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[
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"value": "5. Based on the base $AC$ of the isosceles triangle $ABC$, a circle is constructed with $AC$ as its diameter, intersecting the side $BC$ at point $N$ such that $BN: NC = 7: 2$. Find the ratio of the lengths of the segments $AN$ and $BC$."
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"va... |
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"value": "5. Based on the base $AC$ of the isosceles triangle $ABC$, a circle is constructed with $AC$ as its diameter, intersecting the side $BC$ at point $N$ such that $BN: NC = 5: 2$. Find the ratio of the lengths of the medians $NO$ and $BO$ of triangles $ANC$ and $ABC$."
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... |
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"value": "5. The angle at vertex $B$ of triangle $A B C$ is $130^{\\circ}$. Through points $A$ and $C$, lines perpendicular to line $A C$ are drawn and intersect the circumcircle of triangle $A B C$ at points $E$ and $D$. Find the acute angle between the diagonals of the quadrilateral with... |
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"value": "5. A circle circumscribed around a triangle with sides 3, 5, and 6 is inscribed in a triangle with sides parallel to the sides of the first triangle. Find the lengths of the sides of the second triangle.\n\n# Answers and solutions \n\nTask 1 Answer: -1\n\nTask 2 Answer: 1) 1968 2... |
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"value": "4. Answer: $n=6,15,16,17$"
},
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"value": "<think>\nOkay, so I have this problem where I need to find all positive integers n such that something happens, and the answer is given as n=6,15,16,17. But wait, the user hasn't specified the actual probl... |
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"value": "5. Answer: $R=\\frac{\\rho(1+\\sin (\\pi / n))}{\\sin (\\pi / n)}$"
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"value": "<think>\nOkay, let's see. The problem is asking me to derive the formula for the resistance R, which is given as R equals rho times (1 plus sin(pi/n)) divided by si... |
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"value": "4. Answer: $p=\\frac{112}{405}$"
},
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"value": "<think>\nOkay, so I need to figure out the solution to this problem where the answer is given as p equals 112 over 405. Let me start by recalling what kind of problems result in probabilities or frac... |
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"value": "3. Answer: $S_{b}(n)=b \\cdot n+\\frac{a+b+c}{3}(S(n)-n \\cdot S(1))=4 n+6(S(n)-n \\cdot S(1))$"
},
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"value": "<think>\nOkay, let's see. I have this problem here: I need to figure out how the formula for S_b(n) is derived. The given formula is S_... |
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"value": "4. Answer: $P(A)=\\frac{2}{5!}=\\frac{1}{60}$"
},
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"value": "<think>\nOkay, let's see. The problem here is about finding the probability of event A, denoted as P(A), and the given answer is 2 divided by 5 factorial, which simplifies to 1/60. Hmm,... |
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"value": "5. Answer: 1) 44 integer solutions 2 ) $a=k \\in Z$"
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"value": "<think>\nOkay, let me see. So the problem is asking about integer solutions, right? There are two parts: first, how many integer solutions there are, and second, the value of 'a' ... |
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"value": "1. Answer: $f \\in\\left[(\\sqrt{65}-2)^{2}-20 ;(\\sqrt{65}+2)^{2}-20\\right]=[49-4 \\sqrt{65} ; 49+4 \\sqrt{65}]$"
},
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"value": "<think>\nOkay, let's see. The problem here is to find the interval in which f lies, given by the expression [(\\sqrt... |
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"value": "2. Answer: $x=\\arccos \\frac{2}{\\sqrt{5}}=\\operatorname{arctg} \\frac{1}{2}, y=\\frac{1}{\\sqrt{5}}$"
},
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"value": "<think>\nOkay, let's see. The problem gives me two equations for x and y, and I need to figure out how they arrived at those an... |
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"value": "3. Answer: $x=\\frac{1}{2 \\sqrt{2}}, y=\\frac{1}{\\sqrt{2}}$"
