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 | Problem 4. Let $\left(a_{n}\right)_{n \geq 1}$ be a sequence of positive real numbers such that $a_{1}=1$ and
$$
\frac{1}{a_{1}+a_{2}}+\frac{1}{a_{2}+a_{3}}+\ldots+\frac{1}{a_{n-1}+a_{n}}=a_{n}-1, \text { for all } n \geq 2
$$
Determine the integer part of the number $A_{n}=a_{n} a_{n+1}+a_{n} a_{n+2}+a_{n+1} a_{n+2}... | 3n+2 | 181 | 4 |
math | For every positive integer $n$ determine the least possible value of the expression
\[|x_{1}|+|x_{1}-x_{2}|+|x_{1}+x_{2}-x_{3}|+\dots +|x_{1}+x_{2}+\dots +x_{n-1}-x_{n}|\]
given that $x_{1}, x_{2}, \dots , x_{n}$ are real numbers satisfying $|x_{1}|+|x_{2}|+\dots+|x_{n}| = 1$. | 2^{1-n} | 121 | 7 |
math | 8. Given a six-digit decimal number composed of six positive integers, the digit in the units place is a multiple of 4, the digits in the tens and hundreds places are multiples of 3, and the sum of the digits of the six-digit number is 21. Then the number of six-digit numbers that satisfy the above conditions is $\qqua... | 126 | 74 | 3 |
math | A number is [i]interesting [/i]if it is a $6$-digit integer that contains no zeros, its first $3$ digits are strictly increasing, and its last $3$ digits are non-increasing. What is the average of all interesting numbers? | 308253 | 56 | 6 |
math | 3. Option 1.
In the Ivanov family, both the mother and the father, and their three children, were born on April 1st. When the first child was born, the parents' combined age was 45 years. The third child in the family was born a year ago, when the sum of the ages of all family members was 70 years. How old is the midd... | 5 | 101 | 1 |
math | Question 127, Find the last two digits of $\left[(2+\sqrt{3})^{2^{2020}}\right]$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 53 | 58 | 2 |
math | Determine the integers $n$ such that $n^{5}-2 n^{4}-7 n^{2}-7 n+3=0$. | -1,3 | 31 | 4 |
math | Example 3 Does there exist a positive integer $n$ that satisfies the following two conditions simultaneously:
(1) $n$ can be decomposed into the sum of 1990 consecutive positive integers.
(2) $n$ has exactly 1990 ways to be decomposed into the sum of several (at least two) consecutive positive integers.
(31st IMO Short... | n=5^{180} \times 199^{10}, n=5^{10} \times 199^{180} | 83 | 36 |
math | In Moscow, there are 2000 rock climbers, in St. Petersburg and Krasnoyarsk - 500 each, in Yekaterinburg - 200, and the remaining 100 are scattered across Russia. Where should the Russian Rock Climbing Championship be held to minimize the transportation costs for the participants?
