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 | 12. Given the quadratic function $f(x)$ satisfies $f(-1)=0$, and $x \leqslant f(x) \leqslant \frac{1}{2}\left(x^{2}+1\right)$ for all real numbers $x$. Then, the analytical expression of the function $f(x)$ is $\qquad$ 1. | \frac{1}{4}(x+1)^{2} | 79 | 14 |
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 | Let $n$ be a positive integer, and let $S_n$ be the set of all permutations of $1,2,...,n$. let $k$ be a non-negative integer, let $a_{n,k}$ be the number of even permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$ and $b_{n,k}$ be the number of odd permutations $\sigma$ in $S_n$ such that $\sum_{... | (-1)^k \binom{n-1}{k} | 144 | 14 |
math | 3. The train departed from the initial station with a certain number of passengers. At the next three stations, passengers disembarked in the following order: $\frac{1}{8}$ of the passengers, then $\frac{1}{7}$ of the remaining passengers, and then $\frac{1}{6}$ of the remaining passengers. After that, 105 passengers r... | 168 | 100 | 3 |
math | Example 1. A vector field is given in cylindrical coordinates $\mathbf{a}(M)=\mathbf{e}_{p}+\varphi \mathbf{e}_{\varphi}$. Find the vector lines of this field. We have | C_{1},\rho=C_{2\varphi} | 52 | 13 |
math | For how many integers $a$ with $|a| \leq 2005$, does the system
$x^2=y+a$
$y^2=x+a$
have integer solutions? | 90 | 43 | 2 |
math | Task 1 - 331211 Determine all natural numbers $n$ for which the following conditions are satisfied:
The number $n$ is ten-digit. For the digits of its decimal representation, denoted from left to right by $a_{0}, a_{1}$, $\ldots, a_{9}$, it holds that: $a_{0}$ matches the number of zeros, $a_{1}$ matches the number of... | 6210001000 | 115 | 10 |
math | 11. Among the two hundred natural numbers from 1 to 200, list in ascending order those that are neither multiples of 3 nor multiples of 5. The 100th number in this sequence is | 187 | 47 | 3 |
math | G3.1 Let $m$ be an integer satisfying the inequality $14 x-7(3 x-8)<4(25+x)$. | -3 | 33 | 2 |
math | 33. The students of our class spent the first Sunday of the month in the Carpathians, and on the first Sunday after the first Saturday of this month, they went on an excursion to Kyiv. In the following month, on the first Sunday, they made an excursion to the forest, and on the first Sunday after the first Saturday, th... | February1,February8,March1,March8 | 103 | 11 |
math | 9. (16 points) Given the parabola $y=x^{2}$ and the ellipse $(x-3)^{2}+4 y^{2}=1$, the focus of the parabola is $F$, and $A$ is a point on the ellipse. The line $F A$ intersects the parabola at points $B$ and $C$. Tangents to the parabola are drawn through points $B$ and $C$. Find the locus of the intersection of the t... | -\frac{1}{4}\quad(\frac{-3-\sqrt{33}}{64}\leqslantx\leqslant\frac{-3+\sqrt{33}}{64}) | 109 | 46 |
math | Let $f\colon \mathbb R ^2 \rightarrow \mathbb R$ be given by $f(x,y)=(x^2-y^2)e^{-x^2-y^2}$.
a) Prove that $f$ attains its minimum and its maximum.