},
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"value": "<think>\nOkay, so I have this problem here where I need to find the values of x and y. The answer is supposed to be x = 1/(2√2) and y = 1/√2. Let me try to figure out ho... |
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"value": "4. Answer: $P(A)=\\frac{23}{81}$"
},
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"value": "<think>\nOkay, so I need to figure out why the probability of event A is 23/81. Hmm, the problem isn't stated here, but maybe I can work backwards from the answer. Let me think... 23 and 81 are both... |
[
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"value": "1. Answer: 1) \\( \\left.a=-\\frac{1}{3}, b=\\frac{2}{3}, 2\\right) T(x)=x+1 \\)"
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"value": "<think>\nOkay, so I have this problem here where I need to find the values of a and b, and then determine the transformation T(x). The answer is given... |
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"value": "2. Answer: $x=\\frac{29 \\pi}{12}$"
},
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"value": "<think>\nOkay, so I need to figure out the problem where the answer is given as x = 29π/12. Hmm, let's see. Since the answer is an expression for x, maybe the problem was to solve an equation or f... |
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"value": "4. Answer: $\\Sigma=4^{n}+1$"
},
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"value": "<think>\nOkay, let's see. I need to figure out how to prove that the sum Σ equals 4ⁿ + 1. Hmm, the problem statement is pretty brief. It just says \"Answer: Σ=4^{n}+1\". Wait, maybe I need to figure out... |
[
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"value": "2. Masha chose five digits: $2,3,5,8$ and 9 and used only them to write down all possible four-digit numbers. For example, 2358, 8888, 9235, etc. Then, for each number, she multiplied the digits in its decimal representation, and then added up all the results. What number did Mas... |
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"value": "3. In the line, integers are recorded one after another, starting with 3, and each subsequent number, except the first, is the sum of the two numbers adjacent to it. The number 7 appeared at the 1947th position."
},
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"value": "<think>\nAlright, l... |
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"value": "3. In a string, integers are recorded one after another, starting with 5, and each subsequent number, except the first, is the sum of the two adjacent numbers. The number at the 623rd position turned out to be 6. Find the sum of the first 429 numbers."
},
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"value": "1. According to the problem, Sasha makes one step in 1 second, while Dan makes one step in $\\frac{6}{7}$ seconds. Therefore, after 6 seconds, both Sasha and Dan will have made an integer number of steps, specifically, Sasha will have made 6 steps, and Dan will have made 7 steps.... |
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"value": "5. In a convex quadrilateral $A B C D$, the lengths of sides $B C$ and $A D$ are 2 and $2 \\sqrt{2}$ respectively. The distance between the midpoints of diagonals $B D$ and $A C$ is 1. Find the angle between the lines $B C$ and $A D$."
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"value... |
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"value": "5. In a convex quadrilateral $A B C D$, the lengths of sides $B C$ and $A D$ are 6 and 8, respectively. The distance between the midpoints of diagonals $B D$ and $A C$ is 5. Find the angle between the lines $B C$ and $A D$."
},
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"value": "<think>... |
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"value": "3. For which $y$ does the equation $x^{2}+2 y^{2}+8 z^{2}-2 x y z-9=0$ have no solutions for any $z$?"
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"value": "<think>\nOkay, let me try to figure out for which values of y the equation \\( x^{2} + 2 y^{2} + 8 z^{2} - 2 x y z - 9 = 0 \\) ha... |
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"value": "5. In a convex quadrilateral $A B C D$, the lengths of sides $B C$ and $A D$ are 4 and 6 respectively. The distance between the midpoints of diagonals $B D$ and $A C$ is 3. Find the angle between the lines $B C$ and $A D$."
