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 | ## Task A-3.5.
How many integers can a finite set $S$ contain at most such that among any three elements of the set $S$ there are two different numbers whose sum is also in $S$? | 7 | 47 | 1 |
math |
3. Determine all pairs $(p, m)$ consisting of a prime number $p$ and a positive integer $m$, for which
$$
p^{3}+m(p+2)=m^{2}+p+1
$$
holds.
| (2,5) | 54 | 5 |
math | Points $A$, $B$, $C$, and $D$ lie on a circle. Let $AC$ and $BD$ intersect at point $E$ inside the circle. If $[ABE]\cdot[CDE]=36$, what is the value of $[ADE]\cdot[BCE]$? (Given a triangle $\triangle ABC$, $[ABC]$ denotes its area.) | 36 | 80 | 2 |
math | 4. The numbers $x_{1}, x_{2}, \cdots, x_{100}$ satisfy the following conditions: For $k=1,2, \cdots, 100, x_{k}$ is less than the sum of the other 99 numbers by $k$. Then the value of $x_{25}$ is $\qquad$ . | \frac{650}{49} | 81 | 10 |
math | 5. Let $f(x)=\frac{x^{3}}{1-3 x+3 x^{2}}$, and denote $f_{1}(x)=f(x), f_{n}(x)=f\left(f_{n-1}(x)\right)$. Then $f_{10}(x)=$ $\qquad$ | \frac{x^{3^{10}}}{x^{3^{10}}-(x-1)^{3^{10}}} | 71 | 28 |
math | 8. The function
$$
f(x)=(\sqrt{1+x}+\sqrt{1-x}+2)\left(\sqrt{1-x^{2}}+1\right)
$$
has a range of $\qquad$ . | [2+\sqrt{2},8] | 50 | 9 |
math | Find all functions $f:(1,+\infty) \rightarrow (1,+\infty)$ that satisfy the following condition:
for arbitrary $x,y>1$ and $u,v>0$, inequality $f(x^uy^v)\le f(x)^{\dfrac{1}{4u}}f(y)^{\dfrac{1}{4v}}$ holds. | f(x) = c^{\frac{1}{\ln x}} | 79 | 16 |
math | 1. Let $a_{1}, a_{2}, \cdots, a_{2015}$ be a sequence of numbers taking values from $-1, 0, 1$, satisfying
$$
\sum_{i=1}^{2015} a_{i}=5 \text {, and } \sum_{i=1}^{2015}\left(a_{i}+1\right)^{2}=3040,
$$
where $\sum_{i=1}^{n} a_{i}$ denotes the sum of $a_{1}, a_{2}, \cdots, a_{n}$.
Then the number of 1's in this sequenc... | 510 | 154 | 3 |
math | 91. (8th grade) Find all rectangles that can be cut into 13 equal squares. | 1\times13 | 22 | 5 |
math | On the meadow, there were 45 sheep and several shepherds. After half of the shepherds and a third of the sheep left, the remaining shepherds and sheep together had 126 legs. All the sheep and shepherds had the usual number of legs.
How many shepherds were originally on the meadow?
(L. Hozová)
Hint. How many sheep we... | 6 | 95 | 1 |
math | What are the integers $k$ such that for all real numbers $a, b, c$,
$$
(a+b+c)(a b+b c+c a)+k a b c=(a+b)(b+c)(c+a)
$$ | -1 | 49 | 2 |
math | Example 1 Remove the big and small jokers from a deck of cards, and randomly draw five cards from the remaining 52 cards. The probability that at least two of the cards have the same number (or letter $K, Q, J, A$) is $\qquad$ (require calculating the numerical value of this probability, accurate to 0.01).
(2014, Natio... | 0.49 | 93 | 4 |
math | [ Text problems (other) ]
Sharik and Matroskin milked 10 liters of milk, poured it into two buckets, and carried it home. Sharik got tired and poured some of the milk from his bucket into Matroskin's bucket. As a result, Sharik had three times less milk, and Matroskin had three times more milk. How much milk did Matro... | 7.5 | 87 | 3 |
math | 10. Given $f(x)=x^{5}-10 x^{3}+a x^{2}+b x+c$. If the roots of the equation $f(x)=0$ are all real, and $m$ is the largest of these five real roots, then the maximum value of $m$ is $\qquad$ | 4 | 71 | 1 |
math | 11. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{2}=\frac{1}{4}$, and $a_{n+1}=\frac{(n-1) a_{n}}{n-a_{n}}(n=2,3,4, \cdots)$.