b) Determine all points $(x,y)$ such that $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0$ and determine for which of them $f$ has global or loca... | (1,0) | 114 | 6 |
math | 19. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. The number of different addition equations is ( $\quad$ ). $(a+b$ and $b+a$ are considered the same equation) | 1536 | 86 | 4 |
math | 5. Let $A, B, C, D$ represent four numbers which are $12, 14, 16, 18$, respectively. Substitute these four numbers into the equation $\mathbf{A} \times \mathbf{B} + \mathbf{B} \times \mathbf{C} + \mathbf{B} \times \mathbf{D} + \mathbf{C} \times \mathbf{D}$, and the maximum value is $\qquad$ | 980 | 111 | 3 |
math | 8. A certain intelligence station has four different passwords $A, B, C, D$, and uses one of them each week. Each week, one of the three passwords not used in the previous week is chosen with equal probability. If the first week uses password $A$, what is the probability that the seventh week also uses password $A$? (E... | \frac{61}{243} | 90 | 10 |
math | Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$, where $n$ is either $2012$ or $2013$. | 338 | 56 | 3 |
math | 5. Solution. The original equation is equivalent to
$$
(\sin 7 x+\sin 3 x)^{2}+(\cos 3 x+\cos x)^{2}+\sin ^{2} \pi a=0
$$ | \in\mathbb{Z},\pi/4+\pik/2,k\in\mathbb{Z} | 52 | 26 |
math | 12. (3 points) Xiao Hua and Xiao Jun both have some glass balls. If Xiao Hua gives Xiao Jun 4 balls, the number of Xiao Hua's glass balls will be 2 times that of Xiao Jun; if Xiao Jun gives Xiao Hua 2 balls, then the number of Xiao Hua's glass balls will be 11 times that of Xiao Jun. Xiao Hua originally had $\qquad$ gl... | 20,4 | 105 | 4 |
math | For a positive integer $n$ determine all $n\times n$ real matrices $A$ which have only real eigenvalues and such that there exists an integer $k\geq n$ with $A + A^k = A^T$. | A = 0 | 52 | 4 |
math | Suppose $f$ and $g$ are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real $x$. Moreover, $f$ is nonnegative and $g$ is positive. Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals $a$. Given that $g(f(0))=1$, compute the value o... | e^{-16} | 131 | 5 |
math |
99.1. The function $f$ is defined for non-negative integers and satisfies the condition
$$
f(n)= \begin{cases}f(f(n+11)), & \text { if } n \leq 1999 \\ n-5, & \text { if } n>1999\end{cases}
$$
Find all solutions of the equation $f(n)=1999$.
| 1999=f(6n),ifonlyifn=1,2,\ldots,334 | 94 | 24 |
math | [ Rebus $]$
Rebus system. Decode the numerical rebus - system
$\left\{\begin{array}{c}M A \cdot M A=M H P \\ A M \cdot A M=P h M\end{array}\right.$
(different letters correspond to different digits, and the same 一 the same).
# | 13\times13=169,31\times31=961 | 72 | 21 |
math | Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$ | 120^\circ | 91 | 5 |
math | Solve the following system of equations:
$$
\frac{4}{\sqrt{x+5}}-\frac{3}{\sqrt{y+2}}=1, \quad \frac{2}{\sqrt{x+5}}+\frac{9}{\sqrt{y+2}}=4 .
$$ | -1,7 | 65 | 4 |
math | In the following addition, different letters represent different non-zero digits. What is the 5-digit number $ABCDE$?
$
\begin{array}{ccccccc}
A&B&C&D&E&D&B\\
&B&C&D&E&D&B\\
&&C&D&E&D&B\\
&&&D&E&D&B\\
&&&&E&D&B\\
&&&&&D&B\\
+&&&&&&B\\ \hline
A&A&A&A&A&A&A
\end{array}
$ | 84269 | 104 | 5 |
math | ## 253. Matheknobelei $6 / 86$
Statistische Angaben aus dem Jahre 1981 besagen, dass zu dieser Zeit 11 Prozent der Weltbevölkerung in Afrika lebten. Dieser Kontinent nimmt 20 Prozent des Festlandes der Erde ein. In Europa dagegen lebten 15,5 Prozent aller Menschen auf 7,1 Prozent des Festlandes. Wie viele Mal war Euro... | 3.97 | 124 | 4 |
math | Example 5 If $2 x^{2}+3 x y+2 y^{2}=1$, find the minimum value of $k=x+y+x y$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
Example 5 If $2 x^{2}+3 x y+2 y^{2}=1$, find the minimum value of $k=x+y+x y$. | -\frac{9}{8} | 95 | 7 |
math | It is known that a polynomial $P$ with integer coefficients has degree $2022$. What is the maximum $n$ such that there exist integers $a_1, a_2, \cdots a_n$ with $P(a_i)=i$ for all $1\le i\le n$?