},
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"value": "<think>\... |
[
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"value": "4. Answer: $S=\\left(2^{9}-1\\right) \\cdot \\frac{\\left(3^{4}-1\\right)}{2} \\cdot \\frac{\\left(5^{7}-1\\right)}{4}$"
},
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"value": "<think>\nOkay, so I have this problem here: I need to find the value of S, which is given by the expression \\(... |
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"value": "5. Let $A$ and $E$ be the centers of circles $\\omega_{1}$ and $\\omega_{2}$ with radii $r$ and $R$, respectively $(r<R)$, and by the problem's condition, $A \\in \\omega_{2}$, while $B$ and $C$ are the points of intersection of these circles. Note that their common chord $BC$ is... |
[
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"value": "2. Positive integers $x, y$, for which $\\gcd(x, y)=3$, are the coordinates of a vertex of a square with its center at the origin and an area of $20 \\cdot \\text{lcm}(x, y)$. Find the perimeter of the square."
},
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"value": "<think>\nOkay, so I n... |
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"value": "5. The center of a circle with radius 2 lies on the circumference of a circle with radius 3. Find the area of the intersection of the circles bounded by these circumferences."
},
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"value": "<think>\nOkay, let's tackle this geometry problem. So, w... |
[
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"value": "5. The center of a circle with a radius of 2 lies on the circumference of a circle with a radius of 5. Find the area of the union of the circles bounded by these circumferences."
},
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"value": "<think>\nAlright, so I need to find the area of the u... |
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"value": "2. Positive integers $x, y$, for which $\\gcd(x, y)=5$, are the coordinates of a vertex of a square with its center at the origin and an area of $\\frac{169}{6} \\cdot \\text{lcm}(x, y)$. Find the length of the side of the square."
},
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"value": "... |
[
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"value": "5. In triangle $A B C$, the perpendicular bisectors of sides $A B$ and $A C$ intersect lines $A C$ and $A B$ at points $N$ and $M$ respectively. The length of segment $N M$ is equal to the length of side $B C$ of the triangle. The angle at vertex $C$ of the triangle is $40^{\\cir... |
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"value": "4. The sum $b_{7}+b_{6}+\\ldots+b_{2019}$ of the terms of the geometric progression $\\left\\{b_{n}\\right\\}, b_{n}>0$ is 27, and the sum of their reciprocals $\\frac{1}{b_{7}}+\\frac{1}{b_{6}}+\\ldots+\\frac{1}{b_{2019}}$ is 3. Find the product $b_{7} \\cdot b_{6} \\cdot \\ldot... |
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"value": "4. The sum $b_{6}+b_{7}+\\ldots+b_{2018}$ of the terms of the geometric progression $\\left\\{b_{n}\\right\\}, b_{n}>0$ is 6. The sum of the same terms taken with alternating signs $b_{6}-b_{7}+b_{8}-\\ldots-b_{2017}+b_{2018}$ is 3. Find the sum of the squares of the same terms $... |
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"value": "4. The sum $b_{8}^{2}+b_{9}^{2}+\\ldots+b_{2020}^{2}$ of the squares of the terms of the geometric progression $\\left\\{b_{n}\\right\\}, b_{n}>0$ is 4. The sum of their reciprocals $\\frac{1}{b_{8}^{2}}+\\frac{1}{b_{9}^{2}}+\\ldots+\\frac{1}{b_{2020}^{2}}$ is 1. Find the product... |
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"value": "3. Calculate the value of the expression $\\sin \\frac{b \\pi}{36}$, where $b$ is the sum of all distinct numbers obtained from the number $a=987654321$ by cyclic permutations of its digits (in a cyclic permutation, all digits of the number, except the last one, are shifted one p... |
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"value": "5. Circles $K_{1}$ and $K_{2}$ with centers at points $O_{1}$ and $O_{2}$ have the same radius $r$ and touch the circle $K$ of radius $R$ with center at point $O$ internally. The angle $O_{1} O O_{2}$ is $120^{\\circ}$. Find the radius $q$ of the circle that touches $K_{1}$ and $... |
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"value": "1. Let $x$ be the number of happy subjects; $y$ be the number of redheads, and $z$ be the total number of subjects. According to the condition,\n\n$$\n\\frac{40}{100} x=\\frac{60}{100} y, \\quad z=x+\\frac{40}{100} y=x+\\frac{2}{5} y\n$$\n\nWe have $2 x=3 y$. Express all variable... |
