task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 10 1.39k | answer_tokens int64 1 98 |
|---|---|---|---|---|
math | 3. (USA) Find the integer represented by $\left[\sum_{n=1}^{10^{9}} n^{-2 / 3}\right]$. Here $[x]$ denotes the greatest integer less than or equal to $x$ (e.g. $[\sqrt{2}]=1$ ).
The text is already in English, so no translation is needed. | 2997 | 79 | 4 |
math | 8.2. If $a, b, c, d$ are positive real numbers such that $a b=2$ and $c d=27$, find the minimum value of the expression $E=(a+1)(b+2)(c+3)(d+4)$. | 600 | 61 | 3 |
math | Consider the collection of all 5-digit numbers whose sum of digits is 43. One of these numbers is chosen at random. What is the probability that it is a multiple of 11?
# | \frac{1}{5} | 42 | 7 |
math | 1. Triangle $G R T$ has $G R=5, R T=12$, and $G T=13$. The perpendicular bisector of $G T$ intersects the extension of $G R$ at $O$. Find $T O$. | \frac{169}{10} | 55 | 10 |
math | 7. The set of all natural numbers $n$ that make $3^{2 n+1}-2^{2 n+1}-6^{n}$ a composite number is | n>2 | 36 | 3 |
math | 1.1.9 ** For a subset $S$ of the set $\{1,2, \cdots, 15\}$, if the positive integer $n$ and $n+|S|$ are both elements of $S$, then $n$ is called a "good number" of $S$. If a set $S$ has an element that is a "good number", then $S$ is called a "good set". Suppose 7 is a "good number" of some "good set" $X$. How many suc... | 2^{12} | 122 | 5 |
math | Find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$. | 41 | 28 | 2 |
math | 4. If the complex coefficient equation with respect to $x$
$$
(1+2 \mathrm{i}) x^{2}+m x+1-2 \mathrm{i}=0
$$
has real roots, then the minimum value of the modulus of the complex number $m$ is $\qquad$ | 2 | 65 | 1 |
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 | ## Task 4 - 180934
In a right-angled triangle $A B C$ with the right angle at $C$, the height dropped from $C$ to the hypotenuse $A B$ divides it in the ratio $1: 3$.
Calculate the size of the interior angles at $A$ and $B$ of the triangle $A B C$! | \alpha=30,\beta=60 | 84 | 10 |
math | One, (15 points) Find the maximum value of the function
$$
y=\left[\sin \left(\frac{\pi}{4}+x\right)-\sin \left(\frac{\pi}{4}-x\right)\right] \sin \left(\frac{\pi}{3}+x\right)
$$
and the set of $x$ values at which the maximum value is attained. | \frac{3 \sqrt{2}}{4} | 87 | 12 |
math | ## Task A-2.4.
Determine all real numbers $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{2020} \geqslant 0$ for which
$$
a_{1}+a_{2}+\cdots+a_{2020}=1 \quad \text { and } \quad a_{1}^{2}+a_{2}^{2}+\cdots+a_{2020}^{2}=a_{1}
$$ | a_{1}=\cdots=a_{n}=\frac{1}{n}\quad\text{}\quada_{n+1}=\cdots=a_{2020}=0 | 122 | 40 |
math | 8. Given a six-digit decimal number composed of six positive integers, the digit in the units place is a multiple of 4, the digits in the tens and hundreds places are multiples of 3, and the sum of the digits of the six-digit number is 21. Then the number of six-digit numbers that satisfy the above conditions is $\qqua... | 126 | 74 | 3 |
math | Task B-4.1. Determine the natural number $n \geq 2$ such that the equality
$$
\frac{(n-1)^{2} n(n+1)!}{(n+2)!}=\binom{n}{2}
$$
holds. | 4 | 59 | 1 |
math | 5. (3 points) From the ten digits $0 \sim 9$, the pair of different numbers with the largest product is $\qquad$
multiplied, the pair with the smallest sum is $\qquad$ and $\qquad$ added, pairs that add up to 10 are $\qquad$ pairs. | 9,8,0,1,4 | 68 | 9 |
math | A $5\times5$ grid of squares is filled with integers. Call a rectangle [i]corner-odd[/i] if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?
