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 | Three distinct numbers are given. The average of the average of the two smaller numbers and the average of the two larger numbers is equal to the average of all three numbers. The average of the smallest and the largest number is 2022.
Determine the sum of the three given numbers.
(K. Pazourek)
Hint. Express one of ... | 6066 | 80 | 4 |
math | A technique widely used to calculate summations is the Telescoping Sum. It consists of "decomposing" the terms of a sum into parts that cancel each other out. For example,
$$
\begin{aligned}
& \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}= \\
& \left(\frac{1}{1}-\frac{1}{2}\right)+\le... | 764049 | 279 | 6 |
math | 11. (20 points) Given the function
$$
f(x)=\left(1-x^{2}\right)\left(x^{2}+b x+c\right)(x \in[-1,1]) \text {. }
$$
Let the maximum value of $\mid f(x)$ | be $M(b, c)$. When $b$ and $c$ vary, find the minimum value of $M(b, c)$. | 3-2\sqrt{2} | 94 | 8 |
math |
3. Find all integers $a, b, c$ such that
$$
a^{2}=b c+4, \quad b^{2}=c a+4
$$
| (1,1,-3),(-1,-1,3),(4,4,3),(-4,-4,-3),(2,2,0),(-2,-2,0),(2,-2,0),(-2,2,0),(2,0,-2),(-2,0,2),(0,2,-2),(0,-2,2) | 40 | 78 |
math | Introduce a standard scalar product in $\mathbb{R}^4.$ Let $V$ be a partial vector space in $\mathbb{R}^4$ produced by $\left(
\begin{array}{c}
1 \\
-1 \\
-1 \\
1
\end{array}
\right),\left(
\begin{array}{c}
1 \\-1 \\
1 \\
-1
\end{array}
\right).$
Find a pair of base of orthogonal complement $W$ for $V$ in $\m... | \left\{ \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \\ 1 \end{pmatrix} \right\} | 125 | 55 |
math | 1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=1500$, if $f(0)=1$ and for any $x$ the equality $f(x+3)=f(x)+2 x+3$ holds. | 750001 | 61 | 6 |
math | 13. To obtain phosphorus, calcium phosphate was previously treated with sulfuric acid, the resulting orthophosphoric acid was mixed with carbon and calcined. In this process, orthophosphoric acid turned into metaphosphoric acid, which, upon interaction with carbon, yielded phosphorus, hydrogen, and carbon monoxide. Ill... | 23.4 | 144 | 4 |
math | 238 Test: Can 2007 be expressed in the form
$$
a_{1}^{x_{1}}+a_{2}^{x_{2}}+\cdots+a_{m}^{x_{n}}-b_{1}^{y_{1}}-b_{2}^{y_{2}}-\cdots-b_{n}^{y_{n}}
$$
where $m, n$ are both positive integers greater than 130 and less than 140 (allowing $m$ to equal $n$), $a_{1}, a_{2}, \cdots, a_{m}, b_{1}, b_{2}, \cdots, b_{n}$ are all ... | 2007 | 361 | 4 |
math | 9. 44 Let $k$ be a natural number. Determine for which value of $k$, $A_{k}=\frac{19^{k}+66^{k}}{k!}$ attains its maximum value. | 65 | 51 | 2 |
math | 9.6 There is a red card box and $k$ blue card boxes $(k>1)$, and a deck of cards, totaling $2n$ cards, which are numbered from 1 to $2n$. Initially, this deck of cards is stacked in the red card box in any order. From any card box, the top card can be taken out, either placed in an empty box, or placed on top of a card... | n=k-1 | 117 | 4 |
math | 6. 15 cards, each card has 3 different Chinese characters, any 2 cards do not have exactly the same characters; among any 6 cards, there must be 2 cards that have a common character. Question: What is the maximum number of different Chinese characters on these 15 cards? | 35 | 64 | 2 |
math | 10. Let $P(x)=x^{4}+a x^{3}+b x^{2}+c x+d$, where $a, b, c, d$ are constants, and $P(1)=2004, P(2)=$ $4008, P(3)=6012$, then the value of $\frac{1}{4}[P(11)+P(-7)]$ is $\qquad$ . | 5244 | 100 | 4 |
math | 5. Find all positive integer solutions $x, y, z$ that satisfy the equation:
$$
5(x y+y z+x z)=4 x y z
$$ | (2,4,20), (2,20,4), (4,2,20), (20,2,4), (4,20,2), (20,4,2), (2,5,10), (2,10,5), (5,2,10), (10,2,5), (5,10,2), (10,5,2) | 35 | 96 |
math | Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$. Find $\angle BCA$. | 90^\circ | 82 | 4 |
math | 10. (15 points) Use five different colors to paint the six faces of a cube. If two adjacent faces cannot be painted the same color, how many different ways are there to paint the cube? (Painting methods that are the same after any rotation of the cube are considered the same)
| 15 | 62 | 2 |
math | B2. Integers $a, b, c, d$, and $e$ satisfy the following three properties:
(i) $2 \leq a<b<c<d<e<100$
(ii) $\operatorname{gcd}(a, e)=1$
(iii) $a, b, c, d, e$ form a geometric sequence.
