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"value": "XLII OM - I - Problem 8\n\nDetermine the largest natural number $ n $ for which there exist in space $ n+1 $ polyhedra $ W_0, W_1, \\ldots, W_n $ with the following properties:\n(1) $ W_0 $ is a convex polyhedron with a center of symmetry,\n(2) each of the polyhedra $ W_i $ ($ i ... |
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"value": "XI OM - I - Task 3\n\nEach side of a triangle with a given area $ S $ is divided into three equal parts, and the points of division are connected by segments, skipping one point to form two triangles. Calculate the area of the hexagon that is the common part of these triangles."
... |
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"value": "X OM - I - Task 8\n\nThe sides of a triangle are the medians of another triangle. Calculate the ratio of the areas of both triangles. Can a triangle be constructed from the medians of any triangle?"
},
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"value": "<think>\nOkay, so I need to figur... |
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"value": "XXIV OM - III - Task 2\n\nLet $ p_n $ be the probability that a series of 100 consecutive heads will appear in $ n $ coin tosses. Prove that the sequence of numbers $ p_n $ is convergent and calculate its limit."
},
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"value": "<think>\nOkay, so I... |
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"value": "XXV - I - Problem 11\n\nLet $ X_n $ and $ Y_n $ be independent random variables with the same distribution $ \\left{ \\left(\\frac{k}{2^n}, \\frac{1}{2^n}\\right) : k = 0, 1, \\ldots, 2^n-1\\right} $. Denote by $ p_n $ the probability of the event that there exists a real number ... |
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"value": "XXXVIII OM - II - Task 1\n\nFrom an urn containing one ball marked with the number 1, two balls marked with the number 2, ..., $ n $ balls marked with the number $ n $, we draw two balls without replacement. We assume that drawing each ball from the urn is equally probable. Calcu... |
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"value": "LX OM - I - Task 5\n\nFor each integer $ n \\geqslant 1 $, determine the largest possible number of different subsets of the set $ \\{1,2,3, \\cdots,n\\} $ with the following property: Any two of these subsets are either disjoint or one is contained in the other."
},
{
"f... |
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"value": "LIV OM - I - Problem 9\n\nFind all polynomials $ W $ with real coefficients that have the following property: if $ x+ y $ is a rational number, then\n\n\n\n\n\n\nis a rational number."
},
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"value": "<think>\nAlright, let me try to figure out this... |
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"value": "XXXII - II - Task 4\n\nGiven are natural numbers $ k, n $. We define inductively two sequences of numbers $ (a_j) $ and $ (r_j) $ as follows:\nFirst step: divide $ k $ by $ n $ and obtain the quotient $ a_1 $ and the remainder $ r_1 $,\nj-th step: divide $ k + r_{j-1} $ by $ n $ ... |
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"value": "XXII OM - III - Problem 5\n\nFind the largest integer $ A $ such that for every permutation of the set of natural numbers not greater than 100, the sum of some 10 consecutive terms is at least $ A $."
},
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"from": "assistant",
"value": "<think>\nOkay, so I need to fin... |
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"value": "XV OM - II - Task 3\n\nProve that if three prime numbers form an arithmetic progression with a difference not divisible by 6, then the smallest of these numbers is $3$."
},
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"value": "<think>\nOkay, so I need to prove that if three prime numbers ... |
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"value": "XXXIX OM - III - Problem 6\n\nCalculate the maximum volume of a tetrahedron contained within a hemisphere of radius 1."
},
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"value": "<think>\nOkay, so I need to figure out the maximum volume of a tetrahedron that can fit inside a hemisphere with... |
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"value": "XLVI OM - I - Zadanie 1\n\nWyznaczyć wszystkie pary $ (x,y) $ liczb naturalnych, dla których liczby $ \\frac{x+1}{y} $ oraz $ \\frac{y+1}{x} $ są naturalne.\n"
},
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"value": "<think>\nOkay, let's try to solve this problem. We need to find all pair... |
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"value": "XVIII OM - I - Zadanie 10\n\nTrójka liczb całkowitych tworzy postęp geometryczny o ilorazie całkowitym. Gdy najmniejszą z nich zwiększymy o 9, powstanie postęp arytmetyczny. Jakie to liczby?\n"
},
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"value": "<think>\nOkay, so I've got this proble... |
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"value": "LII OM - III - Task 4\n\nGiven such integers $ a $ and $ b $ that for every non-negative integer $ n $ the number $ 2^na + b $ is a square of an integer. Prove that $ a = 0 $.\n\n"
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"value": "<think>\nOkay, so I have this problem where I need ... |
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"value": "XLIII OM - I - Problem 2\n\nIn square $ABCD$ with side length $1$, point $E$ lies on side $BC$, point $F$ lies on side $CD$, the measures of angles $EAB$ and $EAF$ are $20^{\\circ}$ and $45^{\\circ}$, respectively. Calculate the height of triangle $AEF$ drawn from vertex $A$."