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"value": "2. Let $A=\\overline{a b c b a}$ be a five-digit symmetric number, $a \\neq 0$. If $1 \\leq a \\leq 8$, then the last digit of the number $A+11$ will be $a+1$, and therefore the first digit in the representation of $A+11$ should also be $a+1$. This is possible only with a carry-o... |
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"value": "5. Let the distance between points $A$ and $B$ be $L$ mm, and the length of the flea's jump be $d$ mm. If $L$ were divisible by $d$ exactly, the flea's strategy of jumping in a straight line connecting points $A$ and $B$ would be optimal. It would take $\\frac{L}{d}=n, n \\in \\m... |
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"value": "2. Integers, the decimal representation of which reads the same from left to right and from right to left, we will call symmetric. For example, the number 513315 is symmetric, while 513325 is not. How many six-digit symmetric numbers exist such that adding 110 to them leaves them... |
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"value": "5. Kuzya the flea can make a jump in any direction on a plane for exactly 15 mm. Her task is to get from point $A$ to point $B$ on the plane, the distance between which is 2020 cm. What is the minimum number of jumps she must make to do this?"
},
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... |
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"value": "1. In the confectionery, there were either people with excess weight or sweet lovers (possibly both). It turned out that $80 \\%$ of people with excess weight love sweets, and $70 \\%$ of sweet lovers have excess weight. What fraction of those present love sweets but do not suffe... |
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"value": "2. Integers, the decimal representation of which reads the same from left to right and from right to left, we will call symmetric. For example, the number 5134315 is symmetric, while 5134415 is not. How many seven-digit symmetric numbers exist such that adding 1100 to them leaves... |
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"value": "5. Kuzya the flea can make a jump in any direction on a plane for exactly 17 mm. Its task is to get from point $A$ to point $B$ on the plane, the distance between which is 1947 cm. What is the minimum number of jumps it must make to do this?"
},
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... |
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"value": "2. Integers, whose decimal notation reads the same from left to right and from right to left, we will call symmetric. For example, the number 513151315 is symmetric, while 513152315 is not. How many nine-digit symmetric numbers exist such that adding 11000 to them leaves them sym... |
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"value": "5. Kuzya the flea can make a jump in any direction on a plane for exactly 19 mm. Her task is to get from point $A$ to point $B$ on the plane, the distance between which is 1812 cm. What is the minimum number of jumps she must make to do this?"
},
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... |
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"value": "3. Find four consecutive integers for which the cube of the largest is equal to the sum of the cubes of the others."
},
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"value": "<think>\nOkay, so I have this problem here: Find four consecutive integers where the cube of the largest is equal t... |
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"value": "4. Consider a rectangular parallelepiped, the volume of which is numerically equal to the sum of the lengths of its edges. Find all possible values of its volume given that the lengths of its edges are integers."
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"value": "<think>\nOkay, so I... |
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"value": "6. A plot of land in the form of a right-angled triangle with legs of 4 and 3 needs to be divided by a line $L$ into two plots such that 1) the plots have equal area; 2) the length of the common boundary (fence) of the plots is minimized. Indicate the points on the sides of the t... |
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"value": "1. When buying 2 shirts and 1 tie, the payment was 2600 rubles. Then the second purchase will have a $25 \\%$ discount. The price of the $3-rd$ shirt is $1200 \\cdot 0.75=900$ rubles. The total price of the entire purchase is $2600+900=3500$ rubles."
