problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 2 43 | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 6
values | synthetic bool 1
class |
|---|---|---|---|---|---|---|---|---|
# Task 1. (2 points)
Find the smallest three-digit number ABC that is divisible by the numbers AB and BC (the digit A cannot be 0, and the digit B can be; different letters do not necessarily represent different digits) | Answer: 110
Solution:
The number 110 clearly fits. A smaller answer cannot be, as all three-digit numbers less than 110 start with 10. At the same time, none of them are divisible by 10, except for 100, and 100 is not divisible by 0. | 110 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
# Problem 5. (3 points)
In trapezoid $ABCD$, a point $X$ is taken on the base $BC$ such that segments $XA$ and $XD$ divide the trapezoid into three triangles that are similar to each other but pairwise unequal and non-isosceles. The side $AB$ has a length of 5. Find $XC \cdot BX$.
Answer: 25 | Solution:
The triangles are scalene, meaning all their angles are different.
$\alpha=\angle B X A=\angle X A D \neq \angle A X D=\beta$. The angle $\angle C X D$ together with these angles sums up to $180^{\circ}$, so it is the third angle of the triangle and cannot be equal to either $\alpha$ or $\beta$.
If $\angle... | 25 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Find the largest three-digit number that gives a remainder of 2 when divided by 13, and a remainder of 6 when divided by 15. | Answer: 951
2nd option. Find the largest three-digit number that gives a remainder of 3 when divided by 13, and a remainder of 2 when divided by 14.
Answer: 926
3rd option. Find the largest three-digit number that gives a remainder of 5 when divided by 14, and a remainder of 3 when divided by 15.
Answer: 873
Only ... | 951 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
# Problem 3. (3 points)
All residents of the island are either blondes or brunettes with green or blue eyes. The proportion of brunettes among blue-eyed residents is $65 \%$. The proportion of blue-eyed residents among blondes is $70 \%$. Finally, the proportion of blondes among green-eyed residents is 10\%. What perc... | Answer: 54
## Solution:
Let the number of blue-eyed brunettes be $a$, blue-eyed blondes be $b$, green-eyed blondes be $c$, and green-eyed brunettes be $d$.
Then $\frac{a}{a+b}=0.65$, from which $\frac{a+b}{a}=\frac{20}{13}$ and $\frac{b}{a}=\frac{20}{13}-1=\frac{7}{13}$.
Similarly, $\frac{b}{b+c}=0.7$, from which $... | 54 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
# Problem 5. (3 points)
In the cells of a $7 \times 7$ table, pairwise distinct non-negative integers are written. It turns out that for any two numbers in the same row or column, the integer parts of their quotients when divided by 8 are different. What is the smallest value that the largest number in the table can t... | # Answer: 54
Solution: Each row must contain seven numbers that give different incomplete quotients when divided by 8. This means that in each row, there is at least one number with an incomplete quotient of 6 or more when divided by 8 (6, not 7, since the minimum value of the incomplete quotient is 0). Therefore, the... | 54 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
# Problem 8. (4 points)
In the Magic Country, there are 100 cities, some of which are connected by two-way air routes. Any two cities can be reached from each other with no more than 11 transfers, and there is a unique way to do so. If it is impossible to travel from city $A$ to city $B$ with 10 or fewer transfers, bo... | Answer: 89
## Solution:
Consider two peripheral cities and the route between them. All 11 cities where transfers are made on this route cannot be peripheral. All other cities can be peripheral if there are direct flights from them to the first or last of these 11 intermediate cities.
Thus, the maximum number of peri... | 89 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. Given a cube and 12 colors. Find the number of ways to paint the faces of this cube using these colors (each face in one color) such that adjacent faces are of different colors. Colorings that differ by a rotation are considered different. | Answer: 987360
## Examples of writing the answer:
## 7th grade
# | 987360 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. In a four-digit number, the first digit was crossed out. The resulting number is 9 times smaller than the original. What is the largest value the original number could have had? | Answer: 7875
## Examples of how to write the answer:
1234 | 7875 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. On a plane, a certain number of triangles are drawn, the lengths of whose sides are ten-digit natural numbers, containing only threes and eights in their decimal representation. No segment belongs to two triangles, and the sides of all triangles are distinct. What is the maximum number of triangles that can be drawn... | Answer: 341
## Examples of how to write the answer:
## 17
# | 341 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. At a round table sat raccoons, hedgehogs, and hamsters, a total of 134 animals. When asked: "Are any of your neighbors the same type of animal as you?", everyone answered "No." What is the maximum number of hedgehogs that could have been sitting at the table, given that hamsters and hedgehogs always tell the truth, ... | Answer: 44
## Examples of how to write the answer:
17
# | 44 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
3. The polynomial $P(x)=(x-4)^{4}\left(x^{2}+\frac{3}{2} x-1\right)^{5}$ is represented in the form $\sum_{k=0}^{14} a_{k}(x+1)^{k}$. Find $a_{0}+a_{2}+\ldots+a_{14}$. | Answer: -256
Allowed for input are digits, minus sign, and division sign, a dot or comma as a decimal separator
# | -256 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. How many pairs of natural numbers exist for which the number 392 is the LCM? (The numbers in the pair can be the same, the order of the numbers in the pair does not matter) | Answer: 18
Only digits are allowed as input
# | 18 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
# Problem 1. (2 points)
Let $p_{1}, p_{2}, \ldots, p_{97}$ be prime numbers (not necessarily distinct). What is the largest integer value that the expression
$$
\sum_{i=1}^{97} \frac{p_{i}}{p_{i}^{2}+1}=\frac{p_{1}}{p_{1}^{2}+1}+\frac{p_{2}}{p_{2}^{2}+1}+\ldots+\frac{p_{97}}{p_{97}^{2}+1}
$$
can take? | # Answer: 38
Solution: Note that the function $\frac{x}{x^{2}+1}$ decreases when $x \geqslant 2$, which means $\sum_{i=1}^{97} \frac{p_{i}}{p_{i}^{2}+1} \leqslant 97 \cdot \frac{2}{2^{2}+1}=$ $97 \cdot 0.4=38.8$. Therefore, the answer cannot be greater than 38.
