problem_id
stringlengths 12
21
| task_type
stringclasses 1
value | prompt
stringlengths 0
13.6k
| verification_info
stringlengths 20
1.24k
| metadata
stringclasses 89
values | __index_level_0__
int64 0
776k
|
|---|---|---|---|---|---|
numina_1.5_212
|
verifiable_math
|
II OM - III - Task 1
A beam of length $ a $ has been suspended horizontally by its ends on two parallel ropes of equal length $ b $. We rotate the beam by an angle $ \varphi $ around a vertical axis passing through the center of the beam. By how much will the beam be raised?
|
{"ground_truth": "\\sqrt{b^2-(\\frac{}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 200
|
numina_1.5_213
|
verifiable_math
|
LVII OM - III - Problem 2
Determine all positive integers $ k $ for which the number $ 3^k+5^k $ is a power of an integer with an exponent greater than 1.
|
{"ground_truth": "1"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 201
|
numina_1.5_214
|
verifiable_math
|
XXIV OM - I - Problem 12
In a class of n students, a Secret Santa event was organized. Each student draws the name of the person for whom they are to buy a gift, so student $ A_1 $ buys a gift for student $ A_2 $, $ A_2 $ buys a gift for $ A_3 $, ..., $ A_k $ buys a gift for $ A_1 $, where $ 1 \leq k \leq n $. Assuming that all drawing outcomes are equally probable, calculate the probability that $ k = n $.
|
{"ground_truth": "\\frac{1}{n}"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 202
|
numina_1.5_215
|
verifiable_math
|
VIII OM - I - Task 6
Find a four-digit number, whose first two digits are the same, the last two digits are the same, and which is a square of an integer.
|
{"ground_truth": "7744"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 203
|
numina_1.5_216
|
verifiable_math
|
XXXIV OM - I - Problem 1
$ A $ tosses a coin $ n $ times, $ B $ tosses it $ n+1 $ times. What is the probability that $ B $ will get more heads than $ A $?
|
{"ground_truth": "\\frac{1}{2}"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 204
|
numina_1.5_219
|
verifiable_math
|
LI OM - II - Problem 4
Point $ I $ is the center of the circle inscribed in triangle $ ABC $, where $ AB \neq AC $. Lines $ BI $ and $ CI $ intersect sides $ AC $ and $ AB $ at points $ D $ and $ E $, respectively. Determine all possible measures of angle $ BAC $ for which the equality $ DI = EI $ can hold.
|
{"ground_truth": "60"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 205
|
numina_1.5_220
|
verifiable_math
|
XXXIV OM - II - Problem 6
For a given number $ n $, let $ p_n $ denote the probability that when a pair of integers $ k, m $ satisfying the conditions $ 0 \leq k \leq m \leq 2^n $ is chosen at random (each pair is equally likely), the number $ \binom{m}{k} $ is even. Calculate $ \lim_{n\to \infty} p_n $.
|
{"ground_truth": "1"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 206
|
numina_1.5_221
|
verifiable_math
|
L OM - I - Problem 8
Given a natural number $ n \geq 2 $ and an $ n $-element set $ S $. Determine the smallest natural number $ k $ for which there exist subsets $ A_1, A_2, \ldots, A_k $ of the set $ S $ with the following property:
for any two distinct elements $ a, b \in S $, there exists a number $ j \in \{1, 2, \ldots, k\} $ such that the set $ A_j \cap \{a, b\} $ is a singleton.
|
{"ground_truth": "[\\log_2(n-1)+1]"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 207
|
numina_1.5_222
|
verifiable_math
|
XLVIII OM - II - Problem 3
Given a set of $ n $ points ($ n \geq 2 $), no three of which are collinear. We color all segments with endpoints in this set such that any two segments sharing a common endpoint have different colors. Determine the smallest number of colors for which such a coloring exists.
|
{"ground_truth": "[n/2]"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 208
|
numina_1.5_223
|
verifiable_math
|
LVII OM - III - Problem 6
Determine all pairs of integers $ a $, $ b $, for which there exists a polynomial $ P(x) $ with integer coefficients, such that the product $ (x^2 + ax + b)\cdot P(x) $ is a polynomial of the form
where each of the numbers $ c_0,c_1,\dots ,c_{n-1} $ is equal to 1 or -1.
|
{"ground_truth": "(-2,1),(-1,-1),(-1,1),(0,-1),(0,1),(1,-1),(1,1),(2,1)"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 209
|
numina_1.5_224
|
verifiable_math
|
XXVIII - I - Task 3
Let $ a $ and $ b $ be natural numbers. A rectangle with sides of length $ a $ and $ b $ has been divided by lines parallel to the sides into unit squares. Through the interiors of how many squares does the diagonal of the rectangle pass?
|
{"ground_truth": "\\gcd(,b)"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 210
|
numina_1.5_225
|
verifiable_math
|
XVI OM - I - Problem 4
The school organized three trips for its 300 students. The same number of students participated in each trip. Each student went on at least one trip, but half of the participants in the first trip, one-third of the participants in the second trip, and one-fourth of the participants in the third trip only went on one trip.