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"value": "1. When purchasing goods for an amount of no less than 1000 rubles, the store provides a discount on subsequent purchases of $50 \\%$. Having 1200 rubles in her pocket, Dasha wanted to buy 4 kg of strawberries and 6 kg of sugar. In the store, strawberries were sold at a price of ... |
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"value": "4. Petya wrote in the first column all five-digit numbers, in the notation of which digits from 0 to 7 are used, and in the second - the products of the digits of each such number. Vasya managed to calculate in his mind the sum of the numbers written in the second column. What re... |
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"value": "4. Find a polynomial $P(x)$ of degree 2019, for which $P(2018)=2020$ and $P(2014+x)+P(2022-x)=4040$ for all $x$."
},
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"value": "<think>\nOkay, so I need to find a polynomial P(x) of degree 2019 that satisfies two conditions: P(2018) = 2020 and P(... |
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"value": "5. Points $M, N, P$ are the feet of the altitudes dropped from the vertices of triangle $A B C$ with angles $30^{\\circ}, 75^{\\circ}, 75^{\\circ}$ to its sides. Find the ratio of the areas of triangles $M N P$ and $A B C$."
},
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"value": "<think>... |
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"value": "4. Find a polynomial $P(x)$ of degree 2024, for which $P(2006)=2020, P(2007)=2018 \\quad$ and $\\quad P(2014+x)=P(2000-x)$ for all $x$."
},
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"value": "<think>\nOkay, let me try to figure out how to solve this problem. So, we need to find a polyno... |
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"value": "5. Points $M, N, P$ are the feet of the altitudes dropped from the vertices of triangle $A B C$ with angles $37.5^{\\circ}, 60^{\\circ}, 82.5^{\\circ}$ to its sides. Find the ratio of the areas of triangles $M N P$ and $A B C$."
},
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"from": "assistant",
"value": "<th... |
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"value": "5. Points $M, N, P$ are the feet of the altitudes dropped from the vertices of triangle $A B C$ with angles $37.5^{0}, 67.5^{0}, 75^{0}$ to its sides. Find the ratio of the areas of triangles $M N P$ and $A B C$."
},
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"value": "<think>\nOkay, so ... |
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"value": "1. Pete and Vova want to cross a river 160 (m) wide. They have a small rubber boat, in which only one person can paddle at a speed of 32 (m/min). They started the crossing simultaneously: Pete in the boat, Vova - swimming. At some point, Pete left the boat and continued swimming ... |
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"value": "1. According to the properties of logarithms, after transformations we get\n\n$$\n\\begin{aligned}\n& \\log _{2}\\left(b_{2} b_{3} \\ldots b_{n}\\right)=\\log _{2} b_{1}^{2} \\\\\n& b_{2} b_{3} \\ldots b_{n}=b_{1}^{2}\n\\end{aligned}\n$$\n\nUsing the formula for the general term ... |
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"value": "4. Let $x$ and $y$ be the lengths of two segments on the left side. The length of the third segment is $L-(x+y)$. The sample space consists of all pairs $(x, y)$ for which\n\n$$\n\\left\\{\\begin{array} { l } \n{ 0 \\leq x \\leq L } \\\\\n{ 0 \\leq y \\leq L } \\\\\n{ 0 \\leq L -... |
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"value": "5. The ratio of the lengths of the parallel sides $BC$ and $AD$ of trapezoid $ABCD$ is $1:3$. Diagonals $BD$ and $AC$ intersect at point $O$. Find the ratio of the areas of triangle $AOD$ and the trapezoid.\n\n## Answers and solutions\n\nProblem 1 Answer: 1) it is impossible. 2) ... |
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"value": "3. Note that $(n+1)^{2}-(n-1)^{2}=4 n$, i.e., numbers $a=4 n$ that are multiples of four can be represented as the difference of squares of two integers, and therefore, with the condition of zero, can be represented as the sum or difference of four squares. If $a=4 n \\pm 1$, the... |
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"value": "4. Let's highlight the complete squares: $\\left\\{\\begin{array}{l}(x-2)^{2}+(y+1)^{2}=5, \\\\ (x-2)^{2}+(z-3)^{2}=13, \\\\ (y+1)^{2}+(z-3)^{2}=10 .\\end{array}\\right.$\n\nAdd all the equations $\\left\\{\\begin{array}{c}(x-2)^{2}+(y+1)^{2}=5, \\\\ (x-2)^{2}+(z-3)^{2}=13, \\\\ ... |
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"value": "3. Represent the number 1917 as the sum or difference of the squares of three integers. Prove that any integer can be represented as the sum or difference of the squares of four integers."
