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 | 2.9. (England, 70). Solve the equation
$$
\sqrt{2 \sqrt{3}-3}=\sqrt{x \sqrt{3}}-\sqrt{y \sqrt{3}}
$$
in rational numbers. | 3/2,1/2 | 51 | 7 |
math | Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. If the maximum and minimum values of $x^2+y^2$ are $M$ and $m$ respectively, compute the numerical value of $M-m$. | 8 | 63 | 1 |
math | Regular hexagon $NOSAME$ with side length $1$ and square $UDON$ are drawn in the plane such that $UDON$ lies outside of $NOSAME$. Compute $[SAND] + [SEND]$, the sum of the areas of quadrilaterals $SAND$ and $SEND$.
| \frac{3}{2} + \frac{3\sqrt{3}}{2} | 68 | 20 |
math | Let $a/b$ be the probability that a randomly chosen positive divisor of $12^{2007}$ is also a divisor of $12^{2000}$, where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by $2007$. | 79 | 71 | 2 |
math | Example 9 Find all positive integers $a, b, c$, where $1<a<b<c$, such that $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | (3,5,15),(2,4,8) | 71 | 14 |
math | 4. If the system of inequalities about $x$ $\left\{\begin{array}{l}x^{3}+3 x^{2}-x-3>0, \\ x^{2}-2 a x-1 \leqslant 0\end{array},(a>0)\right.$ has exactly one integer solution, then the range of values for $a$ is $\qquad$ | [\frac{3}{4},\frac{4}{3}) | 86 | 14 |
math | 26. The price of a product was first reduced by $10 \%$, and then by $15 \%$. What is the overall percentage reduction? Would it be the same if the price was first reduced by $15 \%$, and then by $10 \%$? | 23.5 | 58 | 4 |
math | 4 Place $r$ chess pieces on an $m \times n (m>1, n>1)$ chessboard, with at most one piece per square. If the $r$ pieces have the following property $p$: each row and each column contains at least one piece. However, removing any one of the pieces will cause them to no longer have the above property $p$. Find the maximu... | r(m, n)=m+n-2 | 97 | 9 |
math | A robot is standing on the bottom left vertex $(0,0)$ of a $5\times5$ grid, and wants to go to $(5,5)$, only moving to the right $(a,b)\mapsto(a+1,b)$ or upward $(a,b)\mapsto(a,b+1)$. However this robot is not programmed perfectly, and sometimes takes the upper-left diagonal path $(a,b)\mapsto(a-1,b+1)$. As the grid is... | 1650 | 145 | 4 |
math | Problem 1. Little Know-Not has five cards with digits: $1,2,3,4$ and 5. Help him form two numbers - a three-digit number and a two-digit number - from these cards so that the first number is divisible by the second. $\quad[3$ points] (A.V. Shapovalov) | 532:14=38or215:43=5 | 73 | 18 |
math | 8. [6] Which is greater, $\log _{2008}(2009)$ or $\log _{2009}(2010)$ ? | \log_{2008}2009 | 39 | 12 |
math | Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers. | -\frac{3}{2} | 33 | 7 |
math | 5. $n$ is the smallest positive integer satisfying the following condition:
(2) $n$ has exactly 75 positive divisors (including 1 and itself). Find $\frac{n}{75}$. | 432 | 45 | 3 |
math | Two hens sold a total of 110 eggs at the market, and although one had more eggs than the other, they both had the same income. "If I had as many eggs as you did and sold them at the same price as my own," says one hen to the other, "I would have earned 5 K." "If I had sold your eggs at the same price as my own," replie... | 6050 | 120 | 4 |
math | 8.076. $\sin x - \sin 2x + \sin 5x + \sin 8x = 0$. | x_1=\frac{\pin}{3};x_2=\frac{\pi}{7}(2l+1)\quadn,\in\mathbb{Z} | 31 | 35 |
math | 7. A test paper consists of 4 multiple-choice questions, each with three options $(A),(B),(C)$. Several students take the exam, and after grading, it is found that: any 3 students have 1 question whose answers are all different, and no student leaves any question unanswered, then the maximum number of students who can ... | 9 | 80 | 1 |
