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 | 1. (7 points) In a building, on all floors in all entrances there is an equal number of apartments (more than one). Also, in all entrances there are an equal number of floors. The number of floors is greater than the number of apartments per floor, but less than the number of entrances. How many floors are there in the... | 11 | 84 | 2 |
math | ## Task 4 - 330624
In a box, there are balls; each of them has one of the colors blue, yellow, red. There are at least 3 but at most 7 balls of each color. The total number of balls in the box is a prime number.
The number of red balls is divisible by the number of yellow balls. If one yellow and two red balls are ta... | 4 | 138 | 1 |
math | 8. In $\triangle A B C$, $\angle C=\frac{\pi}{3}$, let $\angle B A C=\theta$. If there exists a point $M$ on line segment $B C$ (different from points $B$ and $C$), such that when $\triangle B A M$ is folded along line $A M$ to a certain position to get $\triangle B^{\prime} A M$, it satisfies $A B^{\prime} \perp C M$,... | (\frac{\pi}{6},\frac{2\pi}{3}) | 116 | 16 |
math | 2. (8 points) A shepherd was herding a flock of sheep to graze. After one ram ran out, he counted the number of sheep and found that the ratio of rams to ewes among the remaining sheep was 7:5. After a while, the ram that had run out returned to the flock, but then a ewe ran out. The shepherd counted the sheep again an... | 25 | 108 | 2 |
math | 80. Given that $\mathrm{n}$ is a positive integer, and the tens digit of $n^{2}$ is 7, find $n-100\left[\frac{n}{100}\right]$. (Note: $[x]$ denotes the integer part of $\mathrm{x}$) | 24,26,74,76 | 65 | 11 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty}\left(\frac{2 n-1}{2 n+1}\right)^{n+1}$ | \frac{1}{e} | 44 | 7 |
math | [Mathematical logic (miscellaneous).]
On an island, there live liars and knights, a total of 2001 people. Knights always tell the truth, while liars lie. Each resident of the island stated: "Among the remaining residents of the island, more than half are liars." How many liars are there on the island?
# | 1001 | 75 | 4 |
math |
A1. Find all ordered triples $(x, y, z)$ of real numbers satisfying the following system of equations:
$$
\begin{aligned}
x^{3} & =\frac{z}{y}-2 \frac{y}{z} \\
y^{3} & =\frac{x}{z}-2 \frac{z}{x} \\
z^{3} & =\frac{y}{x}-2 \frac{x}{y}
\end{aligned}
$$
| (1,1,-1),(1,-1,1),(-1,1,1),(-1,-1,-1) | 102 | 27 |
math | # Task 9.1
All three-digit numbers are written in a row: $100101102 \ldots 998$ 999. How many times in this row does a zero follow a two?
## Number of points 7 Answer: 19
# | 19 | 68 | 2 |
math | Problem 9. In a football championship, 16 teams participate, each playing against each other once. What is the minimum number of games that must be played so that among any three teams, there are two that have already played against each other?
# | 56 | 52 | 2 |
math | Bakayev E.V.
Petya places 500 kings on the cells of a $100 \times 50$ board so that they do not attack each other. And Vasya places 500 kings on the white cells (in a chessboard coloring) of a $100 \times 100$ board so that they do not attack each other. Who has more ways to do this? | Vasya | 93 | 3 |
math | ## Task B-4.6.
