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 | 2.1. The sine of the dihedral angle at the lateral edge of a regular quadrilateral pyramid is $\frac{15}{17}$. Find the area of the lateral surface of the pyramid if the area of its diagonal section is $3 \sqrt{34}$. | 68 | 60 | 2 |
math | Kevin is in first grade, so his teacher asks him to calculate $20+1\cdot 6+k$, where $k$ is a real number revealed to Kevin. However, since Kevin is rude to his Aunt Sally, he instead calculates $(20+1)\cdot (6+k)$. Surprisingly, Kevin gets the correct answer! Assuming Kevin did his computations correctly, what was his... | 21 | 93 | 2 |
math | We are given three non-negative numbers $A , B$ and $C$ about which it is known that $A^4 + B^4 + C^4 \le 2(A^2B^2 + B^2C^2 + C^2A^2)$
(a) Prove that each of $A, B$ and $C$ is not greater than the sum of the others.
(b) Prove that $A^2 + B^2 + C^2 \le 2(AB + BC + CA)$ .
(c) Does the original inequality follow from t... | A^2 + B^2 + C^2 \le 2(AB + BC + CA) | 140 | 23 |
math | G3.4 $P$ is a point located at the origin of the coordinate plane. When a dice is thrown and the number $n$ shown is even, $P$ moves to the right by $n$. If $n$ is odd, $P$ moves upward by $n$. Find the value of $d$, the total number of tossing sequences for $P$ to move to the point $(4,4)$. | 38 | 89 | 2 |
math | 2. Find a positive integer $n$ less than 2006 such that $2006 n$ is a multiple of $2006+n$.
(1 mark)求一個小於 2006 的正整數 $n$, 使得 $2006 n$ 是 $2006+n$ 的倍數。 | 1475 | 78 | 4 |
math | 3. The chord $A B$ of the parabola $y=x^{2}$ intersects the y-axis at point $C$ and is divided by it in the ratio $A C: C B=5: 2$. Find the abscissas of points $A$ and $B$, if the ordinate of point $C$ is 20. | (x_{A}=-5\sqrt{2},x_{B}=2\sqrt{2}),(x_{A}=5\sqrt{2},x_{B}=-2\sqrt{2}) | 77 | 43 |
math | We take turns rolling a fair die until the sum $S$ of the numbers obtained exceeds 100. What is the most likely value of $S$? | 101 | 34 | 3 |
math | Example 7 Solve the system of equations: $\left\{\begin{array}{l}x^{4}+y^{2}+4=5 y z, \\ y^{4}+z^{2}+4=5 z x, \\ z^{4}+x^{2}+4=5 x y .\end{array}\right.$ $[6]$
(61st Czech and Slovak Mathematical Olympiad) | (\\sqrt{2},\\sqrt{2},\\sqrt{2}) | 92 | 16 |
math | 7. Let the elements in set $T$ be integers between 1 and $2^{30}$, and their binary representations contain exactly two 1s. If a number is randomly selected from set $T$, then the probability that this number is divisible by 9 is $\qquad$ | \frac{5}{29} | 61 | 8 |
math | 4240 ** Let $a, b \in [0,1]$, find the maximum and minimum values of $S=\frac{a}{1+b}+\frac{b}{1+a}+(1-a)(1-b)$. | 1 | 51 | 1 |
math | Find all positive integers $n=p_1p_2 \cdots p_k$ which divide $(p_1+1)(p_2+1)\cdots (p_k+1)$ where $p_1 p_2 \cdots p_k$ is the factorization of $n$ into prime factors (not necessarily all distinct). | n = 2^r \cdot 3^s | 72 | 13 |
math | 20. When $(1+x)^{38}$ is expanded in ascending powers of $x, N_{1}$ of the coefficients leave a remainder of 1 when divided by 3 , while $N_{2}$ of the coefficients leave a remainder of 2 when divided by 3 . Find $N_{1}-N_{2}$.