math | 256. Palindromic Squares. A certain square is written in the base-6 numeral system using 5 different non-zero digits. If the digit in the units place is moved from the end of this number to the beginning, the square root of the resulting number matches the square root of the original number, written in reverse order. F... | 53241 | 76 | 5 |
math | 5. Find the maximum value of the expression $(\sin 3 x+\sin 2 y+\sin z)(\cos 3 x+\cos 2 y+\cos z)$. (15 points) | 4.5 | 44 | 3 |
math | For each nonzero integer $ n$ find all functions $ f: \mathbb{R} \minus{} \{\minus{}3,0 \} \rightarrow \mathbb{R}$ satisfying:
$ f(x\plus{}3)\plus{}f \left( \minus{}\frac{9}{x} \right)\equal{}\frac{(1\minus{}n)(x^2\plus{}3x\minus{}9)}{9n(x\plus{}3)}\plus{}\frac{2}{n}$ for all $ x \not\equal{} 0,\minus{}3.$
Furthe... | f(x) = \frac{n - 1}{nx} + \frac{1}{n} | 149 | 22 |
math | Example 1 As shown, in Rt $\triangle ABC$, the hypotenuse $AB=5, CD \perp AB$. It is known that $BC, AC$ are the two roots of the quadratic equation $x^{2}-(2 m-1) x+4(m-1)=0$. Then the value of $m$ is $\qquad$. | 4 | 76 | 1 |
math | 6. In triangle $A B C$, it is known that $A B=3, A C=4, \angle B A C=60^{\circ}$. The extension of the angle bisector $A A_{1}$ intersects the circumcircle of triangle $A B C$ at point $A_{2}$. Find the areas of triangles $O A_{2} C$ and $A_{1} A_{2} C$ ( $O$ - the center of the circumcircle of triangle $\left.A B C\ri... | S_{\triangleOA_{2}C}=\frac{13\sqrt{3}}{12};S_{\triangleA_{1}A_{2}C}=\frac{13\sqrt{3}}{21} | 116 | 52 |
math | 12.87 Determine all integer solutions of the indeterminate equation
$$
x^{3}+x^{2} y+x y^{2}+y^{3}=8\left(x^{2}+x y+y^{2}+1\right)
$$
(Luxembourg and four other countries' International Mathematical Competition, 1980) | (x,y)=(8,2)\text{or}(2,8) | 76 | 15 |
math | Task B-3.4. Determine all values of the positive real number $k$ for which the equation
$$
\frac{\log (k x)}{\log (x+1)}=2
$$
has exactly one solution. | 4 | 50 | 1 |
math | Let $D$ be the footpoint of the altitude from $B$ in the triangle $ABC$ , where $AB=1$ . The incircle of triangle $BCD$ coincides with the centroid of triangle $ABC$. Find the lengths of $AC$ and $BC$. | AC = BC = \sqrt{\frac{5}{2}} | 59 | 14 |
math | 1. Let $n$ be the sum of all ten-digit numbers that have each of the digits $0,1, \ldots, 9$ in their decimal representation. Determine the remainder when $n$ is divided by seventy-seven. | 28 | 50 | 2 |
math | ## Task 1 - 270831
During a children's birthday party, they play "Guess the Birthday." Katrin, the birthday girl, explains to each of her play partners:
"Divide your birth date into a day number and a month number! (My birthday today, the 24th of May, would be divided into the day number 24 and the month number 5.) D... | 100x+y | 160 | 5 |
math | # 6. CONDITION
Vasya has three cans of paint of different colors. In how many different ways can he paint a fence consisting of 10 planks so that any two adjacent planks are of different colors and he uses all three colors of paint? Justify your answer. | 1530 | 60 | 4 |
math | 【Question 11】
Mom goes to the supermarket to sell fruits, buying 5 kilograms of apples and 4 kilograms of pears costs 48 yuan, buying 2 kilograms of apples and 3 kilograms of mangoes costs 33 yuan. It is known that each kilogram of mangoes is 2.5 yuan more expensive than pears. If 3 kilograms of apples and 3 kilograms ... | 31.5 | 103 | 4 |
math | Determine the smallest real constant $c$ such that
\[\sum_{k=1}^{n}\left ( \frac{1}{k}\sum_{j=1}^{k}x_j \right )^2\leq c\sum_{k=1}^{n}x_k^2\]
for all positive integers $n$ and all positive real numbers $x_1,\cdots ,x_n$. | 4 | 91 | 1 |