An ellipse $x^{2}+4 y^{2}=4$ has a rhombus inscribed in it, with one vertex at point $A(\sqrt{2}, y>0)$. Determine the coordinates of the remaining vertices and calculate the area of the rhombus. | \frac{10\sqrt{34}}{17} | 67 | 15 |
math | 1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=3000$, if $f(0)=1$ and for any $x$ the equality $f(x+3)=f(x)+2 x+3$ holds. | 3000001 | 61 | 7 |
math | ## Task 4 - 160814
Peter presents his friend Fritz with the following problem:
"Given a circle whose diameter is equal to the Earth's diameter, and a second concentric circle whose circumference is $1 \mathrm{~m}$ longer than the circumference of the first circle. Determine the distance between the two circle lines!"... | \frac{1}{2\pi}\approx16 | 172 | 12 |
math | 11.171. A cube is inscribed in a hemisphere of radius $R$ such that four of its vertices lie on the base of the hemisphere, while the other four vertices are located on its spherical surface. Calculate the volume of the cube. | \frac{2R^{3}\sqrt{6}}{9} | 53 | 15 |
math | (Mid-term training Valbonne 2019 group B)
Let $n \geqslant 1$ be an integer. We have $n$ cubes with sides $1, 2, \ldots, n$. We want to stack them in a certain order, such that a cube of side $k$ can only be placed on a cube of side $l$ with $l \geqslant k-2$. How many such different stackings exist? | 3^{n-2}\times2 | 101 | 8 |
math | Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the ... | 2 | 127 | 1 |
math | Let $A B C D$ be a square, $P$ inside $A B C D$ such that $P A=1, P B=2, P C=3$. Calculate $\widehat{A P B}$. | 135 | 49 | 3 |
math | 13.135. A team of mechanics can complete a certain task of processing parts 15 hours faster than a team of apprentices. If the team of apprentices works for 18 hours on this task, and then the team of mechanics continues working on the task for 6 hours, only 0.6 of the entire task will be completed. How much time does ... | 45 | 89 | 2 |
math | 3.159. $\cos \left(2 \alpha+\frac{7}{4} \pi\right)$, if $\operatorname{ctg} \alpha=\frac{2}{3}$. | \frac{7\sqrt{2}}{26} | 45 | 13 |
math | 10. (20 points) Given the ellipse $C: \frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, and the moving circle $\Gamma: x^{2}+y^{2}=r^{2}(3<r<5)$. If $M$ is a point on the ellipse $C$, and $N$ is a point on the moving circle $\Gamma$, and the line $M N$ is tangent to both the ellipse $C$ and the moving circle $\Gamma$, find the maxi... | 2 | 127 | 1 |
math | XVIII OM - I - Problem 5
Find such natural numbers $ p $ and $ q $, so that the roots of the equations $ x^2 - qx + p = 0 $ and $ x^2 - px + q = 0 $ are natural numbers. | p=4,\p= | 59 | 6 |
math | 1. Find $\log _{35} 28$, if $\log _{14} 7=a$ and $\log _{14} 5=b$. | \frac{2-}{+b} | 38 | 9 |
math | ## Task 4 - 130934
In a plane, regular n-gons (with a uniform number of vertices) are to be laid around a vertex so that the sum of the sizes of the interior angles lying at this vertex is $360^{\circ}$.
Give all natural numbers $n$ for which this is possible; in each case, also give the total number of n-gons requir... | 3,4,6 | 89 | 5 |
math | 1. Solve the equation $9^{x-1}+3^{x+2}=90$. | 2 | 22 | 1 |
math | \section*{Problem 6B - 131236B}
Let \(M\) be the set of all points \(P(x, y)\) in a plane rectangular Cartesian coordinate system, where \(x, y\) are rational integers, and \(0 \leq x \leq 4\) and \(0 \leq y \leq 4\).
Determine the probability that the distance between two different points chosen arbitrarily from \(M... | \frac{9}{25} | 194 | 8 |
math | A sphere is inscribed in an equilateral cone, whose volume is $100 \mathrm{~cm}^{3}$. What is the lateral surface area of the cone? | 156.28\mathrm{~}^{2} | 38 | 14 |
math | ## Task 2 - 190712
Determine all four-digit natural numbers that have the property of being divisible by each of the numbers $2,3,4,5,6,7,8,9,10,12,14,15$! | 2520,5040,7560 | 62 | 14 |
math | 16. Let $\left\{a_{n}\right\}$ be a sequence of positive integers such that $a_{1}=1, a_{2}=2009$ and for $n \geq 1$, $a_{n+2} a_{n}-a_{n+1}^{2}-a_{n+1} a_{n}=0$. Determine the value of $\frac{a_{993}}{100 a_{991}}$. | 89970 | 105 | 5 |
math | 7. Let $A$ be the set of all positive integers not exceeding 2009, i.e., $A=\{1,2, \cdots, 2009\}$, and let $L \subseteq A$, where the difference between any two distinct elements of $L$ is not equal to 4. Then the maximum possible number of elements in the set $L$ is | 1005 | 85 | 4 |
math | 3. Find the linear function $f(x)$ that satisfies the inequality $f[f(x)] \geqslant x-3, x \in \mathbf{R}$. | f(x)=-x+b,b\in{R} | 38 | 12 |
math | 1. A bicycle tire, if installed on the front wheel, will wear out after traveling $5000 \mathrm{~km}$; if installed on the back wheel, it will wear out after traveling $3000 \mathrm{~km}$. If the front and back tires are swapped after traveling a certain distance, so that a pair of new tires wear out simultaneously, th... | 3750 | 95 | 4 |
math | Let $a$ and $b$ be two positive reals such that the following inequality
\[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals $x, y \geq 1$. Determine the smallest possible value of $a^2 + b$.