At the same time, $\frac{3}{3^{2}+1}=0.3$, which is 0.1 ... | 38 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
# Problem 8. (5 points)
In an $8 \times 8$ table, some 23 cells are black, and the rest are white. In each white cell, the sum of the number of black cells on the same row and the number of black cells on the same column is written; nothing is written in the black cells. What is the maximum value that the sum of the n... | Answer: 234
## Solution:
The number in the white cell consists of two addends: "horizontal" and "vertical". Consider the sum of all "horizontal" addends and the sum of all "vertical" addends separately across the entire table. If we maximize each of these two sums separately, the total sum will also be the largest.
... | 234 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. Six natural numbers are written on the board, such that for any two $a$ and $b$ among them, $\log _{a} b$ or $\log _{b} a$ is an integer (the second logarithm does not necessarily exist). What is the smallest value that the maximum of these numbers can take? | The answer can be written in the form of a power of a number: $m^{n}$ is denoted as $\mathrm{m}^{\wedge} \mathrm{n}$.
Answer: 65536 || $2 \wedge 16\left\|4^{\wedge} 8\right\| 16^{\wedge} 4 \| 256^{\wedge} 2$ | 65536 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. $A B C A_{1} B_{1} C_{1}$ - a right triangular prism with a circumscribed sphere. The perimeter of the base $A B C$ is 32 units, and the product of the sides is 896 cubic units. The surface area of the prism is 192 square units. Find the square of the radius of its circumscribed sphere. | Answer: 53
## Examples of answer notations:
7.4
$7 / 14$
714
# | 53 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. (3 points) From the set of three-digit numbers that do not contain the digits $0,1,2$, $3,4,5$ in their notation, several numbers were written on paper in such a way that no two numbers can be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have been wr... | Answer: 40
Solution:
Whether a number can be written or not depends only on numbers consisting of the same set of digits.
Suppose a number ABC, consisting of three different digits, is written. This means that the numbers BAC and ACB cannot be written. Among the numbers consisting of these same digits, BVA, VAB, and... | 40 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
1. (2 points) Does there exist a rectangular parallelepiped with integer sides, for which the surface area is numerically equal to the sum of the lengths of all twelve edges? | Answer: Yes, it exists.
Solution: For example, a $2 \times 2 \times 2$ cube fits. Both the surface area and the sum of the lengths of the edges are 24. | 24 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. (3 points) From the set of three-digit numbers that do not contain the digits 0, 6, 7, 8, 9 in their notation, several numbers were written on paper in such a way that no two numbers can be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have been writt... | Answer: 75
Solution:
Whether a number can be written or not depends only on numbers consisting of the same set of digits.
Suppose a number ABC, consisting of three different digits, is written. This means that the numbers BAC and ACB cannot be written. Among the numbers consisting of these same digits, BVA, VAB, and... | 75 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. The polynomial $P(x)$ of the third degree, as well as its first, second, and third derivatives, take the value 1 at $x=-3$. Find $\mathrm{P}(0)$. | Answer: 13
## Examples of how to write answers:
$1 / 4$
0.25
$-10$
# | 13 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. In space, there are 37 different vectors with integer non-negative coordinates, starting from the point $(0 ; 0 ; 0)$. What is the smallest value that the sum of all their coordinates can take? | Answer: 115
## Examples of answer notation:
239
Answer: $[-1 ; 7]$ | 115 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. In the city of Gentle-city, there live 19 gentlemen, any two of whom are either friends or enemies. At some point, each gentleman asked each of his friends to send a hate card to each of his enemies (gentleman A asks gentleman B to send a card to all enemies of gentleman B). Each gentleman fulfilled all the requests... | Answer: 1538.
## Examples of answer recording:
100
# | 1538 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
4. (3 points) Does there exist a four-digit natural number with the sum of its digits being 23, which is divisible by 23? | Answer: Yes, for example $7682=23 \cdot 334$.
Solution: Generally speaking, the solution is contained in the answer, but let's explain how to find such a number. The sum of the digits of the product gives the same remainder when divided by 9 as the sum of the original numbers. Represent 23 as the difference between a ... | 7682 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
8. (4 points) A triangle is divided into 1000 triangles. What is the maximum number of different points at which the vertices of these triangles can be located? | Answer: 1002
Solution: The sum of the angles of a triangle is $180^{\circ}$, thousands of triangles $-180000^{\circ}$. Where did the extra $179820^{\circ}$ come from? Each internal vertex, where only the angles of triangles meet, adds $360^{\circ}$. Each vertex on the side of the original triangle or on the side of on... | 1002 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
8. (4 points) A triangle is divided into 1000 triangles. What is the minimum number of distinct points at which the vertices of these triangles can be located? | Answer: 503
Solution: The sum of the angles of a triangle is $180^{\circ}$, so for a thousand triangles, it is $180000^{\circ}$. Where did the extra $179820^{\circ}$ come from? Each internal vertex where only the angles of triangles meet adds $360^{\circ}$. Each vertex on the side of the original triangle or on the si... | 503 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
# Problem 4. (3 points)
Three runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.
At some point, the first runner meets the second. Twenty minutes later, the second runner meets the third for the first time. An... | # Answer: 100
Solution: (in general form)
Let the first runner meet the second, then after $a$ minutes the second runner meets the third for the first time, and after another $b$ minutes the third runner meets the first for the first time.
Let the first and second runners meet at point $A$, the second and third at p... | 100 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
# Problem 5. (3 points)
In an isosceles trapezoid $A B C D$, the bisectors of angles $B$ and $C$ intersect on the base $A D$. $A B=50, B C=128$. Find the area of the trapezoid. | Answer: 5472
## Solution:
Let $K$ be the point of intersection of the angle bisectors. Angles $\angle B=\angle C$ are equal as the base angles of an isosceles trapezoid, so their halves are... | 5472 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
# Problem 7. (3 points)
On an island, there live vegetarians who always tell the truth, and cannibals who always lie. 50 residents of the island, including both women and men, gathered around a campfire. Each of them said either "All men at this campfire are cannibals" or "All women at this campfire are vegetarians," ... | Answer: 48
## Solution:
We will prove that if we have $n$ people, the maximum number of vegetarian women is $n-2$.