math | Given that the line $y=mx+k$ intersects the parabola $y=ax^2+bx+c$ at two points, compute the product of the two $x$-coordinates of these points in terms of $a$, $b$, $c$, $k$, and $m$. | \frac{c - k}{a} | 61 | 9 |
math | Example 15 (1989 7th American Invitational Mathematics Examination) If $a<b<c<d<e$ are consecutive positive integers, $b+c+d$ is a perfect square, and $a+b+c+d+e$ is a perfect cube, what is the minimum value of $c$? | 675 | 66 | 3 |
math | Let $1 \leq k \leq n$ be integers. How many $k$-element subsets of the set $\{1,2, \ldots, n\}$ can be chosen such that any two of them consist of the $k$ smallest elements of their union? | n-k+1 | 60 | 4 |
math | 5. Let $f(x)$ be a function defined on $\mathbf{R}$ with a period of 2, which is even, strictly decreasing on the interval $[0,1]$, and satisfies $f(\pi)=1, f(2 \pi)=2$. Then the solution set of the inequality system $\left\{\begin{array}{l}1 \leq x \leq 2, \\ 1 \leq f(x) \leq 2\end{array}\right.$ is $\qquad$. | [\pi-2,8-2\pi] | 113 | 11 |
math | Find all real-coefficient polynomials $P(x)$ such that there exists an infinite sequence of integers $\left\{a_{n}\right\}$, all distinct, and satisfying
$$
P\left(a_{1}\right)=0, P\left(a_{k+1}\right)=a_{k}(k=1,2, \cdots) .
$$ | P(x)=\frac{x+D}{}(\in{Z}\backslash{0,-1},D\in{Z}\backslash{0}) | 78 | 33 |
math | 7. Given that $O$ is the circumcenter of $\triangle A B C$. If $A B=A C$, $\angle C A B=30^{\circ}$, and $\overrightarrow{C O}=\lambda_{1} \overrightarrow{C A}+\lambda_{2} \overrightarrow{C B}$, then $\lambda_{1} \lambda_{2}=$ $\qquad$ . | 7 \sqrt{3}-12 | 88 | 8 |
math | 3.10. (НРБ, 82; Australia, 83). Solve the equation
$$
(y+1)^{x}-1=y!
$$
in natural numbers. | (1;1),(1;2),(2;4) | 42 | 13 |
math | 17. $[\mathbf{1 0}]$ Given positive real numbers $x, y$, and $z$ that satisfy the following system of equations:
$$
\begin{aligned}
x^{2}+y^{2}+x y=1 \\
y^{2}+z^{2}+y z=4, \\
z^{2}+x^{2}+z x=5,
\end{aligned}
$$
find $x+y+z$. | \sqrt{5+2\sqrt{3}} | 100 | 11 |
math | 10.5. Find all triples of real numbers $(x, y, z)$ that satisfy:
$$
\begin{array}{l}
1+x^{4} \leqslant 2(y-z)^{2}, \\
1+y^{4} \leqslant 2(z-x)^{2}, \\
1+z^{4} \leqslant 2(x-y)^{2} .
\end{array}
$$ | (-1,0,1),(-1,1,0),(0,-1,1),(0,1,-1),(1,-1,0),(1,0,-1) | 93 | 38 |
math | 6.083. $\left\{\begin{array}{l}12(x+y)^{2}+x=2.5-y, \\ 6(x-y)^{2}+x=0.125+y .\end{array}\right.$ | (\frac{1}{4};\frac{1}{6}),(\frac{1}{12};\frac{1}{3}),(-\frac{5}{24};-\frac{7}{24}),(-\frac{3}{8};-\frac{1}{8}) | 58 | 61 |
math | Given are sheets and the numbers $00, 01, \ldots, 99$ are written on them. We must put them in boxes $000, 001, \ldots, 999$ so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets? | 34 | 87 | 2 |
math | 4. Let the base $A B C D$ of the right square prism $A B C D-A_{1} B_{1} C_{1} D_{1}$ be a unit square. If the dihedral angle $A_{1}-B D-C_{1}$ is $\frac{\pi}{3}$, then $A A_{1}=$ $\qquad$ . | \frac{\sqrt{6}}{2} | 81 | 10 |
math | Question 2. Given non-negative real numbers $a_{1} \geq a_{2} \geq \ldots \geq a_{n}, b_{1} \leq b_{2} \leq \ldots \leq b_{n}$, satisfying
$$
a_{1} a_{n}+a_{2} a_{n-1}+\ldots+a_{n} a_{1}=b_{1} b_{n}+b_{2} b_{n-1}+\ldots+b_{n} b_{1}=1 \text {, }
$$
Find
$$
S=\sum_{1 \leq i<j \leq n} a_{i} b_{j}
$$
the minimum value. | \frac{n-1}{2} | 164 | 8 |
math | 1. The solution set of the inequality $x^{3}+\left(1-x^{2}\right)^{\frac{3}{2}} \geqslant 1$ is | \{0,1\} | 39 | 7 |
math | ## Task 32/82
We are looking for all prime numbers $p$ for which $z=2^{p}+p^{2}$ is also a prime number. | 3 | 39 | 1 |
math | [Coordinate method on the plane]
Given points $A(6 ; 1), B(-5 ;-4), C(-2 ; 5)$. Form the equation of the line on which the height of triangle $A B C$, drawn from vertex $A$, lies.