# | Moscow | 75 | 2 |
math | 7. A company invested in a project in 2009, with both cash inputs and cash revenues every year. It is known that
(1) In 2009, the company invested 10 million yuan, and the investment will decrease by $20\%$ each subsequent year;
(2) In 2009, the company earned 5 million yuan, and the revenue will increase by $25\%$ eac... | 2013 | 117 | 4 |
math | Problem 4.6. Fourth-grader Vasya goes to the cafeteria every school day and buys either 9 marshmallows, or 2 meat pies, or 4 marshmallows and 1 meat pie. Sometimes Vasya is so busy socializing with classmates that he doesn't buy anything at all. Over 15 school days, Vasya bought 30 marshmallows and 9 meat pies. How man... | 7 | 100 | 1 |
math | 1. Find all values of $p$, for each of which the numbers $p-2$, $2 \cdot \sqrt{p}$, and $-3-p$ are respectively the first, second, and third terms of some geometric progression. | 1 | 51 | 1 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}\left(\sqrt{\left(n^{2}+1\right)\left(n^{2}+2\right)}-\sqrt{\left(n^{2}-1\right)\left(n^{2}-2\right)}\right)
$$ | 3 | 74 | 1 |
math | Problem 5. Petya has 50 balls of three colors: red, blue, and green. It is known that among any 34 balls, there is at least one red; among any 35, there is at least one blue; among any 36, there is at least one green. How many green balls can Petya have? | 15,16,or17 | 77 | 9 |
math | Determine all pairs of positive integers $(x, y)$ satisfying the equation $p^x - y^3 = 1$, where $p$ is a given prime number. | (1, 1) | 37 | 7 |
math | 6.066. $x^{2}-4 x-6=\sqrt{2 x^{2}-8 x+12}$. | x_{1}=-2,x_{2}=6 | 30 | 11 |
math | 7. There are 120 equally distributed spheres inside a regular tetrahedron $A-B C D$. How many spheres are placed at the bottom of this regular tetrahedron? | 36 | 40 | 2 |
math | 13.034. In a store, textbooks on physics and mathematics were received for sale. When $50\%$ of the mathematics textbooks and $20\%$ of the physics textbooks were sold, which amounted to a total of 390 books, the remaining mathematics textbooks were three times as many as the remaining physics textbooks. How many textb... | 720150 | 88 | 6 |
math | 6. Given $a, b, c \in \mathbf{R}_{+}$, and
$$
a+b+c=12, a b+b c+c a=45 \text{. }
$$
Then $\min \max \{a, b, c\}=$ $\qquad$ | 5 | 66 | 1 |
math | Four. (50 points) On a plane, there are $n$ points, no three of which are collinear. Each pair of points is connected by a line segment, and each line segment is colored either red or blue. A triangle with all three sides of the same color is called a "monochromatic triangle." Let the number of monochromatic triangles ... | \frac{k(k-1)(k-2)}{3} | 141 | 14 |
math | 3A. Given are 21 tiles in the shape of a square, of the same size. On four tiles is written the number 1; on two tiles is written the number 2; on seven tiles is written the number 3; on eight tiles is written the number 4. Using 20 of these tiles, Dimitar formed a rectangle with dimensions 4 by 5. For the formed recta... | 1 | 120 | 1 |
math | Nathalie has some quarters, dimes and nickels. The ratio of the number of quarters to the number of dimes to the number of nickels that she has is $9: 3: 1$. The total value of these coins is $\$ 18.20$. How many coins does Nathalie have?
(A) 130
(B) 117
(C) 98
(D) 91
(E) 140 | 91 | 101 | 2 |
math | 3. (7 points) A seller bought a batch of pens and sold them. Some customers bought one pen for 10 rubles, while others bought 3 pens for 20 rubles. It turned out that the seller made the same profit from each sale. Find the price at which the seller bought the pens. | 5 | 67 | 1 |
math | 32nd Putnam 1971 Problem A2 Find all possible polynomials f(x) such that f(0) = 0 and f(x 2 + 1) = f(x) 2 + 1. Solution | f(x)=x | 50 | 4 |
math | Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for La... | 4 | 148 | 1 |
math | 785. Solve the equation in integers
$$
3 x y+y=7 x+3
$$ | (0;3),(-1;2) | 23 | 10 |
math | In the right triangle $ABC$ shown, $E$ and $D$ are the trisection points of the hypotenuse $AB$. If $CD=7$ and $CE=6$, what is the length of the hypotenuse $AB$? Express your answer in simplest radical form.