First, note that if someone has called someone a cannibal, then we definitely have at least one cannibal.
Second, according to the condition, there is at least one man. Therefore, the only case where w... | 48 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
3. On a piece of paper, a square with a side length of 1 was drawn, next to it a square with a side length of 2, then a square with a side length of 3, and so on. It turned out that the area of the entire resulting figure is 42925. How many squares were drawn? | Answer: 50
## Examples of answer notation: 45
# | 50 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. $\mathrm{ABCD}$ is an isosceles trapezoid, $\mathrm{AB}=\mathrm{CD}=25, \mathrm{BC}=40, \mathrm{AD}=60$. $\mathrm{BCDE}$ is also an isosceles trapezoid. Find AE. (Points A and E do not coincide)
If there are multiple possible values, list them in any order separated by a semicolon. | Answer: 44.
## Examples of answer recording:
45
$45 ; 56$
## Problem 7 (3 points). | 44 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. Let $f(x)=\frac{34^{x}}{4^{x}+2}$. Find the sum $f(0)+f\left(\frac{1}{2017}\right) \ldots f\left(\frac{2}{2017}\right) f(1)$. | Answer: 3027
## Examples of answer recording:
45
## Problem 9 (4 points). | 3027 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Two tangents are drawn from point A to a circle. The distance from point A to the point of tangency is 10, and the distance between the points of tangency is 16. Find the greatest possible distance from point A to a point on the circle. | Answer: 30
## Examples of answer recording:
## Problem 10 (2 points). | 30 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. (3 points) From cards with letters, you can form the word KARAKATITSA. How many words (not necessarily meaningful) can be formed from these cards, in which the letters R and T are adjacent? | Answer: $\frac{9!}{4!}=9 \cdot 8 \cdot 7 \cdot 6 \cdot 5=15120$
## Solution:
Let's replace the adjacent letters R and T with one letter, for example, the letter Sh, since they stand together. Then we are left with 9 letters, and among them, 4 letters A and 2 letters K.
Thus, the number of ways to rearrange the lette... | 15120 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. Given a convex hexagon ABCDEF such that AB $\|\mathrm{CF}\| \mathrm{DE}, \mathrm{BC}\|\mathrm{AD}\| \mathrm{EF}$ and $\mathrm{CD} \| \mathrm{BE}$ $\| \mathrm{FA}$ and $\mathrm{AB}=\mathrm{DE}=\mathrm{CD}=\mathrm{AF}=13, \mathrm{BD}=24$. Find the area of the union of triangles $\mathrm{ACE}$ and BDF.
. What is the maximum number of chips that can be on the board? | Answer: 21
## Examples of answer recording:
14
## 9th grade.
# | 21 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
# Problem 1. (2 points)
Let $x, y, z$ be pairwise coprime three-digit natural numbers. What is the greatest value that the GCD $(x+y+z, x y z)$ can take? | Answer: 2994
Solution:
The GCD of two numbers cannot be greater than either of them. The maximum possible value of $x+$ $y+z=997+998+999=2994$ and for these numbers, $x y z$ is indeed divisible by $x+y+z=2994=3 \cdot 998$. | 2994 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
Problem 2. (2 points)
On a circle, 10 points are marked. Any three of them form three inscribed angles. Petya calculated the number of different values that these angles can take. What is the maximum number he could have obtained? | Answer: 80
Solution:
Any two points form two arcs. All inscribed angles subtending the same arc are equal. For two adjacent points, on one of the two arcs between them, no other point lies, meaning a pair of adjacent points gives us one possible angle value, while a pair of non-adjacent points gives two values.
In t... | 80 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
# Problem 5. (3 points)
In triangle $A B C$, the midpoints of sides $A B=40$ and $B C=26$ are marked as points $K$ and $L$ respectively. It turns out that the quadrilateral $A K L C$ is a tangential quadrilateral. Find the area of triangle $A B C$. | # Answer: 264
## Solution:
According to the Midline Theorem, $K L=\frac{1}{2} A C$. In a cyclic quadrilateral, the sums of the opposite sides are equal, that is, $K L+A C=A K+C L=\frac{A B+B C}{2}$, so $\frac{3 A C}{2}=\frac{66}{2}$ and $A C=22$. Knowing the sides of the triangle, the area can be calculated using Her... | 264 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
# Problem 1. (2 points)
A cubic polynomial has three roots. The greatest value of the polynomial on the interval $[4 ; 9]$ is achieved at $x=5$, and the smallest value is achieved at $x=7$. Find the sum of the roots of the polynomial.
# | # Answer: 18
## Solution:
Since in all options the minimum and maximum on the interval are not reached at its ends, they are reached at the roots of the polynomial's derivative. Let the polynomial be of the form $a x^{3}+b x^{2}+c x+d$, and its derivative, respectively, $3 a x^{2}+2 b x+c$. In this case, the sum of t... | 18 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
# Problem 2. (3 points)
Find the sum of natural numbers from 1 to 3000 inclusive that have common divisors with the number 3000, greater than 1. | # Answer: 3301500
## Solution:
$3000=2^{3} \cdot 3 \cdot 5^{3}$, so we are interested in numbers divisible by 2, 3, or 5. First, let's find the number of such numbers. For this, we will use the principle of inclusion and exclusion. There are exactly $\frac{3000}{2}=1500$ even numbers from 1 to 3000, $\frac{3000}{3}=1... | 3301500 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
# Problem 6. (4 points)
Quadrilateral $A B C D$ is circumscribed around a circle with center at point $O, K, L, M, N$ - the points of tangency of sides $A B, B C, C D$ and $A D$ respectively, $K P, L Q, M R$ and $N S$ - the altitudes in triangles $O K B, O L C, O M D, O N A . O P=15, O A=32, O B=64$.
