 12. Grandpa Hua Luogeng was born on November 12, 1910. Arrange these numbers into an integer, and factorize it as 19101112 $=1163 \times 16424$. Are either of the numbers 1163 and 16424 prime? Please explain your reasoning. | 1163 | 84 | 4 |
math | 16. Given $\sin \left(\frac{\pi}{4}-x\right)=\frac{5}{13}$, and $x \in\left(0, \frac{\pi}{4}\right)$, find the value of $\frac{\cos 2 x}{\cos \left(\frac{\pi}{4}+x\right)}$. | \frac{24}{13} | 77 | 9 |
math | 3. Zhenya had 9 cards with numbers from 1 to 9. He lost the card with the number 7. Can the remaining 8 cards be arranged in a row so that any two adjacent cards form a number divisible by 7? | no | 53 | 1 |
math | 1. Points $A$ and $B$ lie on a circle with center $O$ and radius 6, and point $C$ is equidistant from points $A, B$, and $O$. Another circle with center $Q$ and radius 8 is circumscribed around triangle $A C O$. Find $B Q$. Answer: 10 | 10 | 76 | 2 |
math | A positive integer $n$ is defined as a $\textit{stepstool number}$ if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$.
[i]Proposed by [b]Th3Numb3rThr33[/b][/i] | 687 | 114 | 3 |
math | Princeton’s Math Club recently bought a stock for $\$2$ and sold it for $\$9$ thirteen days later. Given that the stock either increases or decreases by $\$1$ every day and never reached $\$0$, in how many possible ways could the stock have changed during those thirteen days?
| 273 | 63 | 3 |
math | 9. Xiao Ming is playing a ring toss game, scoring 9 points for each chicken, 5 points for each monkey, and 2 points for each dog. Xiao Ming tossed 10 times, hitting every time, and each toy was hit at least once. Xiao Ming scored a total of 61 points in 10 tosses. Question: What is the minimum number of times the chick... | 5 | 86 | 1 |
math | On a circle, 2022 points are chosen such that distance between two adjacent points is always the same. There are $k$ arcs, each having endpoints on chosen points, with different lengths. Arcs do not contain each other. What is the maximum possible number of $k$? | 1011 | 61 | 4 |
math | Example 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can be divided by $x^{2}+1$.
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Example 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can be divided by $x^... | 8 | 130 | 1 |
math | Example 6 (2006 National Training Team Test) Find all positive integer pairs $(a, n)$ such that $\frac{(a+1)^{n}-a^{n}}{n}$ is an integer.
Find all positive integer pairs $(a, n)$ such that $\frac{(a+1)^{n}-a^{n}}{n}$ is an integer. | (a, 1) | 79 | 5 |
math | 5. Let $n>2$ be a natural number, $1=a_{1}<\ldots<a_{k}=n-1$ be all numbers from 1 to $n$ that are coprime with $n$. Denote by $f(n)$ the greatest common divisor of the numbers $a_{1}^{3}-1, \ldots, a_{k}^{3}-1$. What values can the function $f(n)$ take? | 1,2,7,26,124 | 97 | 12 |
math | 8. 173 Find all $a_{0} \in R$, such that the sequence
$$a_{n+1}=2^{n}-3 a_{n}, n=0,1,2, \cdots$$
determined by this is increasing. | a_{0}=\frac{1}{5} | 58 | 11 |