[asy]
pair A, B, C, D, E;
A=(0,2.9);
B=(2.1,0);
C=origin;
D=2/3*A+1/3*B;
E=1/3*A+2/3*B;
draw(A--B--C--cycle);
d... | 3\sqrt{17} | 182 | 7 |
math | 2. Quadratic trinomials $P(x)=x^{2}+\frac{x}{2}+b$ and $Q(x)=x^{2}+c x+d$ with real coefficients are such that $P(x) Q(x)=Q(P(x))$ for all $x$. Find all real roots of the equation $P(Q(x))=0$. | x\in{-1,\frac{1}{2}} | 77 | 12 |
math | ## Task 18/71
For which values of $t$ does the system of equations
$$
x^{2 n}+y^{2 n}=1000 \quad, \quad x^{n}+y^{n}=t
$$
with natural $n$ have positive real solutions? | 10\sqrt{10}<\leq20\sqrt{5} | 67 | 18 |
math | 1. In the field of real numbers, solve the system of equations
$$
\begin{aligned}
& x^{2}+6(y+z)=85 \\
& y^{2}+6(z+x)=85 \\
& z^{2}+6(x+y)=85
\end{aligned}
$$ | (5,5,5),(-17,-17,-17),(7,7,-1),(-7,-7,13),(-1,7,7),(13,-7,-7),(7,-1,7),(-7,13,-7) | 67 | 59 |
math | Let $A$ and $B$ be two finite sets such that there are exactly $144$ sets which are subsets of $A$ or subsets of $B$. Find the number of elements in $A \cup B$. | 8 | 48 | 1 |
math |
Problem 10a.1. Find all pairs $(a ; b)$ of integers such that the system
$$
\left\lvert\, \begin{gathered}
x^{2}+2 a x-3 a-1=0 \\
y^{2}-2 b y+x=0
\end{gathered}\right.
$$
has exactly three real solutions.
| (0,1),(0,-1),(-3,2),(-3,-2) | 82 | 19 |
math | B3. A digital clock displays times from 00:00:00 to 23:59:59 throughout a day. Every second of the day, you can add up the digits, resulting in an integer. At 13:07:14, for example, the sum is $1+3+0+7+1+4=16$. If you write down the sum for every possible time on the clock and then take the average of all these numbers... | 18\frac{3}{4} | 111 | 9 |
math | 7.3. Agent 007 wants to encrypt his number using two natural numbers m and n so that $0.07=\frac{1}{m}+\frac{1}{n}$. Can he do it? | 0.07=\frac{1}{50}+\frac{1}{20} | 48 | 20 |
math | Can the values of the letters as digits be chosen in the following two (independent) problems so that the statements are true?
a) $\mathrm{FORTY}+\mathrm{TEN}+\mathrm{TEN}=\mathrm{SIXTY}$
b) DREI+DREI+DREI=NEUN
(Identical letters are to be replaced by the same digits, different letters by different digits. The a) pro... | 29786+850+850=31486,2439\cdot3=7317,1578\cdot3=4734,2716\cdot3=8148 | 126 | 58 |
math | Example 1 If the real number $r$ satisfies $\left[r+\frac{19}{100}\right]+\left[r+\frac{20}{100}\right]+\cdots+\left[r+\frac{91}{100}\right]=$ 546, find $[100 r]$. | 743 | 71 | 3 |
math | $11 \cdot 21$ Arrange all powers of 3 and sums of distinct powers of 3 in an increasing sequence: $1,3,4,9,10,12,13, \cdots$ Find the 100th term of this sequence.
(4th American Mathematical Invitational, 1986) | 981 | 76 | 3 |
math | 4・169 Solve the equation in the set of natural numbers
$$x^{y}=y^{x}(x \neq y) .$$ | x=2, y=4 \text{ or } x=4, y=2 | 32 | 19 |
math | 1. What is the maximum number of lattice points (i.e. points with integer coordinates) in the plane that can be contained strictly inside a circle of radius 1 ? | 4 | 35 | 1 |
math | 11.4. Find the smallest natural number that has exactly 55 natural divisors, including one and the number itself. | 2^{10}\cdot3^{4} | 27 | 10 |
math | Find all polynomials $P$ with real coefficients such that $$\frac{P(x)}{yz}+\frac{P(y)}{zx}+\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x)$$ holds for all nonzero real numbers $x,y,z$ satisfying $2xyz=x+y+z$.
