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math | [ Trigonometric equations ] [ Divisibility of numbers. General properties ]
Authors: Begun $\underline{\text { A.V. }}, \underline{\text { Goryashin D.V. }}$
What is the maximum number of factors of the form $\sin \frac{n \pi}{x}$ that can be crossed out in the left-hand side of the equation
$\sin \frac{\pi}{x} \sin... | 1007 | 140 | 4 |
math | 6.84. Find the curvature, radius of curvature, and coordinates of the center of curvature of the curve $x=\alpha(t)=t-\sin t, y=\beta(t)=1-\cos t$ at the point $t=\pi / 2$. | K(\frac{\pi}{2})=\frac{1}{2\sqrt{2}},R(\frac{\pi}{2})=2\sqrt{2},X(\frac{\pi}{2})=\frac{\pi}{2}+1,Y(\frac{\pi}{2})=-1 | 55 | 61 |
math | In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$ | 110 | 82 | 3 |
math | Consider the quadratic equation $a x^{2}+b x+c=0$. If we substitute $y+k$ for $x$, we obtain a quadratic equation in terms of $y$. Determine $k$ such that the product of the roots of the two equations is equal. Furthermore, calculate the sum of the four roots of the two equations in this case. | -\frac{2b}{}or0 | 74 | 9 |
math | 5. Let $x, y$ be real numbers, then the maximum value of $\frac{2 x+\sqrt{2} y}{2 x^{4}+4 y^{4}+9}$ is $\qquad$ | \frac{1}{4} | 48 | 7 |
math | A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.) | 1 | 66 | 1 |
math | 19.2.1 * Find the sum of all positive rational numbers less than 10 that, when written as reduced fractions, have a denominator of 30. | 400 | 36 | 3 |
math | 6.3. In each of the three chests, Ali-Baba found gold and silver coins; in total, there were 40 gold and 40 silver coins. In the first chest, there were 7 more gold coins than silver coins, in the second chest, there were 15 fewer silver coins than gold coins. Which type of coin is more in the third chest and by how ma... | 22 | 88 | 2 |
math | 74. Let the set $A=\{1,2,3, \cdots, 1997\}$, for any 999-element subset $X$ of $A$, if there exist $x, y \in X$, such that $x<y$ and $x \mid y$, then $X$ is called a good set. Find the largest natural number $a(a \in A)$, such that any 999-element subset containing $a$ is a good set. | 665 | 108 | 3 |
math | 2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group. | 11 | 47 | 4 |
math | ### 5.17. Compute the limit
$$
\lim _{x \rightarrow \infty}\left(\frac{x^{3}+1}{x^{3}-1}\right)^{x^{3}}
$$ | e^2 | 48 | 3 |
math | Example 5. Let $a_{n}=5 n^{4}+2 n^{3}+4 n^{2}+3 n-3$, find $S_{\mathrm{n}}$. | S_{n}=n^{5}+3 n^{4}+4 n^{3}+4 n^{2}-n | 42 | 27 |
math | 27. Find all perfect numbers whose prime factorization includes each prime to an odd power. (Recall that a natural number is called perfect if it is equal to the sum of all its natural divisors, less than the number itself - for example, $28=1+2+4+7+14$.) | 6 | 68 | 1 |
math | 2. Given that the base of a right parallelepiped is a rhombus, and the areas of the two diagonal faces are $Q_{1}$ and $Q_{2}$, find the lateral surface area of this right parallelepiped. | 2\sqrt{Q_{1}^{2}+Q_{2}^{2}} | 51 | 19 |
math | ## Task 4 - 070734
Given the equation
$$
\frac{x}{2}+\frac{x}{3}+7=x-\frac{3}{4}
$$
In this equation, the addend 7 is to be replaced by another number so that $x=11$ satisfies the equation.
