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 | Example 1 Given the function $f_{1}(x)=\frac{2 x-1}{x+1}$, for positive integer $n$, define $f_{n+1}(x)=f_{1}\left[f_{n}(x)\right]$. Find the analytical expression for $f_{1234}(x)$. | \frac{1}{1-x} | 72 | 8 |
math | 64. $\int\left(x^{5}+3 e^{x}\right) d x$
Translation:
64. $\int\left(x^{5}+3 e^{x}\right) d x$ | \frac{1}{6}x^{6}+3e^{x}+C | 46 | 19 |
math | 12. What digits do the decimal representations of the following numbers end with:
1) $135^{x}+31^{y}+56^{x+y}$, if $x \in N, y \in N$
2) $142+142^{2}+142^{3}+\ldots+142^{20}$
3) $34^{x}+34^{x+1}+34^{2 x}$, if $x \in N$. | 6 | 114 | 1 |
math |
2. The roots of the equation
$$
x^{3}-3 a x^{2}+b x+18 c=0
$$
form a non-constant arithmetic progression and the roots of the equation
$$
x^{3}+b x^{2}+x-c^{3}=0
$$
form a non-constant geometric progression. Given that $a, b, c$ are real numbers, find all positive integral values $a$ and $b$.
| (2,9) | 103 | 5 |
math | # Problem №3 (15 points)
Raindrops fall vertically at a speed of 2 m/s. The mass of one drop is 5 mg. There are 400 drops in one cubic meter of air. How long will it take to completely fill a cylindrical vessel with a height of 20 cm and a base area of 10 cm $^{2}$ with water? The density of water $\rho_{B}=1000 \kapp... | 5\cdot10^{4} | 109 | 8 |
math | I2.1 It is given that $m, n>0$ and $m+n=1$. If the minimum value of $\left(1+\frac{1}{m}\right)\left(1+\frac{1}{n}\right)$ is $a$, find the value of $a$.
I2.2 If the roots of the equation $x^{2}-(10+a) x+25=0$ are the square of the roots of the equation $x^{2}+b x=5$, find the positive value of $b$. (Reference: 2001 F... | 15 | 228 | 2 |
math | ## Task 1 - 190611
From a bus station, four buses depart simultaneously at 12:00. The time until each bus returns and departs again from the same bus station is $\frac{3}{4} \mathrm{~h}$ for the first bus, $\frac{1}{2} \mathrm{~h}$ for the second bus, 36 minutes for the third bus, and 1 hour for the fourth bus.
At wh... | 15 | 133 | 2 |
math | \section*{Task 2 - V11122}
The Octavia Touring Sport Car from Škoda Automobile Works Prague reaches a speed of \(80 \frac{\mathrm{km}}{\mathrm{h}}\) in 14 seconds after starting.
a) How many kilometers has it covered in this time (assuming uniform acceleration)? b) In what time, starting from the moment of the start,... | 52\mathrm{~} | 134 | 7 |
math | Example 2 Calculate $[\sqrt{2008+\sqrt{2008+\cdots+\sqrt{2008}}}]$ (2008 appears 2008 times).
