task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | 1. Let $x$ be a positive integer, and $x<50$. Then the number of $x$ such that $x^{3}+11$ is divisible by 12 is $\qquad$. | 5 | 47 | 1 |
math | XXV - I - Task 1
During World War I, a battle took place near a certain castle. One of the shells destroyed a statue of a knight with a spear standing at the entrance to the castle. This happened on the last day of the month. The product of the day of the month, the month number, the length of the spear expressed in f... | 1714 | 119 | 4 |
math | 4. [4] An ant starts at the point $(1,0)$. Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point $(x, y)$ with $|x|+|y| \geq 2$. What is the probability that the ant ends at the point $(1,1)$ ? | \frac{7}{24} | 76 | 8 |
math | 3. On each of 25 sheets of paper, one of the numbers 1 or -1 is written. The number -1 is written 13 times. It is allowed to remove two sheets and replace them with a sheet on which the product of the removed numbers is written. The procedure is repeated 24 times. What number is written on the last sheet of paper? | -1 | 79 | 2 |
math | 10.1. Ivan was walking from Pakhomovo to Vorobyevo. At noon, when he had covered $4 / 9$ of the entire distance, Foma set off after him from Pakhomovo on a bicycle, and Erema set off towards him from Vorobyevo. Foma overtook Ivan at 1 PM, and met Erema at 1:30 PM. When will Ivan and Erema meet? | 14:30 | 93 | 5 |
math | 13 Given $\tan (\alpha+\beta)=-2, \tan (\alpha-\beta)=\frac{1}{2}$, find $\frac{\sin 2 \alpha}{\sin 2 \beta}=$ $\qquad$ . | \frac{3}{5} | 51 | 7 |
math | 10. (15 points) From a point $M$ on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, two tangents are drawn to the circle with the minor axis as its diameter, with points of tangency $A$ and $B$. The line $AB$ intersects the $x$-axis and $y$-axis at points $P$ and $Q$, respectively. Find the minimum value of $|PQ|$. | \frac{10}{3} | 106 | 8 |
math | Find all pairs $ (p, q)$ of prime numbers such that $ p$ divides $ 5^q \plus{} 1$ and $ q$ divides $ 5^p \plus{} 1$. | (2, 2), (2, 13), (3, 3), (3, 7), (13, 2), (7, 3) | 45 | 38 |
math | 6. Let $f(x)$ be a function defined on $\mathbf{R}$, for any $x \in \mathbf{R}$, we have
$$
f(x+3) \leqslant f(x)+3, f(x+2) \geqslant f(x)+2 .
$$
Let $g(x)=f(x)-x$. If $f(4)=2014$, then
$$
f(2014)=
$$
$\qquad$ | 4024 | 108 | 4 |
math | 2.2. Given a convex pentagon $A B C D E$, such that
$$
A B=A E=D C=B C+D E=1 \text { and } \angle A B C=D E A=90^{\circ} .
$$
What is the area of this pentagon? | 1 | 66 | 1 |
math | 3. The minimum value of the function $y=|\cos x|+|\cos 2 x|(x \in \mathbf{R})$ is $\qquad$ | \frac{\sqrt{2}}{2} | 37 | 10 |
math | Let $n \geqslant 2$ be an integer, and let $A_{n}$ be the set
$$ A_{n}=\left\{2^{n}-2^{k} \mid k \in \mathbb{Z}, 0 \leqslant k<n\right\} . $$
Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_{n}$. (Serbia) Answer. $(n-2) 2^{n}+1$. | (n-2) 2^{n}+1 | 119 | 11 |
math | 1. It is known that for three consecutive natural values of the argument, the quadratic function $f(x)$ takes the values 13, 13, and 35, respectively. Find the smallest possible value of $f(x)$. | \frac{41}{4} | 51 | 8 |
math | Example 1 Given a cyclic quadrilateral $ABCD$ with side lengths $AB=2, BC=6, CD=DA=4$, find the area of quadrilateral $ABCD$. | 8\sqrt{3} | 40 | 6 |
math | 6. The bank issued a loan to citizen $N$ on September 9 in the amount of 200 mln rubles. The repayment date is November 22 of the same year. The interest rate on the loan is $25 \%$ per annum. Determine the amount (in thousands of rubles) that citizen N will have to return to the bank. Assume that there are 365 days in... | 210 | 121 | 3 |
math | ## Task A-1.4.