What is this number? | \frac{13}{12} | 76 | 9 |
math | Problem 3. Two equal squares with an area of $100 \mathrm{~cm}^{2}$ are given. The side of one square is increased by $2 \mathrm{~cm}$, and the perimeter of the other square by $16 \mathrm{~cm}$. Which of the resulting squares will have a larger area and by how much? | 52\mathrm{~}^{2} | 77 | 10 |
math | A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$. | 7 | 64 | 1 |
math | # Problem 7. (4 points)
$O A B C$ is a rectangle on the Cartesian plane, with sides parallel to the coordinate axes. Point $O$ is the origin, and point $B$ has coordinates $(11; 8)$. Inside the rectangle, a point $X$ with integer coordinates is taken. What is the smallest value that the area of triangle $O B X$ can ta... | \frac{1}{2} | 96 | 7 |
math | 3. $k \in \mathbf{R}$, the range of real roots $x$ for the equation (in terms of $x$) $x^{4}-2 k x^{2}+k^{2}+2 k-3=0$ is $\qquad$. | [-\sqrt{2},\sqrt{2}] | 61 | 11 |
math | 6.190. $\left\{\begin{array}{l}x y=a, \\ y z=b, \quad a b c>0 . \\ z x=c,\end{array}\right.$ | x_{1,2}=\\sqrt{\frac{}{b}};y_{1,2}=\\sqrt{\frac{}{}};z_{1,2}=\\sqrt{\frac{}{}} | 44 | 42 |
math | A competition involving $n\ge 2$ players was held over $k$ days. In each day, the players received scores of $1,2,3,\ldots , n$ points with no players receiving the same score. At the end of the $k$ days, it was found that each player had exactly $26$ points in total. Determine all pairs $(n,k)$ for which this is possi... | (25,2),(12,4),(3,13) | 88 | 16 |
math | 28. Write down a four-digit number where each subsequent digit is 1 greater than the previous one, then write the number with the same digits but in reverse order and subtract the smaller number from the larger one. Repeat this several times with different numbers and compare the results. Solve the problem in general t... | 3087 | 68 | 4 |
math | 14.28. In how many different ways can 1000000 be represented as a product of three natural numbers? Products that differ only in the order of the factors are considered the same.
## 14.5. Inequalities for binomial coefficients | 139 | 59 | 3 |
math | 1. For which $a$
1) the equation $f(f(x))=x$ has an infinite number of solutions;
2) the equation $f(f(f(f(x))))=x$ has an infinite number of solutions for functions of the form
$$
f(x)=\frac{(2 a+8) x-5}{2 x-a} ?
$$ | =-8,=-2,=-\frac{22}{5} | 75 | 15 |
math | 3B. Given the formula
$$
y=x^{2}-2(k-1) x+k^{2}, k \in \mathbb{R}
$$
a set of parabolas is defined. Determine the geometric locus of the vertices of the parabolas. | 2x+1 | 57 | 4 |
math | Example 3. Point $M$ lies inside a triangle, $k_{1}, k_{2}, k_{3}$ are the distances from $M$ to the sides of the triangle, $h_{1}, h_{2}, h_{3}$ are the corresponding heights. Find the minimum value of the expression
$$
\left(\frac{k_{1}}{h_{1}}\right)^{\alpha}+\left(\frac{k_{2}}{h_{2}}\right)^{\alpha}+\left(\frac{k_... | \frac{1}{3^{\alpha-1}} | 143 | 12 |
math | 7. $\cos ^{2} 10^{\circ}+\cos ^{2} 50^{\circ}-\sin 40^{\circ} \sin 80^{\circ}=$ | \frac{3}{4} | 47 | 7 |
math | Let \[S = 1 + \frac 18 + \frac{1\cdot 5}{8\cdot 16} + \frac{1\cdot 5\cdot 9}{8\cdot 16\cdot 24} + \cdots + \frac{1\cdot 5\cdot 9\cdots (4k+1)}{8\cdot 16\cdot 24\cdots(8k+8)} + \cdots.\] Find the positive integer $n$ such that $2^n < S^{2007} < 2^{n+1}$. | 501 | 138 | 3 |
math | 9.2. On a line, there are blue and red points, with no fewer than 5 red points. It is known that on any segment with endpoints at red points, containing a red point inside, there are at least 4 blue points. And on any segment with endpoints at blue points, containing 3 blue points inside, there are at least 2 red point... | 4 | 104 | 1 |
math | B4. Emile stands in a circle with nine other people. Each of the ten people thinks of an integer (which can also be negative) and whispers this number to both of their neighbors. Then, each person loudly states the average of the two numbers they heard from their neighbors. It turns out that Emile says the number 10, h... | 5 | 119 | 1 |
math | $1 \cdot 19$ For a given $n \in N$, find the number of all different natural number triples whose sum is $6 n$.