},
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"value": "<think>\nOkay, so I need to figure out how ... |
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"value": "4. Solve the system $\\left\\{\\begin{array}{c}x^{2}+y^{2}+2 x+6 y=-5 \\\\ x^{2}+z^{2}+2 x-4 z=8 \\\\ y^{2}+z^{2}+6 y-4 z=-3\\end{array}\\right.$."
},
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"value": "<think>\nOkay, let's see. I need to solve this system of equations with three variab... |
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"value": "3. Represent the number 1947 as the sum or difference of the squares of three integers. Prove that any integer can be represented as the sum or difference of the squares of four integers."
},
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"value": "<think>\nOkay, let's see. The problem has t... |
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"value": "4. Solve the system $\\left\\{\\begin{array}{c}x^{2}+y^{2}+8 x-6 y=-20 \\\\ x^{2}+z^{2}+8 x+4 z=-10 \\\\ y^{2}+z^{2}-6 y+4 z=0\\end{array}\\right.$."
},
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"value": "<think>\nOkay, so I need to solve this system of three equations with three variab... |
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"value": "3. Represent the number 2019 as the sum or difference of the squares of three integers. Prove that any integer can be represented as the sum or difference of the squares of four integers."
},
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"value": "<think>\nOkay, so I need to figure out how ... |
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"value": "3. Let's show that the numbers $a(a+4)$ and $b(b+4)$ cannot be squares. Indeed, suppose, for example,\n\n$$\na(a+4)=c^{2}, c \\in Z, c>0\n$$\n\nRewrite this equation as\n\n$$\n(a+2)^{2}-c^{2}=4 \\quad \\text { or } \\quad(a+c+2)(a-c+2)=4\n$$\n\nConsider the left-hand side of this... |
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"value": "2. Represent the number 1998 as the sum of the cubes of four integers."
},
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"value": "<think>\nOkay, so I need to figure out how to represent the number 1998 as the sum of the cubes of four integers. Hmm, that sounds a bit challenging, but let me... |
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"value": "4. \n\n\n\nThe line $2 y + (x + 4) = 2a$ intersects the line $-x + y = 3$ at a point with abscissa $x = \\frac{2a - 10}{3}$. Considering $x \\in [-3 ; 0... |
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"value": "1. The polynomial $p_{1}=x-a$ can have a root $x=a$ coinciding with one of the roots of the product $p(x)=p_{1}(x) \\cdot p_{2}(x)$.\n\nCase $1 \\quad a=1$\n\nThen the polynomial $p_{2}(x)=(x-1)^{r}(x-2)^{s}(x+3)^{t}$, where $r \\geq 1, s \\geq 1, t \\geq 1-$ are integers, $r+s+t... |
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"value": "4. The discriminant $D=(1-\\xi)^{2}+4 \\xi(\\xi+2)=0 \\rightarrow 5 \\xi^{2}+6 \\xi+1=0 \\rightarrow\\left[\\begin{array}{c}\\xi=-1 \\\\ \\xi=-1 / 5\\end{array}\\right.$\n\nThe probability that $\\xi$ takes one of the values -1 or $-1 / 5$ is $P(A)=\\frac{3}{16}+\\frac{1}{4}=\\fr... |
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"value": "6. \n\n\n\nFig 1\n\n$DM$ is a tangent line drawn from vertex $D$, $ON$ is a perpendicular from point $O$ to $DM$, $PQ$ is a tangent to the circle paralle... |
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"value": "2. Answer: $\\Sigma_{a}=1680$"
},
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"value": "<think>\nOkay, let's see. The problem is to find Σ_a = 1680. Hmm, Σ usually denotes a sum, so maybe it's the sum of some terms involving 'a' that equals 1680. But the problem statement is a bit vague. ... |
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"value": "3. Answer: 1) 167 numbers; 2) $x_{\\max }=975$"
},
