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 | 3. All members of an infinite geometric progression are natural numbers. The sum of the third, fifth, and seventh terms of this progression is equal to $819 \cdot 6^{2016}$. Find the common ratio of the progression. (8 points) | q=1,q=2,q=3,q=4 | 57 | 12 |
math | a) If $A=\underbrace{111 \ldots 111}_{2 m}$ and $B=\underbrace{444 \ldots 444}_{m}$, verify that the sum $A+B+1$ is a perfect square for any positive integer $m$.
b) If $p=\underbrace{111 \ldots 111}_{l}$ is a prime number, verify that $l$ is also a prime number.
c) Verify that if $x=\underbrace{111 \ldots 111}_{n}$ ... | \underbrace{333\ldots33}_{2n\text{times}} | 200 | 20 |
math | 6. Find all pairs of prime numbers $p$ and $q$ such that $p^{2}+5 p q+4 q^{2}$ is a square of a natural number. | (13,3),(7,5),(5,11) | 40 | 15 |
math | A $3 \times 3 \times 3$ cube grid has a mouse placed in one of its corner cubes, and a piece of cheese in the center cube. The mouse wanders in search of the cheese: at each step, it randomly moves to one of the adjacent cubes. On average, in how many steps will the mouse find the cheese? | 21 | 73 | 2 |
math | $\underline{\text { Folklore }}$
For some numbers $a, b, c$ and $d$, different from zero, the equality holds: $\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}$. Determine the sign of the number $a c$. | ac<0 | 65 | 3 |
math | 4. If $2016+3^{n}$ is a perfect square, then the positive integer $n=$ . $\qquad$ | 2 | 30 | 1 |
math | Consider a function $f:\mathbb{R} \to \mathbb{R}$. Given that $f(0)=0$. $\lim_{x\to0} \frac{f(h)}{h}=7$, and $f(x+y)=f(x)+f(y)+3xy$ for all $x,y\in{\mathbb{R}}$. Compute $f(7)$ | 122.5 | 83 | 5 |
math | Consider $n$ lamps clockwise numbered from $1$ to $n$ on a circle.
Let $\xi$ to be a configuration where $0 \le \ell \le n$ random lamps are turned on. A [i]cool procedure[/i] consists in perform, simultaneously, the following operations: for each one of the $\ell$ lamps which are turned on, we verify the number of th... | n = 2^k | 456 | 7 |
math | 9. Given the sequence $\left\{a_{n}\right\}$, where $a_{n+1}=\frac{2005 a_{n}}{2003 a_{n}+2005}, a_{1}=1$, then $a_{2006}=$ $\qquad$ | \frac{1}{2004} | 71 | 10 |
math | 2. The line $\sqrt{3} x-y-\sqrt{3}=0$ intersects the parabola $y^{2}=4 x$ at points $A, B$ ($A$ is above the x-axis), and intersects the x-axis at point $F$. If $\overrightarrow{O F}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}$, then $\mu^{2}-\lambda^{2}=$ $\qquad$ . | \frac{1}{2} | 101 | 7 |
math | Suppose for independent events $A_{2}, A_{3}, \ldots, A_{n}$,
$$
P\left(A_{i}\right)=\frac{1}{2 i^{2}}
$$
What is the probability that an odd number of the events $A_{2}, A_{3}, \ldots, A_{n}$ occur? | \frac{n-1}{4n} | 76 | 9 |
math | Find two simple solutions to the equation: $(x-1)^{2}+(x+1)^{2}=y^{2}+1$, where $x$ and $y$ are non-negative integers.
Find three integers $a, b$, and $c$ such that if $(x, y)$ is a solution to the equation, then $(a x+b y, c x+a y)$ is also a solution.
Deduce that the equation has infinitely many solutions. | (3x+2y,4x+3y) | 99 | 13 |
math | We say that an integer $m$ covers the number 1998 if $1,9,9,8$ appear in this order as digits of $m$. (For instance, 1998 is covered by 215993698 but not by 213326798 .) Let $k(n)$ be the number of positive integers that cover 1998 and have exactly $n$ digits $(n \geqslant 5)$, all different from 0 . What is the remain... | 1 | 132 | 1 |
math | 5. (7 points) Compare the result of the product $1 \cdot 2 \cdot 3 \cdot 4 \cdot \ldots \cdot 98 \cdot 99$ and the number $50^{99}$.