... |
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"value": "XXXI - I - Problem 7\n\nOn a sphere of radius 1, 13 points are chosen. Prove that among these points, there are two whose distance is less than $ \\frac{2\\sqrt{3}}{3} $."
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"value": "<think>\nOkay, so I need to prove that if you have 13 points... |
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"value": "XXV - I - Problem 9\n\nA set of documents has been divided into $ n $ parts, each given to one of $ n $ people, each of whom has a phone. Prove that for $ n \\geq 4 $, it suffices to have $ 2n - 4 $ phone calls, after which each of the $ n $ people knows the content of all the do... |
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"value": "L OM - I - Task 10\n\nGiven is a number $ x_1 > 0 $. The sequence $ (x_n) $ is defined by the formula:\n\n\n\n\n\n\nProve that the limit $ \\lim_{n\\to \\infty} \\frac{x_n}{\\sqrt[3]{n}} $ exists and calculate it."
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"value": "<think>\nOkay, so... |
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"value": "XV OM - I - Problem 10\n\nFind the $ n $-th term of the sequence of numbers $ a_1, a_2, a_n, \\ldots $, where $ a_1 = 1 $, $ a_2 = 3 $, $ a_3 = 6 $, and for every natural number $ k $"
},
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"value": "<think>\nOkay, so I need to find the nth term o... |
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"value": "XLVI OM - III - Problem 1\n\nDetermine the number of subsets of the set $ \\{1,2, \\ldots , 2n\\} $ in which the equation $ x + y = 2n+1 $ has no solutions."
},
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"value": "<think>\nOkay, let's tackle this problem. We need to find the number of su... |
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"value": "LI OM - I - Task 10\n\nIn space, there are three mutually perpendicular unit vectors $ \\overrightarrow{OA} $, $ \\overrightarrow{OB} $, $ \\overrightarrow{OC} $. Let $ \\omega $ be a plane passing through the point $ O $, and let $ A', $ B', $ C' $ be the projections of points $... |
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"value": "XLVI OM - III - Problem 2\n\nThe diagonals of a convex pentagon divide this pentagon into a pentagon and ten triangles. What is the maximum possible number of triangles with equal areas?"
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"value": "<think>\nOkay, so I have this problem about ... |
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"value": "LII OM - I - Problem 3\n\nFind all natural numbers $ n \\geq 2 $ such that the inequality\n\n\n\n\n\n\nholds for any positive real numbers $ x_1,x_2,\\ldots,x_n $."
},
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"value": "<think>\nOkay, so I need to find all natural numbers n ≥ 2 such tha... |
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"value": "XXXVI OM - II - Problem 3\n\nLet $ L $ be the set of all broken lines $ ABCDA $, where $ A, B, C, D $ are distinct vertices of a fixed regular 1985-gon. A broken line is randomly chosen from the set $ L $. Calculate the probability that it is the boundary of a convex quadrilatera... |
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"value": "XXVIII - II - Task 6\n\nWhat is the maximum number of parts that the edges of $ n $ squares can divide a plane into?"
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"value": "<think>\nOkay, so the problem is asking for the maximum number of regions that the edges of n squares can divide a... |
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"value": "XLIV OM - I - Problem 11\n\nIn six different cells of an $ n \\times n $ table, we place a cross; all arrangements of crosses are equally probable. Let $ p_n $ be the probability that in some row or column there will be at least two crosses. Calculate the limit of the sequence $ ... |
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"value": "I OM - B - Task 20\n\nThe sides of a right-angled triangle are expressed by natural numbers. One of the legs is expressed by the number 10. Calculate the remaining sides of this triangle."