The above text has been translated into English, preserving the original text's line breaks and format. | 3n^2 | 53 | 4 |
math | 72. Find, based on geometric considerations, the limit
$$
\lim _{n \rightarrow \infty} 2^{n} \sqrt{\underbrace{2-\sqrt{2+\sqrt{2+\sqrt{2+\ldots+V \sqrt{2}}}}}_{(n-1) \text { twos }}}
$$ | \pi | 75 | 2 |
math | An $\textrm{alien}$ script has $n$ letters $b_1,b_2,\dots,b_n$. For some $k<n/2$ assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$-words. Such a $k$-word is considered $\textbf{sacred}$ if:
i. no letter appears twice and,
ii. if a letter $b_i$ app... | 600 | 273 | 3 |
math | 7. The sum of all positive integers $n$ that satisfy $\frac{1}{4}<\sin \frac{\pi}{n}<\frac{1}{3}$ is $\qquad$ . | 33 | 42 | 2 |
math | ## Problem I - 1
Given a right isosceles triangle $A B C$, with the right angle at $C$, and the legs of length 2. An arc of a circle $l$ with center $A$ divides the triangle into two parts of equal area, while the arc of a circle $m$ with center at $B$ is tangent to the arc $l$ at a point on the hypotenuse $A B$.
Fin... | 2\sqrt{\pi}-\pi | 116 | 8 |
math | Two 10-digit integers are called neighbours if they differ in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbours. | 9 \cdot 10^8 | 73 | 8 |
math | 12. If the sum of the digits of a natural number $a$ equals 7, then $a$ is called a "lucky number". Arrange all lucky numbers in ascending order as $a_{1}, a_{2}, a_{3}, \cdots$, if $a_{n}=$ 2005, then $a_{5 n}=$ $\qquad$ | 52000 | 82 | 5 |
math | 15. Let $f_{1}(x)=\frac{2}{1+x}, f_{n+1}(x)=f_{1}\left(f_{n}(x)\right)$, and $a_{n}=\frac{f_{n}(0)-1}{f_{n}(0)+2}$. Then $a_{2014}=$ | -(\frac{1}{2})^{2015} | 77 | 14 |
math | Find the number of integer solutions of the equation
$x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$ | 0 | 70 | 3 |
math | 1572. Let the germination rate of rye seeds be $90 \%$. What is the probability that out of 7 sown seeds, 5 will germinate? | 0.124 | 39 | 5 |
math | 8 Real numbers $x, y, z$ satisfy $x^{2}+y^{2}+z^{2}=1$, then the maximum value of $xy+yz$ is
$\qquad$ . | \frac{\sqrt{2}}{2} | 45 | 10 |
math | 12.354. The base of the pyramid is an acute isosceles triangle, with the base equal to $a$ and the opposite angle equal to $\alpha$. The lateral edge of the pyramid, passing through the vertex of this angle, forms an angle $\beta$ with the plane of the base. Find the volume of the pyramid if the height of the pyramid p... | \frac{^{3}}{12}\operatorname{ctg}\alpha\operatorname{ctg}\frac{\alpha}{2}\operatorname{tg}\beta | 92 | 36 |
math | **Problem 3** Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for all $x, y \in \mathbf{R}$, we have
$$
f(1+x y)-f(x+y)=f(x) f(y),
$$
and $f(-1) \neq 0$. | f(x)=x-1 | 78 | 6 |
math | ## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(-7 ; 1 ;-4)$
$B(8 ; 11 ;-3)$
$C(9 ; 9 ;-1)$ | x-2y+2z+17=0 | 61 | 12 |
math | Kozhevnikov P.A.