(1) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(2) Prove that for all $n \in \mathbf{N}^{*}$, $\sum_{k=1}^{n} a_{k}^{2}<\frac{7}{6}$. | a_{n}=\frac{1}{3n-2}(n\in{N}^{*}),\sum_{k=1}^{n}a_{k}^{2}<\frac{7}{6} | 147 | 47 |
math | 5. Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, the same letters correspond to the same digits). He got the word "GUATEMALA". How many different numbers could Egor have initially written if his number was divisible by 5? | 114240 | 66 | 6 |
math | 7.2. Petya wrote down all natural numbers from 1 to $n$ in a row on the board and counted the total number of digits written. It turned out to be 777. What is $n$? | 295 | 50 | 3 |
math | Let $x_{1}=1$ and $x_{n+1}=x_{n}+\left\lfloor\frac{x_{n}}{n}\right\rfloor+2$ for $n=1,2,3, \ldots$, where $\lfloor x\rfloor$ denotes the largest integer not greater than $x$. Determine $x_{1997}$. | 23913 | 84 | 5 |
math | ## 69. Math Puzzle $2 / 71$
In a Siemens-Martin furnace, 20 t of steel with a $0.5 \%$ carbon content is melted together with 5 t of gray cast iron with a $5 \%$ carbon content.
What percent of carbon does the mixture contain? | 1.4 | 66 | 3 |
math | 7. Let $P(x)=x^{5}-x^{2}+1$ have five roots $r_{1}$,
$$
\begin{array}{l}
r_{2}, \cdots, r_{5}, Q(x)=x^{2}+1 \text {. Then } \\
\quad Q\left(r_{1}\right) Q\left(r_{2}\right) Q\left(r_{3}\right) Q\left(r_{4}\right) Q\left(r_{5}\right) \\
\quad=
\end{array}
$$ | 5 | 120 | 1 |
math | 3. If positive real numbers $a, b$ satisfy $\frac{1}{a}+\frac{1}{b} \leqslant 2 \sqrt{2},(a-b)^{2}=4(a b)^{3}$, then $\log _{a} b=$ | -1 | 62 | 2 |
math | 4. The sum of the digits of an odd four-digit number is 11. The sum of the digits in the units and tens place is 5. How many such numbers are there? | 18 | 40 | 2 |
math | In a $2 \times 8$ squared board, you want to color each square red or blue in such a way that on each $2 \times 2$ sub-board there are at least $3$ boxes painted blue. In how many ways can this coloring be done?
Note. A $2 \times 2$ board is a square made up of $4$ squares that have a common vertex. | 341 | 85 | 3 |
math | 3. Given an integer $n \geqslant 2$. Let $a_{1}, a_{2}, \cdots, a_{n}, b_{1}, b_{2}, \cdots, b_{n}>0$, satisfying
$$
a_{1}+a_{2}+\cdots+a_{n}=b_{1}+b_{2}+\cdots+b_{n} \text {, }
$$
and for any $i, j(1 \leqslant i<j \leqslant n)$, we have $a_{i} a_{j} \geqslant b_{i}+b_{j}$. Find the minimum value of $a_{1}+a_{2}+\cdot... | 2n | 167 | 2 |
math | Golovanov A.S.
Find the sum
$$
\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\left[\frac{2^{3}}{3}\right]+\cdots+\left[\frac{2^{1000}}{3}\right]
$$ | \frac{2^{1001}-2}{3}-500 | 80 | 17 |
math | 5. Find all positive integers $a, b$ such that $2^{a!}+2^{b!}$ is a cube of some positive integer. | =2,b=2 | 33 | 5 |
math | 9. (This question is worth 16 points) If real numbers $a, b, c$ satisfy $2^{a}+4^{b}=2^{c}, 4^{a}+2^{b}=4^{c}$, find the minimum value of $c$.
| \log_{2}3-\frac{5}{3} | 61 | 13 |
math | 6. Given a sequence $\left\{a_{n}\right\}$ whose terms are all non-zero. Let the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$ be $S_{n}$, and the sum of the first $n$ terms of the sequence $\left\{a_{n}^{2}\right\}$ be $T_{n}$, and $S_{n}^{2}+4 S_{n}-3 T_{n}=0\left(n \in \mathbf{Z}_{+}\right)$. The... | \frac{2^{n+1}}{n+1}-2 | 167 | 15 |
math | ## Task 2 - 320712
Kathrin's aquarium has the shape of an open-top rectangular prism. It is $80 \mathrm{~cm}$ long, $40 \mathrm{~cm}$ wide, and 42 $\mathrm{cm}$ high. The water level is $7 \mathrm{~cm}$ from the top edge.
