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 | Task B-3.1. Determine all four-digit natural numbers divisible by 45 for which the difference of the squares of the hundreds digit and the tens digit is 24. | 6750,3510,1755,7515 | 38 | 19 |
math | The class teacher calculated the class average grades for each subject, and Kati helped her by recalculating the grades based on how many students received each consecutive grade. When comparing the results of the first subject, it turned out that Kati had used the data for consecutive fives in reverse order, taking th... | 6 | 107 | 1 |
math | 298. An unknown polynomial gives a remainder of 2 when divided by $x-1$, and a remainder of 1 when divided by $x-2$. What remainder does this polynomial give when divided by $(x-1)(x-2)$? | -x+3 | 54 | 3 |
math | Find all real numbers$ x$ and $y$ such that $$x^2 + y^2 = 2$$
$$\frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2.$$ | (1, 1) | 51 | 6 |
math | 6. In the sequence $\left\{a_{n}\right\}$, if the adjacent terms $a_{n} 、 a_{n+1}$ are the two roots of the quadratic equation
$$
x_{n}^{2}+3 n x_{n}+c_{n}=0(n=1,2, \cdots)
$$
then when $a_{1}=2$, $c_{100}=$ $\qquad$ | 22496 | 98 | 5 |
math | 5. If the quadratic equation with positive integer coefficients
$$
4 x^{2}+m x+n=0
$$
has two distinct rational roots $p$ and $q$ ($p<q$), and the equations
$$
x^{2}-p x+2 q=0 \text { and } x^{2}-q x+2 p=0
$$
have a common root, find the other root of the equation
$$
x^{2}-p x+2 q=0
$$ | \frac{1}{2} | 109 | 7 |
math | Find all pairs $(k, n)$ of strictly positive integers such that $n$ and $k-1$ are coprime, and $n$ divides $k^{n}-1$. | (1,n) | 40 | 4 |
math | 2.1. How many terms will there be if we expand the expression $\left(4 x^{3}+x^{-3}+2\right)^{2016}$ and combine like terms? | 4033 | 44 | 4 |
math | 102. Find the sum of the squares of the distances from the points of tangency of the inscribed circle with the sides of a given triangle to the center of the circumscribed circle, if the radius of the inscribed circle is $r$, and the radius of the circumscribed circle is $R$. | 3R^{2}-4Rr-r^{2} | 66 | 12 |
math | 1. Find all ordered pairs $(a, b)$ of positive integers such that $a^{2}+b^{2}+25=15 a b$ and $a^{2}+a b+b^{2}$ is prime. | (1,2)(2,1) | 51 | 9 |
math | 11. (5 points) Arrange the natural numbers $1,2,3,4$, in ascending order without any gaps, to get: 1234567891011121314. In this sequence of digits, when the first set of 5 consecutive even digits appears, the position of the first (even) digit from the left is the how many-th digit? | 490 | 88 | 3 |
math | $\left[\frac{\text { Properties of sections }}{\text { Cube }}\right]$
Given a cube ABCDA1B1C1D1 with edge $a$. Find the area of the section of this cube by a plane passing through vertex $C$ and the midpoints of edges $C 1 B 1$ and $C 1 D 1$. | \frac{3}{8}^2 | 78 | 9 |
math | Solve the following equation:
$$
\sin ^{6} x+\cos ^{6} x=\frac{7}{16}
$$ | \\frac{\pi}{6}\k\frac{\pi}{2}=\(3k\1)\frac{\pi}{6}\quad(k=0,1,2,\ldots) | 31 | 40 |
math | 5. In an isosceles trapezoid $A B C D$ with lateral sides $A B$ and $C D$, the lengths of which are 10, perpendiculars $B H$ and $D K$ are drawn from vertices $B$ and $D$ to the diagonal $A C$. It is known that the bases of the perpendiculars lie on segment $A C$ and $A H: A K: A C=5: 14: 15$. Find the area of trapezoi... | 180 | 120 | 3 |
math | 5. let $a$ and $b$ be fixed positive numbers. Depending on $a$ and $b$, find the smallest possible value of the sum
$$
\frac{x^{2}}{(a y+b z)(a z+b y)}+\frac{y^{2}}{(a z+b x)(a x+b z)}+\frac{z^{2}}{(a x+b y)(a y+b x)}
$$
where $x, y, z$ are positive real numbers.