Find the length ... | # Answer: 30
## Solution:
Triangles $O K A$ and $O N A$ are right triangles with a common hypotenuse and a leg equal to the radius of the circle, so they are congruent. Therefore, their altitudes fall on the same point of the common hypotenuse, meaning $K S$ is the altitude in triangle $O K A$. Thus, points $S$ and $... | 30 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
# Problem 7. (4 points)
Two cubes with an edge of $12 \sqrt[4]{\frac{8}{11}}$ share a common face. A section of one of these cubes by a certain plane is a triangle with an area of 16. The section of the other by the same plane is a quadrilateral. What is the maximum value that its area can take? | # Answer: 128
## Solution:
Let our cubes be $A B C D A_{1} B_{1} C_{1} D_{1}$ and $A B C D A_{2} B_{2} C_{2} D_{2}$ with a common face $A B C D$. Let the triangular section of the first cube be $K L M$, where point $K$ lies on $A A_{1}$, point $L$ on $A B$, and point $M$ on $A D$. One side of the quadrilateral sectio... | 128 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
# Problem 8. (4 points)
Hansel and Gretel are playing a game, with Hansel going first. They take turns placing tokens on a $7 \times 8$ grid (7 rows and 8 columns). Each time Gretel places a token, she earns 4 points for each token already in the same row and 3 points for each token already in the same column.
Only o... | # Answer: 700
## Solution:
Let's say that Hansel also earns points according to the same principle as Gretel. In this case, each pair of cells in the same row will ultimately give one of the players 4 points, and each pair of cells in the same column will give 3 points. In one row, there are $\frac{8 \cdot 7}{2}=28$ ... | 700 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
# Problem 2. (2 points)
A natural number $n$ when divided by 12 gives a remainder $a$ and an incomplete quotient $b$, and when divided by 10, it gives a remainder $b$ and an incomplete quotient $a$. Find $n$. | Answer: 119
## Solution:
From the condition, it follows that $n=12 b+a=10 a+b$, from which $11 b=9 a$. Therefore, $b$ is divisible by 9, and $a$ is divisible by 11. On the other hand, $a$ and $b$ are remainders from division by 12 and 10, respectively, so $0 \leqslant a \leqslant 11$ and $0 \leqslant b \leqslant 9$. ... | 119 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
# Problem 6. (4 points)
Six positive numbers, not exceeding 3, satisfy the equations $a+b+c+d=6$ and $e+f=2$. What is the smallest value that the expression
$$
\left(\sqrt{a^{2}+4}+\sqrt{b^{2}+e^{2}}+\sqrt{c^{2}+f^{2}}+\sqrt{d^{2}+4}\right)^{2}
$$
can take? | Answer: 72
In the image, there are three rectangles $2 \times (a+b)$ and three rectangles $2 \times (c+d)$, forming a $6 \times 6$ square, since $a+b+c+d=6$. The segment of length 2 in the c... | 72 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
# Problem 7. (4 points)
In an $8 \times 8$ table, some cells are black, and the rest are white. In each white cell, the total number of black cells on the same row or column is written; nothing is written in the black cells. What is the maximum value that the sum of the numbers in the entire table can take? | Answer: 256
## Solution:
The number in the white cell consists of two addends: a "horizontal" and a "vertical" one. Consider the sum of all "horizontal" addends and the sum of all "vertical" addends separately across the entire table. If we maximize each of these two sums separately, the total sum will also be the gr... | 256 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
# Problem 4. (2 points)
How many negative numbers are there among the numbers of the form $\operatorname{tg}\left(\left(15^{n}\right)^{\circ}\right)$, where $\mathrm{n}$ is a natural number from 1 to 2019? | Answer: 1009
Solution:
$\operatorname{tg} 15^{\circ}>0$.
$15^{2}=225 ; \operatorname{tg} 225^{\circ}>0$.
Further, $225 \cdot 15=3375$, this number gives a remainder of 135 when divided by $360 . \operatorname{tg} 135^{\circ}<0$.
$135 \cdot 15=2025$, this number gives a remainder of 225 when divided by 360. The sequ... | 1009 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
# Problem 7. (4 points)
Can the numbers from 0 to 9999 (each used exactly once) be arranged in a $100 \times 100$ square table so that the sum of the numbers in each $2 \times 2$ square is the same? | Answer: Yes, it can.
Solution:
The desired table is obtained as the sum of two tables.
The first table looks like this:
| 0 | 99 | 0 | 99 | 0 | 99 | $\ldots$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 0 | 99 | 0 | 99 | 0 | 99 | $\ldots$ |
| 1 | 98 | 1 | 98 | 1 | 98 | $\ldots$ |
| 1 | 98 | 1 | 98 | 1 | 9... | 19998 | Combinatorics | proof | Yes | Yes | olympiads | false |
# Problem 4. (2 points)
How many positive numbers are there among the numbers of the form $\operatorname{ctg}\left(\left(15^{n}\right)^{\circ}\right)$, where $\mathrm{n}$ is a natural number from 1 to 2019? | Answer: 1010
Solution:
$\operatorname{ctg} 15^{\circ}>0$.
$15^{2}=225 ; \operatorname{ctg} 225^{\circ}>0$.
Next, $225 \cdot 15=3375$, this number gives a remainder of 135 when divided by $360 . \operatorname{ctg} 135^{\circ}<0$.
$135 \cdot 15=2025$, this number gives a remainder of 225 when divided by 360. The seq... | 1010 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
# Problem 7. (4 points)
Can the numbers from 0 to 999 (each used exactly once) be arranged in a $100 \times 10$ rectangular table so that the sum of the numbers in each $2 \times 2$ square is the same? | Answer: Yes, it can.
Solution:
The desired table is obtained as the sum of two tables.
The first table looks like this:
| 0 | 99 | 0 | 99 | 0 | 99 | $\ldots$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 0 | 99 | 0 | 99 | 0 | 99 | $\ldots$ |
| 1 | 98 | 1 | 98 | 1 | 98 | $\ldots$ |
| 1 | 98 | 1 | 98 | 1 | 9... | 1998 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. Given a regular tetrahedron with edge lengths that are integers. Two of these edges have lengths 9 and 11. What is the greatest possible value of the perimeter of the tetrahedron? | Answer: 68.
