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 | Let's find all right-angled triangles whose sides are integers, and when 6 is added to the hypotenuse, we get the sum of the legs. | (7,24,25),(8,15,17),(9,12,15) | 33 | 25 |
math | Problem 4.7. Vanya goes to the swimming pool every Wednesday and Friday. After one of his visits, he noticed that he had already gone 10 times this month. What will be the date of the third visit in the next month if he continues to go on Wednesdays and Fridays? | 12 | 63 | 2 |
math | 1. (7 points) Compare $3^{76}$ and $5^{50}$. | 3^{76}>5^{50} | 21 | 10 |
math | Five. (20 points) The sequence $\left\{a_{n}\right\}$ is
$$
1,1,2,1,1,2,3,1,1,2,1,1,2,3,4, \cdots \text {. }
$$
Its construction method is:
First, give $a_{1}=1$, then copy this item 1 and add its successor number 2, thus, we get $a_{2}=1, a_{3}=2$;
Then copy all the previous items $1,1,2$, and add the successor numb... | 3961 | 278 | 4 |
math | 31.11. a) Find the remainder of the division of $171^{2147}$ by 52.
b) Find the remainder of the division of $126^{1020}$ by 138. | 54 | 55 | 2 |
math | The Pythagorean school believed that numbers are the origin of all things, and they called numbers such as $1, 3, 6, 10, \cdots$ triangular numbers. Therefore, arranging the triangular numbers in ascending order, the sum of the first 100 triangular numbers is $\qquad$. | 171700 | 67 | 6 |
math | 9.1. At the school for slackers, a competition on cheating and giving hints was organized. It is known that $75 \%$ of the students did not show up for the competition at all, and all the rest participated in at least one of the competitions. When the results were announced, it turned out that $10 \%$ of all those who ... | 200 | 122 | 3 |
math | 2. (6 points) A certain duplicator can print 3600 sheets of paper per hour, so printing 240 sheets of paper requires $\qquad$ minutes. | 4 | 39 | 1 |
math | 18. For any positive integer $n$, let $f(n)=70+n^{2}$ and $g(n)$ be the H.C.F. of $f(n)$ and $f(n+1)$. Find the greatest possible value of $g(n)$.
(2 marks)
對任何正整數 $n$, 設 $f(n)=70+n^{2}$, 而 $g(n)$ 則是 $f(n)$ 與 $f(n+1)$ 的最大公因數。求 $g(n)$ 的最大可能值。 | 281 | 122 | 3 |
math | 5. Find all positive integers $a_{1}, a_{2}, \cdots, a_{n}$, such that
$$
\begin{array}{l}
\quad \frac{99}{100}=\frac{a_{0}}{a_{1}}+\frac{a_{1}}{a_{2}}+\cdots+\frac{a_{n-1}}{a_{n}}, \\
\text { where } a_{0}=1,\left(a_{k+1}-1\right) a_{k-1} \geqslant a_{k}^{2}\left(a_{k}-1\right), k=1,2, \cdots, \\
n-1 .
\end{array}
$$ | a_{1}=2, a_{2}=5, a_{3}=56, a_{4}=78400 | 160 | 28 |
math | If $x$ is a real number satisfying the equation $$9\log_3 x - 10\log_9 x =18 \log_{27} 45,$$ then the value of $x$ is equal to $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
[i]Proposed by pog[/i] | 140 | 99 | 3 |
math | Let $ABC$ be a triangle where$\angle$[b]B=55[/b] and $\angle$ [b]C = 65[/b]. [b]D[/b] is the mid-point of [b]BC[/b]. Circumcircle of [b]ACD[/b] and[b] ABD[/b] cuts [b]AB[/b] and[b] AC[/b] at point [b]F[/b] and [b]E[/b] respectively. Center of circumcircle of [b]AEF[/b] is[b] O[/b]. $\angle$[b]FDO[/b] = ? | 30^\circ | 143 | 4 |
math | $\therefore 、 m$ distinct positive even numbers and $n$ distinct positive odd numbers have a total sum of 1987, for all such $m$ and $n$, what is the maximum value of $3 m+4 n$? Please prove your conclusion.