[Extra: What happens if $P \in \mathbb{Q}[X]$ and $a_i\in \mathbb{Q}$ instead?] | 2022 | 100 | 4 |
math | 11.29 Find the rational roots of the equation
$$
\frac{\sqrt{x+2}}{|x|}+\frac{|x|}{\sqrt{x+2}}=\frac{4}{3} \sqrt{3}
$$
Solve the inequalities (11.30-11.31): | x_{1}=-\frac{2}{3},x_{2}=1 | 70 | 17 |
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 | Find all polynomials $P \in \mathbb{R}[X]$ such that $16 P\left(X^{2}\right)=P(2 X)^{2}$
Hint: use the previous question | P(x)=16(\frac{x}{4})^{i} | 45 | 14 |
math | 115 Let $[x]$ denote the greatest integer not exceeding $x$, referred to as the integer part of $x$. Let $\{x\}=x-[x]$ denote the fractional part of $x$. If the integer part of $x$ is the mean proportional between $x$ and $\{x\}$, then the difference between $x$ and the reciprocal of $x$ is . $\qquad$ | 1 | 88 | 1 |
math | For three numbers $a, b$, and $c$, the following equations hold:
$$
a+b=332, a+c=408 \text{ and } b+c=466 \text{. }
$$
Determine the numbers $a, b$, and $c$. | =137,b=195,=271 | 63 | 14 |
math | 2. Given positive numbers $a, b, c, d$. Find the minimum value of the expression
$$
A=\left(\frac{a^{2}+b^{2}}{c d}\right)^{4}+\left(\frac{b^{2}+c^{2}}{a d}\right)^{4}+\left(\frac{c^{2}+d^{2}}{a b}\right)^{4}+\left(\frac{d^{2}+a^{2}}{b c}\right)^{4}
$$ | 64 | 118 | 2 |
math | 11.5. Three edges of a tetrahedron, emanating from one vertex, are mutually perpendicular, and have lengths 3, 4, and 4. Find the radii of the inscribed and circumscribed spheres of the tetrahedron. | R=\frac{\sqrt{41}}{2},r=\frac{12(10-\sqrt{34})}{33} | 57 | 31 |
math | Below is a square root calculation scheme written out in full detail, where the letter $x$ represents missing digits. What is the number under the root?
$$
\begin{aligned}
& \sqrt{x x x 8 x x}=x x x \\
& \underline{x x} \\
& x x x: x x \cdot x \\
& \quad \underline{x x} \\
& \quad x x x x: x x x \cdot x \\
& \quad \un... | 417^2+1 | 116 | 7 |
math | 4. Given point $P$ on the hyperbola with eccentricity $\sqrt{2}$
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)
$$
$F_{1} 、 F_{2}$ are the two foci of the hyperbola, and $\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}$ $=0$. Then the ratio of the inradius $r$ to the circumradius $R$ of $\triangle P F_... | \frac{\sqrt{6}}{2}-1 | 140 | 11 |
math | \section*{Task 2 - V11122}
The Octavia Touring Sport Car from Škoda Automobile Works Prague reaches a speed of \(80 \frac{\mathrm{km}}{\mathrm{h}}\) in 14 seconds after starting.
a) How many kilometers has it covered in this time (assuming uniform acceleration)? b) In what time, starting from the moment of the start,... | 52\mathrm{~} | 134 | 7 |
math | 29.2.9 Find all positive integers $n$ such that there exists a permutation $a_{1}$, $a_{2}, \cdots, a_{n}$ of $1,2, \cdots, n$ for which $\left|a_{i}-i\right|(i=1,2, \cdots, n)$ are all distinct. | 4k+4or4k+1(k\in{N}) | 79 | 15 |
math | Two squares of a $7 \times 7$ board are painted yellow and the rest is painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many non-equivalent color schemes can we obtain?