Investigate whether it is possible to add 10 liters of water to this aquarium without it overflo... | 22.41 | 101 | 5 |
math | [Example 3.4.6] Let $x_{1}, x_{2}, x_{3}, x_{4}$ be positive real numbers, and $x_{1}+x_{2}+x_{3}+x_{4}=$ $\pi$. Find the minimum value of the expression
$$
\begin{array}{l}
\left(2 \sin ^{2} x_{1}+\frac{1}{\sin ^{2} x_{1}}\right) \cdot\left(2 \sin ^{2} x_{2}+\frac{1}{\sin ^{2} x_{2}}\right) . \\
\left(2 \sin ^{2} x_{3... | 81 | 214 | 2 |
math | $$
\begin{array}{l}
\sqrt{2}(2 a+3) \cos \left(\theta-\frac{\pi}{4}\right)+\frac{6}{\sin \theta+\cos \theta}-2 \sin 2 \theta \\
<3 a+6
\end{array}
$$
For $\theta \in\left[0, \frac{\pi}{2}\right]$, the inequality always holds. Find the range of values for $a$.
(Zhang Hongcheng, provided) | a > 3 | 111 | 4 |
math | 4[ Products and factorials
Calculate with five decimal places (i.e., to an accuracy of 0.00001) the product:
$\left(1-\frac{1}{10}\right)\left(1-\frac{1}{10^{2}}\right)\left(1-\frac{1}{10^{3}}\right) \ldots\left(1-\frac{1}{10^{99}}\right)$
# | 0.89001 | 101 | 7 |
math | 18.4.29 $\star \star$ Find the positive integer solutions to the equation
$$
3^{x}-5^{y}=2
$$ | (x,y)=(3,2) | 34 | 7 |
math | Example 2 Given
$$
\begin{array}{l}
\frac{1}{1 \times \sqrt{2}+2 \sqrt{1}}+\frac{1}{2 \sqrt{3}+3 \sqrt{2}}+\cdots+ \\
\frac{1}{n \sqrt{n+1}+(n+1) \sqrt{n}}
\end{array}
$$
is greater than $\frac{19}{20}$ and less than $\frac{20}{21}$. Then the difference between the maximum and minimum values of the positive integer $n... | 39 | 148 | 2 |
math | ## Task 1 - 310821
In a school class, every student is 13 or 14 years old; both age specifications actually occur in this class. If you add up all these (calculated as whole numbers) age specifications, the sum is 325.
Determine whether the number of students in this class is uniquely determined by these findings! If... | 24 | 93 | 2 |
math | 1. Find all pairs of positive integers $(n . k)$ so that $(n+1)^{k}-1=n$ !. | (1,1),(2,1),(4,2) | 28 | 13 |
math | 11. Let the line $y=a x-4$ be symmetric to the line $y=8 x-b$ with respect to the line $y=x$. Then $a=$ $\qquad$ ,$b=$ $\qquad$ | a=\frac{1}{8}, b=-32 | 49 | 12 |
math | 1. Do there exist natural numbers $a, b$, and $c$ such that each of the equations
$$
\begin{array}{ll}
a x^{2}+b x+c=0, & a x^{2}+b x-c=0 \\
a x^{2}-b x+c=0, & a x^{2}-b x-c=0
\end{array}
$$
has both roots as integers? | =1,b=5,=6 | 93 | 8 |
math | 133. Find all arithmetic progressions for which the sums of the first $n_{1}$ and $n_{2}\left(n_{1} \neq n_{2}\right)$ terms are equal to $n_{1}^{2}$ and $n_{2}^{2}$, respectively. | 1,3,5,7,9,\ldots | 65 | 12 |
math | Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions:
[list]
[*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square.