# | 1\cdot2\cdot3\cdot4\cdot\ldots\cdot98\cdot99<50^{99} | 54 | 30 |
math | Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minim... | 4 | 133 | 1 |
math | 7.095. $\sqrt{\log _{a} x}+\sqrt{\log _{x} a}=\frac{10}{3}$. | \sqrt[9]{};^{9} | 35 | 10 |
math | # 1. Task 1.1
Timur and Alexander are counting the trees growing around their house. Both are moving in the same direction but start counting from different trees. What is the total number of trees growing around the house if the tree that Timur called the 12th, Alexander counted as $33-\mathrm{m}$, and the tree that... | 118 | 98 | 3 |
math | 5. In a certain primary school, 94 students are in the sixth grade. Some students are involved in extracurricular activities: English and German language classes, and sports clubs. Extracurricular English is attended by 40 students, German by 27 students, and sports activities by 60 students. Of the students involved i... | 47 | 171 | 2 |
math | [ Concerning the sphere ] $[$ Regular tetrahedron $]$
Four spheres of radius 1 touch each other pairwise. Find the radius of the sphere that touches all four spheres. | r=\sqrt{\frac{3}{2}}+1orr=\sqrt{\frac{3}{2}}-1 | 39 | 24 |
math | 4.2. The segment connecting the lateral sides of the trapezoid and parallel to its bases, which are 7 and 17, divides the trapezoid into two parts of equal area. Find the length of this segment. | 13 | 51 | 2 |
math | 4. Given that $S$ is a set composed of $n(n \geqslant 3)$ positive numbers. If there exist three distinct elements in $S$ that can form the three sides of a triangle, then $S$ is called a "triangle number set". Consider the set of consecutive positive integers $\{4,5, \cdots, m\}$, all of whose 10-element subsets are t... | 253 | 102 | 3 |
math | 3.342. $\sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\circ}=\frac{1}{16}$. | \frac{1}{16} | 51 | 8 |
math | ## Task Condition
Find the differential $d y$.
$$
y=x(\sin (\ln x)-\cos (\ln x))
$$ | 2\sin(\lnx)\cdot | 29 | 8 |
math | 6. In $\triangle A B C$, the maximum value of $\sin A+\sin B+2 \sqrt{7} \sin C$ is $\qquad$ . | \frac{27}{4} | 36 | 8 |
math | 17 Let $a, b, c, d$ all be prime numbers, and $a>3b>6c>12d, a^{2}-b^{2}+c^{2}-d^{2}=1749$. Find all possible values of $a^{2}+b^{2}+c^{2}+d^{2}$. | 1999 | 79 | 4 |
math | 2. (3 points) Calculate: $99 \times \frac{5}{8}-0.625 \times 68+6.25 \times 0.1=$ $\qquad$ | 20 | 46 | 2 |
math | 4. (15 points) A tanker is being filled with oil at a rate of 3 barrels per minute. Given that 1 barrel equals 159 liters, determine the rate of filling the tanker in m ${ }^{3} /$ hour. | 28.62 | 54 | 5 |
math | 7.2. In 7a grade, there are 33 students. At the beginning of the school year, two clubs were organized in the class. According to school rules, a club can be organized if at least $70 \%$ of all students in the class sign up for it. What is the smallest number of students who could have signed up for both clubs simulta... | 15 | 79 | 2 |
math | 1. Let $f_{1}(x)=\left\{\begin{array}{ll}x, & x \in \mathbf{Q} \text {; } \\ \frac{1}{x}, & x \notin \mathbf{Q} .\end{array}\right.$ For positive integers $n$ greater than 1, define $f_{n}(x)=f_{1}\left(f_{n-1}(x)\right)$.
Then for all positive integers $n$, we have $f_{n}(x)=$ | f_{n}(x)=\left\{\begin{array}{ll} x, & x \in \mathbf{Q} \\ x^{(-1)^{n}}, & x \notin \mathbf{Q} \end{array}\right.} | 116 | 55 |
math | What is the smallest natural number $n$ such that the base 10 representation of $n!$ ends with ten zeros | 45 | 26 | 2 |
math | Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$. | (18, 1) | 45 | 7 |
math | 11. The sequence $a_{0}, a_{1}, a_{2}, \cdots, a_{n}, \cdots$, satisfies the relation $\left(3-a_{n+1}\right)\left(6+a_{n}\right)=18$ and $a_{0}=3$, then $\sum_{i=0}^{n} \frac{1}{a_{i}}=$ $\qquad$ . | \frac{1}{3}(2^{n+2}-n-3) | 91 | 17 |
math | 4. Let the sequence of real numbers $\left(x_{n}\right)_{n \geq 0}$ be such that $x_{0}=a>0$ and $x_{n+1}=x_{n}+\sqrt{1+x_{n}^{2}}, \forall n \in \mathbb{N}$.