# | 300 | 58 | 3 |
math | 109. From the digits $1,2,3,4$, all possible positive decimal fractions with one, two, or three decimal places are formed, each containing each of these digits exactly once. Find the sum of these fractions. | 7399.26 | 49 | 7 |
math | 7. A line is parallel to side $\mathrm{BC}$ of the known $\triangle \mathrm{ABC}$, intersects the other two sides at $D$ and $E$, and makes the area of $\triangle BDE$ equal to the constant $\mathrm{k}^{2}$. What is the relationship between $\mathrm{k}^{2}$ and the area of $\triangle \mathrm{ABC}$ for the problem to be... | S \geqslant 4 k^{2} | 97 | 12 |
math | # 3.1. Condition:
The number 4597 is displayed on the computer screen. In one move, it is allowed to swap any two adjacent digits, but after this, 100 is subtracted from the resulting number. What is the largest number that can be obtained by making no more than two moves? | 8357 | 69 | 4 |
math | 2. Tine was collecting stamps. For his birthday, he received a new album in which he could store many stamps. He took 2002 tolars from his savings and decided to spend all the money on buying stamps. A friend offered him smaller stamps for 10 tolars and larger ones for 28 tolars. Tine decided to buy as many stamps as p... | 193 | 88 | 3 |
math | In an acute triangle $ ABC$, let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[AB\right]$ such that $ \angle ADB\equal{}\angle AEC\equal{}90{}^\circ$. If perimeter of triangle $ AED$ is 9, circumradius of $ AED$ is $ \frac{9}{5}$ and perimeter of triangle $ ABC$ is 15, then $ \left|BC\right|$ i... | \frac{24}{5} | 185 | 8 |
math | 3. A school is hosting a Mathematics Culture Festival. According to statistics, there were more than 980 (no less than 980, less than 990) students visiting the school that day. Each student visits for a period of time and then leaves (and does not return). If, regardless of how these students arrange their visit times... | 32 | 127 | 2 |
math | Example 5 Given the quadratic equation in $x$
$$
\left(k^{2}-8 k+15\right) x^{2}-2(13-3 k) x+8=0
$$
both roots are integers. Find the value of the real number $k$.
Analysis: Since $k$ is a real number, we cannot solve it using the discriminant. We can first find the two roots of the equation $x_{1}=$ $\frac{2}{5-k}, x... | k=4,7, \frac{13}{3} | 187 | 14 |
math | 【Example 4】8 people sit in two rows, with 4 people in each row, where 2 specific people must sit in the front row, and 1 specific person must sit in the back row.