The sum of $m$ distinct positive even numbers and $n$ distinct positive odd numbers is 1987, for all such $m$ a... | 221 | 113 | 3 |
math | 2 [ Arithmetic. Mental calculation, etc. $\quad]$
One tea bag can brew two or three cups of tea. Milla and Tanya divided a box of tea bags equally. Milla brewed 57 cups of tea, and Tanya - 83 cups. How many tea bags could have been in the box? | 56 | 68 | 2 |
math | Let's find the values of $x$ for which the functions
$$
y=\frac{1}{\tan x}+\frac{1}{\cot x} \quad \text{and} \quad y=\frac{1}{\tan x}-\frac{1}{\cot x}
$$
will have their minimum values. | 45 | 71 | 2 |
math | 1. For breakfast, Mihir always eats a bowl of Lucky Charms cereal, which consists of oat pieces and marshmallow pieces. He defines the luckiness of a bowl of cereal to be the ratio of the number of marshmallow pieces to the total number of pieces. One day, Mihir notices that his breakfast cereal has exactly 90 oat piec... | 11 | 114 | 2 |
math | 1. Move everything to the right side
$(x+c)(x+d)-(2 x+c+d)=0$,
$x^{2}+c x+d x+c d-2 x-c-d=0$,
$x^{2}+(c+d-2) x+c d-c-d=0$.
Find the discriminant of the quadratic equation
$D=(c+d-2)^{2}-4(cd-c-d)=c^{2}+d^{2}+4+2cd-4c-4d-4cd+4c+4d=$ $=c^{2}+d^{2}-2cd+4=(c-d)^{2}+4>0$ Therefore, the equation has two distinct roots. | (-)^{2}+4>0 | 154 | 9 |
math | 2. In $\square A B C D$, $\angle B<90^{\circ}, A B<B C$. From point $D$ draw tangents to the circumcircle $\Gamma$ of $\triangle A B C$, the points of tangency are $E$ and $F$. It is known that $\angle E D A=\angle F D C$. Find $\angle A B C$. | 60 | 82 | 2 |
math | ## Task 4 - 140724
Fritz borrowed a book from his friend Max for 6 days. On the morning of the 6th day, he tells his friend Paul, who wants to borrow the book after him:
"On the first day, I read the 12th part of the book, on the following 4 days I read an eighth each, and today, if I want to finish the book, I have ... | 120 | 144 | 3 |
math | Example 1. Solve the Riccati equation
$$
y^{\prime}-y^{2}+2 e^{x} y=e^{2 x}+e^{x}
$$
knowing its particular solution $y_{1}=e^{x}$. | e^{x}+\frac{1}{C-x} | 56 | 12 |
math | It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $n(n+1)/2$. A Pythagorean triple of $\textit{square numbers}$ is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of $\textit{triangular numbers}$ (a PTTN) be an ordered triple of positive integers $(a,b... | 14 | 222 | 2 |
math | 5. Four friends (from 6th grade, 1 point). Masha, Nina, Lena, and Olya are friends. They are all of different heights, but the difference is very small - it's hard to tell by eye. One day they decided to find out who is taller and who is shorter. It turned out that Nina is shorter than Masha, and Lena is taller than Ol... | \frac{1}{6} | 98 | 7 |
math | 62nd Putnam 2001 Problem A5 Find all solutions to x n+1 - (x + 1) n = 2001 in positive integers x, n. Solution | x=13,n=2 | 43 | 7 |
math | 8. (10 points) The largest odd number that cannot be written as the sum of three distinct composite numbers is
保留源文本的换行和格式，翻译结果如下：
8. (10 points) The largest odd number that cannot be written as the sum of three distinct composite numbers is | 17 | 62 | 2 |
math | Four villages are located at the vertices of a square with a side length of 1 km. To allow travel from each village to any other, two straight roads were laid along the diagonals of the square. Can the road network between the villages be laid out in a different way so that the total length of the roads is reduced, but... | 2.75 | 81 | 4 |
math | Four. (20 points) Let $x, y, z$ be the lengths of the sides of a triangle, and $x+y+z=1$. Find the minimum value of the real number $\lambda$ such that
$$
\lambda(x y+y z+z x) \geqslant 3(\lambda+1) x y z+1
$$
always holds. | 5 | 81 | 1 |
math | João has more than 30 and less than 100 chocolates. If he arranges the chocolates in rows of 7, one will be left over. If he arranges them in rows of 10, two will be left over. How many chocolates does he have?