Note: A rectangles must have four distinct corners to be c... | 60 | 113 | 2 |
math | 13. (6 points) There are 6 numbers arranged in a row, their average is 27, it is known that the average of the first 4 numbers is 23, the average of the last 3 numbers is 34, the 4th number is $\qquad$ .
| 32 | 65 | 2 |
math | What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$ | 100 | 40 | 3 |
math | 13.047. A musical theater announced a competition for admission to the orchestra. Initially, it was planned that the number of places for violinists, cellists, and trumpeters would be distributed in the ratio $1.6: 1: 0.4$. However, it was then decided to increase the intake, and as a result, 25% more violinists and 20... | 20 | 117 | 2 |
math | 14. Given the ellipse $C: \frac{x^{2}}{2}+y^{2}=1$ with its left and right foci being $F_{1}, F_{2}$ respectively, let $P$ be a point on the ellipse $C$ in the first quadrant. The extensions of $P F_{1}, P F_{2}$ intersect the ellipse $C$ at points $Q_{1}, Q_{2}$ respectively. Then the maximum value of the difference i... | \frac{2\sqrt{2}}{3} | 133 | 12 |
math | 2.1. Find the smallest value of $a$, for which the sum of the squares of the roots of the equation $x^{2}-3 a x+a^{2}=0$ is $0.28$. | -0.2 | 46 | 4 |
math | Determine the number of all positive ten-digit integers with the following properties:
- The number contains each of the digits 0, 1, 2, ..., 8, and 9 exactly once.
- Each digit, except for the 9, has a neighboring digit that is greater than it.
(Note. For example, in the number 1230, the digits 1 and 3 are the neigh... | 256 | 139 | 3 |
math | 1. Given that $x$ and $y$ are real numbers, and satisfy
$$
\left(x+\sqrt{x^{2}+2008}\right)\left(y+\sqrt{y^{2}+2008}\right)=2008 \text {. }
$$
Then the value of $x^{2}-3 x y-4 y^{2}-6 x-6 y+2008$
is $\qquad$ | 2008 | 98 | 4 |
math | Someone for 10 years deposits $500 \mathrm{~K}-\mathrm{t}$ in the savings bank at the beginning of each year; then for another 10 years, withdraws $500 \mathrm{~K}-\mathrm{t}$ from the savings bank at the beginning of each year. How much money will they have at the end of the 20th year? $p=4$. | 2996.4\mathrm{~K} | 91 | 12 |
math | In a tournament with 10 teams, each of them plays against each other exactly once. Additionally, there are no ties, and each team has a $50 \%$ chance of winning any match. What is the probability that, after tallying the scores from $\frac{10 \cdot 9}{2}=45$ games, no two players have the same number of wins? | \frac{10!}{2^{45}} | 81 | 12 |
math | 17. (1993 3rd Macau Mathematical Olympiad) $x_{1}, x_{2}, \cdots, x_{1993}$ satisfy
$$
\begin{array}{l}
\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{1992}-x_{1993}\right|=1993, \\
y_{k}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k}(k=1,2, \cdots, 1993) .
\end{array}
$$
Then what is the maximum possible value... | 1992 | 205 | 4 |
math | GS. 2 Let $x \geq 0$ and $y \geq 0$. Given that $x+y=18$. If the maximum value of $\sqrt{x}+\sqrt{y}$ is $d$, find the value of $d$. | 6 | 56 | 1 |
math | 11.095. A sphere is circumscribed around a regular triangular prism, the height of which is twice the side of the base. How does its volume relate to the volume of the prism? | \frac{64\pi}{27} | 43 | 11 |
math | 9. For a two-digit number $x$, 6 statements are made:
a) $x$ is divisible by 3; b) $x$ is divisible by 5;
c) $x$ is divisible by 9; d) $x$ is divisible by 15;
e) $x$ is divisible by 25; f) $x$ is divisible by 45.