Here, the 2 people who must sit in the front row and the 1 person who must sit in the back row are all constrained elements, but no specific position is exp... | 5760 | 86 | 4 |
math | Let $x_{1}, x_{2}, \ldots, x_{n}$ be positive numbers whose sum is 1. Let
$$
s=\max \left(\frac{x_{1}}{1+x_{1}}, \frac{x_{2}}{1+x_{1}+x_{2}}, \cdots, \frac{x_{n}}{1+x_{1}+\cdots+x_{n}}\right)
$$
What is the smallest possible value of $s$? For which $x_{1}, x_{2}, \ldots, x_{n}$ does it attain this value? | 1-\sqrt[n]{\frac{1}{2}} | 129 | 12 |
math | Let $n\geq 2$ be a fixed positive integer. Let $\{a_1,a_2,...,a_n\}$ be fixed positive integers whose sum is $2n-1$. Denote by $S_{\mathbb{A}}$ the sum of elements of a set $A$. Find the minimal and maximal value of $S_{\mathbb{X}}\cdot S_{\mathbb{Y}}$ where $\mathbb{X}$ and $\mathbb{Y}$ are two sets with the property ... | 2n - 2 \text{ and } n(n-1) | 279 | 15 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; 0 ; z)$
$B(-18 ; 1 ; 0)$
$C(15 ;-10 ; 2)$ | A(0;0;1) | 63 | 8 |
math | Example 5. Find the length of the arc of the curve $y=\frac{x^{2}}{4}-\frac{\ln x}{2}$, enclosed between the points with abscissas $x=1$ and $x=e$. | \frac{1}{4}(e^{2}+1) | 52 | 14 |
math | 1.011. $\frac{\left(\frac{3}{5}+0.425-0.005\right): 0.1}{30.5+\frac{1}{6}+3 \frac{1}{3}}+\frac{6 \frac{3}{4}+5 \frac{1}{2}}{26: 3 \frac{5}{7}}-0.05$. | 2 | 96 | 1 |
math | 3.2. 4 * Find the general term of the sequence $a_{1}=1, a_{2}=2, \frac{a_{n}}{a_{n-1}}=\sqrt{\frac{a_{n-1}}{a_{n-2}}}(n \geqslant 3)$. | a_{n}=2^{2-(\frac{1}{2})^{n-2}} | 70 | 20 |
math | (a) Let $a, b, c, d$ be integers such that $ad\ne bc$. Show that is always possible to write the fraction $\frac{1}{(ax+b)(cx+d)}$in the form $\frac{r}{ax+b}+\frac{s}{cx+d}$
(b) Find the sum $$\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\frac{1}{7 \cdot 10}+...+\frac{1}{1995 \cdot 1996}$$ | \frac{1995}{3 \cdot 1996} | 123 | 17 |
math | 3. Let $a_{n}$ denote the last digit of the number $n^{4}$. Then $a_{1}+a_{2}+\cdots+a_{2008}=$ $\qquad$ | 6632 | 47 | 4 |
math | One, (20 points) Solve the equation:
$$
\sqrt{x+\sqrt{2 x-1}}+\sqrt{x-\sqrt{2 x-1}}=\sqrt{a}(a)
$$
$0)$.
Discuss the solution of this equation for the value of the positive number $a$.
| (1) \text{ When } a>2, x=\frac{2+a}{4}; (2) \text{ When } a=2, \frac{1}{2} \leqslant x \leqslant 1; (3) \text{ When } a<2, \text{ no solution} | 65 | 72 |
math | 4. Consider triangle $ABC$, where $AC = BC$, $m(ACB) = 90^{\circ}$, and triangle $DAB$, where $DA = DB$, located in perpendicular planes. Let $\quad M \in (BC), \quad BM = 2CM, \quad N \in (AC)$, $AC = 3AN, P \in MN \cap AB$, $T$ be the midpoint of segment $[AB]$, and $G$ be the centroid of triangle $DAB$. Calculate th... | \sqrt{6} | 169 | 5 |
math | Find the largest positive integer $n$ for which the inequality
$$
\frac{a+b+c}{a b c+1}+\sqrt[n]{a b c} \leq \frac{5}{2}
$$
holds for all $a, b, c \in[0,1]$. Here $\sqrt[1]{a b c}=a b c$. | 3 | 79 | 1 |
math | 2. (3 points) There are five gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five gift boxes priced at 3 yuan, 6 yuan, 9 yuan, 12 yuan, and 15 yuan. One gift is paired with one gift box, resulting in $\qquad$ different prices. | 9 | 82 | 1 |
math | How long does a ball dropped from a height of $h$ bounce? The collision number for the collision between the ground and the ball is $k$. For which balls is it useful to measure $k$ with the bouncing time? (The collision number is the ratio of the mechanical energy after and before the collision.) | \frac{1+\sqrt{k}}{1-\sqrt{k}}\cdot\sqrt{\frac{2h_{0}}{}} | 64 | 28 |
math | 8. Given point $A(0,2)$ and two points $B, C$ on the parabola $y^{2}=x+4$ such that $A B \perp B C$, then the range of the y-coordinate of point $C$ is $\qquad$ . | y\leqslant0ory\geqslant4 | 62 | 14 |
math | 8,9 |
In triangle $A B C$, it is known that $\angle B A C=75^{\circ}, A B=1, A C=\sqrt{6}$. A point $M$ is chosen on side $B C$, such that $\angle$ ВАМ $=30^{\circ}$. Line $A M$ intersects the circumcircle of triangle $A B C$ at point $N$, different from $A$. Find $A N$. | 2 | 101 | 1 |
math | Let $ABC$ be an equilateral triangle whose angle bisectors of $B$ and $C$ intersect at $D$. Perpendicular bisectors of $BD$ and $CD$ intersect $BC$ at points $E$ and $Z$ respectively.