# | 92 | 61 | 2 |
math | Exercise 2. Mr. Deschamps owns chickens and cows; the chickens all have two legs, and the cows have four. In preparation for winter, he must make slippers for them. He has 160 animals in total, and he had to make 400 slippers. How many cows does he have?
Only a numerical answer is expected here. | 40 | 77 | 2 |
math | ## Task 2 - 190612
Ulrike wants to specify four natural numbers in a certain order so that the following conditions are met:
The second number is 1 less than double the first number, the third number is 1 less than double the second number, the fourth number is 1 less than double the third number, and the sum of the ... | 6,11,21,41 | 107 | 10 |
math | ## Task 3 - 110613
Four faces of a wooden cube with an edge length of $3 \mathrm{~cm}$ are painted red, the other two remain unpainted. Afterwards, the cube is cut into exactly 27 smaller cubes, each with an edge length of $1 \mathrm{~cm}$. Determine the number of these small cubes that have
no red-painted face, exac... | 3,12,12,0 | 147 | 9 |
math | 7.1. The numbers $p$ and $q$ are chosen such that the parabola $y=p x-x^{2}$ intersects the hyperbola $x y=q$ at three distinct points $A, B$, and $C$, and the sum of the squares of the sides of triangle $A B C$ is 324, while the centroid of the triangle is 2 units away from the origin. Find the product $p q$. | 42 | 96 | 2 |
math | Example 3. Expand the function $y=\alpha+\beta \cos \vartheta+\gamma \cos ^{2} \vartheta$ into a Fourier series on the interval $(0, \pi)$ with respect to the system $P_{0}(\cos \vartheta), P_{1}(\cos \vartheta), P_{2}(\cos \vartheta), \ldots$, where $P_{n}$ are Legendre polynomials. | \alpha+\beta\cos\vartheta+\gamma\cos^{2}\vartheta=(\alpha+\frac{1}{3}\gamma)P_{0}(\cos\vartheta)+\betaP_{1}(\cos\vartheta)+\frac{2}{3}P_{2}(\cos\vartheta),0<\vartheta<\pi | 95 | 77 |
math | 16. There is a target, with 1 ring, 3 ring, 5 ring, 7 ring, and 9 ring indicating the number of rings hit.
甲 says: “I fired 5 shots, each hit the target, totaling 35 rings.”
乙 says: “I fired 6 shots, each hit the target, totaling 36 rings.”
丙 says: “I fired 3 shots, each hit the target, totaling 24 rings.”
丁 says: “I ... | 丙 | 162 | 1 |
math | Example 5 In coin tossing, if Z represents heads and F represents tails, then the sequence of coin tosses is represented by a string composed of Z and F. We can count the number of occurrences of heads followed by tails (ZF), heads followed by heads (ZZ)...... For example, the sequence ZZFFZZZZFZZFFFF is the result of ... | 560 | 144 | 3 |
math | 4. Find the smallest positive integer $k$, such that $\varphi(n)=k$ has no solution; exactly two solutions; exactly three solutions; exactly four solutions (an unsolved conjecture is: there does not exist a positive integer $k$, such that $\varphi(n)=$ $k$ has exactly one solution).
| 3, 1, 2, 4 | 68 | 10 |
math | $$
\begin{aligned}
f(x)= & |a \sin x+b \cos x-1|+ \\
& |b \sin x-a \cos x| \quad(a, b \in \mathbf{R})
\end{aligned}
$$
If the maximum value of the function is 11, then $a^{2}+b^{2}=$ $\qquad$ . | 50 | 85 | 2 |
math | 5. Let $n=1990$, then
$$
\frac{1}{2^{n}}\left(1-3 C_{n}^{2}+3^{2} C_{4}{ }^{n}-3^{3} C_{n}^{6}+\cdots+3^{994} C_{n}^{1988}-3^{995} C_{n}^{1990}\right)
$$
$=$ | -\frac{1}{2} | 103 | 7 |
math | 5. In a certain social event, it was originally planned that each pair of people would shake hands exactly once, but four people each shook hands twice and then left. As a result, there were a total of 60 handshakes during the entire event. Then the number of people who initially participated in the event is $\qquad$ | 15 | 70 | 2 |
math | ## Task 6
A ride on the Pioneer Railway in Küchwald covers a distance of $2.3 \mathrm{~km}$. Normally, there are 5 trips per day.