How many students went on each trip? How many participants in the first trip also participated in the second, and how many of them also participated in the third trip?
|
{"ground_truth": "x=120,y=14,z=27,u=53,w=37"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 211
|
numina_1.5_226
|
verifiable_math
|
LIV OM - III - Task 3
Determine all polynomials $ W $ with integer coefficients that satisfy the following condition: for every natural number $ n $, the number $ 2^n-1 $ is divisible by $ W(n) $.
|
{"ground_truth": "W(x)=1orW(x)=-1"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 212
|
numina_1.5_227
|
verifiable_math
|
VI OM - II - Task 3
What should be the angle at the vertex of an isosceles triangle so that a triangle can be constructed with sides equal to the height, base, and one of the remaining sides of this isosceles triangle?
|
{"ground_truth": "106"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 213
|
numina_1.5_229
|
verifiable_math
|
XXIX OM - II - Problem 4
From the vertices of a regular $2n$-gon, 3 different points are chosen randomly. Let $p_n$ be the probability that the triangle with vertices at the chosen points is acute. Calculate $\lim_{n\to \infty} p_n$.
Note. We assume that all choices of three different points are equally probable.
|
{"ground_truth": "\\frac{1}{4}"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 214
|
numina_1.5_230
|
verifiable_math
|
IX OM - II - Task 2
Six equal disks are placed on a plane in such a way that their centers lie at the vertices of a regular hexagon with a side equal to the diameter of the disks. How many rotations will a seventh disk of the same size make while rolling externally on the same plane along the disks until it returns to its initial position?
|
{"ground_truth": "4"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 215
|
numina_1.5_231
|
verifiable_math
|
XXXVI OM - III - Problem 1
Determine the largest number $ k $ such that for every natural number $ n $ there are at least $ k $ natural numbers greater than $ n $, less than $ n+17 $, and coprime with the product $ n(n+17) $.
|
{"ground_truth": "1"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 216
|
numina_1.5_232
|
verifiable_math
|
LVIII OM - I - Problem 1
Problem 1.
Solve in real numbers $ x $, $ y $, $ z $ the system of equations
|
{"ground_truth": "(x,y,z)=(-2,-2,-2),(\\frac{1}{3},\\frac{1}{3},\\frac{1}{3}),(2,2,-3),(-\\frac{1}{3},-\\frac{1}{3},-\\frac{16}{3}),(2,-3,2),(-\\frac{1}{3}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 217
|
numina_1.5_234
|
verifiable_math
|
XIII OM - I - Problem 6
Factor the quadratic polynomial into real factors
where $ p $ and $ q $ are real numbers satisfying the inequality
Please note that the mathematical expressions and symbols are kept as they are, only the text has been translated.
|
{"ground_truth": "x^4+px^2+(x^2+\\alphax+\\beta)(x^2-\\alphax+\\beta)"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 218
|
numina_1.5_235
|
verifiable_math
|
XXXVIII OM - III - Zadanie 5
Wyznaczyć najmniejszą liczbę naturalną $ n $, dla której liczba $ n^2-n+11 $ jest iloczynem czterech liczb pierwszych (niekoniecznie różnych).
|
{"ground_truth": "132"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 219
|
numina_1.5_236
|
verifiable_math
|
XXV OM - III - Task 2
Salmon swimming upstream must overcome two waterfalls. The probability that a salmon will overcome the first waterfall in a given attempt is $ p > 0 $, and the probability of overcoming the second waterfall in a given attempt is $ q > 0 $. We assume that successive attempts to overcome the waterfalls are independent. Calculate the probability of the event that the salmon will not overcome the first waterfall in $ n $ attempts, given that in $ n $ attempts it will not overcome both waterfalls.
|
{"ground_truth": "\\max(0,1-\\frac{p}{q})"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 220
|
numina_1.5_237
|
verifiable_math
|
LIX OM - I - Task 7
In an $ n $-person association, there are $ 2n-1 $ committees (any non-empty set of association members
forms a committee). A chairperson must be selected in each committee. The following condition must be met: If
committee $ C $ is the union $ C = A\cup B $ of two committees $ A $ and $ B $, then the chairperson of committee $ C $ is also
the chairperson of at least one of the committees $ A $, $ B $. Determine the number of possible selections of chairpersons.
|
{"ground_truth": "n!"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 221
|
numina_1.5_238
|
verifiable_math
|
XVII OM - I - Problem 4
On a plane, a circle and a point $ M $ are given. Find points $ A $ and $ B $ on the circle such that the segment $ AB $ has a given length $ d $, and the angle $ AMB $ is equal to a given angle $ \alpha $.
|
{"ground_truth": "4,3,2,1,\u62160"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 222
|
numina_1.5_239
|
verifiable_math
|
XXXII - I - Problem 10
Determine all functions $ f $ mapping the set of all rational numbers $ \mathbb{Q} $ to itself that satisfy the following conditions:
a) $ f(1)=2 $,
b) $ f(xy) = f(x)f(y)-f(x+y)+1 $ for $ x, y \in \mathbb{Q} $.