[i]Proposed by Fajar Yuliawan[/i] | \frac{2}{3\sqrt{3}} | 82 | 11 |
math | One. (20 points) A factory needs to produce two types of products, A and B. According to the process specifications, each unit of product A requires 2 hours, 3 hours, and 4 hours of processing on three different machines A, B, and C, respectively. Each unit of product B requires 4 hours, 4 hours, and 3 hours of process... | 1 \text{ unit of product A and 3 units of product B} | 176 | 16 |
math | Three cowboys walked into a saloon. One bought 4 sandwiches, a cup of coffee, and 10 donuts for a total of $1.69. The second bought 3 sandwiches, a cup of coffee, and 7 donuts for $1.26. How much did the third cowboy pay for a sandwich, a cup of coffee, and a donut? | 40 | 81 | 2 |
math | 2.1. Find the integer part of the number $a+\frac{9}{b}$, where $a$ and $b-$ are respectively the integer and fractional part of the number $\sqrt{76-42 \sqrt{3}}$. | 12 | 53 | 2 |
math | 6. For any closed interval $I$, let $M_{I}$ denote the maximum value of the function $y=\sin x$ on $I$. If the positive number $a$ satisfies $M_{[0, a]}=2 M_{[a, 2 a]}$, then the value of $a$ is $\qquad$ . | \frac{5\pi}{6} | 73 | 9 |
math | 10. The sequence $\left\{a_{n}\right\}$ of $n$ terms, formed by the permutation of $1,2, \cdots, n$, satisfies: each term is greater than all the terms before it or less than all the terms before it. Then the number of sequences $\left\{a_{n}\right\}$ that satisfy this condition is $\qquad$. | 2^{n-1} | 84 | 6 |
math | Simplify the following expression to its simplest form:
$$
\frac{\cot \alpha+\cot \beta}{\cot \alpha-\cot \beta}+\frac{\sin (\alpha+\beta)}{\sin (\alpha-\beta)}
$$ | 0 | 49 | 1 |
math | 14. For the quadratic equation in $x$, $x^{2}+z_{1} x+z_{2}+m=0$, where $z_{1}, z_{2}, m$ are all complex numbers, and $z_{1}^{2}-4 z_{2}=$ $16+20 \mathrm{i}$. Let the two roots of this equation be $\alpha, \beta$, and $\alpha, \beta$ satisfy $|\alpha-\beta|=2 \sqrt{7}$. Find the maximum and minimum values of $|m|$. | ||_{\max}=\sqrt{41}+7;||_{\}=7-\sqrt{41} | 121 | 25 |
math | 13.078. A tourist traveled $5 / 8$ of the total distance by car and the remaining part by boat. The boat's speed is 20 km/h less than the car's speed. The tourist traveled by car for 15 minutes longer than by boat. What are the speeds of the car and the boat if the total distance of the tourist's journey is 160 km? | 100 | 87 | 3 |
math | 4. At the end of a chess tournament, it was determined that each participant had earned exactly half of their points playing against the competitors who finished in the last three places. How many participants were there in the tournament?