Find all such $x$ for which exactly three of these statements are true. | 15,30,60 | 101 | 8 |
math | 9. Two differentiable real functions $f(x)$ and $g(x)$ satisfy
$$
\frac{f^{\prime}(x)}{g^{\prime}(x)}=e^{f(x)-g(x)}
$$
for all $x$, and $f(0)=g(2003)=1$. Find the largest constant $c$ such that $f(2003)>c$ for all such functions $f, g$. | 1-\ln2 | 97 | 4 |
math | 3: Given the function $f(x)=\sqrt{1-x}(x \leqslant 1)$. Then the coordinates of the intersection point of the function $f(x)$ and its inverse function $f^{-1}(x)$ are $\qquad$ | (1,0),(0,1),\left(\frac{-1+\sqrt{5}}{2}, \frac{-1+\sqrt{5}}{2}\right) | 55 | 37 |
math | It is given an acute triangle $ABC$ , $AB \neq AC$ where the feet of altitude from $A$ its $H$. In the extensions of the sides $AB$ and $AC$ (in the direction of $B$ and $C$) we take the points $P$ and $Q$ respectively such that $HP=HQ$ and the points $B,C,P,Q$ are concyclic.
Find the ratio $\tfrac{HP}{HA}$. | 1 | 104 | 1 |
math | 9. Find the maximum value of the function $f(x)=9 \sin x+12 \cos x$. | 15 | 24 | 2 |
math | 10.401 Find the radius of the circle if the area of the circle is $Q$ square units greater than the area of the inscribed regular dodecagon. | \sqrt{\frac{Q}{\pi-3}} | 38 | 12 |
math | Example 15 (2000 National High School Competition Question) There are $n$ people, and it is known that any 2 of them make at most one phone call. The total number of calls made among any $n-2$ of them is equal, and is equal to $3^{k}$ ($k$ is a positive integer). Find all possible values of $n$.
---
The above text is... | 5 | 104 | 1 |
math | Example 14 Let the three sides of $\triangle ABC$ be $a, b, c$ with corresponding altitudes $h_{a}$, $h_{b}$, $h_{c}$, and the radius of the incircle of $\triangle ABC$ be $r=2$. If $h_{a}+h_{b}+h_{c}=18$, find the area of $\triangle ABC$. | 12\sqrt{3} | 88 | 7 |
math | 3. On a line, consider the points $A_{0}, A_{1}, A_{2}, \ldots, A_{10}$, in this order, such that $A_{1}$ is the midpoint of $\left[A_{0} A_{2}\right]$, $A_{2}$ is the midpoint of $\left[A_{0} A_{3}\right]$, and so on, up to $A_{9}$ being the midpoint of $\left[A_{0} A_{10}\right]$. Knowing that $A_{0} A_{1}=2 \text{~c... | 341 | 207 | 3 |
math | 419. Calculate $\sin \left(-\frac{5 \pi}{3}\right)+\cos \left(-\frac{5 \pi}{4}\right)+\operatorname{tg}\left(-\frac{11 \pi}{6}\right)+$ $+\operatorname{ctg}\left(-\frac{4 \pi}{3}\right)$. | \frac{\sqrt{3}-\sqrt{2}}{2} | 80 | 15 |
math | 51. A task, if worked on by person A and B together, can be completed in 8 days; if worked on by person B and C together, it can be completed in 6 days; if worked on by person C and D together, it can be completed in 12 days; then, if worked on by person A and D together, it will take $\qquad$ days to complete. | 24 | 86 | 2 |
math | 4. Simplify $\sqrt{13+4 \sqrt{3}}+\sqrt{13-4 \sqrt{3}}$. | 4\sqrt{3} | 29 | 6 |
math | 30. Find all integers $n > 1$ such that any of its divisors greater than 1 have the form $a^{r}+1$, where $a \in \mathbf{N}^{*}, r \geqslant 2, r \in \mathbf{N}^{*}$. | S=\left\{10\right. \text{ or primes of the form } a^{2}+1, \text{ where } a \in \mathbf{N}^{*}\} | 70 | 44 |
math | 3. Option 1.
In the Ivanov family, both the mother and the father, and their three children, were born on April 1st. When the first child was born, the parents' combined age was 45 years. The third child in the family was born a year ago, when the sum of the ages of all family members was 70 years. How old is the midd... | 5 | 101 | 1 |
math | $370 \mathrm{~K}$ is to be divided among $A, B$, and $C$ such that $A$ receives as many times more than $B$ as $B$ receives more than $C$. How much does each receive if the share of $B$ and $C$ is $50 \mathrm{~K}$ more than the share of $A$? | A=160\mathrm{~K},B=120\mathrm{~K},C=90\mathrm{~K} | 83 | 32 |
math | 9. A chemistry student conducted an experiment: from a bottle filled with syrup solution, he poured out one liter of liquid, refilled the bottle with water, then poured out one liter of liquid again and refilled the bottle with water. As a result, the percentage of syrup decreased from 9 to 4 percent. Determine the vol... | 3 | 74 | 1 |
math | Question 224, Given a positive integer $n(n \geq 2)$, choose $m$ different numbers from $1, 2, \ldots, 3n$. Among these, there must be four pairwise distinct numbers $a, b, c, d$, satisfying $a=b+c+d$. Find the minimum value of $m$.