Determine all pairs of non-negative integers $(k, m)$ for which
$$
3 m^{3}-m+21=3^{3 k+1}-2 \cdot 3^{2 k+2}+3^{k+3}+3^{k+2}
$$ | (k,)=(0,0) | 69 | 7 |
math |
2. Find the set of all real values of $a$ for which the real polynomial equation $P(x)=x^{2}-2 a x+b=0$ has real roots given that $P(0) \cdot P(1) \cdot P(2) \neq 0$ and $P(0), P(1), P(2)$ form a geometric progression.
| [\frac{2-\sqrt{2}}{2},\frac{2+\sqrt{2}}{2}] | 82 | 24 |
math | 5. Let $a^{b}=\frac{1}{8}$. What is the value of $a^{-3 b}$ ? | 512 | 28 | 3 |
math | Exercise 1. Determine the maximum value of $\sqrt{x}+\sqrt{2 y+2}+\sqrt{3 z+6}$ when $x, y, z$ are strictly positive real numbers satisfying $x+y+z=3$. | 6 | 50 | 1 |
math | 4. Place the natural numbers $1,2,3,4, \cdots, 2 n$ in any order on a circle. It is found that there are $a$ groups of three consecutive numbers that are all odd, $b$ groups where exactly two are odd, $c$ groups where exactly one is odd, and $d$ groups where none are odd. Then $\frac{b-c}{a-d}=$ $\qquad$ . | -3 | 95 | 2 |
math | Example 8 Find the maximum value of the function
$$
f(x)=\sqrt{-x^{2}+10 x-9}+\sqrt{-x^{2}+68 x-256}
$$ | 3\sqrt{35} | 47 | 7 |
math | The sixth question: Given real numbers $x_{1}, x_{2}, \ldots, x_{2021}$ satisfy: $\sum_{i=1}^{2021} x_{i}^{2}=1$, try to find the maximum value of $\sum_{i=1}^{2020} x_{i}^{3} x_{i+1}^{3}$. | \frac{1}{8} | 88 | 7 |
math | Let $n \in \mathbb{N}$. We define
$$
S=\left\{(x, y, z) \in\{0, \ldots, n\}^{3} \mid x+y+z>0\right\}
$$
as a set of $(n+1)^{3}-1$ points in three-dimensional space. Determine the minimum number of planes whose union contains $S$ but not the point $(0,0,0)$.
## Hints:
$\triangleright$ Start by finding a set of plane... | 3n | 323 | 2 |
math | 2. Given $x \in \mathbf{R}, f(x)=x^{2}+12 x+30$. Then the solution set of the equation $f(f(f(f(f(x)))))=0$ is $\qquad$ . | {-6-\sqrt[32]{6},-6+\sqrt[32]{6}} | 53 | 20 |
math | Problem 3. Determine the functions $f:(0, \infty) \rightarrow \mathbb{R}$ with the property
$$
\ln (x y) \leq f(x)+f(y)-x-y \leq f(x y)-x y, \quad(\forall) x, y \in(0, \infty)
$$ | f(x)=\lnx+x | 75 | 7 |
math | Problem 1. We say that a quadruple of nonnegative real numbers $(a, b, c, d)$ is balanced if
$$
a+b+c+d=a^{2}+b^{2}+c^{2}+d^{2}
$$
Find all positive real numbers $x$ such that
$$
(x-a)(x-b)(x-c)(x-d) \geqslant 0
$$
for every balanced quadruple $(a, b, c, d)$.
(Ivan Novak) | x\geqslant\frac{3}{2} | 110 | 13 |
math | 6.1. Sixty students went on a trip to the zoo. Upon returning to school, it turned out that 55 of them had forgotten their gloves, 52 - their scarves, and 50 had managed to forget their hats. Find the smallest number of the most absent-minded students - those who lost all three items. | 37 | 71 | 2 |
math | During a school dance performance, a total of 430 different pairs danced. The first girl danced with 12 different boys, the second girl with 13 different boys, and so on, until the last girl danced with every boy. How many girls and how many boys attended the performance? | 20 | 62 | 2 |
math | 1. A Pythagorean triangle is a right-angled triangle where all three sides are integers. The most famous example is the triangle with legs 3 and 4 and hypotenuse 5.