(2008, International Youth Math Invitational Competition)
【Analysis】Although there are 2008 square root operations, as long as you patiently estimate from the inside out, the pattern will naturally become apparent. | 45 | 91 | 2 |
math | Suppose the polynomial $f(x) = x^{2014}$ is equal to $f(x) =\sum^{2014}_{k=0} a_k {x \choose k}$ for some real numbers $a_0,... , a_{2014}$. Find the largest integer $m$ such that $2^m$ divides $a_{2013}$. | 2004 | 86 | 4 |
math | 11. $[7]$ Define $\phi^{\prime}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when
$$
\sum_{\substack{2 \leq n \leq 50 \\ \operatorname{gcd}(n, 50)=1}} \phi^{!}(n)
$$
is divided by 50 . | 12 | 94 | 2 |
math | Determine all positive integers $n$ with the property that $n = (d(n))^2$. Here $d(n)$ denotes the number of positive divisors of $n$. | 1 \text{ and } 9 | 38 | 8 |
math | At the New Year's school party in the city of Lzheretsark, 301 students came. Some of them always tell the truth, while the rest always lie. Each of the 200 students said: "If I leave the hall, then among the remaining students, the majority will be liars." Each of the other students stated: "If I leave the hall, then ... | 151 | 112 | 3 |
math | 6.42 Given the parabola $y=a x^{2}+b x+c$ has a line of symmetry at $x=-2$, it is tangent to a certain line at one point, this line has a slope of 2, and a y-intercept of 1, and the parabola intersects the $y=0$ at two points, the distance between which is $2 \sqrt{2}$, try to find the equation of this parabola. | x^{2}+4x+2or\frac{1}{2}x^{2}+2x+1 | 101 | 26 |
math | $2 \cdot 86$ Find the largest perfect square, and it is known that such a perfect square, when the last two digits are subtracted, remains a perfect square (assuming the subtracted digits are not all 0). | 1681 | 49 | 4 |
math | 2. Solve the system of equations $\left\{\begin{array}{l}2 y-x-2 x y=-1, \\ 4 x^{2} y^{2}+x^{2}+4 y^{2}-4 x y=61 \text {. }\end{array}\right.$ | (-6;-\frac{1}{2}),(1;3),(1;-\frac{5}{2}),(5;-\frac{1}{2}) | 65 | 34 |
math | ## Problem 2:
Let $\boldsymbol{f}: \mathbb{R}^{*} \rightarrow \boldsymbol{M}, \boldsymbol{M}$ be a subset of $\mathbb{R}, \boldsymbol{f}(\boldsymbol{x})=\frac{a x^{3}+1}{x}, a \in \mathbb{R}$. Determine $a$ and $M$ such that the function $f$ is bijective. | 0,M=\mathbb{R}^{*} | 98 | 11 |
math | For how many integers $n$ between $ 1$ and $2021$ does the infinite nested expression $$\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}}$$ give a rational number? | 44 | 49 | 2 |
math | Let's determine $p$ such that the following equation
$$
3 x^{2}-5(p-1) x+p^{2}+2=0
$$
has roots that satisfy the following relationship:
$$
x_{1}+4 x_{2}=14
$$ | p_{1}=\frac{742}{127}\quad\text{}\quadp_{2}=4 | 60 | 25 |
math | 209. "Fibonacci Tetrahedron". Find the volume of the tetrahedron whose vertices are located at the points with coordinates $\left(F_{n}, F_{n+1}, F_{n+2}\right), \quad\left(F_{n+3}, F_{n+4}, F_{n+5}\right), \quad\left(F_{n+6}, F_{n+7}, F_{n+8}\right)$ and $\left(F_{n+9}, F_{n+10}, F_{n+11}\right)$, where $F_{i}$ is the... | 0 | 160 | 1 |
math | L OM - I - Task 3
In an isosceles triangle $ ABC $, angle $ BAC $ is a right angle. Point $ D $ lies on side $ BC $, such that $ BD = 2 \cdot CD $. Point $ E $ is the orthogonal projection of point $ B $ onto line $ AD $. Determine the measure of angle $ CED $. | 45 | 79 | 2 |
math | 4.39. Integrate the Bernoulli equation
$$
\frac{d y}{d x}+\frac{y}{x}=\frac{1}{x^{2}} \cdot \frac{1}{y^{2}}
$$
using two methods: the Bernoulli method and the method of variation of arbitrary constants, after first reducing it to a linear equation. | y(x)=\sqrt[3]{\frac{3}{2x}+\frac{C_{1}}{x^{3}}} | 80 | 28 |
math | 3 Find all functions $f, g:(0,+\infty) \rightarrow(0,+\infty)$, such that for all $x \in \mathbf{R}^{-}$. we have
$$
f(g(x))=\frac{x}{x f(x)-2}, \text { and } g(f(x))=\frac{x}{x g(x)-2} .
$$ | f(x)=(x)=\frac{3}{x} | 82 | 12 |
math | Let $O$ be the center of the circle $\omega$ circumscribed around the acute-angled triangle $\vartriangle ABC$, and $W$ be the midpoint of the arc $BC$ of the circle $\omega$, which does not contain the point $A$, and $H$ be the point of intersection of the heights of the triangle $\vartriangle ABC$. Find the angle $... | \angle BAC = 60^\circ | 98 | 11 |
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 | 15. ILLUSTRATIVE EXERCISE
Determine all pairs $(x, y)$ of real numbers that satisfy the equation
$$
\frac{4}{x+y}=\frac{1}{x}+\frac{1}{y} .
$$
SOLUTION | y,exceptforthepair(0,0) | 57 | 11 |
math | Exercise 1. Find all integers $p$ such that $p, p+2$ and $p+4$ are all three prime?