What is the value of $c$ ? | 36 | 84 | 2 |
math | 25. (1999 CMO Problem) Find the largest real number $\lambda$, such that when the polynomial $f(x)=x^{3}+a x^{2}+b x+c$ with real coefficients has all non-negative real roots, for any $x \geqslant 0$, we have $f(x) \geqslant \lambda(x-a)^{3}$, and determine when equality holds. | -\frac{1}{27} | 92 | 8 |
math | Given is a positive integer n. What fraction of the non-empty subsets of $\{1,2, \ldots, 2 n\}$ has an odd smallest element?
(Birgit Vera Schmidt) | \frac{2}{3} | 42 | 7 |
math | Example 7.14 Find the number of second-kind circular permutations made from 2 $a$s, 2 $b$s, 2 $c$s. | 11 | 34 | 2 |
math | 5. If $P$ is the circumcenter of $\triangle A B C$, and
$$
\overrightarrow{P A}+\overrightarrow{P B}+\lambda \overrightarrow{P C}=\mathbf{0}, \angle C=120^{\circ} \text {. }
$$
then the value of the real number $\lambda$ is $\qquad$ | -1 | 82 | 2 |
math | $[\underline{\text { U }}$ equations in integers $]$ [ Factorization $]$
Solve the equation $x y+3 x-5 y=-3$ in integers. | (-13,-2),(-4,-1),(-1,0),(2,3),(3,6),(4,15),(6,-21),(7,-12),(8,-9),(11,-6),(14,-5),(23,-4) | 39 | 58 |
math | 44th Putnam 1983 Problem A2 A clock's minute hand has length 4 and its hour hand length 3. What is the distance between the tips at the moment when it is increasing most rapidly? Solution | \sqrt{7} | 48 | 5 |
math | 1. M. V. Lomonosov spent one coin on bread and kvass. When prices increased by $20 \%$, he could buy half a loaf of bread and kvass with that same coin. Would that same coin be enough to buy at least kvass if prices rise by another $20 \%$? | 1.5y>1.44y | 67 | 10 |
math | II. (20 points) Find all positive decimal integers $n$ that satisfy the following conditions: $n$ is an $a$-digit number, and $a^{a}=n$.
| 1,16777216,387420489 | 42 | 20 |
math | 1. There are three consecutive positive integers, the sum of whose reciprocals is $\frac{47}{60}$. Find these three numbers. | 3,4,5 | 32 | 5 |
math | Let $p$ be a prime and let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with integer coefficients such that $0 < a, b, c \le p$. Suppose $f(x)$ is divisible by $p$ whenever $x$ is a positive integer. Find all possible values of $a + b + c$. | 3p | 75 | 2 |
math | 8.2. One side of the rectangle (width) was increased by $10 \%$, and the other (length) - by $20 \%$. a) Could the perimeter increase by more than 20\%? b) Find the ratio of the sides of the original rectangle if it is known that the perimeter of the new rectangle is $18 \%$ larger than the perimeter of the original? | 1:4 | 85 | 3 |
math | Example 6. Solve the system of differential equations
$$
\left\{\begin{array}{l}
x^{\prime}=3 x+4 y+2 z \\
y^{\prime}=x+4 y+z \\
z^{\prime}=4 x+6 y+5 z
\end{array}\right.