---
The translation maintains the original format and line breaks as requested. | 2n+2 | 89 | 4 |
math | Exercise 5
Calculate using a simple method.
1. $682+325$
$573+198$
2. $897+234$
$788+143$
3. $694+367$
$595+698$ | 1007,771,1131,931,1061,1293 | 70 | 27 |
math | Find the smallest positive integer $ n$ such that $ 107n$ has the same last two digits as $ n$. | 50 | 27 | 2 |
math | ## Problem Statement
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=p-3q$
$b=p+2q$
$|p|=\frac{1}{5}$
$|q|=1$
$(\widehat{p, q})=\frac{\pi}{2}$ | 1 | 69 | 1 |
math | 3. Given $A=\left\{x \mid x^{2}-m x+m^{2}-19=0\right\}, B=\left\{x \mid \log _{2}\left(x^{2}-5 x+\right.\right.$ $8)=1\}, C=\left\{x \mid x^{2}+2 x-8=0\right\}$, and $A \cap B \neq \varnothing, A \cap C=\varnothing$, find the value of $m$. | -2 | 114 | 2 |
math | Tsar Gvidon had 5 sons. Among his descendants, 100 each had exactly 3 sons, and the rest died childless.
How many descendants did Tsar Gvidon have
# | 305 | 44 | 3 |
math | 9.48 The total weight of a pile of stones is 100 kilograms, where the weight of each stone does not exceed 2 kilograms. By taking out some of the stones in various ways and calculating the difference between the sum of the weights of these stones and 10 kilograms. Among all these differences, the minimum value of their... | \frac{10}{11} | 99 | 9 |
math | 8.4. Find the natural number $x$ that satisfies the equation
$$
x^{3}=2011^{2}+2011 \cdot 2012+2012^{2}+2011^{3}
$$ | 2012 | 58 | 4 |
math | 3. The sum of positive numbers $a, b, c$ and $d$ does not exceed 4. Find the maximum value of the expression
$$
\sqrt[4]{2 a^{2}+a^{2} b}+\sqrt[4]{2 b^{2}+b^{2} c}+\sqrt[4]{2 c^{2}+c^{2} d}+\sqrt[4]{2 d^{2}+d^{2} a}
$$ | 4\sqrt[4]{3} | 103 | 8 |
math | Solve the following equation:
$$
\frac{x-49}{50}+\frac{x-50}{49}=\frac{49}{x-50}+\frac{50}{x-49}
$$ | 0,99,\frac{4901}{99} | 51 | 15 |
math | Two years ago Tom was $25\%$ shorter than Mary. Since then Tom has grown $20\%$ taller, and Mary has grown $4$ inches taller. Now Mary is $20\%$ taller than Tom. How many inches tall is Tom now? | 45 | 59 | 2 |
math | Example. Compute the limit
$$
\lim _{x \rightarrow 3} \frac{x^{3}-4 x^{2}-3 x+18}{x^{3}-5 x^{2}+3 x+9}
$$ | \frac{5}{4} | 51 | 7 |
math | Example 11 Find the least common multiple of $8127, 11352, 21672$ and 27090. | 3575880 | 38 | 7 |
math | 5. The terms of the sequence $\left\{a_{n}\right\}$ are all positive, and the sum of the first $n$ terms $S_{n}$ satisfies
$$
S_{n}=\frac{1}{2}\left(a_{n}+\frac{1}{a_{n}}\right) .
$$
Then $a_{n}=$ | a_{n}=\sqrt{n}-\sqrt{n-1} | 79 | 14 |
math | 1. The measures of the angles formed around a point $O$ are expressed (in degrees) by powers of the number 5. Find the minimum number of angles under the given conditions. | 8 | 39 | 1 |
math | 2B. In a certain populated place, all telephone numbers consist of six digits arranged in strictly ascending or strictly descending order, with the first digit in the number not being 0. What is the maximum number of telephone numbers that can exist in this place? | 294 | 52 | 3 |
math | Let $t$ be TNYWR.