How many kilometers does the Pioneer Railway travel in 8 days? | 92 | 53 | 2 |
math | The integers $1,2, \ldots, 2018$ are written on the board. Then 2017 operations are performed as follows: choose two numbers $a$ and $b$, erase them, and write $a+b+2 a b$ in their place. At the end, only one integer remains on the board.
What are the possible values that its units digit can take? | 7 | 86 | 1 |
math | I divide 1989 into the sum of 10 positive integers, to maximize their product. | 199^{9} \times 198 | 22 | 12 |
math | B1. Determine all non-zero integers $a$, different from 4, for which the value of the expression $\frac{a}{a-4}+\frac{2}{a}$ is an integer. | =2=-4 | 43 | 4 |
math | 3. Solve the equation
$3 \sqrt{6 x^{2}+13 x+5}-6 \sqrt{2 x+1}-\sqrt{3 x+5}+2=0$ | -1/3;-4/9 | 45 | 8 |
math | B1 A bug moves in the coordinate plane, starting at $(0,0)$. On the first turn, the bug moves one unit up, down, left, or right, each with equal probability. On subsequent turns the bug moves one unit up, down, left, or right, choosing with equal probability among the three directions other than that of its previous mo... | \frac{1}{54} | 125 | 8 |
math | In a regular polygon, a diagonal is a line segment joining a vertex to any non-neighbouring vertex. For example, a regular hexagon has 9 diagonals. If a regular polygon with $n$ sides has 90 diagonals, what is the value of $n$ ? | 15 | 60 | 2 |
math | Katrine has a bag containing 4 buttons with distinct letters M, P, F, G on them (one letter per button). She picks buttons randomly, one at a time, without replacement, until she picks the button with letter G. What is the probability that she has at least three picks and her third pick is the button with letter M?
| \frac{1}{12} | 74 | 8 |
math | 12. (10 points) From five number cards $0, 2, 4, 6, 8$, select 3 different cards to form a three-digit number. How many different three-digit numbers can be formed (6 when flipped is 9)?
| 78 | 56 | 2 |
math | Determine all sequences of non-negative integers $a_1, \ldots, a_{2016}$ all less than or equal to $2016$ satisfying $i+j\mid ia_i+ja_j$ for all $i, j\in \{ 1,2,\ldots, 2016\}$. | a_1 = a_2 = \ldots = a_{2016} | 74 | 20 |
math | Task 3. In the surgical department, there are 4 operating rooms: 1, 2, 3, and 4. In the morning, they were all empty. At some point, a surgery began in operating room 1, after some time - in operating room 2, then after some more time - in operating room 3, and finally in operating room 4.
All four surgeries ended sim... | 13 | 157 | 2 |
math | 3 3.7 A regular dodecahedron has 20 vertices and 30 edges. Each vertex is the intersection of 3 edges, and the other endpoints of these 3 edges are 3 other vertices of the regular dodecahedron. We call these 3 vertices adjacent to the first vertex. Place a real number at each vertex, such that the number placed at each... | 0 | 130 | 1 |
math | ## 265. Math Puzzle $6 / 87$
For a motorcycle, the optimal rotational speed of the engine shaft is $6000 \mathrm{U} / \mathrm{min}$.
What is the gear ratio of the rear axle to the engine shaft produced by the transmission if the optimal rotational speed results in a speed of $60 \mathrm{~km} / \mathrm{h}$?