A prime number is an integer $\geqslant 2$ that is divisible only by 1 and itself. | 3 | 54 | 1 |
math | Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color.... | 3 | 144 | 1 |
math | Find the number of $12$-digit "words" that can be formed from the alphabet $\{0,1,2,3,4,5,6\}$ if neighboring digits must differ by exactly $2$. | 882 | 47 | 3 |
math | Let $S=\{1,2,3,\ldots,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime. | 217 | 49 | 3 |
math | 11.1. In the product of seven natural numbers, each factor was decreased by 3. Could the product have increased exactly 13 times as a result?
( | 1,1,1,1,1,2,16 | 36 | 14 |
math | Julius has a set of five positive integers whose mean is 100. If Julius removes the median of the set of five numbers, the mean of the set increases by 5, and the median of the set decreases by 5. Find the maximum possible value of the largest of the five numbers Julius has. | 269 | 65 | 3 |
math | Find the continuous function $ f(x)$ such that $ xf(x)\minus{}\int_0^x f(t)\ dt\equal{}x\plus{}\ln (\sqrt{x^2\plus{}1}\minus{}x)$ with $ f(0)\equal{}\ln 2$. | f(x) = \ln (1 + \sqrt{x^2 + 1}) | 59 | 19 |
math | G2.4 Given that $\cos 16^{\circ}=\sin 14^{\circ}+\sin d^{\circ}$ and $0<d<90$, find the value of $d$. | 46 | 46 | 2 |
math | In the village, there are 100 houses. What is the maximum number of closed, non-intersecting fences that can be built so that each fence encloses at least one house and no two fences enclose the same set of houses
# | 199 | 52 | 3 |
math | Problem 6. How many solutions does the equation
$$
\arcsin 2x + \arcsin x = \frac{\pi}{3} ?
$$ | 1 | 36 | 1 |
math | For any natural number, let's call the numbers formed from its digits and have the same "digit" arrangement with the initial number as the "partial numbers". For example, the partial numbers of $149$ are ${1, 4, 9, 14,19, 49, 149},$ and the partial numbers of $313$ are ${3, 1, 31,33, 13, 313}.$ Find all natural numbers... | \{2, 3, 5, 7, 23, 37, 53, 73\} | 121 | 31 |
math | 7. (15 points) The plant shooters include Peashooter, Twinshot, Triple-shot, Ice-shooter, Dual-shooter, and Pea Pod, requiring 100, 200, 300, 150, 125, 125 suns respectively to plant one of each type. Feifei planted 10 plant shooters, spending a total of 2500 suns. The number of different possible ways she could have p... | 8 | 141 | 1 |
math | 11. Let $S(k)$ denote the sum of all the digits in the decimal representation of a positive integer $k$. Let $n$ be the smallest positive integer satisfying the condition $S(n)+S(n+1)=$ param1. As the answer to the problem, write down a five-digit number such that its first two digits coincide with the first two digits... | 24 | 956 | 2 |
math | 4. On the plane of rectangle $\mathrm{ABCD}$, $\mathrm{with} \mathrm{AB}=4 \mathrm{~cm}$ and $\mathrm{BC}=8 \mathrm{~cm}$, perpendicular EA is raised. Let $B M \perp E C$ and $D N \perp E C, M, N \in(E C)$. If $M N=3 \mathrm{~cm}$, calculate the length of segment $E C$.
7 points
## NATIONAL MATHEMATICS OLYMPIAD
Loca... | 16\mathrm{~} | 144 | 7 |
math | A10. Amy, Bruce, Chris, Donna and Eve had a race. When asked in which order they finished, they all answered with a true and a false statement as follows:
Amy: Bruce came second and I finished in third place.
Bruce: I finished second and Eve was fourth.