|
{"ground_truth": "f(x)=x+1"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 223
|
numina_1.5_240
|
verifiable_math
|
XXXI - III - Task 1
Calculate the area of an octagon inscribed in a circle, knowing that each of four consecutive sides of this octagon has a length of 1, and each of the remaining four has a length of 2.
|
{"ground_truth": "6+4\\sqrt{2}-1"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 224
|
numina_1.5_241
|
verifiable_math
|
XII OM - II - Task 4
Find the last four digits of the number $ 5^{5555} $.
|
{"ground_truth": "8125"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 225
|
numina_1.5_242
|
verifiable_math
|
XLVII OM - III - Problem 1
Determine all pairs $ (n,r) $, where $ n $ is a positive integer and $ r $ is a real number, for which the polynomial $ (x + 1)^n - r $ is divisible by the polynomial $ 2x^2 + 2x + 1 $.
|
{"ground_truth": "(4k,(-\\frac{1}{4})^k)"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 226
|
numina_1.5_244
|
verifiable_math
|
XXXVIII OM - II - Zadanie 4
Wyznaczyć wszystkie pary liczb rzeczywistych $ a, b $, dla których wielomiany $ x^4 + 2ax^2 + 4bx + a^2 $ i $ x^3 + ax - b $ mają dwa różne wspólne pierwiastki rzeczywiste.
|
{"ground_truth": "(,0),where0"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 227
|
numina_1.5_245
|
verifiable_math
|
VI OM - II - Task 1
Calculate the sum $ x^4 + y^4 + z^4 $ given that $ x + y + z = 0 $ and $ x^2 + y^2 + z^2 = a $, where $ a $ is a given positive number.
|
{"ground_truth": "\\frac{^2}{2}"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 228
|
numina_1.5_248
|
verifiable_math
|
XXI OM - III - Task 5
In how many ways can a set consisting of twelve elements be divided into six disjoint two-element sets?
|
{"ground_truth": "11\\cdot9\\cdot7\\cdot5\\cdot3"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 229
|
numina_1.5_249
|
verifiable_math
|
XXII OM - II - Problem 1
In how many ways can $ k $ fields of a chessboard $ n \times n $ ($ k \leq n $) be chosen so that no two of the selected fields lie in the same row or column?
|
{"ground_truth": "\\binom{n}{k}n(n-1)\\ldots(n-k+1)"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 230
|
numina_1.5_250
|
verifiable_math
|
XXXIII OM - III - Problem 1
Indicate such a way of arranging $ n $ girls and $ n $ boys around a round table so that the number $ d_n - c_n $ is maximized, where $ d_n $ is the number of girls sitting between two boys, and $ c_n $ is the number of boys sitting between two girls.
|
{"ground_truth": "[\\frac{n}{2}]-1"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 231
|
numina_1.5_251
|
verifiable_math
|
III OM - I - Task 4
a) Given points $ A $, $ B $, $ C $ not lying on a straight line. Determine three mutually parallel lines passing through points $ A $, $ B $, $ C $, respectively, so that the distances between adjacent parallel lines are equal.
b) Given points $ A $, $ B $, $ C $, $ D $ not lying on a plane. Determine four mutually parallel planes passing through points $ A $, $ B $, $ C $, $ D $, respectively, so that the distances between adjacent parallel planes are equal.
|
{"ground_truth": "12"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 232
|
numina_1.5_252
|
verifiable_math
|
LX OM - I - Task 1
On some fields of a chessboard of size $ m \times n $, rooks are placed.
It is known that any rook is in the attack range of at most
two other rooks.
Determine, depending on $ m, n \geq 2 $, the maximum number of rooks on the chessboard,
for which such a situation is possible.
|
{"ground_truth": "+n"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 233
|
numina_1.5_253
|
verifiable_math
|
LX OM - II - Task 5
Determine all integers $ n \geqslant 4 $ with the following property: Among any $ n $ different 3-element subsets of an $ n $-element set, one can select two subsets that have exactly one element in common.
|
{"ground_truth": "n\\geq4"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 234
|
numina_1.5_254
|
verifiable_math
|
XXIII OM - I - Problem 6
Determine for which digits $ a $ the decimal representation of a number $ \frac{n(n+1)}{2} $ ($ n\in \mathbb{N} $) consists entirely of the digit $ a $.
|
{"ground_truth": "5,6"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 235
|
numina_1.5_255
|
verifiable_math
|
XXVI - I - Problem 11
A ship is moving at a constant speed along a straight line with an east-west direction. Every $ T $ minutes, the direction of movement is randomly chosen: with probability $ p $, the ship moves in the eastern direction for the next $ T $ minutes, and with probability $ q= 1-p $, it moves in the western direction. At a point outside the line, there is a submarine whose task is to torpedo the ship. The travel time of the torpedo from the point of firing to any point on the ship's track is $ 2T $. The captain of the submarine knows the value of $ p $ and aims to maximize the probability of hitting the ship. How should $ p $ be chosen to minimize the probability of the ship being torpedoed?