The diame... | 1:10 | 115 | 4 |
math | 4. The length and width of a rectangular prism are 20 cm and 15 cm, respectively. If the numerical value of its volume is equal to the numerical value of its surface area, then its height is $\qquad$ cm (write the answer as an improper fraction)
The length and width of a rectangular prism are 20 cm and 15 cm, respecti... | \frac{60}{23} | 116 | 9 |
math | 4.2. A smooth sphere with a radius of 1 cm was dipped in blue paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a blue trail. During its m... | 60.75 | 134 | 5 |
math | 13.389 From two pieces of alloy of the same mass but with different percentage content of copper, pieces of equal mass were cut off. Each of the cut pieces was melted with the remainder of the other piece, after which the percentage content of copper in both pieces became the same. How many times smaller is the cut pie... | 2 | 72 | 1 |
math | Three tired cowboys entered a saloon and hung their hats on a buffalo horn at the entrance. When the cowboys left deep into the night, they were unable to distinguish one hat from another and therefore picked up three hats at random. Find the probability that none of them took their own hat. | \frac{1}{3} | 60 | 7 |
math | 3. Find all solutions of the system of equations: $x+y=4,\left(x^{2}+y^{2}\right)\left(x^{3}+y^{3}\right)=280$. | (3,1)(1,3) | 45 | 9 |
math | 2. In 1986, Janez will be as many years old as the sum of the digits of the year he was born. How old will Janez be in 1986? | 21 | 43 | 2 |
math | 6. Let the non-empty set $A \subseteq\{1,2,3,4,5,6,7\}$, when $a \in A$ it must also have $8-a \in A$, there are $\qquad$ such $A$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 15 | 80 | 2 |
math | Find all real $(x, y)$ such that $x+y=1$ and $x^{3}+y^{3}=19$ | (x,y)=(-2,3)or(x,y)=(3,-2) | 30 | 16 |
math | [ Numerical inequalities. Comparing numbers.]
Arrange the numbers in ascending order: $222^{2}, 22^{22}, 2^{222}$. | 222^{2}<22^{22}<2^{222} | 38 | 18 |
math | What is the smallest natural number $n$ such that the base 10 representation of $n!$ ends with ten zeros | 45 | 26 | 2 |
math | For every positive real number $x$, let
\[
g(x)=\lim_{r\to 0} ((x+1)^{r+1}-x^{r+1})^{\frac{1}{r}}.
\]
Find $\lim_{x\to \infty}\frac{g(x)}{x}$.
[hide=Solution]
By the Binomial Theorem one obtains\\
$\lim_{x \to \infty} \lim_{r \to 0} \left((1+r)+\frac{(1+r)r}{2}\cdot x^{-1}+\frac{(1+r)r(r-1)}{6} \cdot x^{-2}+\dots \ri... | e | 189 | 1 |
math | XXV - I - Problem 11
Let $ X_n $ and $ Y_n $ be independent random variables with the same distribution $ \left{ \left(\frac{k}{2^n}, \frac{1}{2^n}\right) : k = 0, 1, \ldots, 2^n-1\right} $. Denote by $ p_n $ the probability of the event that there exists a real number $ t $ satisfying the equation $ t^2 + X_n \cdot t... | \frac{1}{12} | 129 | 8 |
math | [ Residue arithmetic (other).]
Solve the equation $x^{2}+y^{2}=z^{2}$ in natural numbers. | {x,y}={nk,\frac{1}{2}k(^{2}-n^{2})},\frac{1}{2}k(^{2}+n^{2}) | 30 | 39 |
math | Given $a_n = (n^2 + 1) 3^n,$ find a recurrence relation $a_n + p a_{n+1} + q a_{n+2} + r a_{n+3} = 0.$ Hence evaluate $\sum_{n\geq0} a_n x^n.$ | \frac{1 - 3x + 18x^2}{1 - 9x + 27x^2 - 27x^3} | 69 | 36 |
math | 7. In a race with six runners, $A$ finished between $B$ and $C, B$ finished between $C$ and $D$, and $D$ finished between $E$ and $F$. If each sequence of winners in the race is equally likely to occur, what is the probability that $F$ placed last? | \frac{5}{16} | 70 | 8 |
math | 3. Divide a rectangle with side lengths of positive integers $m, n$ into several squares with side lengths of positive integers, with each square's sides parallel to the corresponding sides of the rectangle. Try to find the minimum value of the sum of the side lengths of these squares. | f(,n)=+n-(,n) | 57 | 11 |
math | Determine the maximum value of $m$, such that the inequality
\[ (a^2+4(b^2+c^2))(b^2+4(a^2+c^2))(c^2+4(a^2+b^2)) \ge m \]
holds for every $a,b,c \in \mathbb{R} \setminus \{0\}$ with $\left|\frac{1}{a}\right|+\left|\frac{1}{b}\right|+\left|\frac{1}{c}\right|\le 3$.