Determine all Pythagorean triangles for which the area is equal to twice the perimeter. | (9,40,41),(10,24,26),(12,16,20) | 59 | 27 |
math | 6. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1}=a_{n}+\frac{1}{2 a_{n}}$, then $\lim _{n \rightarrow \infty}\left(a_{n}-\sqrt{n}\right)=$ | 0 | 69 | 1 |
math | 9. There are four teacups with their mouths facing up. Now, each time three of them are flipped, and the flipped teacups are allowed to be flipped again. After $n$ flips, all the cup mouths are facing down. Then the minimum value of the positive integer $n$ is $\qquad$ . | 4 | 68 | 1 |
math | 18. [10] Over all real numbers $x$ and $y$ such that
$$
x^{3}=3 x+y \quad \text { and } \quad y^{3}=3 y+x
$$
compute the sum of all possible values of $x^{2}+y^{2}$. | 15 | 68 | 2 |
math | 65. Motorcycle with a sidecar. Atkins, Baldwin, and Clark decided to go on a trip. Their journey will be 52 km. Atkins has a motorcycle with a one-person sidecar. He needs to take one of his companions for some distance, drop him off to walk the rest of the way, return, pick up the other companion who started walking a... | 5 | 184 | 1 |
math | Three, let $a>1$ be a positive real number, and $n \geqslant 2$ be a natural number, and the equation $[a x]=x$ has exactly $n$ different solutions. Try to find the range of values for $a$. ( $[x]$ denotes the greatest integer less than or equal to $x$) | 1+\frac{1}{n} \leqslant a<1+\frac{1}{n-1} | 77 | 25 |
math | The integers 390 and 9450 have three common positive divisors that are prime numbers. What is the sum of these prime numbers? | 10 | 32 | 2 |
math | 18. (6 points) A cube with an edge length of 6 is cut into several identical smaller cubes with integer edge lengths. If the total surface area of these smaller cubes is twice the surface area of the original large cube, then the edge length of the smaller cubes is . $\qquad$ | 3 | 62 | 1 |
math | 7.1 In the example of addition and subtraction, the student replaced the digits with letters according to the rule: identical letters are replaced by identical digits, different letters are replaced by different digits. From how many different examples could the record $0<\overline{\overline{Б A}}+\overline{\text { БА ... | 31 | 86 | 2 |
math | 38. Let $x, y, z \in(0,1)$, and satisfy $\sqrt{\frac{1-x}{y z}}+\sqrt{\frac{1-y}{z x}}+\sqrt{\frac{1-z}{x y}}=2$, find the maximum value of $x y z$. (2008 China Western Mathematical Olympiad) | \frac{27}{64} | 77 | 9 |
math | 3A. Find the smallest and largest value of the expression $\left|z-\frac{1}{z}\right|$, if $z$ is a complex number such that $|z|=2$. | \frac{3}{2}\leq|z-\frac{1}{z}|\leq\frac{5}{2} | 42 | 28 |
math | 3. Use three regular polygon tiles with equal side lengths to pave the ground, their vertices fit together perfectly to completely cover the ground. Let the number of sides of the regular polygons be $x$, $y$, and $z$. Then the value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is $\qquad$ . | \frac{1}{2} | 78 | 7 |
math | 3. let $n$ be a positive integer with at least four different positive divisors. Let the four smallest of these divisors be $d_{1}, d_{2}, d_{3}, d_{4}$. Find all such numbers $n$ for which
$$
d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}=n
$$
## Solution | 130 | 95 | 3 |
math | $2.344 A=\sqrt{\frac{1}{6}\left((3 x+\sqrt{6 x-1})^{-1}+(3 x-\sqrt{6 x-1})^{-1}\right)} \cdot|x-1| \cdot x^{-1 / 2}$. | {\begin{pmatrix}\frac{x-1}{3x-1},& | 63 | 17 |
math | 7. (10 points) A toy store sells a type of building blocks: each starship costs 8 yuan, and each mech costs 26 yuan; one starship and one mech can be combined to form an ultimate mech, which is sold for 33 yuan per set. If the store owner sold a total of 31 starships and mechs in a week, and the total revenue was 370 y... | 20 | 100 | 2 |
math | Find all natural numbers $n$ such that when we multiply all divisors of $n$, we will obtain $10^9$. Prove that your number(s) $n$ works and that there are no other such numbers.