# | x+3y-9=0 | 57 | 8 |
math | Let $(a_n)_n\geq 0$ and $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n})$ for every $m\geq n\geq0.$ If $a_1=1,$ then find the value of $a_{2007}.$ | 2007^2 | 79 | 6 |
math | 1. In a row, the numbers $1,2,3, \ldots, 2014,2015$ are written. We will call a number from this row good if, after its removal, the sum of all the remaining 2014 numbers is divisible by 2016. Find all the good numbers. | 1008 | 75 | 4 |
math | One, (40 points) Find all real-coefficient polynomials $p(x)$ satisfying
$$
\begin{array}{l}
\left(x^{3}+2 x^{2}+3 x+2\right) p(x-1) \\
=\left(x^{3}-4 x^{2}+5 x-6\right) p(x)
\end{array}
$$ | p(x)=C(x^{2}+x+2)x(x-1)(x+1)(x-2) | 83 | 25 |
math | Find the smallest integer $n$ satisfying the following condition: regardless of how one colour the vertices of a regular $n$-gon with either red, yellow or blue, one can always find an isosceles trapezoid whose vertices are of the same colour. | n = 17 | 56 | 6 |
math | 2. A collector has $N$ precious stones. If he takes away the three heaviest stones then the total weight of the stones decreases by $35 \%$. From the remaining stones if he takes away the three lightest stones the total weight further decreases by $\frac{5}{13}$. Find $N$.
A collector has $N$ precious stones. If he ta... | 10 | 134 | 2 |
math | 8. Let $[x]$ denote the greatest integer not exceeding the real number $x$. The sequence $\left\{a_{n}\right\}$ satisfies:
$$
x_{1}=1, x_{n+1}=4 x_{n}+\left[\sqrt{11} x_{n}\right] \text {. }
$$
Then the units digit of $x_{2021}$ is | 9 | 87 | 1 |
math | Example 10 Let real numbers $x, y, z$ satisfy
$$
x+y+z=5, \quad xy+yz+zx=3 \text{.}
$$
Then the maximum value of $z$ is $\qquad$. | \frac{13}{3} | 53 | 8 |
math | 8. For any subset $S \subseteq\{1,2, \ldots, 15\}$, a number $n$ is called an "anchor" for $S$ if $n$ and $n+|S|$ are both members of $S$, where $|S|$ denotes the number of members of $S$. Find the average number of anchors over all possible subsets $S \subseteq\{1,2, \ldots, 15\}$. | \frac{13}{8} | 102 | 8 |
math | 2. (5 points) At the World Meteorologists Conference, each participant in turn announced the average monthly temperature in their hometown. At this moment, all the others recorded the product of the temperatures in their and the speaker's cities. In total, 78 positive and 54 negative numbers were recorded. What is the ... | 3 | 78 | 1 |
math | Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 647 | 69 | 3 |
math | 21st APMC 1998 Problem 3 Find all real solutions to x 3 = 2 - y, y 3 = 2 - x. | x=y=1 | 36 | 4 |
math | Task B-4.8. Determine the sum of the reciprocals of all positive divisors of the number $n$, including 1 and $n$, if $n=2^{p-1}\left(2^{p}-1\right)$, and $2^{p}-1$ is a prime number. | 2 | 67 | 1 |
math | 9. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{2}=2, a_{n+2}-3 a_{n+1}+2 a_{n}=1$. Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\left\{\frac{8}{a_{n}}\right\}$, then $\left[T_{2023}\right]=$ $\qquad$ . (The symbol $[x]$ denotes the greatest integer less than or equal to $x$) | 14 | 125 | 2 |
math | (10) How many integer solutions $(x, y)$ does the equation $\sqrt{x}+\sqrt{y}=\sqrt{1968}$ have?