[i]Proposed by Titu Andreescu and Gabriel Dospinescu[/i] | P(x) \equiv c(x^2 + 3) | 92 | 14 |
math | 2. (I. Akulich) Find the largest natural $n$ with the following property: for any odd prime $p$ less than $n$, the difference $n-p$ is also a prime number. | 10 | 45 | 2 |
math | 5. Find all integer pairs $(a, b)$, where $a \geqslant 1, b \geqslant 1$, and satisfy the equation $a^{b^{2}}=b^{a}$.
$$
b=3, a=b^{k}=3^{3}=27 \text {. }
$$
In summary, all positive integer pairs that satisfy the equation are
$$
(a, b)=(1,1),(16,2),(27,3) .
$$ | (a, b)=(1,1),(16,2),(27,3) | 107 | 18 |
math | Determine whether there exist a positive integer $n<10^9$, such that $n$ can be expressed as a sum of three squares of positive integers by more than $1000$ distinct ways? | \text{Yes} | 45 | 5 |
math | Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x, y \), we have:
\[
f\left(x^{2}-y^{2}\right)=x f(x)-y f(y)
\]
(USAMO 2002) | f(X)=Xf(1) | 72 | 8 |
math | Let's determine all non-negative integers $n$ for which there exist integers $a$ and $b$ such that $n^{2}=a+b$ and $n^{3}=a^{2}+b^{2}$. | 0,1,2 | 48 | 5 |
math | 9. Four schools each send 3 representatives to form $n$ groups to participate in social practice activities (each representative can participate in several groups). The rules are: (1) representatives from the same school are not in the same group; (2) any two representatives from different schools are in exactly one gr... | 9 | 78 | 1 |
math | G1.4 When 491 is divided by a two-digit integer, the remainder is 59 . Find this two-digit integer. Let the number be $10 x+y$, where $0<x \leq 9,0 \leq y \leq 9$. | 72 | 61 | 2 |
math | Jeff has a deck of $12$ cards: $4$ $L$s, $4$ $M$s, and $4$ $T$s. Armaan randomly draws three cards without replacement. The probability that he takes $3$ $L$s can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m +n$. | 56 | 86 | 2 |
math | Find a positive integer satisfying the following three clues:
- When multiplied by 2 , it yields a perfect square.
- When multiplied by 3 , it yields a perfect cube.
- When multiplied by 5 , it yields a perfect fifth power. | 2^{15}3^{20}5^{24} | 50 | 15 |
math | 7. (10 points) The last 8 digits of $11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111$ are | 87654321 | 55 | 8 |
math | 1. Let $a, b$ and $c$ be non-zero real numbers and
$$
a+\frac{b}{c}=b+\frac{c}{a}=c+\frac{a}{b}=1
$$
Calculate the value of the expression $a b+b c+c a$. | 0 | 63 | 1 |
math | 303. Find the derivative of the function $y=0.5^{\sin 4 x}$. | 4\cos4x\cdot0.5^{\sin4x}\ln0.5 | 24 | 20 |
math | 6. Let $S_{n}$ be the sum of the elements of all 3-element subsets of the set $A=\left\{1, \frac{1}{2}, \cdots, \frac{1}{2^{n-1}}\right\}$. Then $\lim _{n \rightarrow \infty} \frac{S_{n}}{n^{2}}=$ $\qquad$ . | 1 | 88 | 1 |
math | In a cycling competition with $14$ stages, one each day, and $100$ participants, a competitor was characterized by finishing $93^{\text{rd}}$ each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.) | 2 | 69 | 1 |
math | 1. Baron Munchhausen was told that some polynomial $P(x)=a_{n} x^{n}+\ldots+a_{1} x+a_{0}$ is such that $P(x)+P(-x)$ has exactly 45 distinct real roots. Baron doesn't know the value of $n$. Nevertheless he claims that he can determine one of the coefficients $a_{n}, \ldots, a_{1}, a_{0}$ (indicating its position and v... | a_0 = 0 | 113 | 7 |
math | Kustarev A.A.
The faces of a cube are numbered from 1 to 6. The cube was rolled twice. The first time, the sum of the numbers on the four side faces was 12, the second time it was 15. What number is written on the face opposite the one where the digit 3 is written? | 6 | 73 | 1 |
math | 2. Marina needs to buy a notebook, a pen, a ruler, a pencil, and an eraser to participate in the Olympiad. If she buys a notebook, a pencil, and an eraser, she will spend 47 tugriks. If she buys a notebook, a ruler, and a pen, she will spend 58 tugriks. How much money will she need for the entire set, if the notebook c... | 90 | 98 | 2 |
math | ## Task B-2.2.