math | 2. There were 1009 gnomes with 2017 cards, numbered from 1 to 2017. Ori had one card, and each of the other gnomes had two. All the gnomes knew only the numbers on their own cards. Each gnome, except Ori, said: "I am sure that I cannot give Ori any of my cards so that the sum of the numbers on his two cards would be 20... | 1009 | 109 | 4 |
math | 4. 228 ** Let $x, y, z \in \mathbf{R}^{+}, x+2 y+3 z=1$, find the minimum value of $\frac{16}{x^{3}}+\frac{81}{8 y^{3}}+\frac{1}{27 z^{3}}$. | 1296 | 73 | 4 |
math | Each of the thirty sixth-graders has one pen, one pencil, and one ruler. After their participation in the Olympiad, it turned out that 26 students lost a pen, 23 - a ruler, and 21 - a pencil. Find the smallest possible number of sixth-graders who lost all three items. | 10 | 68 | 2 |
math | 10. (10 points) When the fraction $\frac{1}{2016}$ is converted to a repeating decimal, the repeating block has exactly $\qquad$ digits. | 6 | 39 | 1 |
math | 7. Find the positive integer $x$ that makes $x^{2}-60$ a perfect square
Find the positive integer $x$ that makes $x^{2}-60$ a perfect square | 16 \text{ or } 8 | 43 | 9 |
math | Determine the number of integers $a$ with $1\leq a\leq 1007$ and the property that both $a$ and $a+1$ are quadratic residues mod $1009$. | 251 | 49 | 3 |
math | 14. [8] Let $A B C D$ be a trapezoid with $A B \| C D$ and $\angle D=90^{\circ}$. Suppose that there is a point $E$ on $C D$ such that $A E=B E$ and that triangles $A E D$ and $C E B$ are similar, but not congruent. Given that $\frac{C D}{A B}=2014$, find $\frac{B C}{A D}$. | \sqrt{4027} | 110 | 8 |
math | Exercise 4. We are given five numbers in ascending order, which are the lengths of the sides of a quadrilateral (not crossed, but not necessarily convex, meaning a diagonal is not necessarily inside the polygon) and one of its diagonals $D$. These numbers are 3, 5, 7, 13, and 19. What can be the length of the diagonal ... | 7 | 85 | 1 |
math | Example 9 Calculate
$$
(2+1)\left(2^{2}+1\right)\left(2^{4}+1\right) \cdots \cdot\left(2^{2^{n}}+1\right) .
$$ | 2^{2^{n+1}}-1 | 55 | 10 |
math | The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 481^2 | 69 | 5 |
math | 8. How much does the dress cost? The worker's monthly pay, that is, for thirty days, is ten dinars and a dress. He worked for three days and earned the dress. What is the cost of the dress? | 1\frac{1}{9} | 48 | 8 |
math | 13. B. If five pairwise coprime distinct integers $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ are randomly selected from $1,2, \cdots, n$, and one of these integers is always a prime number, find the maximum value of $n$. | 48 | 68 | 2 |
math | 12. Find the exact value of $\sqrt{\frac{x-2}{6}}$ when $x=2006^{3}-2004^{3}$. | 2005 | 38 | 4 |
math | 1. Riješite sustav jednadžbi
$$
\begin{array}{r}
2\left(x^{2}+y^{2}\right)-3 x y+2(x+y)-39=0 \\
3\left(x^{2}+y^{2}\right)-4 x y+(x+y)-50=0
\end{array}
$$ | \begin{pmatrix}x_{1,2}=\frac{5\\sqrt{13}}{2},\quady_{1,2}=\frac{5\\sqrt{13}}{2}\\x_{3}=3,\quady_{3}=5\\x_{4}=5,\quady_{4}=3\end{pmatrix} | 81 | 77 |
math | Example 3 Find all integer pairs $(x, y)$ that satisfy the equation $y^{4}+2 x^{4}+1=4 x^{2} y$.