When does equality occur? | 729 | 124 | 3 |
math | Solve the following equation if $x$ and $y$ are integers:
$$
x^{2}-2 x y+2 y^{2}-4 y^{3}=0
$$ | 0 | 39 | 1 |
math | 8. 8.1. How many increasing arithmetic progressions of 22 different natural numbers exist, in which all numbers are no greater than 1000? | 23312 | 36 | 5 |
math | 4. (15 points) Two heaters are connected sequentially to the same DC power source. The water in the pot boiled after $t_{1}=3$ minutes from the first heater. The same water, taken at the same initial temperature, boiled after $t_{2}=6$ minutes from the second heater. How long would it take for the water to boil if the ... | 2 | 92 | 1 |
math | 9.1. Two given quadratic trinomials $f(x)$ and $g(x)$ each have two roots, and the equalities $f(1)=g(2)$ and $g(1)=f(2)$ hold. Find the sum of all four roots of these trinomials. | 6 | 64 | 1 |
math | 4・128 Find the relationship between the coefficients $a, b, c$ for the system of equations
$$\left\{\begin{array}{l}
a x^{2}+b x+c=0, \\
b x^{2}+c x+a=0, \\
c x^{2}+a x+b=0 .
\end{array}\right.$$
to have real solutions. | a+b+c=0 | 86 | 5 |
math | ## Task A-4.4.
Determine all triples of natural numbers $(m, n, k)$ for which
$$
D(m, 20)=n, \quad D(n, 15)=k \quad \text { and } \quad D(m, k)=5
$$
where $D(a, b)$ is the greatest common divisor of the numbers $a$ and $b$. | (5a,5,5)forodd,(10a,10,5)forodd,(20b,20,5)forallnaturalb | 86 | 35 |
math | Galperin G.A.
a) For each three-digit number, we take the product of its digits, and then we add up these products, calculated for all three-digit numbers. What will be the result?
b) The same question for four-digit numbers. | 45^4 | 53 | 4 |
math | 6. $n$ is a positive integer, $S_{n}=\left\{\left(a_{1}, a_{2}, \cdots, a_{2^{*}}\right) \mid a_{i}=0\right.$ or $\left.1,1 \leqslant i \leqslant 2^{n}\right\}$, for any two elements $a=\left(a_{1}, a_{2}, \cdots, a_{2^{*}}\right)$ and $b=\left(b_{1}, b_{2}, \cdots, b_{2^{*}}\right)$ in $S_{n}$, let $d(a, b)=\sum_{i=1}... | 2^{n+1} | 247 | 6 |
math | 13.115. Tourist $A$ set off from city $M$ to city $N$ at a constant speed of 12 km/h. Tourist $B$, who was in city $N$, upon receiving a signal that $A$ had already traveled 7 km, immediately set off towards him and traveled 0.05 of the total distance between $M$ and $N$ each hour. From the moment $B$ set off until his... | 140 | 150 | 3 |
math | Example 3: There are $n$ people, and it is known that any two of them make at most one phone call. The total number of phone calls made among any $n-2$ people is equal, and it is $3^{k}$ times, where $k$ is a positive integer. Find all possible values of $n$.
(2000, National High School Mathematics Competition)
Analysi... | 5 | 293 | 1 |
math | Suppose that there exist nonzero complex numbers $a$, $b$, $c$, and $d$ such that $k$ is a root of both the equations $ax^3+bx^2+cx+d=0$ and $bx^3+cx^2+dx+a=0$. Find all possible values of $k$ (including complex values). | 1, -1, i, -i | 76 | 9 |
math | 1.8. In an isosceles triangle, the heights drawn to the base and to the lateral side are equal to 10 and 12 cm, respectively. Find the length of the base. | 15 | 44 | 2 |
math | 2. (10 points) Among all positive integers that are multiples of 20, the sum of those not exceeding 2014 and are also multiples of 14 is $\qquad$ | 14700 | 43 | 5 |
math | 6. Using the four digits 6, 7, 8, and 9, many four-digit numbers without repeated digits can be formed. If they are arranged in ascending order, 9768 is the th. | 21 | 48 | 2 |
math | What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ? | 18 | 28 | 2 |
math | 13. (1993 Putnam Mathematical Competition, 53rd USA) Let $S$ be a set of $n$ distinct real numbers, and let $A_{s}$ be the set of all distinct averages of pairs of elements of $S$. For a given $n \geqslant 2$, what is the least number of elements that $A_{s}$ can have? | 2n-3 | 85 | 4 |
math | Find the largest positive integer $n$ such that for each prime $p$ with $2<p<n$ the difference $n-p$ is also prime. | 10 | 32 | 2 |
math | $5 \cdot 61$ A group of young people went to dance disco. Each dance costs 1 yuan, and each boy danced with each girl exactly once, then they went to another place to dance. Here, they paid with subsidiary coins, and they spent the same amount of money as before. The entry fee for each person is one subsidiary coin, an... | 5 | 119 | 1 |
math | 2. For any point $A(x, y)$ in the plane region $D$:
$$
\left\{\begin{array}{l}
x+y \leqslant 1, \\
2 x-y \geqslant-1, \\
x-2 y \leqslant 1
\end{array}\right.