## Solution | \frac{3}{(+b)^{2}} | 105 | 11 |
math | A number is said to be TOP if it has 5 digits and the product of the $1^{\circ}$ and the $5^{\circ}$ is equal to the sum of the $2^{\circ}, 3^{\circ} \mathrm{and} 4^{\circ}$. For example, 12,338 is TOP, because it has 5 digits and $1 \cdot 8=2+3+3$.
a) What is the value of $a$ so that $23,4 a 8$ is TOP?
b) How many T... | 112 | 148 | 3 |
math | 3. Given $101 \mid a$, and
$$
a=10^{j}-10^{i}(0 \leqslant i<j \leqslant 99) \text {. }
$$
Then the number of $a$ that satisfy the condition is. $\qquad$ . | 1200 | 68 | 4 |
math | 1. Suppose natural numbers $a$ and $b$ are such that the numbers $a x + 2$ and $b x + 3$ are not coprime for any $x \in \mathbb{N}$. What can $a / b$ be equal to? | \frac{b}{}=\frac{3}{2} | 61 | 13 |
math | 6. The sequence of positive integers $\left\{a_{n}\right\}: a_{n}=3 n+2$ and $\left\{b_{n}\right\}: b_{n}=5 n+3, n \in \mathbf{N}$, have a certain number of common terms in $M=\{1,2, \cdots, 2018\}$. Find the number of these common terms. | 135 | 94 | 3 |
math | 9.5 The numbers $x$, $y$, and $z$ satisfy the equations
$$
x y + y z + z x = x y z, \quad x + y + z = 1
$$
What values can the sum $x^{3} + y^{3} + z^{3}$ take? | 1 | 69 | 1 |
math | We define the $x_{i}$ by $x_{1}=a$ and $x_{i+1}=2 x_{i}+1$. We define the sequence $y_{i}=2^{x_{i}}-1$. Determine the largest integer $k$ such that it is possible for $y_{1}, \ldots, y_{k}$ to all be prime.
Theorem 46. $\left(\frac{p}{q}\right)=-1 \frac{(p-1)(q-1)}{4}\left(\frac{q}{p}\right)$
Theorem 47. $\left(\frac... | k_{\max}=2 | 157 | 6 |
math | A triangle with perimeter $7$ has integer sidelengths. What is the maximum possible area of such a triangle? | \frac{3\sqrt{7}}{4} | 25 | 13 |
math | 4. 188 Find all three-digit numbers that satisfy the following condition: the quotient when divided by 11 equals the sum of the squares of its digits. | 550 \text{ and } 803 | 35 | 12 |
math | ## Task 1 - 030721
By what highest power of 2 is the product of four consecutive even natural numbers at least divisible? | 2^7 | 33 | 3 |
math | 2. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, the lengths of edges $A B$ and $A D$ are both 2, and the length of the body diagonal $A C_{1}$ is 3. Then the volume of the rectangular prism is $\qquad$ . | 4 | 76 | 1 |
math | The three medians of a triangle has lengths $3, 4, 5$. What is the length of the shortest side of this triangle? | \frac{10}{3} | 30 | 8 |
math | 6. Given that $a$, $b$, and $c$ are the lengths of the sides of a given triangle. If positive real numbers $x$, $y$, and $z$ satisfy $x+y+z=1$, find the maximum value of $a x y + b y z + c z x$. | \frac{}{2+2+2-^{2}-b^{2}-^{2}} | 65 | 20 |
math | 1. A four-digit number has the digits in the thousands and tens places as 7 and 2, respectively. Which digits should be placed in the hundreds and units places so that the number is divisible by 2 and 3, but not by their higher powers? | 7026,7122,7422,7626,7926 | 55 | 24 |
math | 3. (USA) Find the integer represented by $\left[\sum_{n=1}^{10^{9}} n^{-2 / 3}\right]$. Here $[x]$ denotes the greatest integer less than or equal to $x$ (e.g. $[\sqrt{2}]=1$ ).