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"value": "<think>\nOkay, let's see. The problem mentions two parts: first, finding how many numbers there are, which is 167, and second, determining the maximum value x_max as 975. Hmm, I need t... |
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"value": "4. Answer: $400 \\mathrm{~cm}^{2}$"
},
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"value": "<think>\nOkay, so I need to solve a problem where the answer is 400 cm². Let me start by thinking about what kind of problem would result in an area of 400 square centimeters. Hmm, areas are calcu... |
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"value": "6. A circle can be circumscribed around quadrilateral $A B C D$. The lengths of sides $A B$ and $A D$ are equal.\n\nOn side $C D$ is point $Q$ such that $D Q=1$, and on side $B C$ is point $P$ such that $B P=5$.\n\nFurthermore, $\\measuredangle D A B=2 \\measuredangle Q A P$. Fin... |
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"value": "1. Suppose that while one worker was digging the trench, the other workers did not rest but continued to dig the trench for the next pipes. By the time the fourth worker finished digging 50 m, an additional $4 \\cdot \\frac{3}{4} \\cdot 50=150 \\mathrm{m}$\nof trench had been dug... |
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"value": "2. Adding the first equation to the other two, we get:\n\n$$\nx^{2}+2 x+y^{2}+4 y+z^{2}+6 z=-14\n$$\n\nNext, we complete the square for each variable. As a result, we obtain:\n\n$$\n(x+1)^{2}+(y+2)^{2}+(z+3)^{2}=0\n$$\n\nTherefore, $\\mathrm{x}=-1, y=-2, z=-3$ is the only possibl... |
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"value": "3. Let $p_{1}, p_{2}, \\ldots, p_{m}$ be the prime divisors of $a$. Among them, by condition, there are 3 and 5, for example, $\\quad p_{1}=3, p_{2}=5$. Then $a=3^{s_{1}} \\cdot 5^{s_{2}} \\cdot p_{3}^{s_{3}} \\cdot \\ldots \\cdot p_{m}^{s_{m}}$ and the total number of its diviso... |
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"value": "5. Let's introduce the following notations: $Q$ - the center of the first circle, $F, E, B$ - the points of tangency of the circles and lines, $G$ - the point of tangency of the circles, $AD$ - the common tangent (see the figure). Draw the line $OC$ parallel to the line $FA$. Let... |
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"value": "1. A team of five people is laying a 30 m pipe in the ground. To prepare the trench, they decided to dig in turns with the condition: each person digs for as long as the other four could dig one-third of the trench together. The 30 m trench was dug when the last member of the tea... |
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"value": "2. Solve the system $\\left\\{\\begin{array}{c}x^{2}+y-2 z=-3 \\\\ 3 x+y+z^{2}=14 \\\\ 7 x-y^{2}+4 z=25\\end{array}\\right.$"
},
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"value": "<think>\nOkay, so I have this system of equations to solve. Let me write them down again to make sure I ha... |
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"value": "3. A natural number $a$ is divisible by 21 and has 105 different divisors, including 1 and $a$. Find the smallest such $a$."
},
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"value": "<think>\nOkay, so I need to find the smallest natural number a that is divisible by 21 and has exactly 105 ... |
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"value": "2. Solve the system $\\left\\{\\begin{array}{c}x^{2}-3 y+z=-4 \\\\ x-3 y+z^{2}=-10 \\\\ 3 x+y^{2}-3 z=0\\end{array}\\right.$"
},
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"value": "<think>\nOkay, so I need to solve this system of equations. Let me write them down again to make sure I ha... |
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"value": "3. A natural number $a$ is divisible by 35 and has 75 different divisors, including 1 and $a$. Find the smallest such $a$."