$$ | {\begin{pmatrix}C_{1}e^{}&+7C_{3}e^{9}\\&C_{2}e^{2}+4C_{3}e^{9}\\-C_{1}e^{}-2C_{2}e^{2}+13C_{3}e^{9}\end{pmatrix}.} | 70 | 78 |
math | A1
For any set $A=\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}$ of four distinct positive integers with sum $s_{A}=a_{1}+a_{2}+a_{3}+a_{4}$, let $p_{A}$ denote the number of pairs $(i, j)$ with $1 \leq i<j \leq 4$ for which $a_{i}+a_{j}$ divides $s_{A}$. Among all sets of four distinct positive integers, determine those s... | {,5,7,11} | 137 | 9 |
math | 1. Find all functions $f: \mathbb{R} \backslash\{-1\} \rightarrow \mathbb{R}$, for which
$$
f(x)+f(y)=(x+y+2) f(x) f(y)
$$
for all $x, y \in \mathbb{R} \backslash\{-1\}$. | f(x)=0 | 78 | 4 |
math | Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7. | 4410 | 40 | 4 |
math | 3. For any $x \in[0,1]$, there is $|a x+b| \leqslant 1$. Then the maximum value of $|b x+a|$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 2 | 71 | 1 |
math | II. (50 points) Define a "Hope Set" (Hope Set) abbreviated as HS as follows: HS is a non-empty set that satisfies the condition "if $x \in \mathrm{HS}$, then $2 x \notin \mathrm{HS}$". How many "Hope Subsets" are there in the set $\{1,2, \cdots, 30\}$? Please explain your reasoning. | 26956799 | 90 | 8 |
math | 15. Let $P(x)$ be a polynomial of degree 2010. Suppose $P(n)=\frac{n}{1+n}$ for all $n=0,1,2, \ldots, 2010$. Find $P(2012)$. | 0 | 62 | 1 |
math | 8.4. The steamship "Raritet" after leaving the city moves at a constant speed for three hours, then drifts for an hour, moving with the current, then moves for three hours at the same speed, and so on. If the steamship starts its journey from city A and heads to city B, it takes 10 hours. If it starts from city B and h... | 60 | 110 | 2 |
math | 6. Find the smallest natural number $n$ such that in the decimal representation of $\sqrt{n}$, two nines immediately follow the decimal point. | 2600 | 31 | 4 |
math | 29. In the parking lot, there were passenger cars and motorcycles. The number of motorcycles with sidecars was half the number of those without sidecars. What could be the maximum number of cars if the total number of wheels on these cars and motorcycles was $115?$ | 27 | 57 | 2 |
math | 7. For the function $y=f(x)$, it is known that it is defined and continuous on the entire number line, odd, and periodic with a period of 5, and that $f(-1)=f(2)=-1$. What is the minimum number of roots that the equation $f(x)=0$ can have on the interval [1755; 2017]? Answer: 210. | 210 | 91 | 3 |
math | Example 6 Given an integer array consisting of 121 integers, each integer in this array takes a value between 1 and 1000 (inclusive of 1 and 1000), and repeated values are allowed. The arithmetic mean of these numbers is $m$, and there is a unique "mode" (the number that appears most frequently) $M$ in this set of numb... | 947 | 122 | 3 |
math | 7. Represent the number 531441 as the sum of several addends such that the first one is 1, the second is $k$ times larger, the third is $k$ times larger than the sum of the first two, the fourth is $k$ times larger than the sum of all previous ones, and so on. Find the multiplier $k$ and the number of addends, given th... | k=80;26;8;20n=4;5;7;13 | 95 | 22 |
math | IMO 1974 Problem B2 Determine all possible values of a/(a+b+d) + b/(a+b+c) + c/(b+c+d) + d/(a+c+d) for positive reals a, b, c, d. | (1,2) | 53 | 5 |
math | 9. Several rooks have beaten all the white cells of a $40 \times 40$ chessboard. What is the maximum number of black cells that could remain unbeaten? (A rook beats the cell it stands on.) | 400 | 50 | 3 |
math | 2. If the real number $a$ ensures that for every real number $z$, the system of equations in $x$ and $y$
$$
\left\{\begin{array}{l}
x+a y=2 z, \\
x y=2 z^{2}+3 z+1
\end{array}\right.