(1968 Kyiv Mathematical Olympiad) | 5 | 46 | 1 |
math | 44th Putnam 1983 Problem B3 y 1 , y 2 , y 3 are solutions of y''' + a(x) y'' + b(x) y' + c(x) y = 0 such that y 1 2 + y 2 2 + y 3 2 = 1 for all x. Find constants α, β such that y 1 '(x) 2 + y 2 '(x) 2 + y 3 '(x) 2 is a solution of y' + α a(x) y + βc(x) = 0. Solution | \alpha=\frac{2}{3},\beta=-\frac{2}{3} | 130 | 19 |
math | Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $. | 19 | 45 | 2 |
math | Example 8 (1990 Japanese Spring Selection Problem) Let $x, y, z > 0$, and $x+y+z=1$, find the minimum value of $\frac{1}{x}+\frac{4}{y}+\frac{9}{z}$. | 36 | 59 | 2 |
math | Find all polynomials $p(x)$ with real coefficients that have the following property: there exists a polynomial $q(x)$ with real coefficients such that $$p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n)$$ for all positive integers $n$. | P(x) = \alpha (x - r)(x - r - 1) \dots (x - r - s + 1) | 68 | 31 |
math | 157. The battery fired six shots at the object. The probability of hitting the object with one shot is 0.3. Find: a) the most probable number of hits; b) the probability of the most probable number of hits; c) the probability that the object will be destroyed, if for this at least two hits are required. | 0.58 | 72 | 4 |
math | 2. In $\triangle A B C$, it is known that the three angles $A, B, C$ form an arithmetic sequence. Suppose the sides opposite to these angles are $a, b, c$, respectively, and $c-a$ equals the altitude $h$ from vertex $A$ to side $AC$, then $\sin \frac{C-A}{2}=$ $\qquad$ . | \frac{1}{2} | 83 | 7 |
math | A square number in the decimal system has the form: $\overline{a b 1 a b}$. Which is this square number? | 76176 | 29 | 5 |
math | 4. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=99^{\frac{1}{99}}, a_{n}=$ $\left(a_{n-1}\right)^{a_{1}}$. When $a_{n}$ is an integer, the smallest positive integer $n$ is $\qquad$ | 100 | 77 | 3 |
math | 5 Let real numbers $a_{1}, a_{2}, \cdots, a_{100}$ satisfy:
(1) $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0$;
(2) $a_{1}+a_{2} \leqslant 100$;
(3) $a_{3}+a_{4}+\cdots+a_{100} \leqslant 100$.
Find the maximum value of $a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2}$. | 100^{2} | 161 | 6 |
math | Let $ T \equal{} \int_0^{\ln2} \frac {2e^{3x} \plus{} e^{2x} \minus{} 1} {e^{3x} \plus{} e^{2x} \minus{} e^x \plus{} 1}dx$. Evaluate $ e^T$. | \frac{11}{4} | 72 | 8 |
math | 10. The largest prime $p$ such that $\frac{p+1}{2}$ and $\frac{p^{2}+1}{2}$ are both perfect squares is . $\qquad$ | 7 | 43 | 1 |
math | 5. Among the five-digit numbers formed by the 3 digits $1,2,3$, $1,2,3$ each appear at least once. There are $\qquad$ such five-digit numbers. | 150 | 44 | 3 |
math | Example 6 Solve the system of equations $\left\{\begin{array}{l}x_{1}+x_{2}+\cdots+x_{n}=n \\ x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=n \\ \cdots \\ x_{1}^{n}+x_{2}^{n}+\cdots+x_{n}^{n}=n\end{array}\right.$. | x_{1}=x_{2}=\cdots=x_{n}=1 | 100 | 16 |
math | 41. Given the four-digit number $\overline{4 a b 5}=5 \times x \times 11 \times y$, where $x, y$ are both prime numbers, then the number of such four-digit numbers is $\qquad$, and the largest one is $\qquad$ | 3;4785 | 64 | 6 |
math | 2. Three circles are inscribed in a corner - a small, a medium, and a large one. The large circle passes through the center of the medium one, and the medium one through the center of the small one. Determine the radii of the medium and large circles if the radius of the smaller one is $r$ and the distance from its cen... | \frac{}{-r};\quad\frac{^{2}r}{(-r)^{2}} | 83 | 22 |
math | 11. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1$, and for each $n \in \mathbf{N}_{+}, a_{n} 、 a_{n+1}$ are the two roots of the equation $x^{2}+3 n x+b_{n}=0$. Then $\sum_{k=1}^{20} b_{k}=$ $\qquad$ | 6385 | 95 | 4 |