a) Prove that $BE=EZ=ZC$.
b) Find the ratio of the areas of the triangles $BDE$ to $ABC$ | \frac{1}{9} | 87 | 7 |
math | Three. (20 points) Let real numbers $a, m$ satisfy
$$
a \leqslant 1, 0 < m \leqslant 2 \sqrt{3} \text{, }
$$
The function $f(x)=\frac{a m x - m x^{2}}{a + a(1-a)^{2} m^{2}}(x \in (0, a))$. If there exist $a, m, x$ such that $f(x) \geqslant \frac{\sqrt{3}}{2}$, find all real values of $x$. | x = \frac{1}{2} | 134 | 9 |
math | 3.2. Let's say that number A hides number B if you can erase several digits from A to get B (for example, the number 123 hides the numbers 1, 2, 3, 12, 13, and 23). Find the smallest natural number that hides the numbers 121, 221, 321, ..., 1921, 2021, 2121.
Natural numbers are numbers used for counting objects.
Answ... | 1201345678921 | 127 | 13 |
math | Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible. | 25 | 74 | 2 |
math | 1. The solution set of the inequality $\arccos x<\arctan x$ is
$\qquad$ | \sqrt{\frac{\sqrt{5}-1}{2}} < x \leqslant 1 | 26 | 22 |
math | 6. (8 points) By expanding the expression $(1+\sqrt{11})^{212}$ using the binomial theorem, we obtain terms of the form $C_{212}^{k}(\sqrt{11})^{k}$. Find the value of $k$ for which such a term attains its maximum value. | 163 | 73 | 3 |
math | 6th Australian 1985 Problem B3 Find all real polynomials p(x) such that p(x 2 + x + 1) = p(x) p(x + 1). | (x^2+1)^nforn=0,1,2,3,\ldots | 41 | 19 |
math | Example 3 㷵 On a board, there is a convex 2011-gon, and Betya draws its diagonals one by one. It is known that each diagonal drawn intersects at most one of the previously drawn diagonals at an interior point. Question: What is the maximum number of diagonals Betya can draw? [3] | 4016 | 76 | 4 |
math | 9. Given $f(x)=2^{x} m+x^{2}+n x$. If
$$
\{x \mid f(x)=0\}=\{x \mid f(f(x))=0\} \neq \varnothing,
$$
then the range of values for $m+n$ is $\qquad$ . | [0,4) | 73 | 5 |
math | 19. [11] Let
$$
A=\lim _{n \rightarrow \infty} \sum_{i=0}^{2016}(-1)^{i} \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^{2}}
$$
Find the largest integer less than or equal to $\frac{1}{A}$.