The text is already in English, so no translation is needed. | 2997 | 79 | 4 |
math | 4. Let $n$ be a positive integer greater than 3, and $(n, 3)=1$. Find the value of $\prod_{k=1}^{m}\left(1+2 \cos \frac{2 a_{k} \pi}{n}\right)$, where $a_{1}, a_{2}, \cdots$, $a_{m}$ are all positive integers not exceeding $n$ and coprime with $n$. | 1 | 97 | 1 |
math | Let $\Gamma$ be the circumcircle of a triangle $ABC$ and let $E$ and $F$ be the intersections of the bisectors of $\angle ABC$ and $\angle ACB$ with $\Gamma$. If $EF$ is tangent to the incircle $\gamma$ of $\triangle ABC$, then find the value of $\angle BAC$. | 60^\circ | 73 | 4 |
math | Example 4.22. Find the general solution of the equation
$$
y^{\prime \prime}+6 y^{\prime}+9 y=14 e^{-3 x} .
$$ | (C_{1}+C_{2}x)e^{-3x}+7x^{2}e^{-3x} | 44 | 26 |
math | Problem 2. Calculate:
$$
\left(\frac{1+2}{3}+\frac{4+5}{6}+\frac{7+8}{9}+\ldots+\frac{2017+2018}{2019}\right)+\left(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{673}\right)
$$ | 1346 | 93 | 4 |
math | 18. In a children's park amusement area, there are three types of tickets: Ticket A costs 7 yuan, Ticket B costs 4 yuan, and Ticket C costs 2 yuan. One day, the amusement area sold a total of 85 tickets and earned 500 yuan, with Ticket A selling 31 more tickets than Ticket B. How many Ticket A were sold ( ) tickets | 56 | 84 | 2 |
math | Find the continuous functions $f: \mathbb{R}_{+}^{*} \rightarrow \mathbb{R}_{+}^{*}$ such that for all $x, y$
$$
f(x+y)(f(x)+f(y))=f(x) f(y)
$$ | f(x)=\frac{1}{\alphax} | 60 | 12 |
math | 4. Find all positive integer triples $(x, y, z)$, such that:
$$
x^{3}+y^{3}+z^{3}-3 x y z=2012 \text{. }
$$ | (169,167,167) \text{ and its permutations, } (671,671,670) \text{ and its permutations} | 49 | 41 |
math | 11. (2005 Balkan Mathematical Olympiad) Find the positive integer solutions of the equation $3^{x}=2^{x} y+1$.
untranslated text remains the same as requested. However, if you need any further assistance or a different format, please let me know! | (x,y)=(1,1),(2,2),(4,5) | 62 | 15 |
math | 1. In tetrahedron $ABCD$, $AD \perp$ plane $BCD$, $\angle ABD = \angle BDC = \theta < 45^{\circ}$. It is known that $E$ is a point on $BD$ such that $CE \perp BD$, and $BE = AD = 1$.
(1) Prove: $\angle BAC = \theta$;
(2) If the distance from point $D$ to plane $ABC$ is $\frac{4}{13}$, find the value of $\cos \theta$.
| \cos \theta = \frac{4}{5} | 127 | 12 |
math | A triangle $ABC$ $(a>b>c)$ is given. Its incenter $I$ and the touching points $K, N$ of the incircle with $BC$ and $AC$ respectively are marked. Construct a segment with length $a-c$ using only a ruler and drawing at most three lines. | a - c | 64 | 3 |
math | Find all positive integers $p$, $q$, $r$ such that $p$ and $q$ are prime numbers and $\frac{1}{p+1}+\frac{1}{q+1}-\frac{1}{(p+1)(q+1)} = \frac{1}{r}.$ | \{p, q\} = \{2, 3\} \text{ or } \{3, 2\} | 68 | 30 |
math | 2. A number divided by 20 has a quotient of 10 and a remainder of 10, this number is $\qquad$ | 210 | 31 | 3 |
math | \section*{Task 4 - 121224}
In a city, a network of at least two bus lines is to be established. This network must satisfy the following conditions:
(1) Each line has exactly three stops.
(2) Each line has exactly one stop in common with every other line.
(3) It is possible to reach any stop from any other stop with... | 7 | 108 | 1 |
math | Problem 1. A student started working on an exam between 9 and 10 o'clock and finished between 13 and 14 o'clock. Find the exact time spent by the applicant on completing the task, given that at the beginning and the end of the work, the hour and minute hands, having swapped places, occupied the same positions on the cl... | \frac{60}{13} | 90 | 9 |
math | 3. Person A and Person B go to a discount store to buy goods. It is known that both bought the same number of items, and the unit price of each item is only 8 yuan and 9 yuan. If the total amount spent by both on the goods is 172 yuan, then the number of items with a unit price of 9 yuan is $\qquad$ pieces.