Study the existence of the limit of the sequence $\left(y^{n} x_{n}\right)_{n \geq 1}$, where $y$ is a fixed real number.
Is it possible for the limit of the s... | 2015 | 152 | 4 |
math | In triangle $ABC$ we have $|AB| \ne |AC|$. The bisectors of $\angle ABC$ and $\angle ACB$ meet $AC$ and $AB$ at $E$ and $F$, respectively, and intersect at I. If $|EI| = |FI|$ find the measure of $\angle BAC$. | 60^\circ | 74 | 4 |
math | Three, given that $a$ is an integer, the equation $x^{2}+(2 a+1) x$ $+a^{2}=0$ has integer roots $x_{1}, x_{2}, x_{1}>x_{2}$. Try to find the value of $\sqrt[4]{x_{1}^{2}}-\sqrt[4]{x_{2}^{2}}$. | -1 | 86 | 2 |
math | The function $f: \mathbb{N}\to\mathbb{N}_{0}$ satisfies for all $m,n\in\mathbb{N}$: \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine $f(1982)$. | 660 | 101 | 3 |
math | $a_0, a_1, \ldots, a_{100}$ and $b_1, b_2,\ldots, b_{100}$ are sequences of real numbers, for which the property holds: for all $n=0, 1, \ldots, 99$, either
$$a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n,$$
or
$$a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n.$$
Given $a_{100}\leq a_0$, what i... | 50 | 180 | 2 |
math | 2. Compute
$$
\int_{0}^{\pi / 2} \cos ^{2}(\cos x)+\sin ^{2}(\sin x) d x
$$ | \frac{\pi}{2} | 42 | 7 |
math | 1. (10 points) Calculate: $\frac{3}{4}+\frac{5}{36}+\frac{7}{144}+\frac{9}{400}+\frac{11}{900}+\frac{13}{1764}+\frac{15}{3136}=$ | \frac{63}{64} | 75 | 9 |
math | Example 4 Given that $a$ is an integer, the equation concerning $x$
$$
\frac{x^{2}}{x^{2}+1}-\frac{4|x|}{\sqrt{x^{2}+1}}+2-a=0
$$
has real roots. Then the possible values of $a$ are $\qquad$
(2008, I Love Mathematics Junior High School Summer Camp Mathematics Competition) | 0, 1, 2 | 92 | 7 |
math | The center of the circle inscribed in a quadrilateral lies on its diagonal, which is equal to 5. It is known that the perimeter of the quadrilateral is 14, and the area is 12. Find the second diagonal and the sides of the quadrilateral. | \frac{24}{5},3,4,3,4 | 58 | 15 |
math | 3. In each cell of a $2017 \times 2017$ grid, there is a lamp, and each lamp has only two states: on or off. A lamp is called "bad" if and only if there is an even number of lamps that are on among its adjacent lamps. Find the minimum possible number of bad lamps in this $2017 \times 2017$ grid.