math | $\left[\begin{array}{l}\text { Theory of algorithms (other) }) \\ {[\underline{\text { Estimation + example }}]}\end{array}\right.$
You need to find out a five-digit phone number by asking questions that can be answered with "yes" or "no." What is the minimum number of questions required to guarantee finding the numbe... | 17 | 88 | 2 |
math | B-4. $p(x)$ is a non-zero polynomial of degree less than 1992, and $p(x)$ has no common factor with $x^{3}-x$. Let
$$
\frac{d^{1992}}{d x^{1992}}\left(\frac{p(x)}{x^{3}-x}\right)=\frac{f(x)}{g(x)},
$$
where $f(x), g(x)$ are polynomials. Find the smallest possible degree of $f(x)$. | 3984 | 115 | 4 |
math | Below is a square root calculation scheme written out in full detail, where the letter $x$ represents missing digits. What is the number under the root?
$$
\begin{aligned}
& \sqrt{x x x 8 x x}=x x x \\
& \underline{x x} \\
& x x x: x x \cdot x \\
& \quad \underline{x x} \\
& \quad x x x x: x x x \cdot x \\
& \quad \un... | 417^2+1 | 116 | 7 |
math | 8th CanMO 1976 Problem 3 n+2 students played a tournament. Each pair played each other once. A player scored 1 for a win, 1/2 for a draw and nil for a loss. Two students scored a total of 8 and the other players all had equal total scores. Find n. | 7or14 | 70 | 4 |
math | 10. A lattice point is a point whose coordinates are both integers. Suppose Johann walks in a line from the point $(0,2004)$ to a random lattice point in the interior (not on the boundary) of the square with vertices $(0,0),(0,99),(99,99),(99,0)$. What is the probability that his path, including the endpoints, contains... | \frac{3}{4} | 93 | 7 |
math | 3. Let $a, b, c$ be positive real numbers such that $a b + b c + c a = 1$. Determine the values that the expression
$$
\frac{a\left(b^{2}+1\right)}{a+b}+\frac{b\left(c^{2}+1\right)}{b+c}+\frac{c\left(a^{2}+1\right)}{c+a}
$$
can take.
(Josef Tkadlec \& Patrik Bak) | 2 | 112 | 1 |
math | 1. Alex's friends Boris and Tsane in one round of sports betting placed the following amounts: Alex 600 denars, Boris 900 denars, and Tsane 1500 denars. On the bet, they won a total of 17000 denars. How should they fairly divide the winnings? | =3400,b=5100,=8500 | 73 | 17 |
math | In the triangle $A B C$ the angle $B$ is not a right angle, and $A B: B C=k$. Let $M$ be the midpoint of $A C$. The lines symmetric to $B M$ with respect to $A B$ and $B C$ intersect $A C$ at $D$ and $E$. Find $B D: B E$. | k^2 | 81 | 3 |
math | 4. Several different numbers are written on the blackboard, such that the sum of any three of them is a rational number, while the sum of any two is an irrational number. The maximum number of numbers that can be written on the blackboard is $\qquad$ | 3 | 55 | 1 |
math | 1. Calculate:
$$
\operatorname{arcctg} 5+\operatorname{arctg} \frac{2}{3}
$$
(Solution should be presented in the form of an explicit numerical value expressed in degrees or radians.) | \frac{\pi}{4} | 52 | 7 |
math | 10. Find all four-digit numbers $m$, such that $m<2006$, and there exists a positive integer $n$, such that $m-n$ is a prime number, and $m n$ is a perfect square. | 1156, 1296, 1369, 1600, 1764 | 51 | 28 |
math | 9. Given the sequence $a_{n}=\sqrt{4+\frac{1}{n^{2}}}+\sqrt{4+\frac{2}{n^{2}}}+\cdots+\sqrt{4+\frac{n}{n^{2}}}-2 n, n$ is a positive integer, then the value of $\lim _{n \rightarrow \infty} a_{n}$ is $\qquad$ | \frac{1}{8} | 86 | 7 |
math | Example 8 For a positive integer $n$, find the smallest integer $k$ such that for any given real numbers $a_{1}, a_{2}, \cdots, a_{d}$, we have
$$
\begin{array}{l}
a_{1}+a_{2}+\cdots+a_{d}=n, \\
0 \leqslant a_{i} \leqslant 1(i=1,2, \cdots, d),
\end{array}
$$
these real numbers can be divided into $k$ groups (allowing ... | 2n-1 | 150 | 4 |
math | Problem 4. Three lines intersect to form 12 angles, and $n$ of them turn out to be equal. What is the maximum possible value of $n$? | 6 | 37 | 1 |
math | G7.1 2 apples and 3 oranges cost 6 dollars.