([i]Note[/i]: A natural number $n$ is a positive integer; i.e., $n$ is among the counting numbers $1, 2, 3, \dots$. A [i]divisor[/i] of $n$ is... | 100 | 217 | 3 |
math | 10. For the curve $C: x^{4}+y^{2}=1$, consider the following statements:
(1) The curve $C$ is symmetric with respect to the origin;
(2) The curve $C$ is symmetric with respect to the line $y=x$;
(3) The area enclosed by the curve $C$ is less than $\pi$;
(4) The area enclosed by the curve $C$ is greater than $\pi$.
The... | (1), (4) | 117 | 6 |
math | Let $F:(1,\infty) \rightarrow \mathbb{R}$ be the function defined by
$$F(x)=\int_{x}^{x^{2}} \frac{dt}{\ln(t)}.$$
Show that $F$ is injective and find the set of values of $F$. | (\ln(2), \infty) | 66 | 10 |
math | From point $A$, we launch a heavy object vertically upwards with an initial velocity of $v_{0}$. At the same time, from point $B$, which is $a$ meters above $A$, we drop another heavy object with no initial velocity. After how many seconds will the two objects meet? How much distance will each have traveled? Determine ... | =14\mathrm{~},s_{2}=960.4\mathrm{~} | 170 | 22 |
math | 11. (6 points) Xiaopang bought a coat, a pair of pants, and a pair of leather shoes for 700 yuan. Xiao Ya asked him the price of each item, and Xiaopang told him: The coat is 340 yuan more expensive than the pants, the coat is 180 yuan more expensive than the total price of the shoes and pants, the price of the pants i... | 100 | 95 | 3 |
math | ## Task B-3.1.
Solve the inequality
$$
8 \sin x \cos x \cos 2 x>1
$$ | x\in\langle\frac{\pi}{24}+\frac{k\pi}{2},\frac{5\pi}{24}+\frac{k\pi}{2}\rangle,k\in\mathbb{Z} | 31 | 49 |
math | B2. Which two consecutive terms of a geometric sequence with a common ratio of 3 must be multiplied to get 243 times the square of the first term of this sequence? | 3 | 38 | 1 |
math | 1. Denis housed chameleons that can change color only to two colors: red and brown. Initially, the number of red chameleons was five times the number of brown chameleons. After two brown chameleons turned red, the number of red chameleons became eight times the number of brown chameleons. Find out how many chameleons D... | 36 | 84 | 2 |
math | 4. Consider the segment $[A B]$. We denote $M$ as the midpoint of $[A B]$, $M_{1}$ as the midpoint of $[B M]$, $M_{2}$ as the midpoint of $\left[M_{1} A\right]$, $M_{3}$ as the midpoint of $\left[M_{2} B\right]$, and $M_{4}$ as the midpoint of $\left[A M_{3}\right]$. It is known that $\mathrm{M}_{1} \mathrm{M}_{3}=\mat... | 64 | 280 | 2 |
math | [ Measuring the lengths of segments and the measures of angles. Adjacent angles.]
Points $A, B$ and $C$ are located on the same line, and $A C: B C=m: n$ ( $m$ and $n-$ natural numbers). Find the ratios $A C: A B$ and $B C: A B$. | AC:AB=:(+n),BC:AB=n:(+n)orAC:AB=:(-n),BC:AB=n:(-n)if>n;both1/2if=n | 74 | 42 |
math | Example 5 Given real numbers $x, y$ satisfy $x+y=3, \frac{1}{x+y^{2}}+\frac{1}{x^{2}+y}=\frac{1}{2}$.