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"value": "<think>\nOkay, so I need to find the remain... |
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"value": "XLVI OM - I - Zadanie 2\n\nDana jest liczba naturalna $ n \\geq 2 $. Rozwiązać układ równań\n\n\n\n\n\n\n"
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"value": "<think>\nOkay, so I need to solve this system of equations for a natural number n ≥ 2. Wait, the problem is in Polish, but I ... |
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"value": "XXVI - I - Task 1\n\nAt the ball, there were 42 people. Lady $ A_1 $ danced with 7 gentlemen, Lady $ A_2 $ danced with 8 gentlemen, ..., Lady $ A_n $ danced with all the gentlemen. How many gentlemen were at the ball?"
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"value": "<think>\nOkay... |
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"value": "I OM - B - Task 5\n\n\nA motorboat set off upstream at 9:00, and at the same moment, a ball was thrown from the motorboat into the river. At 9:15, the motorboat turned around and started moving downstream. At what time did the motorboat catch up with the ball?"
},
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"from... |
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"value": "XLVII OM - I - Problem 11\n\nIn a ski jumping competition, 65 participants are involved. They jump one by one according to a predetermined order. Each participant performs one jump. We assume that the scores obtained are all non-zero and that every final ranking is equally likely... |
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"value": "XXXVI OM - I - Problem 11\n\nProvide an example of a convex polyhedron having 1985 faces, among which there are 993 faces such that no two of them share a common edge."
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"value": "<think>\nOkay, so I need to find a convex polyhedron with 1985 ... |
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"value": "XXXV OM - III - Task 4\n\nWe toss a coin $ n $ times and record the result as a sequence $ (a_1, a_2, \\ldots, a_n) $, where $ a_i = 1 $ or $ a_i = 2 $ depending on whether an eagle or a tail appeared in the $ i $-th toss. We assume $ b_j = a_1 + a_2 + \\ldots + a_j $ for $ j = 1... |
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"value": "L OM - II - Task 5\n\nLet $ S = \\{1, 2,3,4, 5\\} $. Determine the number of functions $ f: S \\to S $ satisfying the equation $ f^{50} (x) = x $ for all $ x \\in S $.\nNote: $ f^{50}(x) = \\underbrace{f \\circ f \\circ \\ldots \\circ f}_{50} (x) $."
},
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"value": "XXX OM - I - Task 2\n\nIn the plane, there are $ n $ points ($ n > 4 $), no three of which are collinear. Prove that there are at least $ \\binom{n-3}{2} $ convex quadrilaterals whose vertices are some of the given points."
},
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"value": "<think>\... |
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"value": "XIV OM - I - Problem 4\n\nIn the great circle of a sphere with radius $ r $, a regular polygon with $ n $ sides is inscribed. Prove that the sum of the squares of the distances from any point $ P $ on the surface of the sphere to the vertices of the polygon equals $ 2nr^2 $."
}... |
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"value": "LX OM - I - Task 10\n\nPoint $ P $ is the midpoint of the shorter arc $ BC $ of the circumcircle of\ntriangle $ ABC $, where $ \\measuredangle BAC =60^{\\circ} $. Point $ M $ is the midpoint\nof the segment connecting the centers of two excircles of the given triangle,\ntangent t... |
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"value": "XII OM - III - Task 6\n\nSomeone wrote six letters to six people and addressed six envelopes to them. In how many ways can the letters be placed into the envelopes so that no letter ends up in the correct envelope?"
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"value": "<think>\nOkay, s... |
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"value": "II OM - I - Task 10\n\nA circle is circumscribed around triangle $ ABC $. Given the radius $ R $ of this circle, calculate the radius of the circle passing through the centers of the three excircles (inscribed in the angles) of triangle $ ABC $."
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"from": "assistant",... |
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"value": "XXIII OM - I - Problem 9\n\nHow many natural numbers less than $ 10^n $ have a decimal representation whose digits form a non-decreasing sequence?"
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"value": "<think>\nOkay, so I need to figure out how many natural numbers less than \\(10^n\\)... |
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"value": "XLIII OM - II - Problem 5\n\nDetermine the supremum of the volumes of spheres contained in tetrahedra with all altitudes no longer than $1$."