(Each participant played one game against each of the other participants. A win... | 9 | 77 | 1 |
math | 8.3. To the number 2014, a digit was added on the left and on the right. The six-digit number thus obtained became divisible by 36. Find all such six-digit numbers. | 220140,720144,320148 | 45 | 20 |
math | Solve the system of equations:
$$
\begin{aligned}
& x^{3}-y=6 \\
& y^{3}-z=6 \\
& z^{3}-x=6
\end{aligned}
$$ | 2 | 47 | 1 |
math | For which integers $n \geq 3$ can one find a triangulation of regular $n$-gon consisting only of isosceles triangles? | 2^{}(2^{b}+1) | 33 | 11 |
math | 10. If $x$ is an integer, $3<x<200$, and $x^{2}+(x+1)^{2}$ is a perfect square, then the value of the integer $x$ is $\qquad$. | 20 \text{ or } 119 | 52 | 11 |
math | G2.4 Let $d$ be an odd prime number. If $89-(d+3)^{2}$ is the square of an integer, find the value of $d$. | 5 | 40 | 1 |
math | For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$. | 125 | 92 | 3 |
math | 【Question 7】 80 students stand in a row facing the teacher, and report numbers in sequence from left to right according to the teacher's command: $1, 2, 3 \cdots \cdots$. After reporting, the teacher asks the students who reported numbers that are multiples of 2 to turn around, then asks the students who reported numbe... | 26 | 124 | 2 |
math | In math class, $2 / 3$ of the students had a problem set, and $4 / 5$ of them brought a calculator. Among those who brought a calculator, the same proportion did not have a problem set as those who did not bring a calculator. What fraction of the students had both a problem set and a calculator? | \frac{8}{15} | 70 | 8 |
math | 13. (15 points) In the sequence $\left\{a_{n}\right\}$,
$$
a_{n}=2^{n} a+b n-80\left(a 、 b \in \mathbf{Z}_{+}\right) \text {. }
$$
It is known that the minimum value of the sum of the first $n$ terms $S_{n}$ is obtained only when $n=6$, and $7 \mid a_{36}$. Find the value of $\sum_{i=1}^{12}\left|a_{i}\right|$. | 8010 | 129 | 4 |
math | A set $ E$ of points in the 3D space let $ L(E)$ denote the set of all those points which lie on lines composed of two distinct points of $ E.$ Let $ T$ denote the set of all vertices of a regular tetrahedron. Which points are in the set $ L(L(T))?$ | \text{The entire 3-dimensional space spanned by the vertices of the tetrahedron} | 69 | 21 |
math | 4. Barbara scored 98 points on the second-to-last test of the school year, thus increasing the average of the points she had achieved so far by 1 point. On the last test, she scored 70 points and lowered the new average by 2 points. How many tests did she take in the entire school year? | 10 | 69 | 2 |
math | 2. Find all integer solutions of the equation
$$
x+\frac{1}{y+\frac{1}{z}}=\frac{7}{3}
$$ | (2;2;1);(2;4;-1);(3;-1;-2);(3;-2;2);(1;1;-4) | 34 | 35 |
math | Example 6 (1990 National High School Mathematics League Question) Let $n=1990$, find $\frac{1}{2^{n}}\left(1-3 C_{n}^{2}+3^{2} C_{n}^{4}-3^{3} C_{n}^{6}+\cdots\right.$ $\left.+3^{994} C_{n}^{1988}-3^{995} C_{n}^{1990}\right)$ | -\frac{1}{2} | 113 | 7 |
math | 4. The arithmetic sequence $\left\{a_{n}\right\}$ satisfies $a_{1}^{2}+a_{10}^{2} \leqslant 10$. Then the range of values for $S=a_{10}+a_{11}+\cdots+a_{19}$ is $\qquad$ | -50\leqslantS\leqslant50 | 75 | 16 |
math | Solve the following system of equations:
$$
\begin{gathered}
\frac{x-y}{x+z}=a \\
\frac{x^{2}-y^{2}}{x+z}=b \\
\frac{x^{3}+x^{2} y-x y^{2}-y^{3}}{(x+z)^{2}}=\frac{b^{2}}{a^{2} c}
\end{gathered}
$$ | \frac{^{3}+b}{2},\quad\frac{b-^{3}}{2},\quad\frac{2^{2}-^{3}-b}{2} | 91 | 40 |