Find the value of $x^{5}+y^{5}$. [3]
(2017, National Junior High School Mathematics League) | 123 | 79 | 3 |
math | 6. The maximum value of the function $y=\sin x+\sqrt{3} \cos x-2 \sin 3 x$ is $\qquad$ | \frac{16\sqrt{3}}{9} | 34 | 13 |
math | Example 1. Find the coordinates of the intersection point of the two tangent lines to the ellipse at the points of intersection with the line $x+4 y-2=0$ and the ellipse $\frac{x^{2}}{2}$ $+\mathrm{y}^{2}=1$. | (1,2) | 60 | 5 |
math | For any positive integer $n$ denote $S(n)$ the digital sum of $n$ when represented in the decimal system. Find every positive integer $M$ for which $S(Mk)=S(M)$ holds for all integers $1\le k\le M$. | M = 10^l - 1 | 56 | 11 |
math | ## Task Condition
Find the derivative.
$y=\frac{\sqrt{1-x^{2}}}{x}+\arcsin x$ | -\frac{\sqrt{1-x^{2}}}{x^{2}} | 29 | 15 |
math | 5. For what values of $a$ does the system of equations $\left\{\begin{array}{c}(x-2-\sqrt{5} \cos a)^{2}+(y+1-\sqrt{5} \sin a)^{2}=\frac{5}{4} \text { have two solutions? } \\ (x-2)(x-y-3)=0\end{array}\right.$ | \in(\frac{\pi}{12}+\pik;\frac{\pi}{3}+\pik)\cup(\frac{5\pi}{12}+\pik;\frac{2\pi}{3}+\pik),k\inZ | 89 | 55 |
math | 18. There is a sequence, the first number is 6, the second number is 3, starting from the second number, each number is 5 less than the sum of the number before it and the number after it. What is the sum of the first 200 numbers in this sequence, from the first number to the 200th number? | 999 | 77 | 3 |
math | Determine the number of $8$-tuples $(\epsilon_1, \epsilon_2,...,\epsilon_8)$ such that $\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}$ and $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8$ is a multiple of $3$. | 88 | 87 | 2 |
math | 9.203. $5^{\log _{5}^{2} x}+x^{\log _{5} x}<10$. | x\in(\frac{1}{5};5) | 34 | 12 |
math | 12th Swedish 1972 Problem 3 A steak temperature 5 o is put into an oven. After 15 minutes, it has temperature 45 o . After another 15 minutes it has temperature 77 o . The oven is at a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the ov... | 205 | 85 | 3 |
math | Let $n$ be an given positive integer, the set $S=\{1,2,\cdots,n\}$.For any nonempty set $A$ and $B$, find the minimum of $|A\Delta S|+|B\Delta S|+|C\Delta S|,$ where $C=\{a+b|a\in A,b\in B\}, X\Delta Y=X\cup Y-X\cap Y.$ | n + 1 | 95 | 5 |
math | 15. Given $3 \vec{a}-2 \vec{b}=\{-2,0,4\}, \vec{c}=\{-2,1,2\}, \vec{a} \cdot \vec{c}=2,|\vec{b}|=4$. Find the angle $\theta$ between $\vec{b}$ and $\vec{c}$. | \theta=\pi-\arccos\frac{1}{4} | 81 | 15 |
math | Find all triplets $(k, m, n)$ of positive integers having the following property: Square with side length $m$ can be divided into several rectangles of size $1\times k$ and a square with side length $n$. | (m - n)(m + n) \equiv 0 \pmod{k} | 49 | 19 |
math | What is the sum of the $x$-intercept of the line with equation $20 x+16 y-40=0$ and the $y$-intercept of the line with equation $20 x+16 y-64=0$ ? | 6 | 59 | 1 |
math | 15. (23rd All-Soviet Union Mathematical Olympiad, 1989) Given that $x, y, z$ are positive numbers, and satisfy the equation $x y z(x+y+z)=1$, find the minimum value of the expression $(x+y)(y+z)$. | 2 | 64 | 1 |
math | (11) The function $y=\cos 2x + 2 \sin x, x \in (0, 2\pi)$ is monotonically decreasing in the interval
$\qquad$. | (\frac{\pi}{6},\frac{\pi}{2}),(\frac{5\pi}{6},\frac{3\pi}{2}) | 44 | 32 |
math | [b]Q12.[/b] Find all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respectively. | 6 | 40 | 1 |
math | Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression | 144 | 37 | 3 |
math | 6. Vitya found the smallest possible natural number, which when multiplied by 2 gives a perfect square, and when multiplied by 3 gives a perfect cube. What number did Vitya find? | 72 | 42 | 2 |