4 apples and 7 oranges cost 13 dollars.
16 apples and 23 oranges cost $C$ dollars. Find $C$.
G7.2 If $K=\frac{6 \cos \theta+5 \sin \theta}{2 \cos \theta+3 \sin \theta}$ and $\tan \theta=2$, find $K$. | 47 | 93 | 2 |
math | Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$.
(a) Prove that $f_n$ is well defined.
(b) Find all polynomials $f \in \mathbb Z[X... | f(X) = \pm X + b | 171 | 10 |
math | 4.045. The sum of the first three terms of a geometric progression is 91. If 25, 27, and 1 are added to these terms respectively, the resulting three numbers form an arithmetic progression. Find the seventh term of the geometric progression. | 5103 | 59 | 4 |
math | 10. (20 points) Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms as $S_{n}$, and $S_{n}=2 n-a_{n}\left(n \in \mathbf{N}^{*}\right)$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) If the sequence $\left\{b_{n}\right\}$ satisfies $b_{n}=2^{n-1} a_{n}$, prove that... | \frac{5}{3} | 174 | 7 |
math | Example 2-2 Fibonacci Sequence.
The Fibonacci sequence is another typical problem in combinatorial mathematics, proposed as follows: There is a pair of rabbits, one male and one female. Assuming that two months after birth, they can produce another pair of rabbits, one male and one female, every month thereafter, how m... | 1,1,2,3,5,8,13,21,34,55,89,144,\cdots | 77 | 33 |
math | 8. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{3}$, and for any $n \in \mathbf{N}^{*}, a_{n+1}=a_{n}^{2}+a_{n}$, then the integer part of $\sum_{n=1}^{2016} \frac{1}{a_{n}+1}$ is | 2 | 95 | 1 |
math | 5. (8 points) Through the sides of a regular $2 n$-gon, lines are drawn. Into how many parts do these lines divide the plane | 2n^ | 33 | 3 |
math | 1. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that the equation
$$
f([x] y)=f(x)[f(y)]
$$
holds for all $x, y \in \mathbf{R}$ (where $[z]$ denotes the greatest integer less than or equal to the real number $z$).
(French contribution) | f(x)=c \text{ (constant), where, } c=0 \text{ or } 1 \leqslant c<2 | 85 | 31 |
math | The 277th National Junior High School Mathematics Competition consists of 14 questions (5 multiple-choice questions, 5 fill-in-the-blank questions, 4 problem-solving questions), with a full score of 150 points. Among them, each correct answer for multiple-choice and fill-in-the-blank questions earns 7 points, and a wro... | 127 | 138 | 3 |
math | 3. A natural number $n$ is called pleasant if it satisfies the following properties:
- its decimal representation consists of 4 digits;
- the first and third digits of $n$ are equal;
- the second and fourth digits of $n$ are equal;
- the product of the digits of $n$ divides $n^{2}$.
Determine all pleasant numbers. | 1111,1212,2424,3636,1515 | 77 | 24 |
math | ## Task 4 - 130614
Jörg and Claudia are arguing about whether there are more natural numbers from 0 to 1000 whose decimal representation contains (at least) one 5, or whether there are more that do not contain a 5.
Determine the correct answer to this question! | 271 | 70 | 3 |
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