Kate and Mare often competed in solving math problems. The professor got tired of their competition and gave them a task that they had to solve together. Kate wrote down 2016 numbers on the board, with each number being either $\sqrt{2}-1$ or $\sqrt{2}+1$. Mare then multiplied the first and second numbe... | 505 | 151 | 3 |
math | 10. [60] Let $n$ be a fixed positive integer, and choose $n$ positive integers $a_{1}, \ldots, a_{n}$. Given a permutation $\pi$ on the first $n$ positive integers, let $S_{\pi}=\left\{i \left\lvert\, \frac{a_{i}}{\pi(i)}\right.\right.$ is an integer $\}$. Let $N$ denote the number of distinct sets $S_{\pi}$ as $\pi$ r... | 2^{n}-n | 154 | 5 |
math | Example 5. Find the value of $\sum_{k=1}^{n} k^{2} C_{n}^{k}$. (3rd Putnam Mathematical Competition, USA) | n(n+1) \cdot 2^{n-2} | 40 | 14 |
math | 1. Find the number of points in the $x O y$ plane having natural coordinates $(x, y)$ and lying on the parabola $y=-\frac{x^{2}}{3}+20 x+63$. | 20 | 50 | 2 |
math | ## Task A-3.4.
Determine all pairs of natural numbers $(m, n)$ that satisfy the equation
$$
m n^{2}=100(n+1)
$$ | (200,1),(75,2),(24,5),(11,10) | 40 | 23 |
math | 9. Evaluate: $\sin 37^{\circ} \cos ^{2} 34^{\circ}+2 \sin 34^{\circ} \cos 37^{\circ} \cos 34^{\circ}-\sin 37^{\circ} \sin ^{2} 34^{\circ}$. | \frac{\sqrt{6}+\sqrt{2}}{4} | 78 | 15 |
math | Three. (50 points)
Find all non-military integers $a, b, a \neq b$, such that:
the set of integers can be partitioned into 3 subsets, so that for each $n, n+a, n+b$ belong to these 3 subsets respectively. | ^{\}\cdotb^{\}\equiv2(\bmod3) | 61 | 15 |
math | There are functions $f(x)$ with the following properties:
- $f(x)=a x^{2}+b x+c$ for some integers $a, b$ and $c$ with $a>0$, and
- $f(p)=f(q)=17$ and $f(p+q)=47$ for some prime numbers $p$ and $q$ with $p<q$. For each such function, the value of $f(p q)$ is calculated. The sum of all possible values of $f(p q)$ is $S... | 71 | 129 | 2 |
math | 9 Let $f(x)=\frac{1}{1+2^{\lg x}}+\frac{1}{1+4^{\lg x}}+\frac{1}{1+8^{\lg x}}$, then $f(x)+f\left(\frac{1}{x}\right)=$ | 3 | 64 | 1 |
math | 1. Find all real functions $f, g, h: \mathbf{R} \rightarrow \mathbf{R}$, such that for any real numbers $x, y$, we have
$$
\begin{array}{l}
(x-y) f(x)+h(x)-x y+y^{2} \leqslant h(y) \\
\leqslant(x-y) g(x)+h(x)-x y+y^{2} .
\end{array}
$$
(53rd Romanian Mathematical Olympiad (First Round)) | f(x) = g(x) = -x + a, h(x) = x^{2} - a x + b | 115 | 26 |
math | 3. Team A and Team B each send out 5 players to participate in a chess broadcast tournament according to a pre-arranged order. The two teams first have their No. 1 players compete; the loser is eliminated, and the winner then competes with the No. 2 player of the losing team, …, until all players of one side are elimin... | 252 | 115 | 3 |
math | Exercise 4. Determine all functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that, for all real $x$ and $y$, the following equality holds:
$$
f(x+y)=f(x-y)+f(f(1-x y))
$$ | f(x)=0 | 60 | 4 |
math | 1. Given three positive integers $a$, $b$, and $c$ whose squares sum to 2011, and the sum of their greatest common divisor and least common multiple is 388. Then the sum of the numbers $a$, $b$, and $c$ is $\qquad$ . | 61 | 66 | 2 |
math | For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \le 10^{10}$ that satisfy the equation
\[
S(n) = (S(S(n)))^2.