(1995, Jiangsu Province Junior High School Mathematics Competition) | (x, y)=(\pm 1,1) | 53 | 11 |
math | 3. Find the sum of the first 10 elements that are found both among the members of the arithmetic progression $\{5,8,11,13, \ldots\}$, and among the members of the geometric progression $\{20,40,80,160, \ldots\} \cdot(10$ points) | 6990500 | 77 | 7 |
math | 3. Let $O$ be the circumcenter of acute $\triangle A B C$, with $A B=6, A C=10$. If $\overrightarrow{A O}=x \overrightarrow{A B}+y \overrightarrow{A C}$, and $2 x+10 y=5$, then $\cos \angle B A C=$ $\qquad$ . | \frac{1}{3} | 82 | 7 |
math | 3B. Determine the first two and the last two digits of the decimal representation of the number $x_{1001}$, if $x_{1}=2$ and $x_{n+1}=\frac{1}{\sqrt[10]{2}} x_{n}+\frac{\sqrt[10]{2}-1}{\sqrt[10]{2}}, n \in \mathbb{N}$. | 1025 | 91 | 4 |
math | Given four points $O,\ A,\ B,\ C$ on a plane such that $OA=4,\ OB=3,\ OC=2,\ \overrightarrow{OB}\cdot \overrightarrow{OC}=3.$
Find the maximum area of $\triangle{ABC}$. | 2\sqrt{7} + \frac{3\sqrt{3}}{2} | 57 | 19 |
math | Problem 4. Write the coordinates of point $C$, which divides the segment $A B$ in the ratio $2: 1$, starting from point $A$, if $A(1 ;-1 ; 2)$ and $B(7 ;-4 ;-1)$. | C(5;-3;0) | 57 | 8 |
math | 13.2 設 $n$ 為整數。求 $n^{a}-n$ 除以 30 的稌值 $b$ 。
Let $n$ be an integer. Determine the remainder $b$ of $n^{a}-n$ divided by 30 . | 0 | 66 | 1 |
math | 8. (5 points) According to the rules, Xiaoming's comprehensive evaluation score in mathematics for this semester is equal to the sum of half the average score of 4 tests and half the final exam score. It is known that the scores of the 4 tests are 90, 85, 77, and 96. If Xiaoming wants his comprehensive evaluation score... | 93 | 105 | 2 |
math | 3 Suppose 2005 line segments are connected end-to-end, forming a closed polyline, and no two segments of the polyline lie on the same straight line. Then, what is the maximum number of intersection points where the polyline intersects itself?
| 2007005 | 50 | 7 |
math | $1^{2} / 3 \%$ of the country's students are engaged with math magazines, specifically $2^{1} / 3 \%$ of the boys and $2 / 3 \%$ of the girls. How many boys and how many girls are engaged with the magazines among 75000 students? | u=45000,v=30000 | 68 | 14 |
math | 8. (10 points) In the expression $(x+y+z)^{2024}+(x-y-z)^{2024}$, the brackets were expanded and like terms were combined. How many monomials $x^{a} y^{b} z^{c}$ with a non-zero coefficient were obtained? | 1026169 | 69 | 7 |
math | 14. Three positive integers are such that they differ from each other by at most 6 . It is also known that the product of these three integers is 2808 . What is the smallest integer among them? | 12 | 46 | 2 |
math | Denote by $\mathbb{R}^{+}$ the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$
x f\left(x^{2}\right) f(f(y))+f(y f(x))=f(x y)\left(f\left(f\left(x^{2}\right)\right)+f\left(f\left(y^{2}\right)\right)\right)
$$
for all positive real numbers $x$ and $y$. | f(y) = \frac{1}{y} | 121 | 11 |
math | ## Task 12/76
Mrs. Quidam tells: "My husband, I, and our four children all have the same birthday. On our last birthday, we added up our ages and got our house number. When we multiplied them, we got the mileage on our Trabant before the last major overhaul: $180523.$"
How old are the six Quidams? What is their house... | 104 | 91 | 3 |
math | 1. Given the quadratic function
$$
\begin{aligned}
f(x)= & a(3 a+2 c) x^{2}-2 b(2 a+c) x+ \\
& b^{2}+(c+a)^{2}(a, b, c \in \mathbf{R}).