Chris: I won and Donna came second.
Donna: I was third and Chris c... | Bruce,Donna,Amy,Eve,Chris | 98 | 10 |
math | 10.4. Find the maximum value of the expression $a+b+c+d-ab-bc-cd-da$, if each of the numbers $a, b, c$ and $d$ belongs to the interval $[0 ; 1]$. | 2 | 52 | 1 |
math | 7. In $\triangle A B C$, the sides opposite to $\angle A 、 \angle B 、 \angle C$ are $a 、 b 、 c$ respectively, and $\tan B=\frac{\sqrt{3} a c}{a^{2}+c^{2}-b^{2}}$. Then the size of $\angle B$ is or $\qquad$ | \frac{\pi}{3}, \frac{2 \pi}{3} | 81 | 16 |
math | 2. Given a positive integer $N<10^{2020}$, when 7 is placed at the first position of $N$, the resulting number is 5 times the number formed when 7 is placed at the last position of $N$. Then the number of all different values of $N$ is $\qquad$. | 336 | 70 | 3 |
math | Determine the angles of triangle $ABC$ if we know that two of its altitudes are at least as long as the corresponding base. | \angleACB=90,\angleABC=\angleBAC=45 | 28 | 17 |
math | Anjanan A.
All possible non-empty subsets are taken from the set of numbers $1,2,3, \ldots, n$. For each subset, the reciprocal of the product of all its numbers is taken. Find the sum of all such reciprocal values. | n | 55 | 1 |
math | ## Task 1 - 260721
Anne, Bernd, and Peter help in the garden during the apple harvest. All three use baskets of the same size. Anne needs 10 minutes to fill a basket, Bernd takes 15 minutes, and little Peter even 30 minutes.
How long would it take for the three children to fill a basket together?
We assume that the ... | 5 | 97 | 1 |
math | 6. There are no fewer than 150 boys studying at the school, and there are $15 \%$ more girls than boys. When the boys went on a trip, 6 buses were needed, and each bus had the same number of students. How many people in total study at the school, given that the total number of students is no more than 400? | 387 | 80 | 3 |
math | Solve the following system of equations:
$$
x y\left(x^{2}+y^{2}\right)=78
$$
$$
x^{4}+y^{4}=97 \text {. }
$$ | xy=\6 | 48 | 3 |
math | 185. Find the numbers that give a remainder of 4 when divided by 19, and a remainder of 1 when divided by 11. | 209+23 | 34 | 6 |
math | The sum of three numbers is 100, two are prime and one is the sum of the other two.
(a) What is the largest of the three numbers?
(b) Give an example of such three numbers.
(c) How many solutions exist for this problem? | 4 | 55 | 1 |
math | ## Problem Statement
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(1 ; 2 ; 0) \)
\( A_{2}(1 ;-1 ; 2) \)
\( A_{3}(0 ; 1 ;-1) \)
\( A_{4}(-3 ; 0 ; 1) \) | \frac{19}{6},\sqrt{\frac{19}{2}} | 126 | 18 |
math | 502. Find the smallest fraction, which when divided by each of the fractions $\frac{21}{25}$ and $\frac{14}{15}$, results in a natural number.
The conditions of this problem are unique: here two fractions are mentioned, one of which divides the other exactly. | \frac{42}{5} | 65 | 8 |
math | Problem 2. To number the pages of a textbook, 411 digits are needed. How many pages does the textbook have? | 173 | 28 | 3 |
math | Example 1.32. Form the equation of the plane passing through the line
\[
\begin{aligned}
3 x+2 y+5 z+6 & =0 \\
x+4 y+3 z+4 & =0
\end{aligned}
\]
parallel to the line
\[
\frac{x-1}{3}=\frac{y-5}{2}=\frac{z+1}{-3}
\] | 2x+3y+4z+5=0 | 95 | 12 |
math | Let $P$ be a point outside a circle $\Gamma$ centered at point $O$, and let $PA$ and $PB$ be tangent lines to circle $\Gamma$. Let segment $PO$ intersect circle $\Gamma$ at $C$. A tangent to circle $\Gamma$ through $C$ intersects $PA$ and $PB$ at points $E$ and $F$, respectively. Given that $EF=8$ and $\angle{APB}=60^\... | 12 \sqrt{3} | 128 | 7 |
math | ## Task 1/64
In the sequence of natural numbers, there are two groups of four consecutive prime numbers each, symmetrically arranged around a pair of twin primes. The distance between the smallest and the largest of these prime numbers is 70, and their product is equal to 3959.