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"value": "<think>\nOkay, so I need to find the supremum of the volumes of spheres that can be contai... |
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"value": "LI OM - III - Task 5\n\nFor a given natural number $ n \\geq 2 $, find the smallest number $ k $ with the following property. From any $ k $-element set of fields of the $ n \\times n $ chessboard, one can select a non-empty subset such that the number of fields of this subset in... |
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"value": "IX OM - I - Problem 3\n\nFrom triangle $ ABC $, three corner triangles have been cut off by lines parallel to the sides of the triangle and tangent to the inscribed circle of this triangle. Prove that the sum of the radii of the circles inscribed in the cut-off triangles is equal... |
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"value": "XXII OM - III - Task 3\n\nHow many locks at least need to be placed on the treasury so that with a certain distribution of keys among the 11-member committee authorized to open the treasury, any 6 members can open it, but no 5 can? Determine the distribution of keys among the com... |
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"value": "LVIII OM - I - Problem 4\n\nFor each natural number $ n\\ge 3 $, determine the number of sequences $ (c_1,c_2,\\ldots,c_n) $, where $ {c_i\\in\\{0,1,\\ldots,9\\}} $, with the following property:\\break in every triple of consecutive terms, there are at least two terms that are eq... |
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"value": "XXVI - II - Task 3\n\nIn a certain family, the husband and wife have made the following agreement: If the wife washes the dishes on a certain day, then the husband washes the dishes the next day. If, however, the husband washes the dishes on a certain day, then who washes the dis... |
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"value": "4. Each experience of throwing the circle corresponds to a point $M-$ the position of the center of the circle on the vertical segment $[A ; B]$ of length 1.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_8b002ac84843a2e42fafg-05.jpg?height=315&width=702&top_left_y=1016&top_le... |
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"value": "6. Let's introduce the following notations: $S$ - the area of the trapezoid, $S_{1}$ - the area of triangle $ABC$, $S_{2}$ - the area of triangle $ACD$, $h$ - the height of the trapezoid, $\\gamma=\\frac{q}{p}$, $\\mu=\\frac{n}{m}$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_... |
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"value": "1. Natural numbers $x$ and $y$ are such that their $\\operatorname{GCD}(x, y)=16$, and $\\operatorname{LCM}\\left(\\log _{8} x, \\log _{12} y\\right)=18$. Find these numbers."
},
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"value": "<think>\nOkay, let's see. I need to find natural numbers... |
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"value": "4. On a plane, an infinite number of parallel lines are drawn, each separated from the next by a distance of 1. A circle with a diameter of 1 is randomly thrown onto the plane. Find the probability that a line intersecting the circle divides it into parts such that the ratio of t... |
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"value": "6. Point $N$ divides the diagonal $A C$ of trapezoid $A B C D$ in the ratio $C N: N A=3$. The lengths of the bases $B C$ and $A D$ of the trapezoid are in the ratio $1: 2$. A line is drawn through point $N$ and vertex $D$, intersecting the lateral side $A B$ at point $M$. What fr... |
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"value": "4. On a plane, an infinite number of parallel lines are drawn, each separated from the next by a distance of 1. A circle with a diameter of 1 is randomly thrown onto the plane. Find the probability that a line intersecting the circle divides it into parts such that the ratio of t... |
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"value": "6. Point $N$ divides the diagonal $A C$ of trapezoid $A B C D$ in the ratio $C N: N A=4$. The lengths of the bases $B C$ and $A D$ of the trapezoid are in the ratio $2: 3$. A line is drawn through point $N$ and vertex $D$, intersecting the lateral side $A B$ at point $M$. What fr... |
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"value": "1. Natural numbers $x$ and $y$ are such that their LCM $(x, y)=5^{4} \\cdot 2^{6}$, and LCM $\\left(\\log _{10} x, \\log _{40} y\\right)=4$. Find these numbers."