},
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"value": "<think>\nOkay, so I have this problem here: find the smallest natural number a that is divisible by 35 and h... |
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"value": "2. Solve the system $\\left\\{\\begin{array}{c}x^{2}-y-z=8 \\\\ 4 x+y^{2}+3 z=-11 \\\\ 2 x-3 y+z^{2}=-11\\end{array}\\right.$"
},
{
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"value": "<think>\nOkay, let's try to solve this system of equations. Hmm, there are three equations with three var... |
[
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"value": "3. A natural number $a$ is divisible by 55 and has 117 distinct divisors, including 1 and $a$. Find the smallest such $a$."
},
{
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"value": "<think>\nOkay, so I need to find the smallest natural number 'a' that is divisible by 55 and has 117 distinc... |
[
{
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"value": "5. The border of a square with a side of 9, cut out of white cardboard, is painted red. It is necessary to cut the square into 6 equal-area parts, the boundaries of which contain segments painted red with the same total length.\n\n## Solutions\n\nOption 1\n\nProblem 1"
},
{
... |
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"value": "# 2.3. Qualifying round of the \"Rosatom\" Olympiad, 11th grade\n\n## Answers and solutions\n\nProblem 1 Answer 345"
},
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"value": "<think>\nOkay, let's see. I need to figure out problem 1 from the qualifying round of the \"Rosatom\" Olympiad for ... |
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"value": "Problem 4. Answer: $x=35$"
},
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"value": "<think>\nOkay, so I need to figure out Problem 4 where the answer is supposed to be x = 35. Let me start by thinking about what kind of problem could lead to that answer. Since the answer is a numerical va... |
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"value": "2.11. Final round of the \"Rosatom\" Olympiad, 10th grade\n\nAnswers and solutions\n\n$\\begin{array}{lll}\\text { Problem } 1 \\text { Answer: 1) } a=3, b=12, c=48 & \\text { 2) } a=363, b=462, c=588\\end{array}$\n\nSystem of conditions: $\\left\\{\\begin{array}{c}b^{2}=a c \\\\... |
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"value": "5. Answer: $M P: P B=2: 3$"
},
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"value": "<think>\nOkay, let's see. The problem says \"MP : PB = 2 : 3\". Hmm, I need to figure out what this ratio represents. Are these points on a line segment? Maybe M, P, and B are three points where P divides... |
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"value": "3. In the decimal representation of a six-digit number $a$, there are no zeros and the sum of its digits is 18. Find the sum of all different numbers obtained from the number $a$ by cyclic permutations of its digits. In a cyclic permutation, all digits of the number, except the l... |
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"value": "5. On a sheet of paper, 14 parallel lines $L$ and 15 lines $P$ perpendicular to them are drawn. The distances between adjacent lines from $L$ from the first to the last are given: 2;4;6;2;4;6;2;4;6;2;4;6;2. The distances between adjacent lines from $P$ are also known: 3;1;2;6;3;1... |
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"value": "4. Solve the equation $\\left(\\log _{3}(2 x+1)+\\sqrt{\\log _{3}{ }^{2}(2 x+1)+1}\\right)\\left(\\log _{3}(3 x+2)+\\sqrt{\\log _{3}{ }^{2}(3 x+2)+1}\\right)=1$"
},
{
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"value": "<think>\nOkay, so I have this equation to solve: \n\n\\[\n\\left(\\log... |
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"value": "5. On the cells of a chessboard of size $8 \\times 8$, 5 identical pieces are randomly placed. Find the probability that four of them will be located either on the same row, or on the same column, or on one of the two main diagonals."
},
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"value"... |
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"value": "4. Solve the equation $\\left(\\log _{4}(x+1)+\\sqrt{\\log _{4}^{2}(x+1)+1}\\right)\\left(\\log _{4}(3 x+5)+\\sqrt{\\log _{4}^{2}(3 x+5)+1}\\right)=1$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve this equation: \n\n$\\left(\\log _{4}(x+1)+\\... |
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