Suppose that
$$
\frac{1}{2^{12}}+\frac{1}{2^{11}}+\frac{1}{2^{10}}+\cdots+\frac{1}{2^{t+1}}+\frac{1}{2^{t}}=\frac{n}{2^{12}}
$$
(The sum on the left side consists of $13-t$ terms.)
What is the value of $n$ ? | 127 | 103 | 3 |
math | 5. Three military trains passed through the railway station. The first had 462 soldiers, the second had 546, and the third had 630. How many cars were in each train if it is known that each car had the same number of soldiers and that this number was the largest possible? | 11,13,15 | 66 | 8 |
math | What is the value of $a+b+c+d$, if
$$
\begin{gathered}
6 a+2 b=3848 \\
6 c+3 d=4410 \\
a+3 b+2 d=3080
\end{gathered}
$$ | 1986 | 63 | 4 |
math | We call a positive integer $n$ $\textit{sixish}$ if $n=p(p+6)$, where $p$ and $p+6$ are prime numbers. For example, $187=11\cdot17$ is sixish, but $475=19\cdot25$ is not sixish. Define a function $f$ on the positive integers such that $f(n)$ is the sum of the squares of the positive divisors of $n$. For example, $f(10)... | g(x) = x^2 + 2x + 37 | 266 | 15 |
math | ## Problem 3
For the non-zero natural number $n$, there exists a natural number $k, k \geq 2$, and positive rational numbers $a_{1}, a_{2}, \ldots, a_{k}$ such that $a_{1}+a_{2}+\ldots+a_{k}=a_{1} a_{2} \cdot \ldots \cdot a_{k}=n$. Determine all possible values of the number $n$. | n\in\mathbb{N}^{*}-{1,2,3,5} | 100 | 20 |
math | 9. (16 points) Given the function
$$
\begin{array}{l}
f(x)=a x^{3}+b x^{2}+c x+d(a \neq 0), \\
\text { when } 0 \leqslant x \leqslant 1 \text {, }|f^{\prime}(x)| \leqslant 1 .
\end{array}
$$
Try to find the maximum value of $a$. | \frac{8}{3} | 104 | 7 |
math | One, (20 points) Given that $a$, $b$, and $c$ satisfy the system of equations
$$
\left\{\begin{array}{l}
a+b=8, \\
a b-c^{2}+8 \sqrt{2} c=48 .
\end{array}\right.
$$
Try to find the roots of the equation $b x^{2}+c x-a=0$. | x_{1}=\frac{-\sqrt{2}+\sqrt{6}}{2}, x_{2}=-\frac{\sqrt{2}+\sqrt{6}}{2} | 91 | 40 |
math | 8.1. In the wagon, several kilograms of apple jam were loaded, of which $20 \%$ was good and $80 \%$ was bad. Every day, half of the existing bad jam rotted, and it was thrown away. After several days, it turned out that $20 \%$ of the jam in the wagon was bad and $80 \%$ was good. How many days have passed since the l... | 4 | 92 | 1 |
math | 6. Find the largest ten-digit number of the form $\overline{a_{9} a_{8} a_{7} a_{6} a_{5} a_{4} a_{3} a_{2} a_{1} a_{0}}$, possessing the following property: the digit equal to $\mathrm{a}_{\mathrm{i}}$ appears in its representation exactly $\mathrm{a}_{9-\mathrm{i}}$ times (for example, the digit equal to $\mathrm{a}_... | 8888228888 | 132 | 10 |
math | 15. Let $S=\{1,2,3, \cdots, 65\}$. Find the number of 3-element subsets $\left\{a_{1}, a_{2}, a_{3}\right\}$ of $S$ such that $a_{i} \leq a_{i+1}-(i+2)$ for $i=1,2$. | 34220 | 84 | 5 |
math | 2. Draw two tangent lines to the circle $x^{2}+y^{2}=1$ through the point $(1,2)$. Then the area of the quadrilateral formed by these two tangent lines with the $x$-axis and $y$-axis is $\qquad$ | \frac{13}{8} | 61 | 8 |
math | 3. Let $R$ be the triangular region (including the boundaries of the triangle) in the plane with vertices at $A(4,1), B(-1,-6), C(-3,2)$. Find the maximum and minimum values of the function $4x - 3y$ as $(x, y)$ varies over $R$. (You must prove your assertion)
---
To find the maximum and minimum values of the functio... | 14-18 | 1,386 | 5 |
math | 12. The sum of a set of numbers is the sum of all its elements. Let $S$ be a set composed of positive integers not exceeding 15, such that the sums of any two disjoint subsets of $S$ are not equal, and among all sets with the above property, the sum of $S$ is the largest. Find the set $S$ and its sum. | 61 | 81 | 2 |