(2 marks)
當 $(1+x)^{38}$ 按 $x$ 的升幕序展開時, 其中 $N_{1}$ 個系數除以 3 時餘 $1 、 N_{2}$ 個系數除以 3 時餘 2 。求 $N... | 4 | 153 | 1 |
math | Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \][i]Proposed by Aaron Lin[/i] | 504 | 77 | 3 |
math | 15. Real numbers $x_{1}, x_{2}, \cdots, x_{2001}$ satisfy $\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001$, let $y_{k}=\frac{1}{k}\left(x_{1}+\right.$ $\left.x_{2}+\cdots+x_{k}\right), k=1,2, \cdots, 2001$. Find the maximum possible value of $\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|$. | 2000 | 143 | 4 |
math | 4. A rope, the first time cut off $\frac{1}{3}$ of its total length, the second time cut off $30\%$ of the remaining part. If the parts cut off twice are 0.4 meters more than the remaining part, then the original length of the rope is $\qquad$ meters. | 6 | 70 | 1 |
math | 14. A census taker stands in front of Aunt Wang's house and asks Aunt Wang: “Your age is 40, what are the ages of the three orphans you adopted?” Aunt Wang says: “The product of their ages equals my age, and the sum of their ages equals our house number.” The census taker looks at the house number and says: “I still ca... | 14 | 98 | 2 |
math | 4. In the trapezoid $A B C D$, the following holds
$$
A B=12, B C=7, C D=8 \text { and } \measuredangle A B C=90^{\circ} .
$$
Does the bisector of the interior $\Varangle D A B$ intersect the leg $B C$ or the base $C D$? | BC | 86 | 1 |
math | 4. Let $\Varangle A O D$ with $m(\Varangle A O D)=98^{\circ}$ and $(O B,(O C$ half-lines included in the interior of $\Varangle A O D$, (OC half-line included in the interior of $\Varangle B O D$ such that
$$
a \cdot m(\Varangle A O B)=c \cdot m(\Varangle B O C) \text { and } b \cdot m(\Varangle B O C)=c \cdot m(\Vara... | (\angleAOB)=18,(\angleBOC)=30,(\angleCOD)=50 | 204 | 22 |
math | ## Task A-4.1. (4 points)
The difference of the reciprocal values of two consecutive natural numbers is $0.0 a a a \ldots=0.0 \dot{a}$. What values can the digit $a$ take? | =1or=3 | 54 | 5 |
math | 9 Given $\triangle A B C$, where $D$ is the midpoint of $A C$, $A B=3, B D=B C$, and the area of $\triangle A B C$ is 3, then the size of $\angle A$ is $\qquad$ . | \frac{\pi}{4} | 59 | 7 |
math | 3. Let $S=\{1,2, \cdots, 2005\}$. If any set of $n$ pairwise coprime numbers in $S$ contains at least one prime number, find the minimum value of $n$.
(Tang Lihua) | 16 | 61 | 2 |
math | (solved by Mathieu Aria, Jeanne Nguyen and Thomas Williams). Let $n \geqslant 3$ and $x_{1}, \ldots, x_{n-1}$ be non-negative integers. We assume:
$$
\begin{aligned}
x_{1}+x_{2}+\cdots+x_{n-1} & =n \\
x_{1}+2 x_{2}+\cdots+(n-1) x_{n-1} & =2 n-2
\end{aligned}
$$
Calculate the minimum value of:
$$
\sum_{k=1}^{n-1} k(2... | 3n^{2}-3n | 148 | 7 |
math | 19. Let real numbers $x, y$ satisfy $x^{2}+y^{2} \leqslant 5$, find the extremum of $f(x, y)=3|x+y|+|4 y+9|+|7 y-3 x-18|$. | f(x,y)_{\operatorname{max}}=27+6\sqrt{5},f(x,y)_{\}=27-3\sqrt{10} | 64 | 38 |
math | 264. System of four linear equations. Solve the following system of equations:
$$
\left\{\begin{aligned}
x+7 y+3 v+5 u & =16 \\
8 x+4 y+6 v+2 u & =-16 \\
2 x+6 y+4 v+8 u & =16 \\
5 x+3 y+7 v+u & =-16
\end{aligned}\right.