math | 14. (This problem was proposed by academician A. N. Kolmogorov at one of the summer camp olympiads of the Little Academy of Sciences of Crimea "Iscatel'".) Find all integers \(a\) for which the fraction
\[
\frac{a^{3}+1}{a-1}
\]
takes integer values. | -1,0,2,3 | 78 | 8 |
math | Question 240, Set $S$ satisfies the following conditions: (1) The elements in $S$ are all positive integers not exceeding 100; (2) For any $a, b \in S, a \neq b$, there exists a $c \in S$ different from $a, b$, such that $(a+b, c)=1$; (3) For any $a, b \in S, a \neq b$, there exists a $c \in S$ different from $a, b$, s... | 50 | 164 | 2 |
math | Start with a three-digit positive integer $A$. Obtain $B$ by interchanging the two leftmost digits of $A$. Obtain $C$ by doubling $B$. Obtain $D$ by subtracting $500$ from $C$. Given that $A + B + C + D = 2014$, fi nd $A$. | 344 | 74 | 3 |
math | 6. Solve the equation $\sqrt{2 x^{2}+3 x-5}-\sqrt{2 x^{2}+3 x-8}=1$. | \frac{3}{2},-3 | 35 | 9 |
math | 13. There are 9 people lost in the mountains, and the food they have is only enough to last for 5 days. One day later, these 9 people encounter another group of lost people who have no food at all. After calculation, if the two groups share the food, with each person eating the same amount of food per day, the food wil... | 3 | 96 | 1 |
math | 5. At the Sea Meetings, a competition in games by the sea and in the pool, 8 people from Dubrovnik, 7 from Zadar, 2 from Hvar, and 3 from Split participate. They need to form a five-member team in which there will be at least one competitor from each of the four cities. In how many different ways can the team be formed... | 2688 | 90 | 4 |
math | If the GCD of $a$ and $b$ is $12$ and the LCM of $a$ and $b$ is $168$, what is the value of $a\times b$?
[i]2016 CCA Math Bonanza L1.3[/i] | 2016 | 65 | 4 |
math | Example 1. If $x=\sqrt{19-8 \sqrt{3}}$, then the fraction $\frac{x^{4}-6 x^{3}-2 x^{2}+18 x+23}{x^{2}-8 x+15}=$ $\qquad$ | 5 | 62 | 1 |
math | Question 104, Given complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}$ satisfy the following conditions: $\left|\mathrm{z}_{1}\right|=2,\left|\mathrm{z}_{2}\right|=3,3 \mathrm{z}_{1}-2 \mathrm{z}_{2}=2-\mathrm{i}$, then $\mathrm{z}_{1} \mathrm{z}_{2}=$ $\qquad$ - | -\frac{18}{5}+\frac{24}{5}i | 96 | 17 |
math | 4.5.15 Find the smallest positive integer $k$, such that for all $a$ satisfying $0 \leqslant a \leqslant 1$ and all positive integers $n$, the inequality holds: $a^{k}(1-a)^{n}<\frac{1}{(n+1)^{3}}$. | 4 | 74 | 1 |
math | Question 175, Given point $E(m, n)$ is a fixed point inside the parabola $y^{2}=2 p x(p>0)$, two lines with slopes $k_{1}$ and $k_{2}$ are drawn through $E$, intersecting the parabola at points $A, B$ and $C, D$ respectively, and $M, N$ are the midpoints of segments $A B$ and $C D$ respectively.
(1) When $\mathrm{n}=0$... | p^2 | 194 | 3 |
math | Example 2.3.1 There are three $a$'s, four $b$'s, and two $c$'s. Using these nine letters to form a permutation, if it is required that the same letters cannot all be adjacent in the permutation, how many such permutations are there? | 871 | 63 | 3 |
math | A circle $\omega$ and a point $P$ not lying on it are given. Let $ABC$ be an arbitrary equilateral triangle inscribed into $\omega$ and $A', B', C'$ be the projections of $P$ to $BC, CA, AB$. Find the locus of centroids of triangles $A' B'C'$. | \text{The midpoint of } OP | 71 | 8 |
math | Determine all integers $n \geq 1$ for which there exists a pair $(a, b)$ of positive integers with the following properties:
i) No third power of a prime divides $a^{2}+b+3$.
ii) It holds that $\frac{a b+3 b+8}{a^{2}+b+3}=n$.