math | A train approaching at a speed of $20 \mathrm{~m} / \mathrm{s}$ sounded its horn at the railway crossing. We heard the horn 4 seconds before the train arrived. How far was the train when it started to sound the horn? (The speed of sound is $340 \mathrm{~m} / \mathrm{s}$.) | 85\mathrm{~} | 76 | 7 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(2 ; 3 ; 1)$
$M_{2}(4 ; 1 ;-2)$
$M_{3}(6 ; 3 ; 7)$
$M_{0}(-5 ;-4 ; 8)$ | 11 | 89 | 2 |
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 | Problem 1. Find the last digit of the number $n=2^{0}+2^{1}+2^{2}+\ldots+2^{2014}$. | 7 | 40 | 1 |
math | Example 1 Let $x_{1}, x_{2}, \cdots, x_{n} \geqslant 0$ and $\sum_{i=1}^{n} x_{i} \geqslant k$, where $k(k \geqslant 1)$ is a positive constant. Find
$$
f\left(x_{1}, x_{2}, \cdots, x_{n}\right)=\frac{x_{1} \sqrt{\sum_{i=1}^{n} x_{i}}}{\left(\sum_{i=1}^{n-1} x_{i}\right)^{2}+x_{n}}
$$
the maximum value. | \frac{\sqrt{k}}{2 \sqrt{k}-1} | 150 | 14 |
math | Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum po... | 5 \text{ consecutive primes} | 107 | 7 |
math | 2. A natural number, not ending in zero, had one of its digits erased. As a result, the number decreased by 6 times. Find all numbers for which this is possible. | 108or12awhen=1,2,3,4 | 39 | 16 |
math | Find all functions $f: \mathbb{N} \mapsto \mathbb{N}$ so that for any positive integer $n$ and finite sequence of positive integers $a_0, \dots, a_n$, whenever the polynomial $a_0+a_1x+\dots+a_nx^n$ has at least one integer root, so does \[f(a_0)+f(a_1)x+\dots+f(a_n)x^n.\]
[i]Proposed by Sutanay Bhattacharya[/i] | f(n) = n | 109 | 6 |
math | 9. Given that $\alpha, \beta$ are the two roots of the quadratic equation $2 x^{2}-t x-2=0$ with respect to $x$, and $\alpha<\beta$, if the function $f(x)=$ $\frac{4 x-t}{x^{2}+1}$.
(1) Find the value of $\frac{f(\alpha)-f(\beta)}{\alpha-\beta}$;
(2) For any positive numbers $\lambda_{1}, \lambda_{2}$, prove that $\lef... | 2|\alpha-\beta| | 190 | 6 |
math | Let's determine the sum of the first $n$ terms of the following sequence:
$$
1+7+11+21+\cdots+\left[n(n+1)+(-1)^{n}\right]
$$ | S_{n}=\frac{n(n+1)(n+2)}{3}+\frac{(-1)^{n}-1}{2} | 47 | 31 |
math | 5. Given two propositions, proposition $p$ : the function $f(x)=\log _{a} x(x>0)$ is monotonically increasing; proposition $q$ : the function $g(x)=x^{2}+a x+1>0$ $(x \in \mathbf{R})$. If $p \vee q$ is a true proposition, and $p \wedge q$ is a false proposition, then the range of real number $a$ is $\qquad$ . | (-2,1]\cup[2,+\infty) | 109 | 13 |
math | 1.1.3 $\star \star$ A 4-element real number set $S$ has the sum of the elements of all its subsets equal to 2008 (here the sum of elements of the empty set is considered to be 0). Find the sum of all elements of $S$. | 251 | 64 | 3 |
math | In an urn, there are 5 white and 4 blue balls, and in another, 2 white and 8 blue. We draw one ball from each urn and, without looking at them, put them into a third, empty urn. What is the probability that drawing one ball from the third - containing two balls - urn, it will be white? | \frac{17}{45} | 73 | 9 |
math | 4.1. Given an arithmetic progression $a_{1}, a_{2}, \ldots, a_{100}$. It is known that $a_{3}=9.5$, and the common difference of the progression $d=0.6$. Find the sum $\left\{a_{1}\right\}+\left\{a_{2}\right\}+\ldots+\left\{a_{100}\right\}$. The notation $\{x\}$ represents the fractional part of the number $x$, i.e., t... | 50 | 170 | 2 |
math | Problem 3. Each student's mentor gave them 2 apples, and 19 apples remained in the basket. How many students and how many apples are there, if their total sum is 100? | 27 | 44 | 2 |
math | ## Task 6
All 21 students in class 2b participated in a waste collection. 15 students collected waste paper and 18 students collected glass.