Which are these eight prime numbers and... | 37,41,43,47,71,73,97,101,103,107 | 77 | 32 |
math | 1. Find the largest real number $\theta(\theta<\pi)$ such that
$$
\prod_{k=0}^{10} \cos 2^{k} \theta \neq 0 \text {, and } \prod_{k=0}^{10}\left(1+\frac{1}{\cos 2^{k} \theta}\right)=1
$$ | \frac{2046\pi}{2047} | 85 | 15 |
math | Find the least $k$ for which the number $2010$ can be expressed as the sum of the squares of $k$ integers. | k=3 | 31 | 3 |
math | 10.088. A trapezoid is inscribed in a circle of radius $R$, with the lower base being twice as long as each of the other sides. Find the area of the trapezoid. | \frac{3\sqrt{3}}{4}R^{2} | 48 | 16 |
math | 24. [10] Suppose that $A B C$ is an isosceles triangle with $A B=A C$. Let $P$ be the point on side $A C$ so that $A P=2 C P$. Given that $B P=1$, determine the maximum possible area of $A B C$. | \frac{9}{10} | 70 | 8 |
math | 6. A. If the equation with respect to $x$
$$
(x-2)\left(x^{2}-4 x+m\right)=0
$$
has three roots, and these three roots can exactly serve as the three side lengths of a triangle, then the range of values for $m$ is | 3<m \leqslant 4 | 64 | 9 |
math | 7. (3 points) In a class of 30 people participating in a jump rope competition, at the beginning, 4 people were late and did not participate in the competition, at which point the average score was 20. Later, these 4 students arrived at the competition venue and jumped $26, 27, 28, 29$ times respectively. At this point... | 21 | 99 | 2 |
math | Example 7. Calculate:
$$
\frac{(742-108+321 \times 0.082) \times(37.2-28.7)}{517.4 \times(90.28-37.2)}
$$ | 0.20 | 65 | 4 |
math | There are 2008 red cards and 2008 white cards. 2008 players sit down in circular toward the inside of the circle in situation that 2 red cards and 2 white cards from each card are delivered to each person. Each person conducts the following procedure in one turn as follows.
$ (*)$ If you have more than one red card... | 1004 | 146 | 4 |
math | 883. Solve the equation in natural numbers
$$
1+x+x^{2}+x^{3}=2^{y}
$$ | (1;2) | 29 | 5 |
math | 3.088. $\frac{\operatorname{ctg}\left(270^{\circ}-\alpha\right)}{1-\operatorname{tg}^{2}\left(\alpha-180^{\circ}\right)} \cdot \frac{\operatorname{ctg}^{2}\left(360^{\circ}-\alpha\right)-1}{\operatorname{ctg}\left(180^{\circ}+\alpha\right)}$. | 1 | 105 | 1 |
math | *6. From $1,2, \cdots, 1996$, select $k$ numbers such that the sum of any two numbers cannot be divisible by their difference. Then the maximum possible value of $k$ is $\qquad$ . | 666 | 54 | 3 |
math | IMO 1991 Problem A3 Let S = {1, 2, 3, ... 280}. Find the smallest integer n such that each n-element subset of S contains five numbers which are pairwise relatively prime. Solution | 217 | 50 | 3 |
math | $\frac{n(n + 1)}{2}$ distinct numbers are arranged at random into $n$ rows. The first row has $1$ number, the second has $2$ numbers, the third has $3$ numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers. | \frac{2^n}{(n+1)!} | 74 | 12 |
math | Let $a_1=24$ and form the sequence $a_n$, $n\geq 2$ by $a_n=100a_{n-1}+134$. The first few terms are
$$24,2534,253534,25353534,\ldots$$
What is the least value of $n$ for which $a_n$ is divisible by $99$? | 88 | 102 | 2 |