Note: If two lamps are... | 1 | 111 | 1 |
math | Problem 1. Calculate $\int \frac{12 x+17}{(x+2)(2 x+3)(3 x+4)(6 x+5)+2016} \mathrm{~d} x, x \in(0 ; \infty)$. | \frac{1}{\sqrt{2015}}\cdot\operatorname{arctg}\frac{6x^{2}+17x+11}{\sqrt{2015}}+\mathcal{C} | 61 | 52 |
math | 10.295. The legs of a right triangle are 6 and 8 cm. A circle is drawn through the midpoint of the smaller leg and the midpoint of the hypotenuse, touching the hypotenuse. Find the area of the circle bounded by this circle. | \frac{100\pi}{9} | 58 | 11 |
math | 2. Let $0 \leqslant \alpha, \beta < 2 \pi, \alpha, \beta \neq \frac{\pi}{3}, \frac{5 \pi}{3}$, and
$$
\frac{2 \sin \alpha - \sqrt{3}}{2 \cos \alpha - 1} + \frac{2 \sin \beta - \sqrt{3}}{2 \cos \beta - 1} = 0 \text{. }
$$
Then $\cot \frac{\alpha + \beta}{2} = $ $\qquad$ | -\frac{\sqrt{3}}{3} | 128 | 10 |
math | 一、Fill in the Blanks (20 questions in total, 3 points each, full score 60 points)
1. (3 points) Calculate: $1.25 \times \frac{2}{9}+1 \frac{1}{9} \times 1 \frac{1}{4}-125 \% \times \frac{1}{3}=$ $\qquad$ | \frac{5}{4} | 86 | 7 |
math | Example 5.20. Expand the function $f(x)=e^{3 x}$ into a Taylor series. | e^{3x}=1+\frac{3}{1!}x+\frac{3^{2}}{2!}x^{2}+\frac{3^{3}}{3!}x^{3}+\ldots+\frac{3^{n}}{n!}x^{n}+\ldots | 24 | 66 |
math | 5. The number of positive integers $n$ such that $n+1$ divides $n^{2006}+2006$ is $\qquad$.
Makes the positive integer $n$ such that $n+1$ can divide $n^{2006}+2006$ total $\qquad$.
Note: The second sentence seems to be a repetition or a different phrasing of the first. If it's meant to be a different statement, plea... | 5 | 117 | 1 |
math | Problem 3. All students in the class scored a different number of points (positive integers) on the test, with no duplicate scores. In total, they scored 119 points. The sum of the three lowest scores is 23 points, and the sum of the three highest scores is 49 points. How many students took the test? How many points di... | 10 | 81 | 2 |
math | Three. (20 points) Given
$$
(3 a+5 b-1)^{2}+|a+3 b+1|=0 \text {. }
$$
Find the solution set of the inequality with respect to $x$
$$
a x-b>\frac{x}{3}+6
$$ | x>3 | 67 | 3 |
math | 11.2. In the city of Perpendicularinsk, it was decided to build new multi-story houses (some of them may be single-story), but in such a way that the total number of floors would be 30. The city architect, Parallelnikov, proposed a project according to which, if after construction one climbs to the roof of each new hou... | 112 | 123 | 3 |
math | Determine all positive integers $n \geq 2$ for which there exists a positive divisor $m \mid n$ such that
$$
n=d^{3}+m^{3},
$$
where $d$ is the smallest divisor of $n$ greater than 1. | 16, 72, 520 | 60 | 11 |
math | [ Linear dependence of vectors $]$ [ Angles between lines and planes $]$
The height of a regular hexagonal pyramid is equal to the side of the base. Find the angle between a lateral edge and the plane of the base. | 45 | 48 | 2 |
math | IMO 1979 Problem B2 Find all real numbers a for which there exist non-negative real numbers x 1 , x 2 , x 3 , x 4 , x 5 satisfying: x 1 + 2x 2 + 3x 3 + 4x 4 + 5x 5 = a, x 1 + 2 3 x 2 + 3 3 x 3 + 4 3 x 4 + 5 3 x 5 = a 2 , x 1 + 2 5 x 2 + 3 5 x 3 + 4 5 x 4 + 5 5 x 5 = a 3 . | 1,4,9,16,25 | 153 | 11 |
math | For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n... | \frac{\pi}{2} | 139 | 8 |
math | Question 133, Find the value: $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{m^{2} n}{3^{m}\left(n \cdot 3^{m}+m \cdot 3^{n}\right)}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{9}{32} | 94 | 8 |
math | 4 Find all functions $f$ from the set of positive integers to the set of real numbers, such that: (1) for any $n \geqslant 1, f(n+1) \geqslant f(n)$; (2) for any $m, n, (m, n)=1$, we have $f(m n)=f(m) f(n) . \quad$ (Supplied by Pan Chengbiao) | 0 | 95 | 1 |
math | $\begin{array}{l}\text { 1. If } n \text { satisfies }(n-2003)^{2}+(2004-n)^{2} \\ =1 \text {, then }(2004-n)(n-2003)=\end{array}$ | 0 | 67 | 1 |
math | Richard and Shreyas are arm wrestling against each other. They will play $10$ rounds, and in each round, there is exactly one winner. If the same person wins in consecutive rounds, these rounds are considered part of the same “streak”. How many possible outcomes are there in which there are strictly more than $3$ strea... | 932 | 112 | 3 |
math | 627. Calculate the derivative of the function $y=x^{3} \log _{5} x$. | x^{2}(3\log_{5}x+\frac{1}{\ln5}) | 24 | 20 |
math | ## Task 2 - 070822
Given are a circle $k$ (center $M$, radius of length $r=6 \mathrm{~cm}$) and a circle $k_{1}$ (center $M_{1}$, radius of length $r_{1}=2 \mathrm{~cm}$). Both circles touch each other externally.