Person A a... | 12 | 165 | 2 |
math | 4. Every day, from Monday to Friday, the old man went to the blue sea and cast his net into the water. Each day, the net caught no more fish than the previous day. In total, over the five days, the old man caught exactly 100 fish. What is the smallest total number of fish he could have caught on the three days - Monday... | 50 | 83 | 2 |
math | 4. Given an isosceles triangle with a vertex angle of $20^{\circ}$ and a base length of $a$, the length of the legs is $b$. Then the value of $\frac{a^{3}+b^{3}}{a b^{2}}$ is $\qquad$ | 3 | 66 | 1 |
math | Let $x, y$ be positive numbers and let $s$ denote the smallest of the numbers $x, y+\frac{1}{x}, \frac{1}{y}$. What is the largest possible value of $s$, and for which numbers $x, y$ does it occur? (H) | \sqrt{2} | 65 | 5 |
math | 10. The surface of a convex polyhedron is composed of 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex, one square, one octagon, and one hexagon meet. How many line segments connecting the vertices of the polyhedron lie inside the polyhedron, and not on the faces or edges of the polyhedron? | 840 | 83 | 3 |
math | ## Task B-3.3.
Determine all natural numbers $n$ for which $n^{3}-10 n^{2}+28 n-19$ is a prime number. | 2,3,6 | 42 | 5 |
math | Four. (50 points) Let $n$ be a positive integer, and let the planar point set be
$$
S=\{(x, y) \mid x, y \in\{0,1, \cdots, n\}, x+y \neq 0\} \text {. }
$$
Question: What is the minimum number of lines in the plane whose union can contain $S$, but not include the point $(0,0)$? | 2n | 99 | 2 |
math | On a spherical planet of radius $R$ the intensity of attraction is measured gravitational at a heigh $x$ from the surface, the same measurements is made at a depth equal to the height, achieving both situations the same result.
Find the value of $x$ that satisfies this characteristics of the gravitational field.
[i]Pr... | x = R \left( \dfrac{-1 + \sqrt{5}}{2} \right) | 77 | 24 |
math | 4. On the board, you can either write two ones, or erase two already written identical numbers $n$ and write the numbers $n+k$ and $n-k$, provided that $n-k \geqslant 0$. What is the minimum number of such operations required to obtain the number $2048?$ | 3071 | 68 | 4 |
math | 20. (5 points) Person A and Person B start from points $A$ and $B$ respectively at the same time. If they walk in the same direction, A catches up with B in 30 minutes; if they walk towards each other, they meet in 6 minutes. It is known that B walks 50 meters per minute, then the distance between $A$ and $B$ is $\qqua... | 750 | 92 | 3 |
math | Let $a, b$ be two known constants, and $a>b$. Points $P^{3}, Q$ are on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. If the line connecting point $A(-a, 0)$ and $Q$ is parallel to the line $O P$, and intersects the $y$-axis at point $R$, then $\frac{|A Q| \cdot |A R|}{|O P|^{2}}=$ $\qquad$ (where $O$ is the ... | 2 | 128 | 1 |
math | 129. Infinite Product. Calculate the following infinite product: $3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \ldots$. | \sqrt[4]{27} | 53 | 8 |
math | }
For which $n>3$ can a set of weights with masses $1,2,3, \ldots, n$ grams be divided into three equal-mass piles?
# | 8 | 39 | 1 |
math | In a circle of radius $R$, we draw the chord that, as the circle rotates around a diameter parallel to the chord, generates a cylindrical surface of maximum area. | AB=R\sqrt{2} | 34 | 7 |
math | Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$. | 165 | 35 | 3 |
math | 1. Simplify the fraction $\frac{\sqrt{-x}-\sqrt{-3 y}}{x+3 y+2 \sqrt{3 x y}}$.