There are 2013 cards with the digit 1 written on them, and 2013 cards with the digit 2 written on them. Vasya arranges these cards to form a 4026-digit number. In one move, Petya can swap two cards and pay Vasya 1 ruble. The process ends when Petya gets a number that is divisible by 11. What is the ma... | 5 | 120 | 1 |
math | 9th CanMO 1977 Problem 5 A right circular cone has base radius 1. The vertex is K. P is a point on the circumference of the base. The distance KP is 3. A particle travels from P around the cone and back by the shortest route. What is its minimum distance from K? | \frac{3}{2} | 68 | 7 |
math | Find all continuously differentiable functions $f:[0,1]\to(0,\infty)$ such that $\frac{f(1)}{f(0)}=e$ and
$$\int^1_0\frac{\text dx}{f(x)^2}+\int^1_0f'(x)^2\text dx\le2.$$ | f(x) = \sqrt{2x + \frac{2}{e^2 - 1}} | 75 | 22 |
math | Daddy decided to give his son Mojmir a monthly allowance. Mojmir received his first allowance in January. Daddy increased the allowance by 4 Kč every month. If Mojmir didn't spend any, he would have 900 Kč after the twelfth allowance before Christmas. How many Kč did Mojmir receive for his first allowance in January?
... | 53 | 81 | 2 |
math | 6. In $\triangle A B C$,
$$
\tan A 、(1+\sqrt{2}) \tan B 、 \tan C
$$
form an arithmetic sequence. Then the minimum value of $\angle B$ is $\qquad$ | \frac{\pi}{4} | 54 | 7 |
math | In a regular truncated quadrilateral pyramid with lateral edges $A A 1, B B 1, C C 1, D D 1$, the side of the upper base $A 1 B 1 C 1 D 1$ is 1, and the side of the lower base is 7. A plane passing through the edge $B 1 C 1$ perpendicular to the plane $A D 1 C$ divides the pyramid into two parts of equal volume. Find t... | \frac{38}{\sqrt{5}} | 108 | 11 |
math | ## Task 3 - 070813
Three athletes started simultaneously and ran $100 \mathrm{~m}$. When the first one reached the finish line, the second one still had exactly $10 \mathrm{~m}$ to go. When the second one reached the finish line, the third one still had exactly $10 \mathrm{~m}$ to go.
How far was the third one from t... | 19\mathrm{~} | 119 | 7 |
math | Subject (1). Determine the natural number $a=\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$, where $p, q$, and $r$ are positive prime numbers.
## LUCIAN PETRESCU | 6 | 59 | 1 |
math | 7. Let $a, b, c, d$ all be prime numbers, and $a>3b>6c>12d, a^{2}-b^{2}+c^{2}-d^{2}=1749$. Find all possible values of $a^{2}+$ $b^{2}+c^{2}+d^{2}$. | 1999 | 80 | 4 |
math | 11. (5 points) Place 100 ping-pong balls into 26 boxes arranged in a row from left to right. If the leftmost box contains 4 ping-pong balls, and the sum of the number of ping-pong balls in any 4 consecutive boxes is 15, then the number of ping-pong balls in the rightmost box is $\qquad$. | 6 | 83 | 1 |
math | Problem 8.1. In a box, there are oranges, pears, and apples, a total of 60 fruits. It is known that there are 3 times more apples than non-apples, and there are 5 times fewer pears than non-pears. How many oranges are in the box? | 5 | 66 | 1 |
math | 15. Let the sequence of positive numbers $a_{0}, a_{1}, a_{2}, \cdots, a_{n}, \cdots$ satisfy $\sqrt{a_{n} a_{n-2}}-\sqrt{a_{n-1} a_{n-2}}=2 a_{n-1}(n \geqslant 2)$, and $a_{0}=a_{1}=1$. Find the general term formula for $\left\{a_{n}\right\}$. | a_{n}={\begin{pmatrix}1,n=1,\\\prod_{k=1}^{n}(2^{k}-1)^{2},n\in{N},n\geqslant20\end{pmatrix}.} | 111 | 56 |
math | 6. (8 points) On the board, 34 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 34 minutes? | 561 | 69 | 3 |
math | Three, (20 points) Find all possible values of the positive integer $n$ such that for such $n$, there exist real numbers $a$ and $b$ for which the function $f(x)=\frac{1}{n} x^{2}+a x+b$ is an integer for any integer $x$.