math | 149 Given the quadratic function $f(x)=x^{2}-2 m x+1$. Does there exist a real number $x$, such that for any real numbers $a, b, c$ satisfying $0 \leqslant x \leqslant 1$, $f(a), f(b), f(c)$ can form the lengths of the three sides of a triangle. | 0<m<\frac{\sqrt{2}}{2} | 82 | 13 |
math | 11.4. a) Given a rectangular parallelepiped with a volume of 2017 and integer coordinates of vertices in a Cartesian coordinate system in space. Find the diagonal of the parallelepiped, given that its edges are parallel to the coordinate axes. b) Does there exist a rectangular parallelepiped with a volume of 2017 and i... | \sqrt{4035} | 91 | 8 |
math | There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of diff erent colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ includin... | 4 | 130 | 1 |
math | 5. (1998 National High School Competition Question) In an arithmetic sequence with real number terms, the common difference is 4, and the square of the first term plus the sum of the remaining terms does not exceed 100. The maximum number of terms in such a sequence is $\qquad$. | 8 | 65 | 1 |
math | 3. (15 points) Determine the mass $m$ of helium needed to fill an empty balloon of mass $m_{1}=10$ g so that the balloon will rise. Assume the temperature and pressure of the gas in the balloon are equal to the temperature and pressure of the air. The molar mass of helium $M_{\mathrm{r}}=4$ g/mol, the molar mass of air... | 1.6 | 102 | 3 |
math | I am thinking of several consecutive natural numbers. If we crossed out the numbers 70, 82, and 103, the arithmetic mean of the numbers would not change. If we instead crossed out the numbers 122 and 123, the arithmetic mean would decrease by exactly 1. Which natural numbers am I thinking of?
(L. Šimůnek) | 47to123 | 82 | 6 |
math |
Problem 4. Find all pairs $(x, y)$ of integer numbers such that $x^{3}=$ $y^{3}+2 y^{2}+1$. Nikolay Nikolov and Emil Kolev
| (-2,-3),(1,-2),(1,0) | 48 | 13 |
math | Integers $1,2, \ldots, n$ are written (in some order) on the circumference of a circle. What is the smallest possible sum of moduli of the differences of neighbouring numbers? | 2n-2 | 43 | 4 |
math | ## Task A-3.5.
How many four-digit numbers divisible by 7 do not contain the digits 1, 2, or 7 in their decimal representation? | 294 | 36 | 3 |
math | 10.225. Two circles of different radii touch each other externally. Find the angle determined by the chords connecting the point of contact of the circles with the points of contact of their common external tangent. | 90 | 44 | 2 |
math | ## Task B-3.2.
Determine the zeros of the function $f: \mathbf{R} \rightarrow \mathbf{R}, f(x)=\log _{2}\left(18 \cdot 4^{x}-8 \cdot 2^{x}+1\right)-2 x-1$. | -2 | 70 | 2 |
math | Exercise 1. Fred and Sarah are the eldest of the same large family. Fred has twice as few brothers as sisters, while Sarah has as many sisters as brothers.
How many children are there in this family? | 7 | 43 | 1 |
math | Let $P$ be a cubic monic polynomial with roots $a$, $b$, and $c$. If $P(1)=91$ and $P(-1)=-121$, compute the maximum possible value of \[\dfrac{ab+bc+ca}{abc+a+b+c}.\]
[i]Proposed by David Altizio[/i] | 7 | 78 | 1 |
math | 1. Find all functions $f: \mathbf{Q}_{+} \rightarrow \mathbf{Q}_{+}$ such that for all $x, y \in \mathbf{Q}_{+}$, we have
$$
f\left(x^{2} f^{2}(y)\right)=f^{2}(x) f(y) .