Answer: The only integer with these properties is $n=2$. | 2 | 93 | 1 |
math | 4.004. Find the first three terms $a_{1}, a_{2}, a_{3}$ of an arithmetic progression, given that $a_{1}+a_{3}+a_{5}=-12$ and $a_{1} a_{3} a_{5}=80$. | 2,-1,-4;-10,-7,-4 | 67 | 12 |
math | 13. (3b, 8-11) A coin is tossed 10 times. Find the probability that two heads never appear consecutively.
# | \frac{9}{64} | 35 | 8 |
math | The following is known about the reals $ \alpha$ and $ \beta$
$ \alpha^{3}-3\alpha^{2}+5\alpha-17=0$ and $ \beta^{3}-3\beta^{2}+5\beta+11=0$
Determine $ \alpha+\beta$ | 2 | 72 | 1 |
math | 32. Determine the odd prime number $p$ such that the sum of digits of the number $p^{4}-5 p^{2}+13$ is the smallest possible. | 5 | 39 | 1 |
math | 4- 196 On a horizontal plane, three points are 100 meters, 200 meters, and 300 meters away from the base of an antenna. The sum of the angles of elevation from these three points to the antenna is $90^{\circ}$. What is the height of the antenna? | 100 | 71 | 3 |
math | ## Subject I
a) If $m, n \in N^{*}$, find the smallest number of the form $\left|5^{2 m}-3^{n}\right|$.
b) Compare the numbers: $\frac{\sqrt{2014}^{2014}+\sqrt{2015}^{2015}}{\sqrt{2014}^{2015}+\sqrt{2015}^{2014}}$ and $\frac{\sqrt{2014}}{\sqrt{2015}}$. | 2 | 125 | 1 |
math | 16. We consider all possible broken lines that follow the sides of the cells and connect the two opposite corners of a square sheet of grid paper measuring $100 \times 100$ by the shortest path. What is the smallest number of such broken lines needed so that their union contains all the vertices of the cells? | 101 | 68 | 3 |
math | 7. List all two-element subsets of a five-element set. | 10 | 13 | 2 |
math | 15. Given that $a, b, c$ are distinct integers, the minimum value of $4\left(a^{2}+b^{2}+c^{2}\right)-(a+b+c)^{2}$ is | 8 | 48 | 1 |
math | Call a positive integer $N$ a $\textit{7-10 double}$ if the digits of the base-7 representation of $N$ form a base-10 number that is twice $N.$ For example, 51 is a 7-10 double because its base-7 representation is 102. What is the largest 7-10 double? | 315 | 83 | 3 |
math | Problem 2. a) Determine $(x ; y) \in \mathbb{N} \times \mathbb{N}$ such that $\sqrt{2^{x}}+\sqrt{2^{y}}=33$.
b) Given the number $x=\sqrt{\underbrace{44 . .44}_{\text {2n digits }}-\underbrace{88 \ldots 88}_{\text {n digits }}}$ where $n \in \mathbb{N}^{*}$.
Show that $x$ is a natural number. | (10;0)(0;10) | 121 | 11 |
math | # 9. Solution.
1st method. An elementary outcome in the random experiment is a triplet of positions where the children in green caps stand. Consider the event $A$ "all three green caps are together." This event is favorable in 9 elementary outcomes. The event $B$ "two green caps are together, and the third is separate... | \frac{5}{14} | 462 | 8 |
math | Find the unique 3 digit number $N=\underline{A}$ $\underline{B}$ $\underline{C},$ whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}$ $\underline{B}$ $\underline{C}$ $\times$ $\underline{A}$ $\underline{B}$ $\times$ $\underline{A}$ is divisible by $1000$.
[i]Proposed by Kyle ... | 875 | 102 | 3 |
math | 3. Given $x \in \mathbf{C}, \arg \left(x^{2}-2\right)=\frac{3}{4} \pi, \arg \left(x^{2}\right.$ $+2 \sqrt{3})=\frac{\pi}{6}$. Then the value of $x$ is $\qquad$ . | \pm(1+i) | 74 | 6 |
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