Answer. $\frac{1}{\sqrt{-3 y}-\sqrt{-x}}$. | \frac{1}{\sqrt{-3y}-\sqrt{-x}} | 54 | 16 |
math | Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. | 37 | 73 | 2 |
math | ## Task 1 - 280521
In a restaurant consisting of a dining hall and a grill restaurant, there are exactly 135 seats available for guests in the dining hall. The number of seats in the grill restaurant is one third of the number of seats in the dining hall.
a) How many seats are available in the restaurant in total?
b... | 270 | 117 | 3 |
math |
Problem 8.1. Let three numbers $a, b$ and $c$ be chosen so that $\frac{a}{b}=\frac{b}{c}=\frac{c}{a}$.
a.) Prove that $a=b=c$.
b.) Find the sum $x+y$ if $\frac{x}{3 y}=\frac{y}{2 x-5 y}=\frac{6 x-15 y}{x}$ and the expression $-4 x^{2}+36 y-8$ has its maximum value.
| 2 | 118 | 1 |
math | Let $\triangle ABC$ be a triangle with $AB=3$ and $AC=5$. Select points $D, E,$ and $F$ on $\overline{BC}$ in that order such that $\overline{AD}\perp \overline{BC}$, $\angle BAE=\angle CAE$, and $\overline{BF}=\overline{CF}$. If $E$ is the midpoint of segment $\overline{DF}$, what is $BC^2$?
Let $T = TNYWR$, and let ... | 9602 | 181 | 4 |
math | Bogdanov I.i.
On the plane, the curves $y=\cos x$ and $x=100 \cos (100 y)$ were drawn, and all points of their intersection with positive coordinates were marked. Let $a$ be the sum of the abscissas, and $b$ be the sum of the ordinates of these points. Find $a / b$. | 100 | 84 | 3 |
math | 1. Find a polynomial $P(x)$ for which
$$
x \cdot P(x-1)=(x-1990) \cdot P(x)
$$ | Cx(x-1)(x-2)\ldots(x-1989) | 35 | 18 |
math | The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?
(G. Galperin) | 1990 | 44 | 4 |
math | Problem 3. In the park, there were lindens and maples. Maples among them were $60 \%$. In spring, lindens were planted in the park, after which maples became $20 \%$. In autumn, maples were planted, and maples became $60 \%$ again. By what factor did the number of trees in the park increase over the year?
[6 points] (... | 6 | 96 | 1 |
math | One, (This question is worth 40 points) Given $a^{2}+b^{2}+c^{2}=1$, find the maximum value of $\left(a^{2}-b c\right)\left(b^{2}-c a\right)\left(c^{2}-a b\right)$. | \frac{1}{8} | 67 | 7 |
math | A Math test starts at 12:35 and lasts for $4 \frac{5}{6}$ hours. At what time does the test end? | 17 | 33 | 2 |
math | ## Task B-4.2.
Let $z=x+2i$ and $w=3+yi$ be complex numbers, where $x, y \in \mathbb{R}$. Determine the smallest positive real number $x$ for which the fraction $\frac{z+w}{z-w}$ is an imaginary number. | \sqrt{5} | 69 | 5 |
math | 11.086. A sphere is inscribed in a cone, the axial section of which is an equilateral triangle. Find the volume of the cone if the volume of the sphere is $32 \pi / 3 \mathrm{~cm}^{3}$. | 24\pi\mathrm{}^{3} | 58 | 10 |
math | Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for any positive integers $m, n$ the number
$$(f(m))^2+ 2mf(n) + f(n^2)$$
is the square of an integer.