The following decimal approximation might be useful: $0.6931<\ln (2)<0.6932$, where $\ln$ denotes the natural logarithm f... | 1 | 131 | 1 |
math | 413. A body with a mass of 8 kg moves in a straight line according to the law $s=$ $=2 t^{2}+3 t-1$. Find the kinetic energy of the body $\left(m v^{2} / 2\right)$ 3 seconds after the start of the motion. | 900 | 68 | 3 |
math | Let $x_1=1/20$, $x_2=1/13$, and \[x_{n+2}=\dfrac{2x_nx_{n+1}(x_n+x_{n+1})}{x_n^2+x_{n+1}^2}\] for all integers $n\geq 1$. Evaluate $\textstyle\sum_{n=1}^\infty(1/(x_n+x_{n+1}))$. | 23 | 104 | 2 |
math | ## Task 4 - 300614
The pages of a book are numbered from 1 to 235.
a) How many times was the digit 4 used in total for the numbering?
b) How many times was the digit 0 used in total for the numbering?
c) How many digits in total need to be printed for this numbering? | 44,43,597 | 78 | 9 |
math | 2. The sequence $a_{0}, a_{1}, \cdots$ satisfies:
$$
a_{0}=\sqrt{5}, a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}},
$$
where, $[x]$ denotes the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$. Then $a_{2019}=$ $\qquad$ | 8076+\sqrt{5} | 104 | 9 |
math | 4. Given $f(x)=a \sin x+b \sqrt[3]{x}+c \ln \left(x+\sqrt{x^{2}+1}\right)+1003(a, b, c$ are real numbers $)$, and $f\left(\lg ^{2} 10\right)=1$, then $f(\lg \lg 3)=$ $\qquad$ . | 2005 | 88 | 4 |
math | Let $n$ be a positive integer, $[x]$ be the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$.
(1) Find all positive integers $n$ that satisfy
$$
\sum_{k=1}^{2013}\left[\frac{k n}{2013}\right]=2013+n
$$
(2) Find all positive integers $n$ that maximize $\sum_{k=1}^{2013}\left\{\frac{k n}{2013}\right... | 1006 | 132 | 4 |
math | 6. In a right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A B=1, B C=C C_{1}=\sqrt{3}, \angle A B C=90^{\circ}$, point $P$ is a moving point on the plane $A B C$, then the minimum value of $A_{1} P+\frac{1}{2} P C$ is $\qquad$ . | \frac{5}{2} | 98 | 7 |
math | 1. Positive integers $a, b, c$ satisfy
$$
\log _{6} a+\log _{6} b+\log _{6} c=6 \text {, }
$$
$a, b, c$ form an increasing geometric sequence, and $b-a$ is a perfect square. Then the value of $a+b+c$ is $\qquad$ | 111 | 81 | 3 |
math | Example 1 Given the function $f_{1}(x)=\frac{2 x-1}{x+1}$, for positive integer $n$, define $f_{n+1}(x)=f_{1}\left[f_{n}(x)\right]$. Find the analytical expression for $f_{1234}(x)$. | \frac{1}{1-x} | 72 | 8 |
math | ## Task A-4.1.