\end{aligned}
$$
Assume that for any $x \in \mathbf{R}$, we have $f(x) \leqslant 1$. Find the maximum value of $|a b|$. (Yang Xiaoming, Zhao Bin) | \frac{3\sqrt{3}}{8} | 122 | 12 |
math | In 2012, the Anchuria School Board decided that three questions were too few. Now, one needs to correctly answer six questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of obtaining the Anchurian certificate higher - in 2011 or in 2012? | 2012 | 78 | 4 |
math | Example 3. Suppose 1987 can be represented as a three-digit number $\overline{x y z}$ in base $b$, and $x+y+z=1+9+8+7$. Try to determine all possible values of $x$, $y$, $z$, and $b$. (Canadian 87 Competition Question) | x=5, y=9, z=11, b=19 | 72 | 17 |
math | 1. Verify that the number $7^{20}-1$ is divisible by $10^{3}$ and determine the last three digits of the number $7^{2015}$. | 943 | 41 | 3 |
math | 1. (AUS 3) The integer 9 can be written as a sum of two consecutive integers: $9=4+5$. Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $9=4+5=2+3+4$. Is there an integer that can be written as a sum of 1990 consecutive integers and that can be written as a sum ... | 5^{10}\cdot199^{180} | 111 | 14 |
math | 8. The number of non-negative integer solutions to the equation $2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}=3$ is
$\qquad$ groups. | 174 | 74 | 3 |
math | 4. Solve the equation $\quad \sqrt{x+\sqrt{x}-\frac{71}{16}}-\sqrt{x+\sqrt{x}-\frac{87}{16}}=\frac{1}{2}$. | 4 | 47 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(\frac{\tan 4 x}{x}\right)^{2+x}$ | 16 | 38 | 2 |
math | Question 15: Let the set $M=\{1,2, \ldots, 100\}$ be a 100-element set. If for any n-element subset $A$ of $M$, there are always 4 elements in $A$ that are pairwise coprime, find the minimum value of $\mathrm{n}$.
| 75 | 75 | 2 |
math | 8. given an arbitrary floor plan composed of $n$ unit squares. Albert and Berta want to cover this floor with tiles, each tile having the shape of a $1 \times 2$ domino or a T-tetromino. Albert only has tiles of one color available, while Berta has dominoes of two colors and tetrominoes of four colors. Albert can cover... | 2^{\frac{n}{2}} | 127 | 8 |
math | Simplify the following expression as much as possible:
$$
\left(\frac{\sqrt{1+a}}{\sqrt{1+a}-\sqrt{1-a}}+\frac{1-a}{\sqrt{1-a^{2}}-1+a}\right)\left(\sqrt{\frac{1}{a^{2}}-1}-\frac{1}{a}\right)
$$
where $0<a<1$. | -1 | 87 | 2 |
math | 1. In $\triangle A B C$, it is known that: $a=2, b=2 \sqrt{2}$. Find the range of values for $\angle A$. | A\in(0,\frac{\pi}{4}] | 38 | 12 |
math | 5. Let $N>1$ be a positive integer, and $m$ denote the largest divisor of $N$ that is less than $N$. If $N+m$ is a power of 10, find $N$.
| 75 | 50 | 2 |
math | Find all real numbers $x, y, z$ such that
$$
x+y+z=3, \quad x^{2}+y^{2}+z^{2}=3, \quad x^{3}+y^{3}+z^{3}=3
$$
Distinction between polynomial and polynomial function Here, we explain why it is necessary to make this distinction, starting by defining a polynomial in another way. Here, $\mathbb{K}=\mathbb{Q}, \mathbb{R}... | 1 | 1,246 | 1 |
math | Given a positive integer $n(n \geq 5)$, try to provide a set of distinct positive even numbers $p_{1}, p_{2}, \ldots, p_{n}$, such that
$$
\frac{1}{p_{1}}+\frac{1}{p_{2}}+\ldots+\frac{1}{p_{n}}=\frac{2003}{2002} .