math | 1. Find all values of $x$, for each of which one of the three given numbers $\log _{x}\left(x-\frac{1}{3}\right)$, $\log _{x-\frac{1}{3}}(x-3)$, and $\log _{x-3} x$ is equal to the product of the other two. | \frac{10}{3},\frac{3+\sqrt{13}}{2} | 76 | 21 |
math | A tree has $10$ pounds of apples at dawn. Every afternoon, a bird comes and eats $x$ pounds of apples. Overnight, the amount of food on the tree increases by $10\%$. What is the maximum value of $x$ such that the bird can sustain itself indefinitely on the tree without the tree running out of food? | \frac{10}{11} | 73 | 9 |
math | 5. Find the number of pairs of integers $(x ; y)$ that satisfy the condition $5 x^{2}-6 x y+y^{2}=6^{100}$. | 19594 | 38 | 5 |
math | 10. Koschei is counting his gold coins. When he counts them by tens, there are 7 coins left, and he is 3 coins short of a whole number of dozens. Koschei's wealth is estimated at $300-400$ coins. How many coins does Koschei have? | 357 | 68 | 3 |
math | 277. Combinatorial problem. Find the number of ordered sets $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $n$ natural numbers, in which
$$
1 \leqslant a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n} ; \quad a_{i} \leqslant i \quad(i=1,2, \ldots, n)
$$ | A_{n}=\frac{1}{n+1}C_{2n}^{n}=\frac{(2n!)}{n!(n+1)!} | 111 | 36 |
math | In an acute triangle $ABC$, the points $H$, $G$, and $M$ are located on $BC$ in such a way that $AH$, $AG$, and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$, $AB=10$, and $AC=14$. Find the area of triangle $ABC$. | 12\sqrt{34} | 86 | 8 |
math | 2. Let $\mathrm{i}$ be the imaginary unit, simplify $(\mathrm{i}+1)^{2016}+(\mathrm{i}-1)^{2016}=$ | 2^{1009} | 40 | 7 |
math | [ $\left[\begin{array}{l}\text { The ratio in which the bisector divides the side } \\ {[\text { Law of Cosines }}\end{array}\right]$
In triangle $ABC$, the bisectors $BL$ and $AE$ of angles $ABC$ and $BAC$ respectively are drawn, intersecting at point $O$. It is known that $AB=BL$, the perimeter of triangle $ABC$ is ... | 8 | 109 | 1 |
math | 3. In the Slytherin faculty at Hogwarts, there are 30 students. Some are friends (friendship is mutual, i.e., if A is friends with B, then B is also friends with A), but there are no 3 people who are pairwise friends with each other. On New Year's, each student sent cards to all their friends. What is the maximum numbe... | 450 | 87 | 3 |
math | Solve the following equation:
$$
4^{2 x+1}-\sqrt{29 \cdot 4^{2 x+1}+2^{2 x+1}+2 \cdot 8^{2 x}}=21 \cdot 2^{2 x}-4
$$ | x_1=\frac{3}{2},x_2=-\frac{3}{2} | 62 | 21 |
math | Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by... | 850 | 138 | 3 |
math | 10. In a math competition, there are 5 questions, each with a different natural number score. The smaller the question number, the fewer points the question is worth (for example, the score of question 1 is less than the score of question 2). Xiao Ming answered all the questions correctly, and his total score for the f... | 35 | 107 | 2 |
math | Alex starts with a rooted tree with one vertex (the root). For a vertex $v$, let the size of the subtree of $v$ be $S(v)$. Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects on... | 9901 | 174 | 4 |
math | Example 5 (2003 2004 Swedish Mathematical Competition) Find all real numbers $x$ that satisfy the equation $\left[x^{2}-2 x\right]+2[x]=$ $[x]^{2}$, where $[a]$ denotes the greatest integer less than or equal to $a$.
| x \in \mathbf{Z} \cup\left[n+1, \sqrt{n^{2}+1}+1\right)(n \in \mathbf{N}) | 68 | 40 |
math | 34 Using the digits $0,1,2,3$ and 4, find the number of 13 -digit sequences that can be written so that the difference between any two consecutive digits is 1 .