Construct all circles with a radius of length $2 \mathrm{~cm}$ that touch the two given circles!
Also c... | 3 | 116 | 1 |
math | 3. (6 points) A car, after starting, first traveled 18 kilometers in 24 minutes, then continued at a speed of 72 kilometers per hour for another 35 minutes to reach its destination. The car's average speed per minute is approximately $\qquad$ kilometers (round the answer to two decimal places). | 1.02 | 70 | 4 |
math | 12. If the inequality $a b+b^{2}+c^{2} \geqslant \lambda(a+b) c$ holds for all positive real numbers $a, b, c$ satisfying $b+c \geqslant a$, then the maximum value of the real number $\lambda$ is $\qquad$. | \sqrt{2}-\frac{1}{2} | 70 | 12 |
math | Example 20. Solve the equation
$$
3 x^{\log _{5} 2}+2^{\log _{5} x}=64
$$ | 625 | 38 | 3 |
math | 4. Calculate $\lim _{x \rightarrow 1} \frac{\sqrt{\operatorname{arctg} x}-\sqrt{\arccos \left(\frac{\sqrt{2}}{2} x\right)}}{\sqrt[3]{1+\ln [x(2 x-1)]}-\sqrt[3]{1+\ln \left(x^{3}-4 x+4\right)}}$.
Iulian Danielescu, Brăila | \frac{9}{8\sqrt{\pi}} | 100 | 11 |
math | 2. (7p) Consider the positive real numbers $a, b, c, d$, such that $a b c d=1$. Calculate
$$
E=\frac{7+a}{1+a+a b+a b c}+\frac{7+b}{1+b+b c+b c d}+\frac{7+c}{1+c+c d+c d a}+\frac{7+d}{1+d+d a+d a b}
$$
GM11/2015 | 8 | 101 | 1 |
math | Example 3. Find the real roots of the equation
$$
\mathrm{f}(\mathrm{x})=\mathrm{x}^{41}+\mathrm{x}^{3}+1=0
$$ | x=-0.9524838 | 44 | 11 |
math | 5. Let the first two terms of the sequence $\left\{a_{n}\right\}$ be $a_{1}=2$, $a_{2}=4$. Let $b_{n}=a_{n+1}-a_{n}$. If the sequence $\left\{b_{n}\right\}$ is an arithmetic sequence with a common difference of 1, then $a_{2020}=$ $\qquad$ | 2041211 | 93 | 7 |
math | A $0 \leq t \leq \pi$ real parameter for which values does the equation $\sin (x+t)=1-\sin x$ have no solution? | \frac{2\pi}{3}<\leq\pi | 36 | 14 |
math | One, (40 points) Find all prime numbers $p, q$ such that $p^{2}-p+1=q^{3}$.
| (19,7) | 31 | 6 |
math | 4. (7 points) Solve the equation
$$
x+\sqrt{x+\frac{1}{2}+\sqrt{x+\frac{1}{4}}}=n^{2} \quad n>1
$$ | n^2-n | 45 | 4 |
math | 12.191. The base of the triangle is equal to $a$, and the angles at the base are $\alpha$ and $\beta$ radians. From the opposite vertex of the triangle, a circle is drawn with a radius equal to the height of the triangle. Find the length of the arc of this circle that is enclosed within the triangle. | (\pi-\alpha-\beta)\cdot\frac{\sin\alpha\sin\beta}{\sin(\alpha+\beta)} | 73 | 26 |
math | Example 7 Let $S=\{1,2, \cdots, 98\}$. Find the smallest positive integer $n$, such that in any $n$-element subset of $S$, one can always select 10 numbers, and no matter how these 10 numbers are divided into two groups, there is always one group in which there is a number that is coprime with the other four numbers, a... | 50 | 113 | 2 |
math | 4. let $q(n)$ be the sum of the digits of the natural number $n$. Determine the value of
$$
q\left(q\left(q\left(2000^{2000}\right)\right)\right)
$$
## Solution | 4 | 57 | 1 |
math | 8. There are three identical red balls, three identical yellow balls and three identical green balls. In how many different ways can they be split into three groups of three balls each?