|
{"ground_truth": "\\frac{4}{9}"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 236
|
numina_1.5_256
|
verifiable_math
|
LI OM - I - Task 5
Determine all pairs $ (a,b) $ of natural numbers for which the numbers $ a^3 + 6ab + 1 $ and $ b^3 + 6ab + 1 $ are cubes of natural numbers.
|
{"ground_truth": "(1,1)"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 237
|
numina_1.5_257
|
verifiable_math
|
XXXV OM - I - Problem 9
Three events satisfy the conditions:
a) their probabilities are equal,
b) any two of them are independent,
c) they do not occur simultaneously.
Determine the maximum value of the probability of each of these events.
|
{"ground_truth": "\\frac{1}{2}"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 238
|
numina_1.5_258
|
verifiable_math
|
VII OM - I - Task 3
In a square $ABCD$ with area $S$, vertex $A$ is connected to the midpoint of side $BG$, vertex $B$ is connected to the midpoint of side $CD$, vertex $C$ is connected to the midpoint of side $DA$, and vertex $D$ is connected to the midpoint of side $AB$. Calculate the area of the part of the square that contains its center.
|
{"ground_truth": "\\frac"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 239
|
numina_1.5_259
|
verifiable_math
|
XLVIII OM - III - Problem 2
Find all triples of real numbers $ x $, $ y $, $ z $ satisfying the system of equations
|
{"ground_truth": "(x,y,z)=(\\1/3,\\1/3,\\1/3)(x,y,z)=(\\1/\\sqrt{3},0,0),(0,\\1/\\sqrt{3},0),(0,0,\\1/\\sqrt{3})"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 240
|
numina_1.5_260
|
verifiable_math
|
L OM - I - Task 3
In an isosceles triangle $ ABC $, angle $ BAC $ is a right angle. Point $ D $ lies on side $ BC $, such that $ BD = 2 \cdot CD $. Point $ E $ is the orthogonal projection of point $ B $ onto line $ AD $. Determine the measure of angle $ CED $.
|
{"ground_truth": "45"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 241
|
numina_1.5_263
|
verifiable_math
|
XXX OM - I - Task 8
Among the cones inscribed in the sphere $ B $, the cone $ S_1 $ was chosen such that the sphere $ K_1 $ inscribed in the cone $ S_1 $ has the maximum volume. Then, a cone $ S_2 $ of maximum volume was inscribed in the sphere $ B $, and a sphere $ K_2 $ was inscribed in the cone $ S_2 $. Determine which is greater: the sum of the volumes of $ S_1 $ and $ K_1 $, or the sum of the volumes of $ S_2 $ and $ K_2 $.
|
{"ground_truth": "V(\\frac{\\sqrt{3}}{3})>V(\\frac{1}{2})"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 242
|
numina_1.5_264
|
verifiable_math
|
XXXI - I - Task 1
Determine for which values of the parameter $ a $ a rhombus with side length $ a $ is a cross-section of a cube with edge length 2 by a plane passing through the center of the cube.
|
{"ground_truth": "[2,\\sqrt{5}]"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 243
|
numina_1.5_265
|
verifiable_math
|
VIII OM - I - Zadanie 5
Jaki warunek powinna spełniać liczbą $ q $, aby istniał trójkąt, którego boki tworzą postęp geometryczny o ilorazie $ q $?
|
{"ground_truth": "\\frac{-1+\\sqrt{5}}{2}<\\frac{1+\\sqrt{5}}{2}"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 244
|
numina_1.5_266
|
verifiable_math
|
LV OM - I - Task 7
Find all solutions to the equation $ a^2+b^2=c^2 $ in positive integers such that the numbers $ a $ and $ c $ are prime, and the number $ b $ is the product of at most four prime numbers.
|
{"ground_truth": "(3,4,5),(5,12,13),(11,60,61)"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 245
|
numina_1.5_267
|
verifiable_math
|
LX OM - III - Zadanie 2
Let $ S $ be the set of all points in the plane with both coordinates being integers. Find
the smallest positive integer $ k $ for which there exists a 60-element subset of the set $ S $
with the following property: For any two distinct elements $ A $ and $ B $ of this subset, there exists a point
$ C \in S $ such that the area of triangle $ ABC $ is equal to $ k $.
|
{"ground_truth": "210"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 246
|
numina_1.5_268
|
verifiable_math
|
I OM - B - Task 1
Determine the coefficients $ a $ and $ b $ of the equation $ x^2 + ax + b=0 $ such that the values of $ a $ and $ b $ themselves are roots of this equation.
|
{"ground_truth": "1,0"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 247
|
numina_1.5_269
|
verifiable_math
|
I OM - B - Zadanie 1
Wyznaczyć współczynniki $ a $ i $ b $, równania $ x^2 + ax +b=0 $ tak, aby same wartości $ a $ i $ b $ były pierwiastkami tego równania.