[i]Proposed by Fedir Yudin[/i] | f(n) = n | 74 | 6 |
math | 11. Let $\triangle A B C$ be inscribed in a circle $\odot O$ with radius $R$, and $A B=A C, A D$ be the altitude from $A$ to the base $B C$. Then the maximum value of $A D + B C$ is $\qquad$ . | R+\sqrt{5} R | 68 | 7 |
math | Let $k(a)$ denote the number of points $(x, y)$ in the plane coordinate system for which $1 \leqq x \leqq a$ and $1 \leqq y \leqq a$ are relatively prime integers. Determine the following sum:
$$
\sum_{i=1}^{100} k\left(\frac{100}{i}\right)
$$ | n^2 | 84 | 3 |
math | 14. (15 points) Four people, A, B, C, and D, go fishing and catch a total of 25 fish. Ranked by the number of fish caught from most to least, it is A, B, C, and D. It is also known that the number of fish A caught is the sum of the number of fish B and C caught, and the number of fish B caught is the sum of the number ... | 11,7,4,3 | 114 | 8 |
math | 8. If $n$ is a natural number less than 50, find all values of $n$ such that the values of the algebraic expressions $4n+5$ and $7n+6$ have a common divisor greater than 1. | 7,18,29,40 | 54 | 10 |
math | 16. Let $f(x)=\frac{x^{2010}}{x^{2010}+(1-x)^{2010}}$. Find the value of
$$
f\left(\frac{1}{2011}\right)+f\left(\frac{2}{2011}\right)+f\left(\frac{3}{2011}\right)+\ldots+f\left(\frac{2010}{2011}\right)
$$ | 1005 | 110 | 4 |
math | Example: 39 passengers enter 4 garages, with no garage being empty. How many ways are there to distribute the passengers? | 186480 | 28 | 6 |
math | Example 3 In a finite sequence of real numbers, the sum of any seven consecutive terms is negative, while the sum of any eleven consecutive terms is positive. How many terms can such a sequence have at most?
Translate the above text into English, please keep the original text's line breaks and format, and output the t... | 16 | 68 | 2 |
math | A5. Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_{1}0$ such that the $\frac{1}{2} n(n-1)$ differences $a_{j}-a_{i}$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^{1}, r^{2}, \ldots, r^{\frac{1}{2} n(n-1)}$. | n\in{2,3,4} | 113 | 10 |
math | 4. Given that $a$, $b$, and $c$ are positive integers, and $a<b<c$. If the product of any two of these numbers minus 1 is divisible by the third number, then $a^{2}+b^{2}+c^{2}=$ $\qquad$ | 38 | 64 | 2 |
math | 7. Let $\left\{a_{n}\right\}$ be a monotonically increasing sequence of positive integers, satisfying
$$
a_{n+2}=3 a_{n+1}-a_{n}, a_{6}=280 \text {. }
$$
Then $a_{7}=$ | 733 | 66 | 3 |
math | 12. Let $S=\{1,2,3, \cdots, 100\}$, find the smallest positive integer $n$, such that every $n$-element subset of $S$ contains 4 pairwise coprime numbers. | 75 | 55 | 2 |
math | 4. For any positive integer $n$, let $a_{n}$ be the smallest positive integer such that $n \mid a_{n}$!. If $\frac{a_{n}}{n}=\frac{2}{5}$, then $n=$ $\qquad$ . | 25 | 59 | 2 |
math | 10.8. Find all pairs of distinct real numbers \(x\) and \(y\) such that \(x^{100} - y^{100} = 2^{99}(x - y)\) and \(x^{200} - y^{200} = 2^{199}(x - y)\).
(I. Bogdanov) | (x,y)=(2,0)(x,y)=(0,2) | 81 | 14 |
math | ## 4. From Bordeaux to Saint-Jean-de-Luz
Two cyclists set out simultaneously from Bordeaux to Saint-Jean-de-Luz (the distance between these cities is approximately 195 km). One of the cyclists, whose average speed is 4 km/h faster than the second cyclist, arrives at the destination 1 hour earlier. What is the speed of... | 30 | 78 | 2 |
math | 8.5. In the list $1,2, \ldots, 2016$, two numbers $a<b$ were marked, dividing the sequence into 3 parts (some of these parts might not contain any numbers at all). After that, the list was shuffled in such a way that $a$ and $b$ remained in their places, and no other of the 2014 numbers remained in the same part where ... | 508536 | 112 | 6 |