$$ | p_{1}, p_{2}, \ldots, p_{5} = 2, 4, 6, 12, 2002 | 92 | 35 |
math | 4. In the geometric sequence $\left\{a_{n}\right\}$, if $a_{1}=1, a_{3}=3$, then $\left(\sum_{k=0}^{10} \mathrm{C}_{10}^{k} a_{k+1}\right)\left[\sum_{k=0}^{10}(-1)^{k} \mathrm{C}_{10}^{k} a_{k+1}\right]=$ $\qquad$ | 1024 | 106 | 4 |
math | 9. (6 points) The sum of the ages of A and B this year is 43 years old. In 4 years, A will be 3 years older than B. A is $\qquad$ years old this year. | 23 | 50 | 2 |
math | $\because 、\left(20\right.$ points) For a quadratic equation with real coefficients $a x^{2}+2 b x$ $+ c=0$ having two real roots $x_{1}, x_{2}$. Let $d=\left|x_{1}-x_{2}\right|$. Find the range of $d$ when $a>b>c$ and $a+b+c=0$. | \sqrt{3}<d<2 \sqrt{3} | 89 | 13 |
math | 11. Let $f(x)$ be a function defined on $\mathbf{R}$. If $f(0)=2008$, and for any $x \in \mathbf{R}$, it satisfies
$$
\begin{array}{l}
f(x+2)-f(x) \leqslant 3 \times 2^{x}, \\
f(x+6)-f(x) \geqslant 63 \times 2^{x},
\end{array}
$$
then $f(2008)=$ | 2^{2008}+2007 | 120 | 12 |
math | (19) Let $P$ be a moving point on the major axis of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$. A line passing through $P$ with slope $k$ intersects the ellipse at points $A$ and $B$. If $|P A|^{2}+|P B|^{2}$ depends only on $k$ and not on $P$, find the value of $k$.
Let $P$ be a moving point on the major axis ... | \\frac{4}{5} | 201 | 7 |
math | Let's construct a $12.75^{\circ}$ angle as simply as possible. | 12.75=\frac{36+15}{2^{2}} | 20 | 18 |
math | [ Measuring the lengths of segments and the measures of angles. Adjacent angles.]
One of two adjacent angles is $30^{\circ}$ greater than the other. Find these angles.
# | 105,75 | 40 | 6 |
math | 10. Fill in each box in the following equation with “+” or “-”, what is the sum of all different calculation results?
$$
625 \square 125 \square 25 \square 5 \square 1
$$ | 10000 | 55 | 5 |
math | Given $100$ quadratic polynomials $f_1(x)=ax^2+bx+c_1, ... f_{100}(x)=ax^2+bx+c_{100}$. One selected $x_1, x_2... x_{100}$ - roots of $f_1, f_2, ... f_{100}$ respectively.What is the value of sum $f_2(x_1)+...+f_{100}(x_{99})+f_1(x_{100})?$
---------
Also 9.1 in 3rd round of Russian National Olympiad | 0 | 137 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(2-3^{\operatorname{arctg}^{2} \sqrt{x}}\right)^{\frac{2}{\sin x}}$ | \frac{1}{9} | 55 | 7 |
math | $ 2^n $ coins are given to a couple of kids. Interchange of the coins occurs when some of the kids has at least half of all the coins. Then from the coins of one of those kids to the all other kids are given that much coins as the kid already had. In case when all the coins are at one kid there is no possibility for in... | n | 94 | 2 |
math | Task No. 1.1
## Condition:
Kirill Konstantinovich's age is 48 years 48 months 48 weeks 48 days 48 hours. How many full years old is Kirill Konstantinovich? | 53 | 54 | 2 |
math | Example 5.20. Expand the function $f(x)=e^{3 x}$ into a Taylor series. | e^{3x}=1+\frac{3}{1!}x+\frac{3^{2}}{2!}x^{2}+\frac{3^{3}}{3!}x^{3}+\ldots+\frac{3^{n}}{n!}x^{n}+\ldots | 24 | 66 |
math | ## 43. Rue Saint-Nicaise