$$ | -2,2,v=-2,u=2 | 99 | 10 |
math | 3. How many distinct triangles satisfy all the following properties:
(i) all three side-lengths are a whole number of centimetres in length;
(ii) at least one side is of length $10 \mathrm{~cm}$;
(iii) at least one side-length is the (arithmetic) mean of the other two side-lengths? | 17 | 72 | 2 |
math | 10.111. The side of a regular triangle inscribed in a circle is equal to $a$. Calculate the area of the square inscribed in the same circle. | \frac{2a^2}{3} | 37 | 10 |
math | 9.8 The commission consists of a chairman, his deputy, and five other people. In how many ways can the commission members distribute among themselves the duties of chairman and deputy? | 42 | 36 | 2 |
math | 5. If a non-negative integer $m$ and the sum of its digits are both multiples of 6, then $m$ is called a "Lucky Six Number". Find the number of Lucky Six Numbers among the non-negative integers less than 2012. | 168 | 56 | 3 |
math | Find the integers $n$ such that 6 divides $n-4$ and 10 divides $n-8$. | 30k-2 | 26 | 5 |
math | 3.5.10 ** Find the largest real number $k$ such that the inequality
$$
(a+b+c)^{3} \geqslant \frac{5}{2}\left(a^{3}+b^{3}+c^{3}\right)+k a b c
$$
holds for the side lengths $a, b, c$ of any triangle. | \frac{39}{2} | 81 | 8 |
math | 1. The natural odd numbers are arranged as follows:
Line 1:
1
Line 2:
3, 5
Line 3:
7, 9, 11
Line 4:
13, 15, 17, 19
a) Write the numbers on the seventh line.
b) Find the first and last number on the 101st line.
c) Show that the sum of the numbers in the first 101 lines is a perfect square. | 5151^2 | 107 | 6 |
math | 2. Solve the equation $\sin ^{4}(2025 x)+\cos ^{2019}(2016 x) \cdot \cos ^{2018}(2025 x)=1$.
# | \frac{\pi}{4050}+\frac{\pin}{2025},n\inZ,\frac{\pik}{9},k\inZ | 53 | 36 |
math | 5. Let $z$ be a complex number with modulus 1. Then the maximum value of $\left|\frac{z+\mathrm{i}}{z+2}\right|$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{2\sqrt{5}}{3} | 67 | 12 |
math | 5. Given $a_{1}, a_{2}, \cdots, a_{10}$ and $b_{1}, b_{2}, \cdots, b_{10}$ are both permutations of $1,2, \cdots, 10$, i.e., the sets
$$
\begin{array}{l}
\left\{a_{1}, a_{2}, \cdots, a_{10}\right\} \\
=\left\{b_{1}, b_{2}, \cdots, b_{10}\right\} \\
=\{1,2, \cdots, 10\} .
\end{array}
$$
Let $c_{k}=a_{k}^{2}+b_{k}^{2}(k... | 101,61 | 248 | 6 |
math | ## Task 2
The students of a 4th grade class are sewing their own work aprons. For ten students, the teacher buys 6 meters of fabric. However, there are 30 students in the class.
a) How many meters of fabric will be needed for all 30 students?
b) How many more meters of fabric does the teacher need to buy? | 18 | 79 | 2 |
math | ## Task 4 - 090524
Determine all natural numbers $z$ for which the following conditions hold simultaneously:
(a) $z$ is odd;
(b) $z$ is divisible by 3, 5, and 7;
(c) $500<z<1000$. | 525,735,945 | 69 | 11 |
math | A4. What is the value of $\frac{10^{5}}{5^{5}}$ ? | 32 | 23 | 2 |
math | In some foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies in the government (friendship/enemyship is a symmetric relation). A triplet of ministers is called [b]uniform[/b] if all three of them are friends with each other, or all three of them are enemies. How many uniform t... | 40 | 78 | 2 |
math | For how many positive numbers less than 1000 is it true that among the numbers $2,3,4,5,6,7,8$ and 9 there is exactly one that is not its divisor?
(E. Semerádová) | 4 | 56 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow-2} \frac{x^{3}+5 x^{2}+8 x+4}{x^{3}+3 x^{2}-4}
$$ | \frac{1}{3} | 54 | 7 |
math | 17. Let $x$, $y$, $z$ be the lengths of the sides of a right triangle, with $z$ being the hypotenuse. When the inequality $x^{2}(y+z)+y^{2}(z+x)+z^{2}(x+y)>k x y z$ always holds, find the maximum value of the parameter $k$, and indicate when equality holds. | 2+3\sqrt{2} | 83 | 8 |
math | Solve the following system of equations:
$$
\begin{aligned}
x^{y} & =y^{x} \\
100^{x} & =500^{y}
\end{aligned}
$$ | 3.1815,2.3575 | 46 | 13 |
math | Problem 9.4. Find all integers $a$ such that the equation
$$
x^{4}+2 x^{3}+\left(a^{2}-a-9\right) x^{2}-4 x+4=0
$$
has at least one real root.