$$ | f(x)=1 | 76 | 4 |
math | 7. Let $f:[0,1) \rightarrow \mathbb{R}$ be a function that satisfies the following condition: if
$$
x=\sum_{n=1}^{\infty} \frac{a_{n}}{10^{n}}=. a_{1} a_{2} a_{3} \ldots
$$
is the decimal expansion of $x$ and there does not exist a positive integer $k$ such that $a_{n}=9$ for all $n \geq k$, then
$$
f(x)=\sum_{n=1}^{\i... | 0 | 168 | 1 |
math | 4. In the number $2016 * * * * 02 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,2,4,7,8,9$ (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done? | 1728 | 78 | 4 |
math | 4. 103 Real numbers $\alpha, \beta$ satisfy the system of equations
$$\left\{\begin{array}{l}
\alpha^{3}-3 \alpha^{2}+5 \alpha-17=0 \\
\beta^{3}-3 \beta^{2}+5 \beta+11=0
\end{array}\right.$$
Find $\alpha+\beta$. | 2 | 87 | 1 |
math | $2 \cdot 111$ two-digit number set $\{00,01, \cdots, 98,99\}$ has a subset $X$ with the following property: in any infinite sequence of digits, there are two adjacent digits that form an element of $X$. How many elements does $X$ have at minimum? | 55 | 75 | 2 |
math | Example \ Solve the equation $3 x^{3}-[x]=3$ | \sqrt[3]{\frac{4}{3}} | 16 | 12 |
math | One of the boxes that Joshua and Wendy unpack has Joshua's collection of board games. Michael, Wendy, Alexis, and Joshua decide to play one of them, a game called $\textit{Risk}$ that involves rolling ordinary six-sided dice to determine the outcomes of strategic battles. Wendy has never played before, so early on Mi... | 17 | 256 | 2 |
math | The sweeties shop called "Olympiad" sells boxes of $6,9$ or $20$ chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of $15=6+9$ students, $38=2*9+20... | 43 | 178 | 2 |
math | 6. A real estate agent is trying to sell the last apartment in a building for $482,100 \mathrm{kn}$, which was the price of the penultimate apartment, and by doing so, the average price of the apartments he sold in that building would be $519,500 \mathrm{kn}$. However, due to market saturation, he sells the apartment f... | 17 | 139 | 2 |
math | 3. A two-digit number has the following property: if it is added to the number with the same digits but in reverse order, the result is a perfect square.
Find all such two-digit numbers. | 29,38,47,56,65,74,83,92 | 41 | 23 |
math | 3. The perimeter of a right-angled triangle is $2 p$ ( $p>0$ ), and its height dropped to the hypotenuse has a length of $h$. Determine its sides. | \begin{aligned}&=\frac{p}{+2p}(+p+\sqrt{(p-)^{2}-2^{2}}),\\&b=\frac{p}{+2p}(+p-\sqrt{(p-)^{2}-2^{2}}),\\&=\frac{2p^{2}}{+2p}\end{aligned} | 42 | 77 |
math | 15. Given the sequence $\left\{a_{n}\right\}, a_{n}+a_{n+1}=n \cdot(-1)^{\frac{n(n+1)}{2}}$, the sum of the first $n$ terms is $S_{n}$, and $m+S_{2015}=-1007, a_{1} m>0$. Then the minimum value of $\frac{1}{a_{1}}+\frac{4}{m}$ is . $\qquad$ | 9 | 112 | 1 |
math | 1537. In an urn, there are 5 white and 4 black balls. Two balls are drawn in succession. Find the probability that both balls are white. | \frac{5}{18} | 36 | 8 |
math | 11. Let $A$ be a set composed of any 100 distinct positive integers, and let
$$
B=\left\{\left.\frac{a}{b} \right\rvert\, a 、 b \in A \text { and } a \neq b\right\},
$$
$f(A)$ denotes the number of elements in set $B$. Then the sum of the maximum and minimum values of $f(A)$ is $\qquad$ . | 10098 | 101 | 5 |
math | 4. The sequence $\left\{x_{n}\right\}$ is defined as follows: $x_{1}=\frac{1}{2}, x_{k+1}=x_{k}^{2}+x_{k}, k=1,2, \cdots$. Find the integer part of the following sum:
$$
\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{2005}}
$$ | 1 | 108 | 1 |
math | 8. $\frac{\tan 96^{\circ}-\tan 12^{\circ}\left(1+\frac{1}{\sin 6^{\circ}}\right)}{1+\tan 96^{\circ} \tan 12^{\circ}\left(1+\frac{1}{\sin 6^{\circ}}\right)}=$ $\qquad$ | \frac{\sqrt{3}}{3} | 85 | 10 |
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