How many students collected both waste paper and glass? | 12 | 46 | 2 |
math | Find positive reals $a, b, c$ which maximizes the value of $a+ 2b+ 3c$ subject to the constraint that $9a^2 + 4b^2 + c^2 = 91$ | \frac{1}{3} + 2 \cdot \frac{3}{2} + 3 \cdot 9 = \frac{1}{3} + 3 + 27 = 30.333 | 53 | 49 |
math | How many $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$? (The leading digit of $N$ cannot be zero.)
[i]Based on a proposal by Yannick Yao[/i] | 256 | 74 | 3 |
math | G3.2 Let $n$ be the integral part of $\frac{1}{\frac{1}{1980}+\frac{1}{1981}+\cdots+\frac{1}{2009}}$. Find the value of $n$. | 66 | 59 | 2 |
math | 10.5. Find all pairs $(x ; y)$ of real numbers that satisfy the conditions: $x^{3}+y^{3}=1$ and $x^{4}+y^{4}=1$. | (0;1);(1;0) | 46 | 10 |
math | One. (20 points) A batch of goods is prepared to be transported to a certain place, and there are three trucks, A, B, and C, available for hire. It is known that the cargo capacity of trucks A, B, and C remains constant each time, and trucks A and B would need $2a$ and $a$ trips, respectively, to transport this batch o... | 2160 \text{ yuan, } 4320 \text{ yuan, } 4320 \text{ yuan} | 195 | 31 |
math | 4A. Calculate the value of the expression
$$
\cos x \cos 2 x \cos 3 x \ldots \cos 1010 x \text {, for } x=\frac{2 \pi}{2021}
$$ | -\frac{1}{2^{1010}} | 56 | 12 |
math | Find all polynomials $P(x)$ with integral coefficients whose values at points $x = 1, 2, . . . , 2021$ are numbers $1, 2, . . . , 2021$ in some order. | P(x) = x + (x-1)(x-2)\cdots(x-2021)R(x) | 56 | 28 |
math | 6. Let $a, b, c, d \in\{-1,0,1\}$. If the ordered array $(a, b, c, d)$ satisfies that $a+b$, $c+d$, $a+c$, and $b+d$ are all distinct, then $(a, b, c, d)$ is called a "good array". The probability that an ordered array $(a, b, c, d)$ is a good array among all possible ordered arrays $(a, b, c, d)$ is $\qquad$ | \frac{16}{81} | 114 | 9 |
math | I2.1 Determine the positive real root, $\alpha$, of $\sqrt{(x+\sqrt{x})}-\sqrt{(x-\sqrt{x})}=\sqrt{x}$. | \frac{4}{3} | 37 | 7 |
math | Let $A, B, C$ be unique collinear points$ AB = BC =\frac13$. Let $P$ be a point that lies on the circle centered at $B$ with radius $\frac13$ and the circle centered at $C$ with radius $\frac13$ . Find the measure of angle $\angle PAC$ in degrees. | 30^\circ | 75 | 4 |
math | Let $a_0, a_1,\dots, a_{19} \in \mathbb{R}$ and $$P(x) = x^{20} + \sum_{i=0}^{19}a_ix^i, x \in \mathbb{R}.$$ If $P(x)=P(-x)$ for all $x \in \mathbb{R}$, and $$P(k)=k^2,$$ for $k=0, 1, 2, \dots, 9$ then find $$\lim_{x\rightarrow 0} \frac{P(x)}{\sin^2x}.$$ | -(9!)^2 + 1 | 138 | 8 |
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