---
Please note that the translation retains the original formatting and structure of the text, ... | n=1 \text{ or } 2 | 93 | 10 |
math | The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$, find $n$. | 648 | 43 | 3 |
math | $2 \cdot 84$ For any positive integer $q_{0}$, consider the sequence $q_{1}, q_{2}, \cdots, q_{n}$ defined by
$$
q_{i}=\left(q_{i-1}-1\right)^{3}+3 \quad(i=1,2, \cdots, n)
$$
If each $q_{i}(i=1,2, \cdots, n)$ is a power of a prime, find the largest possible value of $n$. | 2 | 115 | 1 |
math | 8.6. The altitudes of an acute-angled scalene triangle $ABC$ intersect at point $H$. $I$ is the incenter of triangle $ABC$, $O$ is the circumcenter of triangle $BHC$. It is known that point $I$ lies on the segment $OA$. Find the angle $BAC$. | 60 | 71 | 2 |
math | One, (20 points) The side lengths of squares $A B C D$ and $A E F G$ are $a$ and $b$ respectively, with $a > b$, and $A$ being the common vertex. $D C$ intersects $E F$ at $P$, and $A P \perp F C$.
Find $\angle E A D$. | 45^{\circ} | 81 | 6 |
math | 3. Given an integer $n \geqslant 2$. Find the smallest positive real number $c$, such that for any complex numbers $z_{1}, z_{2}, \cdots, z_{n}$, we have
$$
\left|\sum_{i=1}^{n} z_{i}\right|+c \sum_{1 \leqslant i<j \leqslant n}\left|z_{i}-z_{j}\right| \geqslant \sum_{i=1}^{n}\left|z_{i}\right| .
$$
(Supplied by Zhang D... | \frac{2}{n} | 138 | 7 |
math | 6. Given in $\triangle A B C$, $\tan A,(1+\sqrt{2}) \tan B, \tan C$ form an arithmetic sequence, then the minimum value of $\angle B$ is $\qquad$ | \frac{\pi}{4} | 47 | 7 |
math | 4. In the equality $a+b=c+d=e+f$ the letters represent different prime numbers less than 20. Determine at least one solution. | 5+19=7+17=11+13 | 31 | 15 |
math | The perimeter of a certain right-angled triangle is $30 \mathrm{~cm}$, the height corresponding to the hypotenuse is $m_{c}=6 \mathrm{~cm}$. What are the lengths of the sides of the triangle? | 10 | 53 | 2 |
math | 7. If real numbers $x, y, z$ satisfy $x^{2}+y^{2}+z^{2}=3, x+2 y-2 z=4$, then $z_{\text {max }}+z_{\text {min }}=$ | -\frac{16}{9} | 58 | 8 |
math | 33. It is given that $a, b$ and $c$ are three positive integers such that
$$
a^{2}+b^{2}+c^{2}=2011 .