},
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"value": "<think>\nOkay, let me try to figure out this problem. Hmm, so we have ... |
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"value": "4. On a plane, an infinite number of parallel lines are drawn, each at a distance of 1 from each other. A circle with a diameter of 1 is randomly thrown onto the plane. Find the probability that a line intersecting the circle divides it into parts, the ratio of the areas of which... |
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"value": "6. Point $N$ divides the diagonal $A C$ of trapezoid $A B C D$ in half. The lengths of the bases $B C$ and $A D$ of the trapezoid are in the ratio $1: 4$. A line is drawn through point $N$ and vertex $D$, intersecting the lateral side $A B$ at point $M$. What fraction of the area... |
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"value": "1. Solution. According to the problem, the sum of the original numbers is represented by the expression:\n\n$$\n\\begin{aligned}\n& \\left(a_{1}+2\\right)^{2}+\\left(a_{2}+2\\right)^{2}+\\ldots+\\left(a_{50}+2\\right)^{2}=a_{1}^{2}+a_{2}^{2}+\\ldots+a_{50}^{2} \\rightarrow \\\\\n... |
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"value": "1. A set of 60 numbers is such that adding 3 to each of them does not change the value of the sum of their squares. By how much will the sum of the squares of these numbers change if 4 is added to each number?"
},
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"value": "<think>\nOkay, let's ... |
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"value": "5. The length of the diagonal $AC$ of the rhombus $ABCD$ with an acute angle at vertex $A$ is 4. Points $M$ and $N$ on sides $DA$ and $DC$ are the feet of the altitudes of the rhombus dropped from vertex $B$.\n\nThe height $BM$ intersects the diagonal $AC$ at point $P$ such that ... |
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"value": "1. A set of 70 numbers is such that adding 4 to each of them does not change the magnitude of the sum of their squares. By how much will the sum of the squares of these numbers change if 5 is added to each number?"
},
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"from": "assistant",
"value": "<think>\nAlright,... |
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"value": "5. The length of the diagonal $AC$ of the rhombus $ABCD$ with an acute angle at vertex $A$ is 12. Points $M$ and $N$ on sides $DA$ and $DC$ are the bases of the heights of the rhombus dropped from vertex $B$. The height $BM$ intersects the diagonal $AC$ at point $P$ such that $AP... |
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"value": "1. A set of 80 numbers is such that adding 5 to each of them does not change the magnitude of the sum of their squares. By how much will the sum of the squares of these numbers change if 6 is added to each number?"
},
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"from": "assistant",
"value": "<think>\nOkay, le... |
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"value": "5. The length of the diagonal $AC$ of the rhombus $ABCD$ with an acute angle at vertex $A$ is 20. Points $M$ and $N$ on sides $DA$ and $DC$ are the bases of the heights of the rhombus dropped from vertex $B$. The height $BM$ intersects the diagonal $AC$ at point $P$ such that $AP... |
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"value": "3. The sum of two natural numbers is 2013. If you erase the last two digits of one of them, add one to the resulting number, and then multiply the result by five, you get the other number. Find these numbers. Enter the largest of them in the provided field."
},
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"from": ... |
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"value": "4. The sum of two natural numbers is 2014. If you strike out the last two digits of one of them, multiply the resulting number by three, you get a number that is six more than the other number. Find these numbers. Enter the smallest of them in the provided field."
},
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"f... |
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"value": "5. Find the fraction $\\frac{p}{q}$ with the smallest possible natural denominator, for which $\\frac{1}{2014}<\\frac{p}{q}<\\frac{1}{2013}$. Enter the denominator of this fraction in the provided field"
},
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"value": "<think>\nAlright, so I need ... |
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"value": "6. How many pairs of natural numbers $(a ; b)$ exist such that the number $5 a-3$ is divisible by $b$, and the number $5 b-1$ is divisible by $a$? Enter the number of such pairs of numbers in the provided field."
},
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"value": "<think>\nAlright, l... |
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"value": "7. The coordinates $(x ; y ; z)$ of point $M$ are consecutive terms of a geometric progression, and the numbers $x y, y z, x z$ in the given order are terms of an arithmetic progression, with $z \\geq 1$ and $x \\neq y \\neq z$. Find the smallest possible value of the square of t... |
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"value": "8. The residents of the village Razumevo, located 3 km away from the river, love to visit the village Vkusnotevo, situated 3.25 km downstream on the opposite bank of the river, 1 km away from the shore. The width of the river is 500 m, the speed of the current is 1 km/h, and the ... |
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"value": "9. Find the last two digits of the number $14^{14^{14}}$. Enter your answer in the provided field."