|
{"ground_truth": "0,0\u62161,-2"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 248
|
numina_1.5_270
|
verifiable_math
|
XXI OM - II - Problem 6
If $ A $ is a subset of set $ X $, then we define $ A^1 = A $, $ A^{-1} = X - A $. Subsets $ A_1, A_2, \ldots, A_k $ are called mutually independent if the product $ A_1^{\varepsilon_1} \cap A_2^{\varepsilon_2} \ldots A_k^{\varepsilon_k} $ is non-empty for every system of numbers $ \varepsilon_1, \varepsilon_2, \ldots, \varepsilon_k $, such that $ |\varepsilon_i| = 1 $ for $ i = 1, 2, \ldots, k $.
What is the maximum number of mutually independent subsets of a $ 2^n $-element set?
|
{"ground_truth": "n"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 249
|
numina_1.5_271
|
verifiable_math
|
X OM - III - Task 3
Given is a pyramid with a square base $ABCD$ and apex $S$. Find the shortest path that starts and ends at point $S$ and passes through all the vertices of the base.
|
{"ground_truth": "k++3a"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 250
|
numina_1.5_272
|
verifiable_math
|
XIX OM - I - Problem 4
In the plane, given are parallel lines $ a $ and $ b $ and a point $ M $ lying outside the strip bounded by these lines. Determine points $ A $ and $ B $ on lines $ a $ and $ b $, respectively, such that segment $ AB $ is perpendicular to $ a $ and $ b $, and the angle $ AMB $ is the largest.
|
{"ground_truth": "ABlieonthecirclewithdiameterMN"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 251
|
numina_1.5_273
|
verifiable_math
|
XIII OM - III - Task 4
In how many ways can a set of $ n $ items be divided into two sets?
|
{"ground_truth": "2^{n-1}-1"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 252
|
numina_1.5_274
|
verifiable_math
|
LV OM - III - Task 5
Determine the maximum number of lines in space passing through a fixed point and such that any two intersect at the same angle.
|
{"ground_truth": "6"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 253
|
numina_1.5_278
|
verifiable_math
|
XVI OM - III - Task 5
Points $ A_1 $, $ B_1 $, $ C_1 $ divide the sides $ BC $, $ CA $, $ AB $ of triangle $ ABC $ in the ratios $ k_1 $, $ k_2 $, $ k_3 $. Calculate the ratio of the areas of triangles $ A_1B_1C_1 $ and $ ABC $.
|
{"ground_truth": "\\frac{k_1k_2k_3}{(1+k_1)(1+k_2)(1+k_3)}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 254
|
numina_1.5_279
|
verifiable_math
|
LVII OM - I - Problem 1
Determine all non-negative integers $ n $ for which the number
$ 2^n +105 $ is a perfect square of an integer.
|
{"ground_truth": "4,6,8"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 255
|
numina_1.5_280
|
verifiable_math
|
XVI OM - II - Task 4
Find all prime numbers $ p $ such that $ 4p^2 +1 $ and $ 6p^2 + 1 $ are also prime numbers.
|
{"ground_truth": "5"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 256
|
numina_1.5_281
|
verifiable_math
|
XXXVIII OM - II - Problem 5
Determine all prime numbers $ p $ and natural numbers $ x, y $, for which $ p^x - y^3 = 1 $.
|
{"ground_truth": "(2,1,1)(3,2,2)"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 257
|
numina_1.5_282
|
verifiable_math
|
XXVI - I - Problem 5
Determine all integers $ m $ for which the polynomial $ x^3-mx^2+mx-(m^2+1) $ has an integer root.
|
{"ground_truth": "=0=-3"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 258
|
numina_1.5_283
|
verifiable_math
|
XL OM - I - Task 6
Calculate the sum of the series $ \sum_{n\in A}\frac{1}{2^n} $, where the summation runs over the set $ A $ of all natural numbers not divisible by 2, 3, 5.
|
{"ground_truth": "\\frac{1}{1-2^{-30}}(\\frac{1}{2^1}+\\frac{1}{2^7}+\\frac{1}{2^{11}}+\\frac{1}{2^{13}}+\\frac{1}{2^{17}}+\\frac{1}{2^{19}}+\\frac{1}{2^{23}}+\\frac{1}{2^{}"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 259
|
numina_1.5_284
|
verifiable_math
|
III OM - III - Task 6
In a round tower with an internal diameter of $2$ m, there are spiral stairs with a height of $6$ m. The height of each step is $0.15$ m. In a horizontal projection, the steps form adjacent circular sectors with an angle of $18^\circ$. The narrower ends of the steps are attached to a round pillar with a diameter of $0.64$ m, whose axis coincides with the axis of the tower. Calculate the maximum length of a straight rod that can be moved up these stairs from the bottom to the top (do not take into account the thickness of the rod or the thickness of the plates from which the stairs are made).