Examples of such 13-digit sequences are 0123432123432,2323432321234 and 3210101234323. | 3402 | 98 | 4 |
math | 1. In a game, three types of tokens are used, each with a different value expressed in denars. The value of each token is a natural number. Bojan, Ace, and Sasha each have at least one token of each type. Bojan has 4 tokens with a total value of 28 denars, Ace has 5 tokens with a total value of 21 denars, and Sasha has... | 17 | 100 | 2 |
math | 21.1.13 ** There are $n$ parking spaces along a road, and $n$ drivers each driving a car. Each driver parks their car in front of their favorite parking space. If that space is already occupied, they park in the nearest available space down the road. If that space and all the spaces below it are occupied, they drive aw... | (n+1)^{n-1} | 159 | 9 |
math | 6. Given that the radius of the inscribed sphere of a regular tetrahedron is 1, then the volume of the regular tetrahedron is | 8\sqrt{3} | 33 | 6 |
math | An integer consists of 7 different digits, and is a multiple of each of its digits.
What digits are in this nubmer? | 1, 2, 3, 6, 7, 8, 9 | 28 | 19 |
math | 2. Find the value of the expression $\cos ^{4} \frac{7 \pi}{24}+\sin ^{4} \frac{11 \pi}{24}+\sin ^{4} \frac{17 \pi}{24}+\cos ^{4} \frac{13 \pi}{24}$. | \frac{3}{2} | 76 | 7 |
math | Determine all possible values of $m+n$, where $m$ and $n$ are positive integers satisfying \[\operatorname{lcm}(m,n) - \gcd(m,n) = 103.\] | 21, 105, 309 | 45 | 12 |
math | a) Solve the equation:
$$
x+\sqrt{(x+1)(x+2)}=3
$$
b) Solve the equation:
$$
x+\sqrt{(x-1) x}+\sqrt{x(x+1)}+\sqrt{(x+1)(x-1)}=3
$$ | \frac{7}{9} | 65 | 7 |
math | ## SUBJECT III
On the sides $A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{n-1} A_{n}, A_{n} A_{1}$ of a regular polygon with side length $a$, consider the points $B_{1}, B_{2}, \ldots, B_{n}$, respectively, in the same direction and such that $\left[A_{1} B_{1}\right] \equiv\left[A_{2} B_{2}\right] \equiv \ldots \equiv\left[A... | \frac{}{2} | 174 | 6 |
math | Three friends wish to divide five different tasks among themselves, such that every friend must handle at least one task. In how many different ways can this be done? | 150 | 32 | 3 |
math | Problem 3. Jana asked Philip how much the fish his father caught weighed. Philip replied: The head together with the tail weighs $3 \mathrm{~kg}$, the head together with the body weighs $7 \mathrm{~kg}$, and the body together with the tail weighs $8 \mathrm{~kg}$. How much does the fish weigh? | 9 | 76 | 1 |
math | 【Question 17】
For any positive integer $\mathrm{n}$, let $\mathrm{f}(\mathrm{n})$ denote the last digit of $1+2+3+\cdots \cdots+\mathrm{n}$, such as $\mathrm{f}(1)=1, \mathrm{f}(2)=3, \mathrm{f}(5)=5$, etc. Find the value of $f(2)+f(4)+f(6)+\cdots+f$ (2012). | 3523 | 108 | 4 |
math | 6. (10 points) There are two docks, $A$ and $B$, on a river, with $A$ upstream and $B$ downstream. Two people, A and B, start from $A$ and $B$ respectively at the same time, rowing towards each other, and meet after 4 hours. If A and B start from $A$ and $B$ respectively at the same time, rowing in the same direction, ... | 10 | 143 | 2 |
math | ## Task A-3.3.