On December 24, 1800, First Consul Bonaparte was heading to the Opera along Rue Saint-Nicaise. A bomb exploded on his route with a delay of a few seconds. Many were killed and wounded. Bonaparte accused the Republicans of the plot; 98 of them were exiled to the Seychelles and Guiana. Several p... | 9 | 198 | 1 |
math | 1. (2 points) In a puddle, there live three types of amoebas: red, blue, and yellow. From time to time, any two amoebas of different types can merge into one amoeba of the third type. It is known that in the morning, there were 47 red, 40 blue, and 53 yellow amoebas in the puddle, and by evening, only one amoeba remain... | blue | 99 | 1 |
math | 6. For all non-negative values of the real variable $x$, the function $f(x)$ satisfies the condition $f(x+1)+1=f(x)+\frac{43}{(x+1)(x+2)}$. Calculate $\frac{101}{f(2020)}$, if $\quad f(0)=2020$. | 2.35 | 77 | 4 |
math | ## Problem T-1
Determine the smallest and the greatest possible values of the expression
$$
\left(\frac{1}{a^{2}+1}+\frac{1}{b^{2}+1}+\frac{1}{c^{2}+1}\right)\left(\frac{a^{2}}{a^{2}+1}+\frac{b^{2}}{b^{2}+1}+\frac{c^{2}}{c^{2}+1}\right)
$$
provided $a, b$, and $c$ are non-negative real numbers satisfying $a b+b c+c ... | \frac{27}{16}\leq\frac{9}{4}-(x-\frac{3}{2})^{2}\leq2 | 170 | 33 |
math | Given a positive integer $n(n \geqslant 3)$. Try to find the largest constant $\lambda(n)$, such that for any $a_{1}, a_{2}, \cdots, a_{n} \in \mathbf{R}_{+}$, we have
$$
\prod_{i=1}^{n}\left(a_{i}^{2}+n-1\right) \geqslant \lambda(n)\left(\sum_{i=1}^{n} a_{i}\right)^{2} .
$$ | n^{n-2} | 120 | 6 |
math | 6. In the Thrice-Tenth Kingdom, there are 17 islands, each inhabited by 119 people. The inhabitants of the kingdom are divided into two castes: knights, who always tell the truth, and liars, who always lie. During the census, each person was first asked: "Excluding you, are there an equal number of knights and liars on... | 1013 | 180 | 4 |
math | 7.5. If $a, b, c, d, e, f, g, h, k$ are all 1 or -1, try to find the maximum possible value of
$$
a e k - a f h + b f g - b d k + c d h - c e g
$$ | 4 | 68 | 1 |
math | 9. (16 points) In the geometric sequence $\left\{a_{n}\right\}$ where all terms are positive, what is the maximum number of terms that can be integers between $100 \sim 1000$? | 6 | 53 | 1 |
math | 7. It is known that the number $a$ satisfies the equation param1, and the number $b$ satisfies the equation param2. Find the smallest possible value of the sum $a+b$.
| param1 | param2 | |
| :---: | :---: | :--- |
| $x^{3}-3 x^{2}+5 x-17=0$ | $x^{3}-6 x^{2}+14 x+2=0$ | |
| $x^{3}+3 x^{2}+6 x-9=0$ | $x^{3}+6 x^{2}+15... | -3 | 239 | 2 |
math | 7. Find the maximum value of the function
$$
f(x)=\sin (x+\sin x)+\sin (x-\sin x)+\left(\frac{\pi}{2}-2\right) \sin \sin x
$$ | \frac{\pi-2}{\sqrt{2}} | 51 | 12 |
math | 29*. a) How many different positive integer solutions does the equation
$$
x_{1}+x_{2}+x_{3}+\ldots+x_{m}=n
$$
have?
b) How many different non-negative integer solutions does the equation
$$
x_{1}+x_{2}+x_{3}+\ldots+x_{m}=n
$$
have?
Note. A particular case of problem 29a) (corresponding to $m=3$) is problem 25.
... | C_{n+-1}^{-} | 380 | 8 |
math | Source: 2017 Canadian Open Math Challenge, Problem A4
-----
Three positive integers $a$, $b$, $c$ satisfy
$$4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}.$$
Determine the sum of $a + b + c$.