math | $(NET 3)$ Let $x_1, x_2, x_3, x_4,$ and $x_5$ be positive integers satisfying
\[x_1 +x_2 +x_3 +x_4 +x_5 = 1000,\]
\[x_1 -x_2 +x_3 -x_4 +x_5 > 0,\]
\[x_1 +x_2 -x_3 +x_4 -x_5 > 0,\]
\[-x_1 +x_2 +x_3 -x_4 +x_5 > 0,\]
\[x_1 -x_2 +x_3 +x_4 -x_5 > 0,\]
\[-x_1 +x_2 -x_3 +x_4 +x_5 > 0\]
$(a)$ Find the maximum of $(x_1 + x_3)^{... | (a + c)^{b + d} = 499^{499} | 244 | 21 |
math | The circle $\omega$ is drawn through the vertices $A$ and $B$ of the triangle $A B C$. If $\omega$ intersects $A C$ at point $M$ and $B C$ at point $P$. The segment $M P$ contains the center of the circle inscribed in $A B C$. Given that $A B=c, B C=a$ and $C A=b$, find $M P$. | \frac{(+b)}{+b+} | 91 | 11 |
math | # 5. Solve the inequality
$$
\left(3 x+4-2 \sqrt{2 x^{2}+7 x+3}\right)\left(\left|x^{2}-4 x+2\right|-|x-2|\right) \leq 0
$$ | x\in(-\infty;-3]\cup[0;1]\cup{2}\cup[3;4] | 63 | 26 |
math | II. (50 points)
From an $n$-term non-constant rational arithmetic sequence, what is the maximum number of terms that can be selected to form a geometric sequence $(n \in \mathbf{N})$? | 1+[\log_{2}n] | 49 | 9 |
math | # Task 4. (12 points)
Find the smallest positive integer in which the product of the digits is 5120. | 25888 | 29 | 5 |
math | We define the sets $A_1,A_2,...,A_{160}$ such that $\left|A_{i} \right|=i$ for all $i=1,2,...,160$. With the elements of these sets we create new sets $M_1,M_2,...M_n$ by the following procedure: in the first step we choose some of the sets $A_1,A_2,...,A_{160}$ and we remove from each of them the same number of elemen... | 8 | 205 | 1 |
math | For any real number $x$, we let $\lfloor x \rfloor$ be the unique integer $n$ such that $n \leq x < n+1$. For example. $\lfloor 31.415 \rfloor = 31$. Compute \[2020^{2021} - \left\lfloor\frac{2020^{2021}}{2021} \right \rfloor (2021).\]
[i]2021 CCA Math Bonanza Team Round #3[/i] | 2020 | 127 | 4 |
math | 11.3. Find all solutions of the system of equations in real numbers:
$$
\left\{\begin{array}{l}
x^{5}=y^{3}+2 z \\
y^{5}=z^{3}+2 x \\
z^{5}=x^{3}+2 y
\end{array}\right.
$$ | (0,0,0),\(\sqrt{2},\sqrt{2},\sqrt{2}) | 73 | 23 |
math | 6. Given $0 \leqslant 6 x 、 3 y 、 2 z \leqslant 8$,
$$
\sqrt{12 x}+\sqrt{6 y}+\sqrt{4 z}=6 \text {. }
$$
Then the function
$$
f(x, y, z)=\frac{1}{1+x^{2}}+\frac{4}{4+y^{2}}+\frac{9}{9+z^{2}}
$$
has a maximum value of $\qquad$ . | \frac{27}{10} | 115 | 9 |
math | At the elementary school U Tří lip, where Lukáš also attends, they are holding a knowledge competition with pre-arranged tasks. Each correctly solved task is scored as many points as its order. Each unsolved or partially solved task is not scored at all. Lukáš correctly solved the first 12 tasks. If he had correctly so... | 71 | 104 | 2 |
math | 5. Let $x_{1}, x_{2}, \cdots, x_{51}$ be natural numbers, $x_{1}<x_{2}$ $<\cdots<x_{51}$, and $x_{1}+x_{2}+\cdots+x_{51}=1995$. When $x_{26}$ reaches its maximum value, the maximum value that $x_{51}$ can take is | 95 | 94 | 2 |
math | 33. [15] The polynomial $a x^{2}-b x+c$ has two distinct roots $p$ and $q$, with $a, b$, and $c$ positive integers and with $0<p, q<1$. Find the minimum possible value of $a$. | 5 | 61 | 1 |
math | ## Problem 5
Let $p(x)$ be the polynomial $(1-x)^{a}\left(1-x^{2}\right)^{b}\left(1-x^{3}\right)^{c} \ldots\left(1-x^{32}\right)^{k}$, where $a, b, \ldots, k$ are integers. When expanded in powers of $x$, the coefficient of $x^{1}$ is -2 and the coefficients of $x^{2}, x^{3}, \ldots, x^{32}$ are all zero. Find $\mathr... | 2^{27}-2^{11} | 128 | 10 |
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