},
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"value": "<think>\nOkay, so I need to find the last two digits of 14 raised to the power of 14^14. That seems really big, so calculating it di... |
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"value": "10. Find the number of twos in the prime factorization of the number $2011 \\cdot 2012 \\cdot 2013 \\cdot \\ldots .4020$. Enter your answer in the provided field."
},
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"value": "<think>\nOkay, so I need to find the number of twos in the prime fac... |
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"value": "11. For what values of $a$ does the equation $|x|=a x-2$ have no solutions? Enter the length of the interval of values of the parameter $a$ in the provided field."
},
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"value": "<think>\nOkay, so I need to figure out for which values of 'a' the e... |
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"value": "12. For which $a$ does the equation $|x-3|=a x-1$ have two solutions? Enter the midpoint of the interval of values for the parameter $a$ in the provided field. Round the answer to three significant digits according to rounding rules and record it in the provided field."
},
{
... |
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"value": "13. For what value of $a$ does the equation $|x-2|=a x-2$ have an infinite number of solutions? Enter the answer in the provided field"
},
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"value": "<think>\nOkay, so I need to find the value of 'a' such that the equation |x - 2| = a x - 2 has a... |
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"value": "1. Answer: 1) 2 pairs; 2$)(4 ; 1)$"
},
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"value": "<think>\nOkay, so I need to solve this problem where the answer is given as two parts: 1) 2 pairs; and 2) (4 ; 1). Hmm, let's try to figure out what the original problem might have been. Since the... |
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"value": "3. Answer: 1) $\\left.x_{n}^{1}=3 / 2^{n-1}, x_{n}^{2}=-1 / 2^{n-2} ; 2\\right) \\frac{3\\left(2^{n}-1\\right)}{4^{n}-1}$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem here with two parts. Let me try to understand what's being asked. The f... |
[
{
"from": "user",
"value": "4. Answer: $p=\\frac{1}{70}$"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I have to figure out this problem where the answer is given as p equals 1 over 70. Hmm, so I need to work through how to arrive at that answer. Let me start by thinking about w... |
[
{
"from": "user",
"value": "6. Answer: $2 \\pi r-4 \\varphi R$, where $\\varphi-$ is the acute angle, $\\sin \\varphi=\\frac{r}{R-r}$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem here where I need to find an expression, and the answer is given as \\( 2 \\pi r - 4 \\... |
[
{
"from": "user",
"value": "# 2.4. Final round of the \"Rosatom\" Olympiad, 11th grade, set 4\n\n## Answers and solutions\n\nProblem 1 Answer: $x=2$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve this problem from the Rosatom Olympiad. The answer is given as x=2, but I ne... |
[
{
"from": "user",
"value": "4. Answer: $P(G)=0.117376$"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's tackle this probability problem. Hmm, the user mentioned that the answer is P(G) = 0.117376, but they want me to figure out how to get there. Since the problem isn't stated here, I ne... |
[
{
"from": "user",
"value": "2. Answer: $S_{\\min }=\\frac{4 \\pi^{2}}{7}$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the minimum value of S, which is given as S_min = (4π²)/7. Hmm, let me think. The problem probably involves some optimization, maybe calculus? But since... |
[
{
"from": "user",
"value": "3. Square the numbers $a=101, b=10101$. Extract the square root of the number $c=102030405060504030201$."
},
{
"from": "assistant",
"value": "<think>\nAlright, so I have to square the numbers a=101 and b=10101, and then find the square root of c=102030405060504030201.... |
[
{
"from": "user",
"value": "3. Square the numbers $a=1001, b=1001001$. Extract the square root of the number $c=1002003004005004003002001$."