|
{"ground_truth": "4.47\\textrm{}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 260
|
numina_1.5_285
|
verifiable_math
|
XXXIX OM - II - Problem 3
Inside an acute triangle $ ABC $, consider a point $ P $ and its projections $ L, M, N $ onto the sides $ BC, CA, AB $, respectively. Determine the point $ P $ for which the sum $ |BL|^2 + |CM|^2 + |AN|^2 $ is minimized.
|
{"ground_truth": "PisthecircumcenteroftriangleABC"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 261
|
numina_1.5_286
|
verifiable_math
|
XXXVI OM - I - Zadanie 9
W urnie jest 1985 kartek z napisanymi liczbami 1,2,3,..., 1985, każda lczba na innej kartce. Losujemy bez zwracania 100 kartek. Znaleźć wartość oczekiwaną sumy liczb napisanych na wylosowanych kartkach.
|
{"ground_truth": "99300"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 262
|
numina_1.5_287
|
verifiable_math
|
LII OM - I - Task 4
Determine whether 65 balls with a diameter of 1 can fit into a cubic box with an edge of 4.
|
{"ground_truth": "66"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 263
|
numina_1.5_288
|
verifiable_math
|
II OM - III - Task 2
What digits should be placed instead of zeros in the third and fifth positions in the number $ 3000003 $ to obtain a number divisible by $ 13 $?
|
{"ground_truth": "3080103,3040203,3020303,3090503,3060603,3030703,3000803"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 264
|
numina_1.5_289
|
verifiable_math
|
L OM - I - Task 5
Find all pairs of positive integers $ x $, $ y $ satisfying the equation $ y^x = x^{50} $.
|
{"ground_truth": "8"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 265
|
numina_1.5_290
|
verifiable_math
|
VIII OM - I - Problem 8
In the rectangular prism $ ABCDA_1B_1C_1D_1 $, the lengths of the edges are given as $ AA_1 = a $, $ AB = b $, $ AD = c $. On the face $ A_1B_1C_1D_1 $, a point $ M $ is chosen at a distance $ p $ from the side $ A_1B_1 $, and at a distance $ q $ from the side $ A_1D_1 $, and a parallelepiped is constructed with the base $ ABCD $ and the lateral edge $ AM $. Calculate the area of the lateral faces of this parallelepiped.
|
{"ground_truth": "\\sqrt{^2+}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 266
|
numina_1.5_292
|
verifiable_math
|
XIII OM - I - Problem 2
How many digits do all natural numbers with at most $ m $ digits have in total?
|
{"ground_truth": "9\\cdot\\frac{10^-1}{9}\\cdot\\frac{(+1)}{2}"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 267
|
numina_1.5_294
|
verifiable_math
|
XV OM - I - Problem 8
On three pairwise skew edges of a cube, choose one point on each in such a way that the sum of the squares of the sides of the triangle formed by them is minimized.
|
{"ground_truth": "\\frac{9}{2}^2"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 268
|
numina_1.5_295
|
verifiable_math
|
XLVII OM - I - Problem 1
Determine all integers $ n $ for which the equation $ 2 \sin nx = \tan x + \cot x $ has solutions in real numbers $ x $.
|
{"ground_truth": "8k+2"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 269
|
numina_1.5_296
|
verifiable_math
|
XLII OM - I - Problem 5
Given a segment $ AD $. Find points $ B $ and $ C $ on it, such that the product of the lengths of the segments $ AB $, $ AC $, $ AD $, $ BC $, $ BD $, $ CD $ is maximized.
|
{"ground_truth": "\\frac{1}{5}\\sqrt{5}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 270
|
numina_1.5_297
|
verifiable_math
|
LX OM - I - Task 12
Given a prime number $ p $. On the left side of the board, the numbers $ 1, 2, 3, \cdots, p - 1 $ are written,
while on the right side, the number $ 0 $ is written. We perform a sequence of $ p - 1 $ moves, each of which proceeds
as follows: We select one of the numbers written on the left side of the board, add it to all the
remaining numbers on the board, and then erase the selected number.
Determine for which values
of $ p $ it is possible to choose the numbers in such a way that the number remaining on the board after
all moves have been performed is divisible by $ p $.
|
{"ground_truth": "Forallpexcept23"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 271
|
numina_1.5_300
|
verifiable_math
|
XXX OM - I - Task 12
To qualify for the team, a player must pass at least four out of seven tests with a positive result. If the player prepares for all tests, the probability of success in each of these tests is the same and equals $ \frac{1}{2} $. If the player prepares for five tests, the corresponding probability of success in each of these five tests will be $ \frac{3}{4} $. If, however, the player prepares for four tests, the probability of success in each of these four tests will be $ \frac{4}{5} $. Which of the three mentioned ways of preparing is the most advantageous for the player?
|
{"ground_truth": "\\frac{81}{128}"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 272
|
numina_1.5_301
|
verifiable_math
|
XXXIX OM - I - Problem 1
For each positive number $ a $, determine the number of roots of the polynomial $ x^3+(a+2)x^2-x-3a $.