| 36 | 82 | 2 |
math | 11. Given $f(x)=\frac{a x+1}{3 x-1}$, and the equation $f(x)=-4 x+8$ has two distinct positive roots, one of which is three times the other. Let the first $n$ terms of the arithmetic sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ be $S_{n}$ and $T_{n}$ respectively, and $\frac{S_{n}}{T_{n}}=f(n)(n=1,2, \cd... | \frac{5}{2} | 279 | 7 |
math | \section*{Problem 2 - 021032}
Calculate:
\[
\log _{2} \frac{1}{256}+\log _{2} \frac{1}{128}+\log _{2} \frac{1}{64}+\log _{2} \frac{1}{32}+\ldots+\log _{2} \frac{1}{2}+\log _{2} 1+\log _{2} 2+\ldots+\log _{2} 64+\log _{2} 128
\] | -8 | 135 | 2 |
math | 8. Let the function $f: \mathbf{R} \rightarrow \mathbf{R}$, satisfy $f(0)=1$, and for any $x, y \in \mathbf{R}$, there is $f(x y+1)=$ $f(x) f(y)-f(y)-x+2$, then $f(x)=$ $\qquad$ | f(x)=x+1 | 81 | 6 |
math | 8.4. Can the numbers from 1 to 9 be placed in the cells of a $3 \times 3$ square so that the sum of any two numbers in adjacent cells (cells sharing a side) is a prime number? | No | 50 | 1 |
math | Mr. Anderson has more than 25 students in his class. He has more than 2 but fewer than 10 boys and more than 14 but fewer than 23 girls in his class. How many different class sizes would satisfy these conditions?
(A) 5
(B) 6
(C) 7
(D) 3
(E) 4 | 6 | 78 | 1 |
math | 8.021. $2 \operatorname{ctg}^{2} x \cos ^{2} x+4 \cos ^{2} x-\operatorname{ctg}^{2} x-2=0$. | \frac{\pi}{4}(2k+1),k\inZ | 51 | 16 |
math | Recently, the following game has become popular: $A$ says to $B$: "Write down your shoe size (as a whole number only!), add 9 to it. Multiply the resulting number by 2, and then add 5. Multiply the resulting sum by 50, and add 1708. Subtract your birth year from this sum. Tell me the final result!" $B$ (who has been ca... | 43,50 | 222 | 5 |
math | The $42$ points $P_1,P_2,\ldots,P_{42}$ lie on a straight line, in that order, so that the distance between $P_n$ and $P_{n+1}$ is $\frac{1}{n}$ for all $1\leq n\leq41$. What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.) | 861 | 94 | 3 |
math | Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$.
Find the greatest value of $M = abc$ | 6171 | 39 | 4 |
math | Tokarev S.i.
Can one of the factorials be erased from the product $1!\cdot 2!\cdot 3!\cdot \ldots \cdot 100!$ so that the product of the remaining ones is a square of an integer? | 50! | 55 | 3 |
math | 2. Let $A B C$ be a triangle with $\angle B A C=90^{\circ}$. Let $D, E$, and $F$ be the feet of altitude, angle bisector, and median from $A$ to $B C$, respectively. If $D E=3$ and $E F=5$, compute the length of $B C$. | 20 | 80 | 2 |
math | Example 8 Find the odd prime $p$ that satisfies the following condition: there exists a permutation $b_{1}, b_{2}, \cdots, b_{p-1}$ of $1,2, \cdots, p-1$, such that $1^{b_{1}}, 2^{b_{2}}, \cdots,(p-1)^{b_{p-1}}$ forms a reduced residue system modulo $p$. | 3 | 94 | 1 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty}\left(\frac{n+3}{n+5}\right)^{n+4}$ | e^{-2} | 41 | 4 |
math | Let's simplify the following fraction:
$$
\frac{x y^{2}+2 y z^{2}+y z u+2 x y z+2 x z u+y^{2} z+2 z^{2} u+x y u}{x u^{2}+y z^{2}+y z u+x u z+x y u+u z^{2}+z u^{2}+x y z}
$$ | \frac{y+2z}{u+z} | 92 | 11 |
math | XXVII OM - I - Zadanie 4
Samolot leci bez zatrzymywania się po najkrótszej drodze z Oslo do miasta $ X $ leżącego na równiku w Ameryce Południowej. Z Oslo startuje dokładnie w kierunku zachodnim. Wiedząc, że współrzędne geograficzne Oslo są : $ 59^{\circ}55 szerokości północnej i $ 10^{\circ}43 długości wschodniej, ob... | Quito,10000 | 212 | 8 |
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