## Examples of answer recording: 45
# | 68 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. Given a cubic polynomial $p(x)$. It is known that $p(-6)=30, p(-3)=45, p(-1)=15, p(2)=30$.
Find the area of the figure bounded by the lines $x=-6, y=0, x=2$ and the graph of the given polynomial, if it is also known that on the interval from -6 to 2 the given polynomial takes only positive values. | Answer: 240.
## Examples of answer recording:
45 | 240 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. In triangle $\mathrm{ABC}$, angle $\mathrm{C}$ is twice as small as angle $\mathrm{B}$. $\mathrm{BB}_{1}$ is the bisector of angle $\mathrm{B}, \mathrm{D}$ is the point of intersection of the circumcircle of triangle $\mathrm{ABB}_{1}$ and side $\mathrm{AB} . \mathrm{BD}=14$, $\mathrm{CD}=18$. Find the length of $\m... | Answer: 30.
## Examples of answer notation:
45
# | 30 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
2. $A B C D E$ - a pentagon inscribed in circle $S$. Lines $D E$ and $A B$ are parallel, $B C=C D$, $A D=D E=20, \cos C B D=5 / 8$. Find the radius of circle $S$. | Answer: 16.
## Examples of how to write answers:
## $1 / 4$
0.25
1
# | 16 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. (3 points) Anya, Vanya, Danya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of apples collected, and all these numbers are different and greater than zero. What is the minimum number of apples that could have been collected?
Answer: 20 | Solution:
Example: there are many different examples, for instance $1+2+3+4+10$ or $2+3+4+5+6=20$
Estimate (proof that there are no fewer than 20 apples):
Let Anna have collected the fewest apples, and this number is at least 2. Then we get that the total collected is no less than $2+3+4+5+6=20$ apples.
If Anna has... | 20 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
8. (4 points) In an $8 \times 8$ table, integers from 0 to 10 are arranged (naturally, numbers can repeat, not all specified numbers necessarily appear). It is known that in each rectangle $3 \times 2$ or $2 \times 3$, the sum of the numbers is 10. Find the smallest possible value of the sum of the numbers in the entir... | # Answer: 105
## Solution:
Let's look at the left part of the figure below. Consider the rectangles $3 \times 2$ and $2 \times 3$, which contain the left upper $2 \times 2$ square. The remaining parts of these two rectangles are the leftmost upper rectangles $1 \times 2$ and $2 \times 1$, shaded in gray. Since the su... | 105 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
5. (3 points) Anya, Vanya, Danya, Manya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of apples collected, and all these numbers are different and greater than zero. What is the minimum number of apples that could have been collected?
Answer:... | Solution:
Example: there are many different examples, for instance $1+2+3+4+5+10$.
Evaluation (proof that there are no fewer than 25 apples):
Let Anya have collected the fewest apples, and this number is at least 2. Then we get that the total collected is no less than $2+3+4+5+6+7=27$ apples.
If Anya has 1 apple an... | 25 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
8. (4 points) In an $11 \times 11$ table, integers from 0 to 10 are arranged (naturally, numbers can repeat, not all specified numbers necessarily appear). It is known that in each $3 \times 2$ or $2 \times 3$ rectangle, the sum of the numbers is 10. Find the smallest possible value of the sum of the numbers in the ent... | # Solution:
The right part of the figure below shows the table divided into 20 rectangles and one cell. This means that the sum of the numbers in the entire table is no less than the sum of the numbers in these 20 rectangles, which is 200.
An example is provided in the left part of the figure, cells filled with zeros... | 200 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. (2 points) Anya was making pancakes, planning for each of the three members of her family to get an equal number of pancakes. But something went wrong: every third pancake Anya couldn't flip; $40 \%$ of the pancakes that Anya managed to flip burned; and $\frac{1}{5}$ of the edible pancakes Anya dropped on the floor.... | Solution:
In total, Anya managed to save $\frac{2}{3} \cdot 0.6 \cdot 0.8=0.32$ of the number of pancakes. | 32 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
8. (5 points) From a $12 \times 12$ grid, a $4 \times 4$ square was cut out, lying at the intersection of the third to sixth horizontal lines and the same vertical lines. What is the maximum number of non-attacking rooks that can be placed on this field, if the rooks do not attack through the cut-out cells?
## Answer:... | # Solution:
The available part of the board is divided into the following four areas, in each of which no more than the specified number of rooks can be placed (see the left figure), as in each rectangle, no more rooks can be placed than the length of its shorter side. Therefore, the total number of rooks does not exc... | 14 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. (2 points) Anya was making pancakes, planning for each of the five members of her family to get an equal number of pancakes. But something went wrong: every fifth pancake Anya couldn't flip; $49\%$ of the pancakes that Anya managed to flip burned; and $\frac{1}{6}$ of the edible pancakes Anya dropped on the floor. W... | # Solution:
In total, Anya managed to save $0.8 \cdot 0.51 \cdot \frac{5}{6}=0.34$ of the pancakes. | 34 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
8. (5 points) From a $12 \times 12$ grid, a $4 \times 4$ square was cut out, lying at the intersection of the fourth to seventh horizontal and vertical lines. What is the maximum number of non-attacking rooks that can be placed on this field, if the rooks do not attack through the cut-out cells?
## Answer: 15
# | # Solution:
The available part of the board is divided into the following four areas, in each of which no more than the specified number of rooks can stand (see the left figure), as in each rectangle, no more rooks can stand than the length of its shorter side. Therefore, the total number of rooks does not exceed 15.