\]
Find the remainder when $K$ is divided by $1000$. | 632 | 83 | 3 |
math | ## Task 2 - 230822
A class is divided into learning brigades in such a way that the number of members in each brigade is 2 more than the number of brigades. If one fewer brigade had been formed, each brigade could have had 2 more members.
Show that the number of students in this class can be uniquely determined from ... | 24 | 84 | 2 |
math | 8. A number plus 79 becomes a perfect square, and when this number plus 204 is added, it becomes another perfect square. Then this number is $\qquad$ . | 21 | 40 | 2 |
math | 1. Given real numbers $x, y$ satisfy the system of equations
$$
\left\{\begin{array}{l}
x^{3}+y^{3}=19, \\
x+y=1 .
\end{array}\right.
$$
then $x^{2}+y^{2}=$ $\qquad$ | 13 | 71 | 2 |
math | Call a rational number short if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m$-tastic if there exists a number $c \in\{1,2,3, \ldots, 2017\}$ such that $\frac{10^{t}-1}{c \cdot m}$ is short, and such that $\frac{10^{k}-1}{c \cdot m}$ is not sh... | 807 | 179 | 3 |
math | 9.3. What is the smallest number of digits that need to be appended to the right of the number 2014 so that the resulting number is divisible by all natural numbers less than $10?$ | 4 | 44 | 1 |
math | 7. If the inequality
$$
2 \sin ^{2} C+\sin A \cdot \sin B>k \sin B \cdot \sin C
$$
holds for any $\triangle A B C$, then the maximum value of the real number $k$ is . $\qquad$ | 2\sqrt{2}-1 | 62 | 7 |
math | $1 \cdot 39$ Find all natural numbers $n, k$ such that $n^{n}$ has $k$ digits, and $k^{k}$ has $n$ digits. | n=k=1,n=k=8,n=k=9 | 42 | 12 |
math | In triangle $ABC$, the altitude and median from vertex $C$ divide the angle $ACB$ into three equal parts. Determine the ratio of the sides of the triangle. | 2:\sqrt{3}:1 | 36 | 7 |
math | 1. On an island, there live knights who always tell the truth, and liars who always lie. In a room, 15 islanders gathered. Each of those in the room said two phrases: "Among my acquaintances in this room, there are exactly six liars" and "among my acquaintances in this room, there are no more than seven knights." How m... | 9 | 87 | 1 |
math | 8. An angle can be represented by two capital letters on its two sides and the letter of its vertex, the symbol for $\angle A O B$ (" $\angle$ " represents an angle), and it can also be represented by $\angle O$ (when there is only one angle at the vertex). In the triangle $\mathrm{ABC}$ below, $\angle B A O=\angle C A... | 20 | 126 | 2 |
math | 9. Given $O$ as the origin, $N(1,0)$, and $M$ as a moving point on the line $x=-1$, the angle bisector of $\angle M O N$ intersects the line $M N$ at point $P$, and the locus of point $P$ is curve $E$.
(1) Find the equation of curve $E$;
(2) A line $l$ with slope $k$ passes through the point $Q\left(-\frac{1}{2},-\frac... | (-\frac{1}{3},1]\cup{\frac{1+\sqrt{3}}{2}} | 147 | 23 |
math | 3. Given that the function $f(x)$ is an increasing function defined on $[-4,+\infty)$, to ensure that for all real numbers $x$ in the domain, the inequality $f\left(\cos x-b^{2}\right) \geqslant f\left(\sin ^{2} x-b-3\right)$ always holds, the range of the real number $b$ is $\qquad$. | [\frac{1}{2}-\sqrt{2},1] | 92 | 14 |
math | What is the last digit of $2023^{2024^{2025}}$? | 1 | 24 | 1 |
math | ## Task B-2.6.