$$
always has real solutions, then the range of values for $a$ is . $\qquad$ | -4 \leqslant a < 0 | 93 | 11 |
math | 7. Given the sequence $\left\{a_{n}\right\}$ with the general term
$$
a_{n}=n^{4}+6 n^{3}+11 n^{2}+6 n \text {. }
$$
Then the sum of the first 12 terms $S_{12}=$ $\qquad$ | 104832 | 74 | 6 |
math | # Problem 7. ( **points** )
A number written on the board is allowed to be multiplied by 8, have 14 added to it, or have 14 subtracted from it. In each case, the old number is erased and the new number is written in its place. Initially, a single digit was written on the board. After several operations, the number 777... | 2;9 | 118 | 3 |
math | On the board, an example of dividing two positive numbers was given. David noticed that if he increased the dividend by two and the divisor by seven, the quotient would remain unchanged.
By how much would the divisor need to increase so that increasing the dividend by three would again result in the same quotient?
(M... | 10.5 | 66 | 4 |
math | For the positive constant number $ a$, let $ D$ be the part surrounded by the curve $ \sqrt{x}\plus{}\sqrt{y}\equal{}\sqrt{a}$ and the line $ x\plus{}y\equal{}a$.
(1) Draw the outline of $ D$ and find the area.
(2) Find the volume of the solid by rotating $ D$ about the line $ x\plus{}y\equal{}a$ as the rotation... | \frac{\pi \sqrt{2}}{15} a^3 | 100 | 16 |
math | Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$.
[i](Karl Czakler)[/i] | p = 7, q = 3, r = 2 | 39 | 15 |
math | 5. The subset $X$ of the set $\{00,01, \cdots, 98,99\}$ satisfies: in any infinite sequence of digits, there are two adjacent digits that form an element of $X$. What is the minimum number of elements that $X$ should contain?
（52nd Moscow Olympiad problem） | 55 | 75 | 2 |
math | ## Problema 3
Fie matricea $A=\left(\begin{array}{cc}7 & 18 \\ -4 & 7\end{array}\right) \in M_{2}(\mathbb{R})$. Rezolvați ecuația $X^{2}=A, X \in M_{2}(\mathbb{R})$.
| X_{1}=(\begin{pmatrix}3&3\\-\frac{2}{3}&3\end{pmatrix}),X_{2}=(\begin{pmatrix}-3&-3\\\frac{2}{3}&-3\end{pmatrix}) | 84 | 59 |
math | A basin can be filled with water from three taps. The first tap alone would fill the basin in 3 hours, the second in 6 hours, and the third in 12 hours. How long will it take to fill the basin if all three taps are turned on? | \frac{12}{7} | 57 | 8 |
math | Solve the following equation:
$$
\sqrt{x^{2}+x+2}=x^{2}+x
$$ | x_{1}=1,\quadx_{2}=-2 | 27 | 13 |
math | ## Problem 1
Find the natural numbers that satisfy the relation:
$9 \leq \sqrt{\sqrt{2 n+2013}+\sqrt{2 n+2009}}+\sqrt{\sqrt{2 n+2013}-\sqrt{2 n+2009}}<10$. | n\in{0,1,\ldots,145} | 71 | 15 |
math | 3. Misha calculated the products $1 \times 2, 2 \times 3$, $3 \times 4, \ldots, 2017 \times 2018$. For how many of them is the last digit zero? | 806 | 56 | 3 |
math | Let $a_n =\sum_{d|n} \frac{1}{2^{d+ \frac{n}{d}}}$. In other words, $a_n$ is the sum of $\frac{1}{2^{d+ \frac{n}{d}}}$ over all divisors $d$ of $n$.