$$
Let the highest common factor (HCF) and the least common multiple (LCM) of the three numbers $a, b, c$ be denoted by $x$ and $y$ respectively. Suppose that $x+y=388$. Find the value of $a+b+c$. (Remark: The highest... | 61 | 123 | 2 |
math | ## Task $7 / 71$
Let $p_{n}$ be the n-th prime number. Find all $i$ such that $p_{i}=2 i+1$. | i=5i=6 | 39 | 6 |
math | Suppose $x,y$ are nonnegative real numbers such that $x + y \le 1$. Prove that $8xy \le 5x(1 - x) + 5y(1 - y)$
and determine the cases of equality. | 8xy \le 5x(1 - x) + 5y(1 - y) | 56 | 23 |
math | Example 6 Given $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{1}+x_{2}+\cdots+x_{n}=2008$. Find the maximum value of the product $x_{1} x_{2} \cdots x_{n}$. ${ }^{[3]}$
(2008, National Junior High School Mathematics Competition, Tianjin Preliminary) | 2^{2} \times 3^{668} | 98 | 13 |
math | Let $n$ be a natural number, with the prime factorisation
\[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define
\[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ w... | P(x) = \frac{x}{k} | 141 | 11 |
math | 3. (3 points) Equilateral triangles $A B C$ and $A_{1} B_{1} C_{1}$ with side length 12 are inscribed in circle $S$ such that point $A$ lies on arc $B_{1} C_{1}$, and point $B$ lies on arc $A_{1} B_{1}$. Find $A A_{1}^{2}+B B_{1}^{2}+C C_{1}^{2}$. | 288 | 108 | 3 |
math | Find all the natural numbers $a,b,c$ such that:
1) $a^2+1$ and $b^2+1$ are primes
2) $(a^2+1)(b^2+1)=(c^2+1)$ | (a, b, c) = (2, 1, 3) | 54 | 18 |
math | 6. If non-negative real numbers $x, y$ satisfy
$$
x^{2}+4 y^{2}+4 x y+4 x^{2} y^{2}=32 \text {, }
$$
then the maximum value of $\sqrt{7}(x+2 y)+2 x y$ is $\qquad$ | 16 | 73 | 2 |
math | On planet $X$, there are 100 alien countries in conflict with each other. To prevent a world war, these countries organize themselves into military alliance groups for mutual protection. We know that the alliances follow these rules:
1) No alliance contains more than 50 countries.
2) Any two countries belong to at lea... | 6 | 105 | 1 |
math | 5. Suppose $f$ is a second-degree polynomial for which $f(2)=1, f(4)=2$, and $f(8)=3$. Find the sum of the roots of $f$. | 18 | 44 | 2 |
math | ## Task 2 - 020822
According to the plans outlined at the XXII Congress of the CPSU, coal production in 1980 should be 687 million tons higher than in 1960. Coal production in 1980 is 234 percent of that in 1960.
Calculate the planned coal production for the year 1960! Round to the nearest million tons! | 513 | 97 | 3 |
math | 2. Simplify the fraction: $\frac{x^{14}+x^{13}+\ldots+x+1}{x^{5}+x^{4}+x^{3}+x^{2}+x}$. | \frac{x^{10}+x^{5}+1}{x} | 50 | 17 |
math | Father is $42$ years old, and son has $14$ years. In how many years father will be twice as old as his son? | 14 | 32 | 2 |
math | Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with $2$ tokens. In each round, they are given the same MafsCounts question. The first person to solve the MafsCounts question wins the round and steals one token from each of the other players in the game. They all have the same pro... | \frac{1}{2} | 145 | 7 |
math | 1. Which whole numbers from 1 to $4 \cdot 10^{25}$ (inclusive) are there more of, and by how many: those containing only even digits or those containing only odd digits? | \frac{5^{26}-5}{4} | 45 | 12 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1}\left(\frac{e^{\sin \pi x}-1}{x-1}\right)^{x^{2}+1}
$$ | \pi^2 | 52 | 4 |
math | Task 2. Let's call a year interesting if a person turns as many years old as the sum of the digits of their birth year in that year. A certain year turned out to be interesting for Ivan, who was born in the 20th century, and for Vovochka, who was born in the 21st century. What is the difference in their ages?