Stoyan Atanasov | -2,-1,0,1,2,3 | 67 | 12 |
math | While writing his essay, Sanyi glanced at his watch and realized that five times as much time had already passed as the time that was still left. $M$ minutes later, this ratio had already increased to 8. What will the ratio be after another $M$ minutes? | 17 | 58 | 2 |
math | 1. Given the lines $l_{1}: a x+2 y+6=0$, $l_{2}: x+(a-1) y+a^{2}-1=0$.
If $l_{1} \perp l_{2}$, then $a=$ $\qquad$ | \frac{2}{3} | 63 | 7 |
math | Given a set of points in space, a [i]jump[/i] consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in $10$-dimensional-space, it is possible to perform a finite number of jumps so that some two points ... | 1025 | 99 | 6 |
math | 2. Given in $\triangle A B C$, $\overrightarrow{A B} \cdot \overrightarrow{B C}=2 \overrightarrow{B C} \cdot \overrightarrow{C A}=4 \overrightarrow{C A} \cdot \overrightarrow{A B}$, then the cosine value of the largest angle in $\triangle A B C$ is | \frac{\sqrt{15}}{15} | 77 | 12 |
math | 7. Given a sequence of positive numbers $a_{1}, a_{2}, \ldots, a_{10}$, satisfying the relation $a_{n}\left(a_{n-1}+a_{n+1}\right)=2 a_{n-1} a_{n+1}\left(a_{n}+1\right)$ for $n=2,3, \ldots, 9$. Find $a_{5}$ if it is known that $a_{1}=1$ and $a_{10}=0.01$ | 0.04 | 119 | 4 |
math | 1. A bag contains counters, of which ten are coloured blue and $Y$ are coloured yellow. Two yellow counters and some more blue counters are then added to the bag. The proportion of yellow counters in the bag remains unchanged before and after the additional counters are placed into the bag.
Find all possible values of ... | 1,2,4,5,10,20 | 66 | 13 |
math | 4. The perimeter of a cardboard sheet in the shape of a square is 28 dm. How many square centimeters does its area contain? | 4900 | 30 | 4 |
math | Find the largest value of n, such that there exists a polygon with n sides, 2 adjacent sides of length 1, and all his diagonals have an integer length. | n = 4 | 36 | 5 |
math | Find all non-negative integers $ x,y,z$ such that $ 5^x \plus{} 7^y \equal{} 2^z$.
:lol:
([i]Daniel Kohen, University of Buenos Aires - Buenos Aires,Argentina[/i]) | (0, 0, 1), (0, 1, 3), (2, 1, 5) | 56 | 28 |
math | 5. (10 points) There is a special calculator. When a number is input, the calculator multiplies this number by 2, then reverses the order of the digits of the result. After adding 2, it displays the final result. If a two-digit number is input, and the final result displayed is 27, then the number initially input was .... | 26 | 81 | 2 |
math | 10. Given that
$$
\begin{aligned}
\alpha+\beta+\gamma & =14 \\
\alpha^{2}+\beta^{2}+\gamma^{2} & =84 \\
\alpha^{3}+\beta^{3}+\gamma^{3} & =584
\end{aligned}
$$
find $\max \{\alpha, \beta, \gamma\}$. | 8 | 87 | 1 |
math | Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ that satisfy
$$
f(f(f(n)))+f(f(n))+f(n)=3 n
$$
for all $n \in \mathbb{Z}_{>0}$. | f(n)=n | 65 | 4 |
math | 1. If $z=\frac{1+i}{\sqrt{2}}$, calculate the sum
$$
1+z+z^{2}+z^{3}+\ldots+z^{2006}
$$ | \frac{-1+i}{\sqrt{2}} | 45 | 11 |
math | 14. Let \( f(x) = A(x^2 - 2x)e^x - e^x + 1 \). For any \( x \leq 0 \), \( f(x) \geq 0 \) holds. Determine the range of the real number \( A \). | [-\frac{1}{2},+\infty) | 64 | 12 |
math | \section*{Problem 1 - 261011}
What is the last digit of the number
\[
z=4444^{444^{44^{4}}} ?