Find
$$\frac{\sum_{k=1} ^{\infty}ka_k}{\sum_{k=1}^{\infty} a_k} =\frac{a_1 + 2a_2 + 3a_3 + ....}{a_1 + a_2 + a_3 +....}$$ | 4 | 140 | 1 |
math | 10. Find the positive integer solutions of the equation $3^{x}=2^{x} y+1$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | (x, y)=(1,1),(2,2),(4,5) | 50 | 16 |
math | Example 1. Write the equations of the tangent and normal to the curve $f(x)=x^{3}$ at the point $M_{0}(2,8)$. | 12x-y-16=0 | 36 | 9 |
math | 28.30. Let $(x+a)^{n}=A_{0}+A_{1} x+\ldots+A_{n} x^{n}$. Find the coefficients $A_{0}, \ldots, A_{n}$ by successive differentiation. | n(n-1)\ldots(n-+1)^{n-}=(-1)\cdot\ldots\cdot1\cdotA_{} | 56 | 31 |
math | Example 3 Let the temperature $T$ of an object be a function of time $t$: $T(t)=a t^{3}+b t^{2}+c t+d(a \neq 0)$, where the unit of temperature is ${ }^{\circ} \mathrm{C}$, the unit of time is hours, and $t=0$ represents $12: 00$, with positive $t$ values indicating times after $12: 00$. If the temperature of the objec... | 62\mathrm{C} | 291 | 7 |
math | ## Problem 4
Find all complex numbers $x, y, z$ which satisfy $x+y+z=x^{2}+y^{2}+z^{2}=x^{3}+y^{3}+z^{3}=3$.
| 1,1,1 | 53 | 5 |
math | Task B-2.4. Let $f(x)=x^{2}+b x+c, b, c \in \mathbb{R}$. If $f(0)+f(1)=\frac{1}{2}$, calculate $f\left(\frac{1}{2}\right)$. | 0 | 66 | 1 |
math | 6. Boys are dividing the catch. The first one took $r$ fish and a seventh part of the remainder; the second - $2 r$ fish and a seventh part of the new remainder; the third - $3 r$ fish and a seventh part of the new remainder, and so on. It turned out that in this way all the caught fish were divided equally. How many b... | 6 | 83 | 1 |
math | A factory produced an original calculator that performs two operations:
- the usual addition +
- the operation $\circledast$
We know that for any natural number $a$ we have:
$$
\text { (i) } a \circledast a=a \quad \text { and (ii) } a \circledast 0=2 a
$$
and, for any four naturals $a, b, c$ and $d$
$$
\text { (i... | 7 | 169 | 1 |
math | 1. Determine a four-digit number that is a square of a natural number and where the first two digits and the last two digits are equal. | 7744 | 29 | 4 |
math | 6.155. $a x^{4}-x^{3}+a^{2} x-a=0$.
6.155. $a x^{4}-x^{3}+a^{2} x-a=0$. | x_{1}=\frac{1}{},x_{2}=-\sqrt[3]{}when\neq0;0when=0 | 53 | 31 |
math | 45th Putnam 1984 Problem A5 V is the pyramidal region x, y, z >= 0, x + y + z <= 1. Evaluate ∫ V x y 9 z 8 (1 - x - y - z) 4 dx dy dz. Solution | \frac{9!8!4!}{25!} | 65 | 14 |
math | What is the measure, in degrees, of the smallest positive angle $x$ for which $4^{\sin ^{2} x} \cdot 2^{\cos ^{2} x}=2 \sqrt[4]{8}$ ? | 60 | 51 | 2 |
math | 2. Let $a$ and $b$ be real numbers that satisfy the equations
$$
\frac{a}{b}+\frac{b}{a}=\frac{5}{2} \quad \text { and } \quad a-b=\frac{3}{2} \text {. }
$$
Find all possible values of $a^{2}+2 a b+b^{2}+2 a^{2} b+2 a b^{2}+a^{2} b^{2}$. | 081 | 107 | 3 |
math | ## Problem 6.