... | 15 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. A five-digit number ABCDE, consisting of different digits, is divisible by both the three-digit number CDE and the two-digit number AB. Find the smallest possible value of ABCDE. | Answer: 12480
## Examples of answer recording:
12345
# | 12480 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. On Misfortune Island, a table tennis tournament was held in a round-robin format (i.e., everyone played against everyone else once). After each match, both participants approached the Chief Referee separately and reported the result. Among the participants, there were only three types of people: knights, who always ... | Answer: 16
## Examples of answer recording:
17
# | 16 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
1. A circle with radius 12 and center at point $O$ and a circle with radius 3 touch internally at point $H$. The line $X H$ is their common tangent, and the line $O X$ is tangent to the smaller circle. Find the square of the length of the segment $O X$. | Answer: 162
2.. A circle with radius 12 and center at point $O$ and a circle with radius 4 touch internally at point $H$. Line $X H$ is their common tangent, and line $O X$ is tangent to the smaller circle. Find the square of the length of segment $O X$.
Answer: 192 | 162 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. A circle of radius 20 with center at point $O$ and a circle of radius 8 touch internally at point $H$. The line $X H$ is their common tangent, and the line $O X$ is tangent to the smaller circle. Find the square of the length of the segment $O X$. | Answer: 720
## Examples of how to write answers:
17
$1 / 7$
1.7 | 720 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. There are 12 students in the class. Each of them was asked how many friends they have in the class. Each number \( n \) was either mentioned exactly \( n \) times or not at all. What is the maximum value that the sum of all the numbers mentioned can take? | Answer: 90
Only digits are allowed as input.
## Problem 5. (3 points) | 90 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. Each vertex of a tetrahedron with a volume of 216 was symmetrically reflected relative to the point of intersection of the medians of the opposite face. The four resulting points formed a new tetrahedron. Find its volume. | Answer: 1000
Allowed for input are digits, a dot or comma, and a division sign
# | 1000 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. A $10 x 10$ grid is filled with non-negative numbers. It is known that in each (vertical or horizontal) strip of 1 x 3, the sum of the numbers is 9. What is the maximum value that the sum of all numbers in the grid can take? | Answer: 306
Examples of writing answers:
14 | 306 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
1. The graphs of a quadratic trinomial with a positive leading coefficient 2 and its derivative intersect at the vertex of the parabola with abscissa $x_{0}$ and at another point with abscissa $x_{1}$. Find the total area of both regions bounded by the graphs of the trinomial, its derivative, and the line symmetric to ... | Answer: 16
## Examples of answer notations:
17
$-1.7$
$1 / 7$
# | 16 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Around a circle, 129 (not necessarily integer) numbers from 5 to 25 inclusive were written. From each number, the logarithm to the base of the next number in the clockwise direction was taken, after which all the obtained logarithms were added. What is the maximum value that the sum of these logarithms can take? | Answer: 161
## Examples of answer notations:
17
$-1.7$
$1 / 7$
## 10th grade
# | 161 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. (3 points) In a certain country, there are 100 cities. The Ministry of Aviation requires that every two cities be connected by a two-way flight operated by exactly one airline, and that it should be possible to travel from any city to any other city using the flights of each airline (possibly with layovers). What is... | Solution: Evaluation: in order to connect all cities, the airline must have at least 99 flights. In total, there are $100 \cdot 99 / 2$ pairs of cities, which means there are no more than 50 airlines.
Example: let's denote the cities as $a_{1}, a_{2}, \ldots a_{50}, b_{1}, b_{2}, \ldots, b_{50}$ in some way. Then the ... | 50 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. (3 points) In a certain country, there are 50 cities. The Ministry of Aviation requires that every two cities be connected by a two-way flight operated by exactly one airline, and that it should be possible to travel from any city to any other using the flights of each airline (possibly with layovers). What is the m... | Solution: Evaluation: in order to connect all cities, the airline must have at least 49 flights. In total, there are $50 \cdot 49 / 2$ pairs of cities, so there are no more than 25 airlines.
Example: let the cities be denoted as $a_{1}, a_{2}, \ldots a_{25}, b_{1}, b_{2}, \ldots, b_{25}$ in some way. Then the airline ... | 25 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. (3 points) In a certain country, there are 200 cities. The Ministry of Aviation requires that every two cities be connected by a two-way flight operated by exactly one airline, and that it should be possible to travel from any city to any other using the flights of each airline (possibly with layovers). What is the ... | # Answer: 100
Solution: Evaluation: for the airline to connect all cities, it must have at least 199 flights. The total number of city pairs is $200 \cdot 199 / 2$, so there are no more than 100 airlines.
Example: let's denote the cities as $a_{1}, a_{2}, \ldots a_{100}, b_{1}, b_{2}, \ldots, b_{100}$ in some way. Th... | 100 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. (3 points) In a certain country, there are 120 cities. The Ministry of Aviation requires that every two cities be connected by a two-way flight operated by exactly one airline, and that it should be possible to travel from any city to any other using the flights of each airline (possibly with layovers). What is the ... | Solution: Evaluation: in order to connect all cities, the airline must have at least 119 flights. The total number of city pairs is $120 \cdot 119 / 2$, so there are no more than 60 airlines.
Example: let's denote the cities as $a_{1}, a_{2}, \ldots a_{60}, b_{1}, b_{2}, \ldots, b_{60}$ in some way. Then the airline w... | 60 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. (2 points) A chess piece called a quadraliner attacks two verticals and two horizontals adjacent to the square it occupies. What is the maximum number of non-attacking quadraliners that can be placed on a $10 \times 10$ board? | Answer: 25
## Solution:
Example: place the four-line rulers at the intersections of even columns and even rows.
Evaluation: In each pair of rows, the four-line rulers can only stand on one row. At the same time, they cannot stand on two adjacent cells. | 25 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
7. (4 points) Prove that if a convex 102-gon can be divided into triangles by non-intersecting diagonals such that from each vertex either exactly three diagonals or none emanate, then the number of resulting triangles, all three sides of which are diagonals of the original polygon, will be exactly 34.