\] | 6 | 46 | 1 |
math | 8. Given that $n$ is a positive integer, and $7^{n}+2 n$ is divisible by 57, then the minimum value of $n$ is | 25 | 38 | 2 |
math | 6. A circle passes through vertices $A$ and $C$ of triangle $ABC$ and intersects its sides $AB$ and $BC$ at points $K$ and $T$ respectively, such that $AK: KB = 3: 2$ and $BT: TC = 1: 2$. Find $AC$, if $KT = \sqrt{6}$. | 3\sqrt{5} | 80 | 6 |
math | Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be sequences of integers such that $a_1 = 20$, $b_1 = 15$, and for $n \ge 1$, \[\left\{\begin{aligned} a_{n+1}&=a_n^2-b_n^2, \\ b_{n+1}&=2a_nb_n-b_n^2 \end{aligned}\right.\] Let $G = a_{10}^2-a_{10}b_{10}+b_{10}^2$. Determine the number of positive integer facto... | 525825 | 166 | 6 |
math | Task 5. (20 points) Solve the equation $674 \frac{1}{3}\left(x^{2}+\frac{x^{2}}{(1-x)^{2}}\right)=2023$. | -\frac{1+\sqrt{5}}{2};\frac{\sqrt{5}-1}{2} | 49 | 23 |
math | Solve the following equation:
$$
\sqrt{x+2 \sqrt{x-1}}+\sqrt{x-2 \sqrt{x-1}}=2
$$ | 1\leqqx\leqq2 | 34 | 9 |
math | 7.3. Petya bought one cupcake, two muffins, and three bagels, Anya bought three cupcakes and a bagel, and Kolya bought six muffins. They all paid the same amount of money for their purchases. Lena bought two cupcakes and two bagels. How many muffins could she have bought for the same amount she spent? | 5 | 76 | 1 |
math | 13.012. The working day has been reduced from 8 to 7 hours. By what percentage does labor productivity need to increase to ensure that, with the same rates, the wage increases by $5 \%$? | 20 | 48 | 2 |
math | Oldjuk meg az egyenletrendszert:
$$
\begin{aligned}
x^{2}-x y+y^{2} & =49(x-y) \\
x^{2}+x y+y^{2} & =76(x+y)
\end{aligned}
$$
| 0,40,-24 | 62 | 7 |
math | 2. (16 points) A truck left the village of Mirny at a speed of 40 km/h. At the same time, a car left the city of Tikhaya in the same direction as the truck. In the first hour of the journey, the car traveled 50 km, and in each subsequent hour, it traveled 5 km more than in the previous hour. How many hours will it take... | 6 | 112 | 1 |
math | 12.16. Solve the equation $2^{n} + 1 = 3^{m}$ in natural numbers. | n=1,=1orn=3,=2 | 27 | 12 |
math | For nonnegative real numbers $x,y,z$ and $t$ we know that $|x-y|+|y-z|+|z-t|+|t-x|=4$.
Find the minimum of $x^2+y^2+z^2+t^2$.
[i]proposed by Mohammadmahdi Yazdi, Mohammad Ahmadi[/i] | 2 | 79 | 1 |
math | 6. If $a^{2}+b^{2}+c^{2}=a b+b c+c a$, what is $(a+2 b-3 c)^{2009}$? | 0 | 43 | 1 |
math | 3. Usually, schoolboy Gavriil takes a minute to go up a moving escalator by standing on its step. But if Gavriil is late, he runs up the working escalator and thus saves 36 seconds. Today, there are many people at the escalator, and Gavriil decides to run up the adjacent non-working escalator. How much time will such a... | 40 | 101 | 2 |
math | \section*{Task 3 - 301013}
If the production of a company decreases by \(50 \%\) (e.g., due to the failure of part of the plant), it must subsequently be doubled, i.e., increased by \(100 \%\) to return to the initial value.