A number consisting only of the digits 2 and 3 is called a happy number. Thus, happy numbers are in sequence $2,3,22,23, \ldots$. Determine the 2050th happy number. | 22222222233 | 59 | 11 |
math | Aliens from Lumix have one head and four legs, while those from Obscra have two heads and only one leg. If 60 aliens attend a joint Lumix and Obscra interworld conference, and there are 129 legs present, how many heads are there? | 97 | 60 | 2 |
math | Example 2 (1992 "Friendship Cup" International Mathematics Invitational for Grade 9) Let $a, b, c, x, y, z \in R, a^{2}+b^{2}+c^{2}=25, x^{2}+$ $y^{2}+z^{2}=36, a x+b y+c z=30$, find the value of $\frac{a+b+c}{x+y+z}$. | \frac{5}{6} | 100 | 7 |
math | 9.2. What digit can the number $f(x)=[x]+[3 x]+[6 x]$ end with, where $x$ is an arbitrary positive real number? Here $[x]$ denotes the integer part of the number $x$, that is, the greatest integer not exceeding $x$.
| 0,1,3,4,6,7 | 65 | 11 |
math | For any real number sequence $A=\left(a_{1}, a_{2}, a_{3}, \cdots\right)$, define $\Delta A$ as the sequence $\left(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3}, \cdots\right)$, where its $n$-th term is $a_{n+1}-a_{n}$. Assume that all terms of the sequence $\Delta (\Delta A)$ are 1, and $a_{19}=a_{92}=0$. Try to find $a_{1}$... | 819 | 147 | 3 |
math | Robin goes birdwatching one day. he sees three types of birds: penguins, pigeons, and robins. $\frac23$ of the birds he sees are robins. $\frac18$ of the birds he sees are penguins. He sees exactly $5$ pigeons. How many robins does Robin see? | 16 | 76 | 2 |
math | 10. (10 points) During the Spring Festival promotion, customers receive a 50 yuan voucher for every 100 yuan paid in cash. These vouchers cannot be exchanged for cash but can be used to purchase goods, with the following rules: vouchers received in a single purchase cannot be used in the same purchase; the cash paid fo... | 2300 | 129 | 4 |
math | 9. From the 10 numbers $0,1,2,3,4,5,6,7,8,9$, choose 3 numbers such that their sum is an even number not less than 10. The number of different ways to do this is $\qquad$.
| 51 | 62 | 2 |
math | 4. Find all the options for writing the number 2003 as the sum of at least 2 consecutive natural numbers. | 2003=1001+1002 | 27 | 14 |
math | Problem 10.4. Consider the sequence
$$
a_{n}=\cos (\underbrace{100 \ldots 0^{\circ}}_{n-1})
$$
For example, $a_{1}=\cos 1^{\circ}, a_{6}=\cos 100000^{\circ}$.
How many of the numbers $a_{1}, a_{2}, \ldots, a_{100}$ are positive? | 99 | 103 | 2 |
math | 6. Given a positive integer $n(n \geqslant 2)$. Find the largest real number $\lambda$, such that the inequality $a_{n}^{2} \geqslant \lambda\left(a_{1}+a_{2}+\cdots+a_{n-1}\right)+2 a_{n}$ holds for any positive integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying $a_{1}<a_{2}<\cdots<a_{n}$. | \frac{2 n-4}{n-1} | 112 | 12 |
math | Example 5.26. Compute $\int_{0}^{1} \frac{\sin x}{x} d x$ with an accuracy of 0.01. | 0.94 | 37 | 4 |
math | 9.042. $0.3^{2+4+6+\ldots+2x}>0.3^{72}(x \in N)$. | 1,2,3,4,5,6,7 | 36 | 13 |
math | Let $n$ be a positive integer. How many polynomials $P$ with coefficients in the set $\{0,1,2,3\}$ are there such that $P(2)=n$? | \lfloorn/2\rfloor+1 | 44 | 11 |
math | 372. Solve the system of equations:
$$
\left\{\begin{array}{l}
\frac{1}{x}+\frac{1}{y}=1 \\
\frac{1}{3-x}+\frac{1}{3-y}=2
\end{array}\right.
$$ | (2;2) | 64 | 5 |
math | Example 1 Solve the system of congruences
$$
\left\{\begin{array}{l}
x \equiv 1(\bmod 3) \\
x \equiv -1(\bmod 5) \\
x \equiv 2(\bmod 7) \\
x \equiv -2(\bmod 11)
\end{array}\right.
$$ | x\equiv394(\bmod1155) | 79 | 14 |
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