|
{"ground_truth": "3"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 273
|
numina_1.5_302
|
verifiable_math
|
XLVIII OM - II - Problem 4
Determine all triples of positive integers having the following property: the product of any two of them gives a remainder of $ 1 $ when divided by the third number.
|
{"ground_truth": "(5,3,2)"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 274
|
numina_1.5_303
|
verifiable_math
|
LVIII OM - I - Problem 2
Determine all pairs of positive integers $ k $, $ m $, for which each of the numbers $ {k^2+4m} $, $ {m^2+5k} $ is a perfect square.
|
{"ground_truth": "(1,2),(9,22),(8,9)"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 275
|
numina_1.5_305
|
verifiable_math
|
XX OM - II - Task 2
Find all four-digit numbers in which the thousands digit is equal to the hundreds digit, and the tens digit is equal to the units digit, and which are squares of integers.
|
{"ground_truth": "7744"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 276
|
numina_1.5_306
|
verifiable_math
|
XXXVIII OM - III - Problem 6
A plane is covered with a grid of regular hexagons with a side length of 1. A path on the grid is defined as a sequence of sides of the hexagons in the grid, such that any two consecutive sides have a common endpoint. A path on the grid is called the shortest if its endpoints cannot be connected by a shorter path. Find the number of shortest paths on the grid with a fixed starting point and a length of 60.
|
{"ground_truth": "6\\cdot2^{30}-6"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 277
|
numina_1.5_307
|
verifiable_math
|
XXXV OM - II - Problem 1
For a given natural number $ n $, find the number of solutions to the equation $ \sqrt{x} + \sqrt{y} = n $ in natural numbers $ x, y $.
|
{"ground_truth": "n-1"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 278
|
numina_1.5_308
|
verifiable_math
|
V OM - I - Task 2
Investigate when the sum of the cubes of three consecutive natural numbers is divisible by $18$.
|
{"ground_truth": "3"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 279
|
numina_1.5_310
|
verifiable_math
|
XLVI OM - III - Problem 5
Let $ n $ and $ k $ be natural numbers. From an urn containing 11 slips numbered from 1 to $ n $, we draw slips one by one, without replacement. When a slip with a number divisible by $ k $ appears, we stop drawing. For a fixed $ n $, determine those numbers $ k \leq n $ for which the expected value of the number of slips drawn is exactly $ k $.
|
{"ground_truth": "k"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 280
|
numina_1.5_311
|
verifiable_math
|
LIV OM - I - Task 7
At Aunt Reni's, $ n \geq 4 $ people met (including Aunt Reni). Each of the attendees gave at least one gift to at least one of the others. It turned out that everyone gave three times as many gifts as they received, with one exception: Aunt Reni gave only $ \frac{1}{6} $ of the number of gifts she received. Determine, depending on $ n $, the smallest possible number of gifts Aunt Reni could have received.
|
{"ground_truth": "12[(n+3)/5]"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 281
|
numina_1.5_312
|
verifiable_math
|
XXXV OM - III - Problem 1
Determine the number of functions $ f $ mapping an $ n $-element set to itself such that $ f^{n-1} $ is a constant function, while $ f^{n-2} $ is not a constant function, where $ f^k = f\circ f \circ \ldots \circ f $, and $ n $ is a fixed natural number greater than 2.
|
{"ground_truth": "n!"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 282
|
numina_1.5_314
|
verifiable_math
|
XXI OM - I - Problem 12
Find the smallest positive number $ r $ with the property that in a regular tetrahedron with edge length 1, there exist four points such that the distance from any point of the tetrahedron to one of them is $ \leq r $.
|
{"ground_truth": "\\frac{\\sqrt{6}}{8}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 283
|
numina_1.5_315
|
verifiable_math
|
LVI OM - I - Task 2
Determine all natural numbers $ n>1 $ for which the value of the sum $ 2^{2}+3^{2}+\ldots+n^{2} $ is a power of a prime number with a natural exponent.
|
{"ground_truth": "2,3,4,7"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 284
|
numina_1.5_316
|
verifiable_math
|
XL OM - II - Task 3
Given is a trihedral angle $ OABC $ with vertex $ O $ and a point $ P $ inside it. Let $ V $ be the volume of the parallelepiped with two vertices at points $ O $ and $ P $, whose three edges are contained in the rays $ OA^{\rightarrow} $, $ OB^{\rightarrow} $, $ OC^{\rightarrow} $. Calculate the minimum volume of the tetrahedron, whose three faces are contained in the faces of the trihedral angle $ OABC $, and the fourth face contains the point $ P $.
|
{"ground_truth": "\\frac{9}{2}V"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 285
|
numina_1.5_318
|
verifiable_math
|
XLVIII OM - I - Problem 7
Calculate the upper limit of the volume of tetrahedra contained within a sphere of a given radius $ R $, one of whose edges is the diameter of the sphere.