(1 mark)
現有三個一模一樣的紅球、三個一模一樣的黃球和三個一模一樣的綠球。有多少種不同的方法把球分成三組使得每組各有三個球？ | 10 | 82 | 2 |
math | ## Task 11/78
A mathematician writes to another: "This year I will be 100 years old, next year 200."
Obviously, the age references are not in the decimal system. How old is the mathematician? | 49_{(10)} | 55 | 7 |
math | ## Task B-4.3.
Determine $f^{2018}(2018)$ if for the function $f: \mathbb{R} \rightarrow \mathbb{R}$ it holds that
$$
(x-1) f(x)+f\left(\frac{1}{x}\right)=\frac{1}{x-1}
$$
Note: $f^{2018}(x)=(\underbrace{f \circ f \circ \ldots \circ f}_{2018})(x)$. | \frac{2017}{2018} | 117 | 13 |
math | 610. Decompose the number 100 into two addends so that their product is the largest possible. | 50,50 | 25 | 5 |
math | Define mutually externally tangent circles $\omega_1$, $\omega_2$, and $\omega_3$. Let $\omega_1$ and $\omega_2$ be tangent at $P$. The common external tangents of $\omega_1$ and $\omega_2$ meet at $Q$. Let $O$ be the center of $\omega_3$. If $QP = 420$ and $QO = 427$, find the radius of $\omega_3$.
[i]Proposed by Tan... | 77 | 123 | 2 |
math | 3.2.6 * If the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=5$, and $a_{n+1}=\frac{3 a_{n}-1}{-a_{n}+3}, n=1,2, \cdots$, find the general term $a_{n}$. | a_{n}=\frac{3+2^{n}}{3-2^{n}} | 74 | 20 |
math | 8.5. There are 100 boxes numbered from 1 to 100. One of the boxes contains a prize, and the host knows where it is. The audience can send the host a batch of notes with questions that require a "yes" or "no" answer. The host shuffles the notes in the batch and, without reading the questions aloud, honestly answers all ... | 99 | 105 | 2 |
math | Find all polynomials $P(x)$ with real coefficients that satisfy
$$
P(x \sqrt{2})=P\left(x+\sqrt{1-x^{2}}\right)
$$
for all real numbers $x$ with $|x| \leq 1$. | P(x)=f(U(x/\sqrt{2})) | 59 | 11 |
math | Let $f: N \to N$ satisfy $n=\sum_{d|n} f(d), \forall n \in N$. Then sum of all possible values of $f(100)$ is? | 40 | 47 | 2 |
math | 8.5. Given a convex quadrilateral $A B C D$, where $A B=A D=1, \angle A=80^{\circ}$, $\angle C=140^{\circ}$. Find the length of the diagonal $A C$. | 1 | 57 | 1 |
math | ## Aufgabe 3
a) $4000+88 ; \quad 570+380 ; \quad 2800+900$
b) $980-650 ; \quad 4600-800 ; \quad 10000-4500$
c) $60: 3 ; \quad 96: 4 ; \quad 720: 9$
d) $9 \cdot 88 ; \quad 300 \cdot 7 ; \quad 60 \cdot 8$
| 4088,950,3700,330,3800,5500,20,24,80,792,2100,480 | 136 | 49 |
math | 4. If $x, y, z \in \mathbf{R}_{+}$, satisfy $x y+y z+z x=1$, then the maximum value of the function $f(x, y, z)=\sqrt{x y+5}+\sqrt{y z+5}+\sqrt{z x+5}$ is $\qquad$ . | 4\sqrt{3} | 75 | 6 |
math | ## Subiectul IV.(20 puncte)
Determinaţi toate funcţiile $f: R \rightarrow(0, \infty)$, primitivabile, ce verifică relaţia: $F(x)+\ln (f(x))=\ln \left(1+\frac{x}{\sqrt{1+x^{2}}}\right)$, $\forall x \in R$, unde $F: R \rightarrow R$ este o primitivă a lui $f$ şi $F(0)=0$.
prof. Cristian Petru Pop,ISJ Cluj
Toate subiec... | f(x)=\frac{1}{\sqrt{x^{2}+1}} | 198 | 17 |
math | # Problem 6.