# | # Solution:
A 102-gon is divided into 100 triangles by 99 diagonals. Each diagonal has two ends, so the sum of the number of diagonals emanating from the vertices of the polygon is 198. Therefore, the number of vertices from which three diagonals emanate is 66, and the number of remaining vertices from which no diagon... | 34 | Combinatorics | proof | Yes | Yes | olympiads | false |
3. In country Gamma, there are 101 cities. It is known that at least 2 roads lead out of each city. It is also known that if there is a road from city A to city B and from city B to city C, then there is also a road from city A to city C. What is the minimum number of roads that can be in the country? | Answer: 205
## Examples of answer notation:
17
# | 205 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. It is known that the number $\sqrt{5}+\sqrt{7}$ is a root of a polynomial of the fourth degree with integer coefficients, the leading coefficient of which is 1. What is the sum of the coefficients of this polynomial? | Answer: -19
## Examples of how to write answers:
17
$1 / 7$
1.7
## Problem 8 | -19 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. (3 points) On the Island of Misfortune, there live knights who always tell the truth, and liars who always lie. One day, $n$ islanders gathered in a room.
The first one said: "Exactly 1 percent of those present in this room are liars."
The second one said: "Exactly 2 percent of those present in this room are liars... | Answer: 100
## Solution:
There must be exactly one knight, as all other islanders in the room contradict each other. This means that the knights are $\frac{1}{n}$ of the total number of people. Let them be $k$ percent of the total number of people present, which means the person with the number $100-k$ is telling the... | 100 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
3. (3 points) Anya, Vanya, Danya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of apples collected, and all these numbers were different and greater than zero. Then Tanya, who collected the most apples, ate her apples. After this, it turned out that ... | # Solution:
Let Anya have collected the fewest apples, which is $k$, and the total number of apples is $n$. Then we get that the total number of apples collected is no less than $4k + 6$. For $k \geq 5$, we already get more than 20 apples. For $k < 5$, we need to consider the cases where $k$ is 1, 2, 3, or 4.
If $k ... | 20 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
7. (3 points) Given a convex pentagon $A B C D E$. Point $P$ is the intersection of $B E$ and $A C$, point $Q$ is the intersection of $C E$ and $A D$, point $O$ is the intersection of $A D$ and $B E$. It turns out that triangles $A B P$ and $D E Q$ are isosceles triangles with the angle at the vertex (exactly at the ve... | Answer: $60^{\circ}$ or $105^{\circ}$
Solution: Consider triangle $A B P$. The angle $\angle A P B$ in it can be $80^{\circ}$ or $50^{\circ}$. Then the adjacent $\angle A P O$ is $100^{\circ}$ or $130^{\circ}$. Therefore, in triangle $A P O$, this is the angle at the vertex and $\angle A O P$ is $40^{\circ}$ or $25^{\... | 60 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
8. (4 points) On the board, all three-digit natural numbers are written, the first digits of which are odd and greater than 1. What is the maximum number of quadratic equations of the form $a x^{2}+b x+c=0$ that can be formed using these numbers as $a, b$ and $c$, each no more than once, such that all these equations h... | # Answer: 100
## Solution:
In each equation, there must be a coefficient less than five hundred. Otherwise, the discriminant $b^{2}-4 a c$ 2 variant | 100 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. (3 points) Anya, Vanya, Danya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of collected apples, and all these numbers were different and greater than zero. Then Tanya, who collected the most apples, ate her apples. After that, it turned ou... | Answer: 20, for example $1+2+3+4+10$
## Solution:
Let Anna have collected the fewest apples, and this number is $k$, and the total number of apples is $n$. Then we get that the total number of apples collected is no less than $4 k+10$. For $k \geq 3$, we already get more than 20 apples. For $k=2$, we get $n \geq 18=9... | 20 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. (3 points) Given a rectangle $A B C D, A B=5, B C=6$. Point $K$ lies on side $C D$, point $L$ lies on side $A B$, point $M$ lies on side $B C$, point $N$ lies on side $C D$. Prove that the length of the closed broken line $A K L M N A$ is not greater than 26. | Solution:
Let's add several more rectangles equal to the given one, as shown in the figure. Take point $L^{\prime}$ such that $A L=A L^{\prime}$, point $M$ such that $B^{\prime} M^{\prime}=... | 26 | Geometry | proof | Yes | Yes | olympiads | false |
7. (3 points) Given a convex pentagon $A B C D E$. Point $P$ is the intersection of $B E$ and $A C$, point $Q$ is the intersection of $C E$ and $A D$, point $O$ is the intersection of $A D$ and $B E$. It turns out that triangles $A B P$ and $D E Q$ are isosceles triangles with the angle at the vertex (exactly at the ve... | Answer: $120^{\circ}$ or $75^{\circ}$
Solution: Consider triangle $A B P$. The angle $\angle A P B$ in it can be $40^{\circ}$ or $70^{\circ}$. Then the adjacent $\angle A P O$ is $140^{\circ}$ or $110^{\circ}$. Therefore, in triangle $A P O$, this is the angle at the vertex and $\angle A O P$ is $20^{\circ}$ or $35^{\... | 120 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
8. (4 points) On the board, all three-digit natural numbers are written, the first digits of which are odd and greater than 1. What is the maximum number of quadratic equations of the form $a x^{2}+b x+c=0$ that can be formed using these numbers as $a, b$, and $c$, each no more than once, such that all these equations ... | Answer: 100.
Solution:
In each equation, there must be a coefficient less than five hundred. Otherwise, the discriminant $b^{2}-4 a c<1000^{2}-4 \cdot 500^{2}=0$ and the equation has no roots. Thus, we get no more than 100 equations.
Now for an example: Take $b$ as numbers starting with 9, and for each equation, tak... | 100 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. The train was supposed to travel 300 km. Having traveled 212 km at a certain speed, it then reduced its speed by 50 km/h. As a result, the train arrived 40 minutes later than the scheduled time. Find the initial speed of the train. Answer in km/h. | Answer: 110
3rd option. The train was supposed to travel 300 km. After traveling 50 km at a certain speed, it then reduced its speed by 40 km/h. As a result, the train arrived 1 hour and 40 minutes later than the scheduled time. Find the initial speed of the train. Record your answer in km/h.