$$
and a fixed point $B(a, b)$ satisfying $\overrightarrow{O A} \cdot \overrightarrow{O B} \leqslant 1$. Then the maximum value of $a+b$ is $\qquad$ | 2 | 124 | 1 |
math | 9-3-1. The numbers from 1 to 217 are divided into two groups: one group has 10 numbers, and the other has 207. It turns out that the arithmetic means of the numbers in the two groups are equal. Find the sum of the numbers in the group of 10 numbers. | 1090 | 71 | 4 |
math | 8. Find all integer values of the parameter $a$ for which the system $\left\{\begin{array}{l}x-2 y=y^{2}+2, \\ a x-2 y=y^{2}+x^{2}+0.25 a^{2}\end{array}\right.$. has at least one solution. In your answer, specify the sum of the found values of the parameter $a$. | 10 | 90 | 2 |
math | 3. In the complex plane, the points $0, z, \frac{1}{z}, z+\frac{1}{z}$ form a parallelogram with an area of $\frac{4}{5}$. Then the minimum value of $\left|z+\frac{1}{z}\right|$ is | \frac{2\sqrt{5}}{5} | 65 | 12 |
math | 26 On the $x O y$ coordinate plane, the curve
$$
y=(3 x-1)\left(\sqrt{9 x^{2}-6 x+5}+1\right)+(2 x-3)\left(\sqrt{4 x^{2}-12 x+13}+1\right)
$$
intersects the $x$-axis at the point with coordinates $\qquad$ . | (\frac{4}{5},0) | 89 | 9 |
math | Let's determine the relatively prime numbers $a$ and $b$ if
$$
\frac{a^{2}-b^{2}}{a^{3}-b^{3}}=\frac{49}{1801}
$$ | =25,b=24or=24,b=25 | 50 | 15 |
math | 27. Someone took $\frac{1}{13}$ from the treasury. From what was left, another took $\frac{1}{17}$, and he left 150 in the treasury. We want to find out how much was originally in the treasury? | 172\frac{21}{32} | 57 | 12 |
math | 3. Let $R$ be the triangular region (including the boundaries of the triangle) in the plane with vertices at $A(4,1), B(-1,-6), C(-3,2)$. Find the maximum and minimum values of the function $4x - 3y$ as $(x, y)$ varies over $R$. (You must prove your assertion)
---
To find the maximum and minimum values of the functio... | 14-18 | 1,386 | 5 |
math | ## Task 4
Arrange the products in order of size. $\quad 27 \cdot 4 ; \quad 52 \cdot 6 ; \quad 17 \cdot 0 ; \quad 81 \cdot 3$ | 0;108;243;312 | 53 | 13 |
math | 3. The function $f(x)=2 \sqrt{x-1}+\sqrt{6-2 x}$ achieves its maximum value when $x=$ | \frac{7}{3} | 31 | 7 |
math | A function $f$ is given by $f(x)+f\left(1-\frac{1}{x}\right)=1+x$ for $x \in \mathbb{R} \backslash\{0,1\}$.
Determine a formula for $f$. | \frac{-x^{3}+x^{2}+1}{2 x(1-x)} | 59 | 21 |
math | SG. 1 The sum of two numbers is 50 , and their product is 25 . If the sum of their reciprocals is $a$, find $a$. | 2 | 38 | 1 |
math | 4. Solve the system of equations: $\left\{\begin{array}{l}x^{2}-y=z^{2}, \\ y^{2}-z=x^{2}, \\ z^{2}-x=y^{2} .\end{array}\right.$
(2013, Croatian Mathematical Competition) | (1,0,-1),(0,0,0),(0,-1,1),(-1,1,0) | 66 | 26 |
math | Determine all pairs $(a, b)$ of strictly positive integers such that $a b-a-b=12$. | (2,14),(14,2) | 24 | 11 |
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