[*]There exists infinitely many positive integers $n$ such that $a_n=4\time... | 4043^2 | 178 | 8 |
math | 13. Teacher Li and three students, Xiao Ma, Xiao Lu, and Xiao Zhou, set off from the school one after another and walk along the same road to the cinema. The three students have the same walking speed, and Teacher Li's walking speed is 1.5 times that of the students. Now, Teacher Li is 235 meters away from the school, ... | 42 | 149 | 2 |
math | 1. Let $A B C$ be a right triangle, with hypotenuse $A C$, and let $H$ be the foot of the altitude from $B$ to $A C$. Given that the lengths $A B, B C$, and $B H$ form the sides of a new right triangle, determine the possible values of $\frac{A H}{C H}$. | \frac{\sqrt{5}+1}{2} | 81 | 12 |
math | 454. Several identical boxes together weigh 10 tons, with each of them weighing no more than 1 ton. What is the minimum number of three-ton trucks needed to haul away all this cargo in one trip? | 5 | 46 | 1 |
math | Find all positive integers $k>1$, such that there exist positive integer $n$, such that the number $A=17^{18n}+4.17^{2n}+7.19^{5n}$ is product of $k$ consecutive positive integers. | k = 2 | 60 | 5 |
math | 3. Given an equilateral triangle $ABC$ with side length 4, points $D$, $E$, and $F$ are on $BC$, $CA$, and $AB$ respectively, and $|AE|=|BF|=|CD|=1$. Connect $AD$, $BE$, and $CF$, intersecting to form $\triangle RQS$. Point $P$ moves within $\triangle RQS$ and on its sides, and the distances from $P$ to the three sides... | \frac{648\sqrt{3}}{2197} | 165 | 17 |
math | 14. $\overrightarrow{O A} 、 \overrightarrow{O B}$ are both unit vectors, $\overrightarrow{O A}=\{p, q, 0\}, \overrightarrow{O B}=\{r, s, 0\}$, and both form a $45^{\circ}$ angle with $\overrightarrow{O C}=$ $\{1,1,1\}$.
(1) Find the values of $p+q$ and $p q$; (2) Find the size of $\angle A O B$ (acute angle). | p+q=\frac{\sqrt{6}}{2},pq=\frac{1}{4},\angleAOB=60 | 125 | 28 |
math | Given a permutation $\sigma = (a_1,a_2,a_3,...a_n)$ of $(1,2,3,...n)$ , an ordered pair $(a_j,a_k)$ is called an inversion of $\sigma$ if $a \leq j < k \leq n$ and $a_j > a_k$. Let $m(\sigma)$ denote the no. of inversions of the permutation $\sigma$. Find the average of $m(\sigma)$ as $\sigma$ varies over all permutati... | \frac{n(n-1)}{4} | 107 | 10 |
math | 7. In $\triangle A B C$, it is known that
$$
\begin{array}{l}
|\overrightarrow{A B}|=\sqrt{3},|\overrightarrow{B C}|=1, \\
|\overrightarrow{A C}| \cos B=|\overrightarrow{B C}| \cos A \text {. } \\
\text { Then } \overrightarrow{A C} \cdot \overrightarrow{A B}=
\end{array}
$$ | 2 | 100 | 1 |
math | 109.
For each of two people A and B, it is known that they are either a knight or a liar. Suppose A makes the following statement: "If I am a knight, then B is a knight."
Can it be determined who A and B are: who is the knight and who is the liar?
110
A is asked: "Are you a knight?" He replies: "If I am a knight, ... | A | 110 | 1 |
math | 8. Given $x, y \in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right], a \in \mathbf{R}$, and
$$
\left\{\begin{array}{l}
x^{3}+\sin x-2 a=0, \\
4 y^{3}+\frac{1}{2} \sin 2 y+a=0 .
\end{array}\right.
$$
then the value of $\cos (x+2 y)$ is | 1 | 109 | 1 |
math | Problem 1. Determine the prime numbers $p$ for which the number $a=7^{p}-p-16$ is a perfect square.