Students Marko and Luka worked on a part-time basis, Marko in a tourist office, and Luka in a hotel. Marko earned a total of 200 kn for his work. Luka worked 5 hours less than Marko and earned 125 kn. If Marko had worked as many hours as Luka, and Luka as many hours as Marko, then Luka would have earned... | 10, | 132 | 3 |
math | 5. In a regular tetrahedron $ABCD$, $E$ and $F$ are on edges $AB$ and $AC$, respectively, such that $BE=3$, $EF=4$, and $EF$ is parallel to plane $BCD$. Then the area of $\triangle DEF$ is $\qquad$ | 2\sqrt{33} | 69 | 7 |
math | Solve the following system of equations:
$$
\begin{aligned}
& x^{2}-4 \sqrt{3 x-2}+10=2 y \\
& y^{2}-6 \sqrt{4 y-3}+11=x
\end{aligned}
$$ | 2,3 | 61 | 3 |
math | 10. (5 points) A rectangular prism, if the length is reduced by 2 cm, the width and height remain unchanged, the volume decreases by 48 cubic cm; if the width is increased by 3 cm, the length and height remain unchanged, the volume increases by 99 cubic cm; if the height is increased by 4 cm, the length and width remai... | 290 | 109 | 3 |
math | A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\sqrt7$.
Notes: ''Using a ruler construct points $\ldots$'' means: Newly constructed points arise only as the intersecti... | \sqrt{7} | 93 | 6 |
math | C1. How many three-digit multiples of 9 consist only of odd digits? | 11 | 17 | 2 |
math | 2. A school organized three extracurricular activity groups in mathematics, Chinese, and foreign language. Each group meets twice a week, with no overlapping schedules. Each student can freely join one group, or two groups, or all three groups simultaneously. A total of 1200 students participate in the extracurricular ... | 80 | 159 | 2 |
math | 12. Arrange all positive integers $m$ whose digits are no greater than 3 in ascending order to form a sequence $\left\{a_{n}\right\}$. Then $a_{2007}=$ $\qquad$ . | 133113 | 52 | 6 |
math | Sersa * Does there exist a positive integer $n$, such that the complex number $z=\left(\frac{3-\sqrt{3} \mathrm{i}}{2}\right)^{n}$ is a pure imaginary number? | \(\sqrt{3})^{n}\mathrm{i} | 49 | 12 |
math | # 2. Clone 1
The teacher wanted to write an example for calculation on the board:
$$
1,05+1,15+1,25+1,4+1,5+1,6+1,75+1,85+1,95=?
$$
but accidentally forgot to write one comma. After this, Kolya went to the board and, correctly performing all the operations, obtained an integer result. What is it? | 27 | 103 | 2 |
math | 13. (6 points) Given the five-digit number $\overline{54 \mathrm{a} 7 \mathrm{~b}}$ can be simultaneously divisible by 3 and 5, the number of such five-digit numbers is $\qquad$. | 7 | 55 | 1 |
math | 22. Given that $169(157-77 x)^{2}+100(201-100 x)^{2}=26(77 x-157)(1000 x-2010)$, find the value of $x$. | 31 | 68 | 2 |
math | 6.150. $\frac{u^{2}}{2-u^{2}}+\frac{u}{2-u}=2$. | u_{1}=1;u_{2,3}=\frac{1\\sqrt{33}}{4} | 29 | 25 |
math | 1、When Ma Xiaohu was doing a subtraction problem, he mistakenly wrote the unit digit of the minuend as 5 instead of 3, and the tens digit as 0 instead of 6. He also wrote the hundreds digit of the subtrahend as 2 instead of 7. The resulting difference was 1994. What should the correct difference be? $\qquad$ | 1552 | 85 | 4 |
math | 2. Points $X, Y$, and $Z$ lie on a circle with center $O$ such that $X Y=12$. Points $A$ and $B$ lie on segment $X Y$ such that $O A=A Z=Z B=B O=5$. Compute $A B$. | 2\sqrt{13} | 65 | 7 |
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