Find all real solutions of the equation
$$
3^{x^{2}-x-y}+3^{y^{2}-y-z}+3^{z^{2}-z-x}=1
$$ | (x,y,z)=(1,1,1) | 47 | 10 |
math | ## Task B-4.1.
Solve the inequality $5 \cdot\binom{n+2}{n-1}>\binom{n+4}{4}$ in the set of natural numbers. | n\in{2,3,\ldots,11} | 43 | 14 |
math | Suppose that $X_1, X_2, \ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S=\sum_{i=1}^kX_i/2^i,$ where $k$ is the least positive integer such that $X_k<X_{k+1},$ or $k=\infty$ if there is no such integer. Find the expected value of $S.$ | 2\sqrt{e} - 3 | 97 | 9 |
math | 31 If $x, y, z>0$, and $x^{2}+y^{2}+z^{2}=1$, then $S=\frac{(z+1)^{2}}{2 x y z}$ reaches its minimum value when $x$ equals | \sqrt{\sqrt{2}-1} | 58 | 9 |
math | [Coordinate method on the plane $]$
Form the equation of the line passing through the point $M(-3 ; 2)$ parallel to the line $2 x-3 y+4=0$.
# | 2x-3y+12=0 | 44 | 10 |
math | 9.167. $\frac{1}{x+1}-\frac{2}{x^{2}-x+1} \leq \frac{1-2 x}{x^{3}+1}$. | x\in(-\infty;-1)\cup(-1;2] | 47 | 16 |
math | 3. What is the largest three-digit number that needs to be added to the number 184952 so that the sum is divisible by 2, 3, and 7? | 982 | 41 | 3 |
math | Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$ | 3 | 84 | 1 |
math | # Problem 4. (3 points)
Circles $O_{1}$ and $O_{2}$ touch circle $O_{3}$ with radius 13 at points $A$ and $B$ respectively and pass through its center $O$. These circles intersect again at point $C$. It is known that $O C=12$. Find $A B$. | 10 | 77 | 2 |
math | 1. Answer: $1+\frac{1}{11}+\frac{1}{13}+\frac{1}{15}=\frac{2648}{2145}$ | \frac{2648}{2145} | 43 | 13 |
math | Problem 9. In a football championship, 20 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? | 90 | 51 | 2 |
math | ## Task 5 - V00905
In an electrical circuit, a voltage of 120 V is applied. If the resistance is increased by 10 Ohms, the current decreases by 1 Ampere.
What are the current and resistance? | 4,30 | 56 | 4 |
math | 1. The function $f(x)=-\log _{\frac{1}{2}}\left(x^{2}-a x-a\right)$ is decreasing on the interval $(-\infty, 1-\sqrt{3})$, the range of values for $a$ is $\qquad$ . | 2(1-\sqrt{3})\leqslant\leqslant2 | 64 | 19 |
math | 21. Let $P$ be a 30 -sided polygon inscribed in a circle. Find the number of triangles whose vertices are the vertices of $P$ such that any two vertices of each triangle are separated by at least three other vertices of $P$. | 1900 | 56 | 4 |
math | 2. A parallelogram has 3 of its vertices at $(1,2),(3,8)$, and $(4,1)$. Compute the sum of the possible $x$-coordinates for the 4 th vertex. | 8 | 48 | 1 |
math | Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292 = 444_{\text{eight}}.$ | 585 | 63 | 3 |
math | 1.1. Masha thought of a 10-digit number and told Vasya that the remainder of dividing this number by 9 is 3. Then Masha crossed out one digit and told Vasya that the remainder of dividing the resulting 9-digit number by 9 is 7. Help Vasya guess the digit that Masha crossed out. Write this digit in the answer. | 5 | 83 | 1 |
math | 12. Let $n \in \mathbf{N}_{+}$, and $(\sqrt{2}+\sqrt{3})^{2 n-1}=a_{n} \sqrt{2}+b_{n} \sqrt{3}$. If $a_{n+1}=p a_{n}+q b_{n}$, then $p+q=$ $\qquad$ , $2 a_{n}^{2}-3 b_{n}^{2}$ is $\qquad$ | 11,-1 | 108 | 4 |
math | 4. Let $A$ and $B$ be the two foci of a hyperbola, and point $C$ lies on the hyperbola. It is known that the three sides of $\triangle A B C$ form an arithmetic sequence, and $\angle A C B=120^{\circ}$. Then the eccentricity of the hyperbola is $\qquad$ | \frac{7}{2} | 81 | 7 |
math | Exercise 1. Calculate
$$
\frac{1 \times 2 \times 4+2 \times 4 \times 8+3 \times 6 \times 12+4 \times 8 \times 16}{1 \times 3 \times 9+2 \times 6 \times 18+3 \times 9 \times 27+4 \times 12 \times 36}
$$
Only a numerical answer is expected here. The answer should be given as an irreducible fraction (i.e., in the form $... | \frac{8}{27} | 147 | 8 |
math | 3. Find all mappings $f: \mathbf{Z} \rightarrow \mathbf{Z}$, such that for any $m, n \in \mathbf{Z}$, we have
$$
f(f(m+n))=f(m)+f(n) .