## Lucian Petrescu | 3 | 38 | 1 |
math | 7. (10 points) For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "Six-Union Number". The smallest "Six-Union Number" greater than 2000 is $\qquad$ . | 2016 | 70 | 4 |
math |
Problem 1. Find all value of the real parameter $a$ such that the equation
$$
9^{t}-4 a 3^{t}+4-a^{2}=0
$$
has an unique root in the interval $(0,1)$.
| \in(-13,-5)\cup{\frac{2\sqrt{5}}{5}} | 56 | 21 |
math | 3.- A Tourism Office is going to conduct a survey on the number of sunny and rainy days that occur in a year. For this purpose, it turns to six regions that provide the following data:
| Region | Sunny or Rainy | Unclassifiable |
| :---: | :---: | :---: |
| A | 336 | 29 |
| B | 321 | 44 |
| C | 335 | 30 |
| D | 343 | ... | F | 192 | 1 |
math | Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares. | \sqrt{2} - \frac{1}{2} | 31 | 13 |
math | 1. Find the smallest four-digit number $\overline{a b c d}$ such that the difference $(\overline{a b})^{2}-(\overline{c d})^{2}$ is a three-digit number written with three identical digits. | 2017 | 54 | 4 |
math |
NT1 Determine all positive integer numbers $k$ for which the numbers $k+9$ are perfect squares and the only prime factors of $k$ are 2 and 3 .
| {16,27,72,216,432,2592} | 39 | 23 |
math | 3. Variant 1.
The height $A H$ and the bisector $C L$ of triangle $A B C$ intersect at point $O$. Find the angle $B A C$, if it is known that the difference between the angle $C O H$ and half the angle $A B C$ is $46^{\circ}$. | 92 | 74 | 2 |
math | 4. If the cotangents of the three interior angles $A, B, C$ of triangle $\triangle A B C$, $\cot A, \cot B, \cot C$, form an arithmetic sequence, then the maximum value of angle $B$ is $\frac{\pi}{3}$. | \frac{\pi}{3} | 62 | 7 |
math | Example 2 Let $f(x)=x^{n}, x \in D, n \in \mathbf{N}^{+}$, determine whether $f(x)$ is a solution to the functional inequality
$$
f(x)+f(1-x)>1
$$
If it is, find the domain $D$; if not, explain the reason. | (-\infty,0)\cup(1,+\infty) | 76 | 15 |
math | 13. (3 points) The judging panel of the Young Singers Grand Prix consists of several people. Each judge gives a score to the singer, with a maximum of 10 points. After the first singer performs, the scoring situation is as follows: the average score given by all judges is 9.64 points; if the highest score is removed, t... | 9.28,10 | 145 | 7 |
math | 6.35. Find the second derivative of the function $y(x)=$ $=x^{2} \ln (1+\sin x)$. | 2\ln(1+\sinx)+\frac{4x\cdot\cosx-x^2}{1+\sinx} | 32 | 28 |
math | 【Question 7】
The quotient of the two septenary (base-7) integers 454 and 5, expressed in septenary, is $\qquad$. | (65)_{7} | 37 | 7 |
math | Determine all polynomials $P(x)$ with integer coefficients which satisfies $P(n)\mid n!+2$ for all postive integer $n$. | P(x) = \pm 1 | 32 | 9 |
math | The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\pi$. Find $n$.
[asy]
size(150);
defaultpen(linewidth(0.8));
int i;
for(i=5;i>=1;i=i-1)
{
if (floor(i/2)==i/2)
{
filldraw(circle(origin,4*i),white);
}
else
{
filldraw(circle(origin,4*i),red... | 60 | 125 | 2 |
math | 9. (16 points) Let the function $f(x)=\frac{a}{x}-x$. If for any $x \in\left(\frac{1}{4}, 1\right)$, we have $f(x)\left|x-\frac{1}{2}\right| \leqslant 1$, find the range of the real number $a$. | (-\infty,\frac{17}{16}) | 80 | 13 |
math | 18. Given the quadratic function $f(x)=4 x^{2}-4 a x+\left(a^{2}-2 a+2\right)$ has a minimum value of 2 on $0 \leqslant x \leqslant 1$, find the value of $a$.
| 0 | 64 | 1 |
math | ## Task A-2.2.