Note. Fo... | 18 | 104 | 2 |
math | 9. $[7]$ Find the remainder when $1^{2}+3^{2}+5^{2}+\cdots+99^{2}$ is divided by 1000 . | 650 | 43 | 3 |
math | Is there such an $n$, and if there is, how large is it, for which the sum of the first $n$ natural numbers is a three-digit number with equal digits? | 36 | 38 | 2 |
math | Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$. | (8, 3), (2, 9), (-7, 9), (-7, 3), (2, -6), (8, -6) | 50 | 36 |
math | 2. Find all pairs of natural numbers $a$ and $b$ such that
$$
\operatorname{HCF}(a+1, b+1)=a^{2}-b^{2}
$$ | (,b)=(7,5) | 44 | 8 |
math | 2. Determine all sextuples ( $p, q, r, x, y, z$ ) such that $p, q, r$ are prime numbers, $x, y, z$ are natural numbers, and
$$
p^{2 x}=q^{y} r^{z}+1
$$ | (2,3,5,2,1,1),(2,5,3,2,1,1),(3,2,2,1,1,2),(3,2,2,1,2,1),(3,2,5,2,4,1),(3,5,2,2,1,4),(5,2,3,1,3,1),(5,3,2, | 66 | 91 |
math | 8. The function $f(x)$ is defined on $\mathbf{R}$, and for any real number $x$, it holds that $f(1+x)=f(3-x)$, and $f(2+x)=$ $-f(4-x)$, find the value of $f(1)+f(2)+\cdots+f(100)$. | 0 | 80 | 1 |
math | The base of the pyramid is a square $ABCD$ with a side length of 1.5, the lateral edge $SC$ is perpendicular to the base plane and equals 1.75. Points $S, B$, and $D$ lie on the lateral surface of a cone with the vertex at point $A$, and point $C$ lies in the base plane of this cone. Find the lateral surface area of th... | \frac{30\pi}{\sqrt{22}} | 91 | 14 |
math | Simplify the following expression as much as possible.
$$
\left[x(1-x)^{-\frac{2}{3}}+\frac{x^{2}}{(1-x)^{\frac{5}{3}}}\right]:\left[(1-x)^{\frac{1}{3}}\left(1-2 x+x^{2}\right)^{-1}\right] .
$$ | x | 80 | 1 |
math | ## Task B-1.5.
In the new exhibition of a certain museum, each exhibit is numbered in sequence with the numbers $1, 2, 3, 4, \ldots$ If a total of 2022 digits were used to number all the exhibits, how many exhibits are there in the new exhibition of the museum? | 710 | 73 | 3 |
math | 131. Find $\lim _{x \rightarrow \infty} \frac{2 x^{3}-3 x^{2}+5 x+7}{3 x^{3}+4 x^{2}-x+2}$. | \frac{2}{3} | 51 | 7 |
math | ## Task B-3.5.
Which of the right-angled triangles with integer sides and one leg of length $1000 \mathrm{~cm}$ has the largest possible perimeter? | =250001\mathrm{~},b=249999\mathrm{~} | 40 | 25 |
math | Example 11 (9th American Invitational Mathematics Examination) How many real numbers $a$ are there such that $x^{2}+a x+6 a=0$ has only integer solutions.
保留了原文的换行和格式。 | 10 | 52 | 2 |
math | 9. (1990 Hungarian Mathematical Olympiad) For any positive integer $q_{0}$, consider the sequence $q_{1}, q_{2}, \cdots, q_{n}$ defined by $q_{i}=\left(q_{i-1}-1\right)^{3}+$ $3(i=1,2, \cdots, n)$. If each $q_{i}(i=1,2, \cdots, n)$ is a prime, find the largest possible value of $n$. | 2 | 112 | 1 |
math | [ $[$ Dihedral Angle ]
Find the volume of a regular quadrilateral prism if its diagonal forms an angle of $30^{\circ}$ with the plane of a lateral face, and the side of the base is equal to $a$.
# | ^3\sqrt{2} | 53 | 7 |
math | Example 8. Find $\int x^{2} \sin \left(x^{3}+1\right) d x$. | -\frac{1}{3}\cos(x^{3}+1)+C | 27 | 16 |
math | 4 Given the complex number $z=\cos \frac{2 \pi}{5}+i \sin \frac{2 \pi}{5}$, then $(1-z)\left(1-z^{2}\right) \cdot\left(1-z^{3}\right)\left(1-z^{4}\right)=$ | 5 | 68 | 1 |
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