The product of the second and fourth terms of an arithmetic sequence with difference $d$ is $-d^{2}$. Determine the product of the third and fifth terms of this sequence. | 0 | 46 | 1 |
math | Colin has $900$ Choco Pies. He realizes that for some integer values of $n \le 900$, if he eats n pies a day, he will be able to eat the same number of pies every day until he runs out. How many possible values of $n$ are there? | 27 | 67 | 2 |
math | 45. Find the scalar product of vectors $\vec{a}=(3 ; 5)$ and $\vec{b}=(-2 ; 7)$. | 29 | 33 | 2 |
math | 8.2. Find the value of the expression $a^{3}+12 a b+b^{3}$, given that $a+b=4$. | 64 | 33 | 2 |
math | 11. Insert the sum of each pair of adjacent terms between them in a sequence, forming a new sequence, which is called a "Z-extension" of the sequence. Given the sequence $1,2,3$, after the first "Z-extension" it becomes $1,3,2,5,3$; after the second "Z-extension" it becomes $1,4,3,5,2,7,5,8,3 ; \cdots$; after the $n$-t... | a_{n}=4\cdot3^{n}+2 | 239 | 13 |
math | Find for which values of $n$, an integer larger than $1$ but smaller than $100$, the following expression has its minimum value:
$S = |n-1| + |n-2| + \ldots + |n-100|$ | n = 50 | 57 | 6 |
math | Let $ABCDEF$ regular hexagon of side length $1$ and $O$ is its center. In addition to the sides of the hexagon, line segments from $O$ to the every vertex are drawn, making as total of $12$ unit segments. Find the number paths of length $2003$ along these segments that star at $O$ and terminate at $O$. | \frac{3}{\sqrt{7}} \left( (1 + \sqrt{7})^{2002} - (1 - \sqrt{7})^{2002} \right) | 83 | 45 |
math | 8. For a regular hexagon $A B C D E F$, the diagonals $A C$ and $C E$ are internally divided by points $M$ and $N$ in the following ratios: $\frac{A M}{A C}=$ $\frac{C M}{C E}=r$. If points $B$, $M$, and $N$ are collinear, find the ratio $r$. | \frac{\sqrt{3}}{3} | 87 | 10 |
math | 2. [4] Let $A B C$ be a triangle, and let $M$ be the midpoint of side $A B$. If $A B$ is 17 units long and $C M$ is 8 units long, find the maximum possible value of the area of $A B C$. | 68 | 65 | 2 |
math | 4. Find all five-digit numbers consisting of non-zero digits such that each time we erase the first digit, we get a divisor of the previous number.
## Solutions
Problem 1. | 91125,53125,95625 | 38 | 17 |
math | ## Task Condition
Find the derivative.
$$
y=\cos ^{2}(\sin 3)+\frac{\sin ^{2} 29 x}{29 \cos 58 x}
$$ | \frac{\operatorname{tg}58x}{\cos58x} | 45 | 18 |
math | 5. In the store "Third is Not Excessive," there is a promotion: if a customer presents three items at the cash register, the cheapest of them is free. Ivan wants to buy 11 items costing $100, 200, 300, \ldots, 1100$ rubles. For what minimum amount of money can he buy these items? | 4800 | 84 | 4 |
math | 4. $A$ and $B$ are playing a game on a $25 \times 25$ grid. At the beginning, $A$ can mark some cells. After marking is completed, they start taking turns to place gold coins on the grid, with $B$ going first. The rules for placing gold coins are as follows: (1) gold coins cannot be placed in marked cells; (2) once a g... | 25 | 177 | 2 |
math | Let $a, b, c$ be positive real numbers such that: $$ab - c = 3$$ $$abc = 18$$ Calculate the numerical value of $\frac{ab}{c}$ | 2 | 42 | 1 |
math | 2. If $a^{2}+a+1=0$, determine the value of the expression
$$
a^{1987}+\frac{1}{a^{1987}}
$$ | -1 | 44 | 2 |
math | Find all real functions $f(x)$ that satisfy the following conditions:
a) there exists a finite interval in which $f(x)$ is bounded,
b) for all $x_{1}, x_{2}$,
$$
f\left(x_{1}+x_{2}\right)=f\left(x_{1}\right)+f\left(x_{2}\right)
$$
c) $f(1)=1$. | f(x)=x | 89 | 4 |
math | Jay has a $24\times 24$ grid of lights, all of which are initially off. Each of the $48$ rows and columns has a switch that toggles all the lights in that row and column, respectively, i.e. it switches lights that are on to off and lights that are off to on. Jay toggles each of the $48$ rows and columns exactly once, s... | 9408 | 180 | 4 |
math | How many pairs of positive integers $(x, y)$ are there such that
$$
\frac{x y}{x+y}=144 ?
$$ | 45 | 31 | 2 |
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