Determine all pairs $(p, q)$ of prime numbers for which the quadratic equation $x^{2}+p x+q=0$ has two distinct solutions in the set of integers. | (p,q)=(3,2) | 48 | 7 |
math | To be calculated
$$
a^{3}+b^{3}+3\left(a^{3} b+a b^{3}\right)+6\left(a^{3} b^{2}+a^{2} b^{3}\right)
$$
value, if $a+b=1$. | 1 | 64 | 1 |
math | 4. Let $z$ be a complex number. If $\frac{z-2}{z-\mathrm{i}}$ (where $\mathrm{i}$ is the imaginary unit) is a real number, then the minimum value of $\mid z+3$ | is $\qquad$ .
| \sqrt{5} | 59 | 5 |
math | 13. What is the value of $\sqrt[4]{2^{20}+2^{27}+2^{31}+2^{32}+2^{37}+2^{40}}$? | 1056 | 50 | 4 |
math | 1. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Given a sequence of positive integers $\left\{a_{n}\right\}$ satisfying $a_{1}=a$, and for any positive integer $n$,
$$
a_{n+1}=a_{n}+2\left[\sqrt{a_{n}}\right] \text {. }
$$
(1) If $a=8$, find the smallest positive integer $n$ such that $a_{n}... | 5 | 157 | 1 |
math | Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$. | 942 | 69 | 3 |
math | [Coordinate method in space] [equation of a plane]
Given points $A(1 ; 0 ; 1), B(-2 ; 2 ; 1), C(2 ; 0 ; 3)$. Form the equation of the plane $ABC$.
# | 2x+3y-z-1=0 | 58 | 10 |
math | ## Task A-2.3.
How many natural numbers less than 1000000 are squares of natural numbers and give a remainder of 4 when divided by 8? | 250 | 40 | 3 |
math | 20. We can find sets of 13 distinct positive integers that add up to 2142. Find the largest possible greatest common divisor of these 13 distinct positive integers. | 21 | 40 | 2 |
math | A chessboard $1000 \times 1000$ is covered by dominoes $1 \times 10$ that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few $N$ cells of the chessboard, for which we know the position of the dominoes that cover them.
Which is the minimum $N$ such that after t... | 100,000 | 126 | 7 |
math | Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"
Claire asks, "If you picked a digit of your favorite number at random and re... | 13 | 153 | 2 |
math | The decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A$. | 9 | 32 | 1 |
math | # Problem No. 5 (15 points)
Two people are moving in the same direction. At the initial moment, the distance between them is $S_{0}=200 \mathrm{~m}$. The speed of the first, faster, pedestrian is $v_{1}=7 \mathrm{~m} / \mathrm{c}$. Determine the speed $v_{2}$ of the second pedestrian, given that after $t=5$ minutes, t... | 6or6.67 | 111 | 6 |
math | 5. Let any real numbers $x_{0}>x_{1}>x_{2}>x_{3}>0$, to make $\log _{\frac{x_{0}}{x_{1}}} 1993+$ $\log _{\frac{x_{1}}{x_{2}}} 1993+\log _{\frac{x_{2}}{x_{3}}} 1993 \geqslant k \log _{\frac{x_{0}}{x_{3}}} 1993$ always hold, then the maximum value of $k$ is | 9 | 123 | 1 |
math | Draw a circle $k$ around the center of a regular hexagon, with a radius no larger than that of the circle tangent to the sides of the hexagon, then draw a circle around each vertex of the hexagon that touches $k$ and has a diameter no larger than the side of the hexagon. The 7 circular segments together cover a part of... | \frac{2a}{3} | 109 | 8 |
math | 10,11
A plane passes through the vertex $A$ of the triangular pyramid $S A B C$, bisects the median $S K$ of triangle $S A B$, and intersects the median $S L$ of triangle $S A C$ at a point $D$ such that $S D: D L=1: 2$. In what ratio does this plane divide the volume of the pyramid? | 1:14 | 89 | 4 |
math | Task B-4.1. The lengths of the sides of 5 equilateral triangles form an arithmetic sequence. The sum of the perimeters of these triangles is $120 \mathrm{~cm}$. The sum of the areas of the smallest and the largest triangle is $10 \sqrt{3} \mathrm{~cm}^{2}$ less than the sum of the areas of the remaining three triangles... | 90\sqrt{3}\mathrm{~}^{2} | 109 | 14 |
math | \section*{Exercise 3 - 261023}
We typically represent numbers in the decimal positional system (using the base 10 and the digits \(0,1, \ldots, 9\)).