B-1
Find the minimum value of the expression $4 x+9 y+\frac{1}{x-4}+\frac{1}{y-5}$ given that $x>4$ and $y>5$. | 71 | 53 | 2 |
math | The remainder when $x^{100} -x^{99} +... -x +1$ is divided by $x^2 -1$ can be written in the form $ax +b$. Find $2a +b$.
[i]Proposed by Calvin Garces[/i] | -49 | 63 | 3 |
math | 27.8*. In a convex $n$-gon $(n \geqslant 4)$, all diagonals are drawn, and no three of them intersect at the same point. Find the number of intersection points of the diagonals. | \frac{n(n-1)(n-2)(n-3)}{24} | 52 | 19 |
math | Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and
\[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\]
for all natural $n \geq 2$.
[i]Peter Boyvalenkov[/i] | a_n = n | 131 | 4 |
math | $n$ students take a test with $m$ questions, where $m,n\ge 2$ are integers. The score given to every question is as such: for a certain question, if $x$ students fails to answer it correctly, then those who answer it correctly scores $x$ points, while those who answer it wrongly scores $0$. The score of a student is th... | m(n-1) | 128 | 6 |
math | 10. (20 points) Given $f(x)=\frac{2 x}{x+1}$. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=2$, and
$$
a_{n}=\frac{1}{2} f\left(a_{n-1}\right)\left(n \in \mathbf{N}_{+}, n \geqslant 2\right) \text {, }
$$
(1) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(2) Prove: For all positive integ... | a_{n}=\frac{2}{2 n-1}\left(n \in \mathbf{N}_{+}\right) | 193 | 28 |
math | 1. 200 people are standing in a circle. Each of them is either a liar or a conformist. Liars always lie. A conformist who stands next to two conformists always tells the truth. A conformist who stands next to at least one liar can either tell the truth or lie. 100 of those standing said: "I am a liar," and the other 10... | 150 | 123 | 3 |
math | 6. Two Skaters
Allie and Bllie are located
at points $A$ and $B$ on
a smooth ice surface, with $A$ and $B$
being 100 meters apart. If Allie starts from $A$
and skates at 8 meters/second
along a line that forms a $60^{\circ}$
angle with $A B$, while Bllie starts from $B$ at 7 meters/second
along the shortest line to me... | 160 | 122 | 3 |
math | 7.2. Harry, Ron, and Hermione wanted to buy identical waterproof cloaks. However, they lacked the money: Ron was short by a third of the cloak's price, Hermione by a quarter, and Harry by one fifth of the cloak's price. When the price of the cloak dropped by 9.4 sickles during a sale, the friends pooled their savings a... | 36 | 100 | 2 |
math | 3. Define the function on $\mathbf{R}$
$$
f(x)=\left\{\begin{array}{ll}
\log _{2}(1-x), & x \leqslant 0 ; \\
f(x-1)-f(x-2), & x>0 .
\end{array}\right.
$$
Then $f(2014)=$ | 1 | 82 | 1 |
math | 9. (i) (Grade 11) Given $\sin \theta+\cos \theta=\frac{\sqrt{2}}{5}$ $\left(\frac{\pi}{2}<\theta<\pi\right)$. Then $\tan \theta-\cot \theta=$ $\qquad$ .
(ii) (Grade 12) The function $f(x)=\frac{1}{\sin ^{2} x}+\frac{2}{\cos ^{2} x}$ $\left(0<x<\frac{\pi}{2}\right)$ has a minimum value of $\qquad$ . | 3 + 2 \sqrt{2} | 128 | 9 |
math | $1 \cdot 29$ Let $P$ be an odd prime. Consider the set $\{1,2, \cdots, 2 p\}$ and its subsets $A$ that satisfy the following two conditions:
(1) $|A|=p$;
(2) The sum of all elements in $A$ is divisible by $p$. Find the number of all such subsets $A$. | \frac{1}{p}(C_{2p}^{p}-2)+2 | 86 | 18 |
math | 1.1. If the 200th day of some year is Sunday and the 100th day of the following year is also Sunday, then what day of the week was the 300th day of the previous year? Enter the number of this day of the week (if Monday, then 1, if Tuesday, then 2, etc.). | 1 | 78 | 1 |
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