Answer: 100
# | 110 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Four different numbers $a, b, c, d$, greater than one and not divisible by 5, are such that $\gcd(a, b) = \gcd(c, d)$ and $\operatorname{lcm}(a, b) = \operatorname{lcm}(c, d)$. What is the smallest possible value of $a + b + c + d$? | Answer: 24
## Examples of how to write the answer:
17
# | 24 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. At a round table, raccoons, hedgehogs, and hamsters were sitting, a total of 134 animals. When asked: "Are there any animals of the same kind as you among your neighbors?", everyone answered "No." What is the maximum number of hedgehogs that could have been sitting at the table, given that hamsters and hedgehogs alw... | Answer: 44
## Examples of how to write the answer:
17
# | 44 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
# Problem 8. (5 points)
A figure called a "half-bishop" moves one square diagonally. What is the maximum number of non-attacking half-bishops that can be placed on a $7 \times 7$ chessboard?
Answer: 28 | Solution:
Consider all diagonals of one direction. On each diagonal of even length, no more than half of the cells can be occupied by semi-bishops, as it can be divided into pairs of cells that cannot be simultaneously occupied.
On each diagonal of odd length $2n+1$, no more than $n+1$ semi-bishops can be placed for ... | 28 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
# Problem 8. (5 points)
A figure called a "half-bishop" moves one square diagonally. What is the maximum number of non-attacking half-bishops that can be placed on a $9 \times 9$ chessboard? | Answer: 45
Solution:
Consider all diagonals of one direction. On each diagonal of even length, no more than half of the cells can be occupied by pseudo-bishops, as it can be divided into pairs of cells that cannot be simultaneously occupied.
On each diagonal of odd length $2n+1$, no more than $n+1$ pseudo-bishops ca... | 45 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
1. (2 points) In trapezoid $A B C D$ with bases $A D=16$ and $B C=10$, the circles constructed on sides $A B, B C$ and $C D$ as diameters intersect at one point. The length of diagonal $A C$ is 10. Find the length of $B D$. | Answer: 24
## Solution:
Let the intersection point of the three circles be $O$. Then, since the circles are constructed on the sides of the trapezoid $AB, BC$, and $CD$ as diameters, the angles $\angle AOB, \angle BOC$, and $\angle COD$ are right angles. Therefore, points $A, O, C$ lie on the same line and points $B,... | 24 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. (2 points) In trapezoid $A B C D$ with bases $A D=12$ and $B C=8$, the circles constructed on sides $A B, B C$, and $C D$ as diameters intersect at one point. The length of diagonal $A C$ is 12. Find the length of $B D$. | Answer: 16
## Solution:
Let the intersection point of the three circles be $O$. Then, since the circles are constructed on the sides of the trapezoid $AB, BC$, and $CD$ as diameters, the angles $\angle AOB, \angle BOC$, and $\angle COD$ are right angles. Therefore, points $A, O, C$ lie on the same line and points $B,... | 16 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
6. (3 points) Let $x, y, z$ and $t$ be non-negative numbers such that $x+y+z+t=4$. Prove the inequality
$$
\sqrt{x^{2}+t^{2}}+\sqrt{z^{2}+1}+\sqrt{z^{2}+t^{2}}+\sqrt{y^{2}+x^{2}}+\sqrt{y^{2}+64} \geqslant 13
$$ | # Solution:
Consider the following points on the plane: $A(0,0) ; B(x, t) ; C(x+z, t+1) ; D(x+z+t, t+1+z)$; $E(x+z+t+y, t+1+z+x) ; F(x+z+t+y+8, t+1+z+x+y)$. Then the length of the broken line $A B C D E F$ coincides with the expression that needs to be evaluated. By the triangle inequality, the length of the broken li... | 13 | Inequalities | proof | Yes | Yes | olympiads | false |
1. (2 points) In trapezoid $A B C D$ with bases $A D=20$ and $B C=14$, the circles constructed on sides $A B, B C$ and $C D$ as diameters intersect at one point. The length of diagonal $A C$ is 16. Find the length of $B D$. | Answer: 30
## Solution:
Let the intersection point of the three circles be $O$. Then, since the circles are constructed on the sides $AB$, $BC$, and $CD$ of the trapezoid as diameters, the angles $\angle AOB$, $\angle BOC$, and $\angle COD$ are right angles. Therefore, points $A$, $O$, and $C$ lie on the same line, a... | 30 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
6. (3 points) Let $x, y, z$ and $t$ be non-negative numbers such that $x+y+z+t=7$. Prove the inequality
$$
\sqrt{x^{2}+y^{2}}+\sqrt{x^{2}+1}+\sqrt{z^{2}+y^{2}}+\sqrt{t^{2}+64}+\sqrt{z^{2}+t^{2}} \geqslant 17
$$ | Solution:
Consider the following points on the plane: $A(0,0) ; B(x, y) ; C(x+1, y+x) ; D(x+1+y, y+x+z) ; E(x+1+y+t, y+x+z+8) ; F(x+1+y+t+z, y+x+z+8+t)$. Then the length of the broken line $A B C D E F$ coincides with the expression that needs to be evaluated. By the triangle inequality, the length of the broken line ... | 17 | Inequalities | proof | Yes | Yes | olympiads | false |
1. (2 points) In trapezoid $A B C D$ with bases $A D=20$ and $B C=10$, circles constructed on sides $A B, B C$, and $C D$ as diameters intersect at one point. The length of diagonal $A C$ is 18. Find the length of $B D$. | Answer: 24
## Solution:
Let the intersection point of the three circles be $O$. Then, since the circles are constructed on the sides of the trapezoid $AB, BC$, and $CD$ as diameters, the angles $\angle AOB, \angle BOC$, and $\angle COD$ are right angles. Therefore, points $A, O, C$ lie on the same line and points $B,... | 24 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. A four-digit number ABCD, consisting of different digits, is divisible by both the two-digit number CD and the two-digit number AB. Find the smallest possible value of ABCD. | Answer: 1248
## Examples of answer recording:
1234
# | 1248 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. In country Gamma, there are 83 cities. It is known that at least 2 roads lead out of each city. It is also known that if there is a road from city A to city B and from city B to city C, then there is also a road from city A to city C. What is the minimum number of roads that can be in the country? | Answer: 87
## Examples of answer notation:
17
# | 87 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.