Determine a formula that allows you to calculate the percentage \(b\) from a given percentage... | \frac{100a}{100-} | 127 | 13 |
math | (12) (13 points) Given a tetrahedron $P-ABC$ with three lateral edges $PA$, $PB$, and $PC$ that are mutually perpendicular, the dihedral angles formed by the lateral faces $PAB$, $PBC$, and $PCA$ with the base $ABC$ have plane angles of $\theta_{1}$, $\theta_{2}$, and $\theta_{3}$, respectively. The area of the base $\... | \frac{8\sqrt{2}}{3} | 203 | 12 |
math | In rectangle $ABCD$, $AB=100$. Let $E$ be the midpoint of $\overline{AD}$. Given that line $AC$ and line $BE$ are perpendicular, find the greatest integer less than $AD$. | 141 | 51 | 3 |
math | ## Task 2 - 160522
Two young pioneers in their rowing boat covered a distance of $1 \mathrm{~km}$ and $200 \mathrm{~m}$ downstream in 10 minutes.
How much time did they need to row the same distance upstream, if they covered on average $40 \mathrm{~m}$ less per minute than on the way downstream? | 15 | 87 | 2 |
math | 5. Xiao Ming and Xiao Hong independently and repeatedly roll a fair die until the first 6 appears. The probability that the number of rolls by Xiao Ming and Xiao Hong differs by no more than 1 is $\qquad$ | \frac{8}{33} | 46 | 8 |
math | 4. An infinite geometric progression consists of natural numbers. It turned out that the product of the first four terms equals param1. Find the number of such progressions.
The infinite geometric progression consists of positive integers. It turned out that the product of the first four terms equals param1. Find the ... | 442 | 241 | 3 |
math | Example 7. Find all real numbers $a$ such that there exist non-negative real numbers $x_{\mathrm{k}}, k=1,2,3,4,5$, satisfying the relations
$$\begin{array}{c}
\sum_{k=1}^{5} k x_{\mathrm{k}}=a, \quad \sum_{k=1}^{5} k^{3} x_{k}=a^{2} \\
\sum_{k=1}^{5} k^{5} x_{k}=a^{3}
\end{array}$$ | 0,1,4,9,16,25 | 123 | 13 |
math | 6.71 Determine all pairs of positive integers $(n, p)$ satisfying:
$p$ is a prime, $n \leqslant 2 p$, and $(p-1)^{n}+1$ is divisible by $n^{p-1}$.
(40th International Mathematical Olympiad, 1999) | (n,p)=(1,p),(2,2),(3,3) | 72 | 14 |
math | A store in Quixajuba only sells items at prices of R $0.99, R$ 1.99, R$ 2.99, and so on. Uncle Mané made a purchase for a total of R $125.74. How many items could he have bought? | 26or126 | 67 | 6 |
math | 5. A subset $M$ of the set $S=\{1928,1929, \cdots, 1949\}$ is called "red" if and only if the sum of any two distinct elements in $M$ is not divisible by 4. Let $x$ and $y$ represent the number of red four-element subsets and red five-element subsets of $S$, respectively. Try to compare the sizes of $x$ and $y$, and ex... | x<y | 119 | 2 |
math | 6.277 \sqrt{x-1}+\sqrt{x+3}+2 \sqrt{(x-1)(x+3)}=4-2 x
$$
6.277 \sqrt{x-1}+\sqrt{x+3}+2 \sqrt{(x-1)(x+3)}=4-2 x
$$ | 1 | 75 | 1 |
math | ## SUBJECT 1
a) Show that the equality $\frac{1}{x+1}+\frac{2}{y+2}=\frac{1}{10}$ is satisfied for $x=14$ and $y=58$.
Give another example of natural numbers for which the equality is satisfied.