|
{"ground_truth": "\\frac{1}{3}R^3"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 286
|
numina_1.5_319
|
verifiable_math
|
XXI OM - I - Problem 10
Find the largest natural number $ k $ with the following property: there exist $ k $ different subsets of an $ n $-element set, such that any two of them have a non-empty intersection.
|
{"ground_truth": "2^{n-1}"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 287
|
numina_1.5_320
|
verifiable_math
|
LIX OM - II - Task 1
Determine the maximum possible length of a sequence of consecutive integers, each of which can be expressed in the form $ x^3 + 2y^2 $ for some integers $ x, y $.
|
{"ground_truth": "5"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 288
|
numina_1.5_321
|
verifiable_math
|
XV OM - I - Problem 11
In triangle $ ABC $, angle $ A $ is $ 20^\circ $, $ AB = AC $. On sides $ AB $ and $ AC $, points $ D $ and $ E $ are chosen such that $ \measuredangle DCB = 60^\circ $ and $ \measuredangle EBC = 50^\circ $. Calculate the angle $ EDC $.
|
{"ground_truth": "30"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 289
|
numina_1.5_322
|
verifiable_math
|
XLVI OM - II - Problem 6
A square with side length $ n $ is divided into $ n^2 $ unit squares. Determine all natural numbers $ n $ for which such a square can be cut along the lines of this division into squares, each of which has a side length of 2 or 3.
|
{"ground_truth": "nisdivisible2or3"}
|
{"source": "olympiads", "problem_type": "Combinatorics"}
| 290
|
numina_1.5_323
|
verifiable_math
|
XXVI - III - Task 2
On the surface of a regular tetrahedron with edge length 1, a finite set of segments is chosen in such a way that any two vertices of the tetrahedron can be connected by a broken line composed of some of these segments. Can this set of segments be chosen so that their total length is less than $1 + \sqrt{3}$?
|
{"ground_truth": "\\sqrt{7}"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 291
|
numina_1.5_324
|
verifiable_math
|
XVI OM - I - Problem 7
On the side $ AB $ of triangle $ ABC $, a point $ K $ is chosen, and points $ M $ and $ N $ are determined on the lines $ AC $ and $ BC $ such that $ KM \perp AC $, $ KN \perp BC $. For what position of point $ K $ is the area of triangle $ MKN $ the largest?
|
{"ground_truth": "KisthemidpointofsideAB"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 292
|
numina_1.5_327
|
verifiable_math
|
XVII OM - I - Problem 1
Present the polynomial $ x^5 + x + 1 $ as a product of two polynomials of lower degree with integer coefficients.
|
{"ground_truth": "x^5+x+1=(x^2+x+1)(x^3-x^2+1)"}
|
{"source": "olympiads", "problem_type": "Algebra"}
| 293
|
numina_1.5_328
|
verifiable_math
|
XLVI OM - III - Zadanie 3
Dana jest liczba pierwsza $ p \geq 3 $. Określamy ciąg $ (a_n) $ wzorami
Wyznaczyć resztę z dzielenia liczby $ a_{p^3} $ przez $ p $.
|
{"ground_truth": "p-1"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 294
|
numina_1.5_329
|
verifiable_math
|
XXVIII - II - Task 3
In a hat, there are 7 slips of paper. On the $ n $-th slip, the number $ 2^n-1 $ is written ($ n = 1, 2, \ldots, 7 $). We draw slips randomly until the sum exceeds 124. What is the most likely value of this sum?
|
{"ground_truth": "127"}
|
{"source": "olympiads", "problem_type": "Number Theory"}
| 295
|
numina_1.5_330
|
verifiable_math
|
I OM - B - Task 12
How many sides does a polygon have if each of its angles is $ k $ times larger than the adjacent angle? What values can $ k $ take?
|
{"ground_truth": "\\frac{1}{2},1,\\frac{3}{2},2,\\dots"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 296
|
numina_1.5_331
|
verifiable_math
|
IV OM - III - Task 3
Through each vertex of a tetrahedron of a given volume $ V $, a plane parallel to the opposite face of the tetrahedron has been drawn. Calculate the volume of the tetrahedron formed by these planes.
|
{"ground_truth": "27V"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 297
|
numina_1.5_332
|
verifiable_math
|
XLII OM - I - Problem 8
Determine 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:
(1) $ W_0 $ is a convex polyhedron with a center of symmetry,
(2) each of the polyhedra $ W_i $ ($ i = 1,\ldots, n $) is obtained from $ W_0 $ by a translation,
(3) each of the polyhedra $ W_i $ ($ i = 1,\ldots, n $) has a point in common with $ W_0 $,
(4) the polyhedra $ W_0, W_1, \ldots, W_n $ have pairwise disjoint interiors.
|
{"ground_truth": "26"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 298
|
numina_1.5_333
|
verifiable_math
|
XI OM - I - Task 3
Each 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.
|
{"ground_truth": "\\frac{2}{9}S"}
|
{"source": "olympiads", "problem_type": "Geometry"}
| 299
|
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