$$ | f(n)=n+(\in{Z})orf(n)=0 | 59 | 13 |
math | 9. Let $A B C$ be a triangle, and let $B C D E, C A F G, A B H I$ be squares that do not overlap the triangle with centers $X, Y, Z$ respectively. Given that $A X=6, B Y=7$, and $C Z=8$, find the area of triangle $X Y Z$. | \frac{21\sqrt{15}}{4} | 79 | 14 |
math | 426. Solve the equation:
$$
\sqrt{4 x-y^{2}}=\sqrt{y+2}+\sqrt{4 x^{2}+y}
$$ | (\frac{1}{2},-1) | 38 | 10 |
math | The teacher said to Xu Jun: "Two years ago, my age was three times your age." Xu Jun said to the teacher: "In eight years, your age will be twice mine." Xu Jun is $\qquad$ years old this year. | 12 | 51 | 2 |
math | 6. Determine all ordered triples $(x, y, z)$ of distinct real numbers that satisfy the set equation
$$
\{x, y, z\}=\left\{\frac{x-y}{y-z}, \frac{y-z}{z-x}, \frac{z-x}{x-y}\right\} \text {. }
$$
(J. Šimša) | (,-\frac{1}{1+},-\frac{1+}{}),(1,-2,-\frac{1}{2}),(-\frac{1}{2},1,-2),(-2,-\frac{1}{2},1) | 80 | 53 |
math | Let's determine $a, b, c$, and $d$ such that the fraction
$$
\frac{a x^{2}-b x+c}{x^{2}-d x-a}
$$
is maximal at $x=2$ and minimal at $x=5$, and that the maximum value of the fraction is 1, and the minimum value is 2.5. | =3,b=10,=5,=2 | 81 | 12 |
math | ## Task $7 / 82$
Let $a ; b ; c ; d$ be four non-zero real numbers for which the equations
$$
\frac{a+b}{c}=\frac{a+c}{b}=\frac{b+c}{a}=d
$$
hold. Determine all possible values of $d$. | d_1=-1,d_2=2 | 70 | 10 |
math | 26. Find the minimum value of the function $f(x)=|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|, x \in \mathbf{R}$. | 2\sqrt{2}-1 | 45 | 7 |
math | 8. In two weeks three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks. In four weeks, two cows eat all the grass on two hectares of land, together with all the grass that regrows there during the four weeks.
How many cows will eat all the grass on six ... | 5 | 145 | 1 |
math | Example 8 Given 3 non-negative numbers $a, b, c$ satisfying $3a+2b+c=5$ and $2a+b-3c=1$. If $m=$ $3a+b-7c$, then the minimum value of $m$ is $\qquad$, and the maximum value of $m$ is $\qquad$
(14th Junior High School Mathematics Competition in Jiangsu Province) | -\frac{5}{7}, -\frac{1}{11} | 90 | 16 |
math | Find all positive integers that have as many divisors divisible by six as divisors not divisible by six. | 72n | 21 | 3 |
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 |
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