},
{
"from": "assistant",
"value": "<think>\nOkay, let me try to figure out how to solve these problems. So, the first part is squaring the numbers a = 100... |
[
{
"from": "user",
"value": "3. Square the numbers $a=10001, b=100010001$. Extract the square root of the number $c=1000200030004000300020001$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to square the numbers a = 10001 and b = 100010001, and then find the square root of c = 100... |
[
{
"from": "user",
"value": "2. Transform the equation, writing the right side as a fraction\n\n$$\n\\begin{gathered}\n\\left(1+1:(1+1:(1+1:(2 x-3)))=\\frac{1}{x-1}\\right. \\\\\n1:\\left(1+1:(1+1:(2 x-3))=\\frac{2-x}{x-1}\\right. \\\\\n\\left(1+1:(1+1:(2 x-3))=\\frac{x-1}{2-x^{\\prime}}\\right. \\\\\n1:\\le... |
[
{
"from": "user",
"value": "5. Let's make a drawing and carry out additional constructions.\n\n\n\nExtend the line $P D$ until it intersects with the line $B C$ at point $F$; Constr... |
[
{
"from": "user",
"value": "2. Solve the equation $2+2:(1+2:(1+2:(1+2:(3 x-8))))=x$.\n\nTranslate the text above into English, please keep the original text's line breaks and format, and output the translation result directly."
},
{
"from": "assistant",
"value": "<think>\nAlright, let's try to s... |
[
{
"from": "user",
"value": "5. Point $P$ is located on side $AB$ of square $ABCD$ such that $AP: PB=2: 3$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ: QC=3$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $AE: EQ$."
},
{
"from": "assistant",
... |
[
{
"from": "user",
"value": "2. Solve the equation $3+3:(1+3:(1+3:(1+3:(4 x-15))))=x$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this equation to solve: 3 + 3 divided by (1 + 3 divided by (1 + 3 divided by (1 + 3 divided by (4x - 15)))) equals x. That looks pretty complicated ... |
[
{
"from": "user",
"value": "5. Point $P$ is located on side $AB$ of square $ABCD$ such that $AP: PB=1: 2$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ: QC=2$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $PE: ED$."
},
{
"from": "assistant",
... |
[
{
"from": "user",
"value": "2. Solve the equation $4+4:(1+4:(1+4:(1+4:(5 x-24))))=x$.\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to solve this equation: 4 + 4 divided by (1 + 4 divided by (1 + 4 divided by (1 + 4 divided by (5x - 24)))) equals x. Hmm, that looks co... |
[
{
"from": "user",
"value": "5. Point $P$ is located on side $AB$ of square $ABCD$ such that $AP: PB=1: 4$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ: QC=5$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $AE: EQ$."
},
{
"from": "assistant",
... |
[
{
"from": "user",
"value": "Task 1. Answer: 10 boys, 13 girls."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The problem states that there are 10 boys and 13 girls. But wait, the user just gave me \"Task 1. Answer: 10 boys, 13 girls.\" Hmm, maybe the actual problem was presented... |
[
{
"from": "user",
"value": "Problem 4. Answer: $(1 ; 1)$"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to figure out how to solve Problem 4, which has the answer (1; 1). Hmm, the problem isn't stated here, just the answer. That's a bit tricky. Maybe the original problem w... |
[
{
"from": "user",
"value": "Problem 5. Answer: 1.\n\n## Variant 0\n\nCircles $K_{1}$ and $K_{2}$ have a common point $A$. Through point $A$, three lines are drawn: two pass through the centers of the circles and intersect them at points $B$ and $C$, the third is parallel to $BC$ and intersects the circles a... |
[
{
"from": "user",
"value": "4. Let $x_{1}, x_{2}$ be the integer solutions of the equation $x^{2} + p x + 3 q = 0$ with prime numbers $p$ and $q$. Then, by Vieta's theorem,\n\n$$\nx_{1} + x_{2} = -p\n$$\n\nFrom this, we conclude that both roots must be negative. Further, since the roots are integers and the... |
[
{
"from": "user",
"value": "5. Let $N C=x$, then $B N=3 x$. Triangle $A B C$ is isosceles, so $A B=B C=4 x$. Angle $A N C$ is a right angle because it subtends the diameter. Therefore, angle $A N B$ is also a right angle, as $A N C$ and $A N B$ are adjacent.\n\nIn the right triangle $A N B$, by the Pythagor... |
[
{
"from": "user",
"value": "2. By what natural number can the numerator and denominator of the ordinary fraction of the form $\\frac{5 n+3}{7 n+8}$ be reduced? For which integers $n$ can this occur?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out by what natural numbe... |
[
{
"from": "user",
"value": "2. By what natural number can the numerator and denominator of the ordinary fraction of the form $\\frac{4 n+3}{5 n+2}$ be reduced? For which integers $n$ can this occur?"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to figure out by what natur... |
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