b) If $x$ and $y$ are positive rational numbers and $\frac{1}{x+1}+\frac{2}{y+2}=\frac{1}{10}$, calculate the value of the... | \frac{19}{10} | 133 | 9 |
math | 40. In a football tournament, each of the participating teams played against each other once. The teams scored $16,14,10,10,8,6,5,3$ points. How many teams participated in the tournament and how many points did the teams that played and finished in the top 4 positions lose? (A team gets 2 points for a win and 1 point f... | 9 | 89 | 1 |
math | Given are the sequences
\[ (..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...); (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)\]
of positive real numbers. For each integer $n$ the following inequalities hold:
\[a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})\]
\[b_n \geq \frac{1}{2} (c_{n+1} + a_{... | (2004, 26, 6) | 245 | 13 |
math | In a survey involving 500 people, it was found that $46 \%$ of those surveyed like apple, $71 \%$ like vanilla, and $85 \%$ like chocolate ice cream. Are there at least six people among those surveyed who like all three types of ice cream? | 10 | 62 | 2 |
math | 8. Arrange the $n$ consecutive positive integers from 1 to $n$ ($n>1$) in a sequence such that the sum of every two adjacent terms is a perfect square. The minimum value of $n$ is $\qquad$ . | 15 | 53 | 2 |
math | $\begin{array}{l} \text { Given } a>0, b>0, a+b=1, \\ \sqrt{2}<\sqrt{a+\frac{1}{2}}+\sqrt{b+\frac{1}{2}} \leqslant 2 .\end{array}$ | \sqrt{a+\frac{1}{2}}+\sqrt{b+\frac{1}{2}}>\frac{\sqrt{2}}{2}+\frac{\sqrt{6}}{2} | 66 | 42 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 4} \frac{\sqrt[3]{16 x}-4}{\sqrt{4+x}-\sqrt{2 x}}$ | -\frac{4\sqrt{2}}{3} | 47 | 12 |
math | Let the set $S = \{P_1, P_2, \dots, P_{12}\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called "communal" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that... | 134 | 109 | 3 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(7 ; 2 ; 4)$
$M_{2}(7 ;-1 ;-2)$
$M_{3}(-5 ;-2 ;-1)$
$M_{0}(10 ; 1 ; 8)$ | 3 | 88 | 1 |
math | 4. Determine the maximum product of natural numbers whose sum is equal to a given natural number $n$. | P=\begin{cases}3^{k},&n=3k,k\in\mathbb{N}\\2^{2}3^{k-1},&n=3k+1,k\in\mathbb{N}\\2\cdot3^{k},&n=3k+2,k\in\mathbb{N}\cup{0}\end{cases} | 21 | 81 |
math | 3. What is the smallest number, $n$, which is the product of 3 distinct primes where the mean of all its factors is not an integer? | 130 | 32 | 3 |
math | 2. $[\mathbf{3}]$ Let $f$ be a function such that $f(0)=1, f^{\prime}(0)=2$, and
$$
f^{\prime \prime}(t)=4 f^{\prime}(t)-3 f(t)+1
$$
for all $t$. Compute the 4th derivative of $f$, evaluated at 0 . | 54 | 84 | 2 |
math | Each of the three cutlets needs to be fried on a pan for five minutes on each side. Only two cutlets fit on the pan. Can all three cutlets be fried in less than 20 minutes (neglecting the time for flipping and moving the cutlets)? | 15 | 57 | 2 |
math | 10. From any point $P$ on the parabola $y^{2}=2 x$, draw a perpendicular to its directrix $l$, with the foot of the perpendicular being $Q$. The line connecting the vertex $O$ and $P$ and the line connecting the focus $F$ and $Q$ intersect at point $R$. Then the equation of the locus of point $R$ is | y^{2}=-2x^{2}+x | 85 | 12 |
math | Given positive integers $a, b, c$ satisfying $a<b<c$, find all triples $(a, b, c)$ that satisfy the equation
$$
\frac{(a, b)+(b, c)+(c, a)}{a+b+c}=\frac{1}{2}
$$
where $(x, y)$ denotes the greatest common divisor of the positive integers $x$ and $y$. | (,b,)=(,2d,3d)or(,3d,6d) | 84 | 21 |
math | 1. For the system of inequalities about real numbers $x, y$
$$
\left\{\begin{array}{l}
y \geqslant x^{2}+2 a, \\
x \geqslant y^{2}+2 a
\end{array}\right.
$$
to have a unique solution, the values of the parameter $a$ are | \frac{1}{8} | 80 | 7 |
math | 7.083. $x^{\log _{4} x-2}=2^{3\left(\log _{4} x-1\right)}$.
7.083. $x^{\log _{4} x-2}=2^{3\left(\log _{4} x-1\right)}$. | 2;64 | 75 | 4 |
math | 1.23 In triangle $ABC$, $AB=33$, $AC=21$, and $BC=m$, where $m$ is a positive integer. If there exists a point $D$ on $AB$ and a point $E$ on $AC$ such that $AD=DE=EC=n$, where $n$ is a positive integer, what value must $m$ take?
(Swedish Mathematical Competition, 1982) | 30 | 96 | 2 |
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