Numbers can also be represented in the binary positional system (or binary system) using the base 2 and the digits 0 and 1. To distinguish this binary representation of... | 33=[100001]_{2}\quad;\quad99=[1100011]_{2} | 199 | 30 |
math | 4. Find all positive real numbers $a$ such that there exists a positive integer $n$ and $n$ pairwise disjoint infinite sets $A_{1}, A_{2}, \cdots, A_{n}$ satisfying $A_{1} \cup A_{2} \cup \cdots \cup A_{n}=\mathbf{N}^{*}$, and for any two numbers $b>c$ in each $A_{i}$, we have $b-c \geqslant a^{i}$. | 2 | 110 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 3}\left(\frac{\sin x}{\sin 3}\right)^{\frac{1}{x-3}}$ | e^{\cot3} | 45 | 6 |
math | Example 5 Given $x, y, z \in R^{+}, x^{2}+y^{2}+z^{2}=$ 1 , find the minimum value of $\frac{x^{5}}{y^{2}+z^{2}-y z}+\frac{y^{5}}{x^{2}+y^{2}-x y}+\frac{z^{5}}{x^{2}+y^{2}-x y}$. (2005 Jiangsu Province Mathematical Olympiad Winter Camp) | \frac{\sqrt{3}}{3} | 113 | 10 |
math | 12. Let $x \in \mathbf{R}$, then the minimum value of the function $f(x)=|2 x-1|+|3 x-2|+|4 x-3|+|5 x-4|$ is | 1 | 54 | 1 |
math | 4. In $\triangle A B C$, $a+c=2 b$. Then $\tan \frac{A}{2} \cdot \tan \frac{C}{2}=$ $\qquad$ | \frac{1}{3} | 42 | 7 |
math | Let $x, y$ be positive real numbers. If \[129-x^2=195-y^2=xy,\] then $x = \frac{m}{n}$ for relatively prime positive integers $m, n$. Find $100m+n$.
[i]Proposed by Michael Tang | 4306 | 68 | 4 |
math | ## Task $8 / 79$
Determine all prime numbers $a, b, c, d$ for which the following equations hold: $a+b=c$ and $2a+b=d$. | (2,3,5,7) | 42 | 9 |
math | $C D$ is the median of triangle $A B C$. The circles inscribed in triangles $A C D$ and $B C D$ touch the segment $C D$ at points $M$ and $N$. Find $M N$, if $A C - B C = 2$. | 1 | 63 | 1 |
math | $1 \cdot 15$ Choose a 1962-digit number that is divisible by 9, and let the sum of its digits be $a$, the sum of the digits of $a$ be $b$, and the sum of the digits of $b$ be $c$. What is $c$? | 9 | 68 | 1 |
math | Example 10 Calculation
$$
\begin{array}{l}
\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2006}\right)\left(1+\frac{1}{2}+\cdots+\frac{1}{2005}\right) \\
\left(1+\frac{1}{2}+\cdots+\frac{1}{2006}\right)\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2005}\right)
\end{array}
$$ | \frac{1}{2006} | 134 | 10 |
math | A Mathlon is a competition in which there are $M$ athletic events. Such a competition was held in which only $A$, $B$, and $C$ participated. In each event $p_1$ points were awarded for first place, $p_2$ for second and $p_3$ for third, where $p_1 > p_2 > p_3 > 0$ and $p_1$, $p_2$, $p_3$ are integers. The final score... | M = 5 | 165 | 5 |
math | 34. If $a+x^{2}=2015, b+x^{2}=2016, c+x^{2}=2017$, and $a b c=24$, then the value of $\frac{a}{b c}+\frac{b}{a c}+\frac{c}{a b}-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}$ is . $\qquad$ | \frac{1}{8} | 99 | 7 |
math | 20. (2002 National High School Competition) Real numbers $a$, $b$, $c$ and a positive number $\lambda$ make $f(x)=x^{3}+a x^{2}+b \dot{x}+c$ have three real roots $x_{1}$, $x_{2}$, $x_{3}$, and satisfy
(1) $x_{2}-x_{1}=\lambda$;
(2) $x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)$.
Find the maximum value of $\frac{2 a^{3}+2... | \frac{3\sqrt{3}}{2} | 152 | 12 |
math | 11、(20 points) Given the hyperbola
$$
C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)
$$
$F_{1}, F_{2}$ are the left and right foci of the hyperbola $C$, respectively, and $P$ is a point on the right branch of the hyperbola $C$ such that $\angle F_{1} P F_{2}=\frac{\pi}{3}$. The area of $\triangle F_{1} P F_{2}$... | e=2,\lambda=2 | 258 | 7 |
math | Let's determine those prime numbers $p$ and $q$ for which $p^{q}+q^{p}$ is also a prime number. | p=2,q=3 | 31 | 6 |
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