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olympiads | 6.139. $(x+1)^{5}+(x-1)^{5}=32 x$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
6.139. $(x+1)^{5}+(x-1)^{5}=32 x$. | <think>
Okay, so I have this equation to solve: (x + 1)^5 + (x - 1)^5 = 32x. Hmm, let's see. I need to find the values of x that satisfy this equation.
First, maybe expanding the left side would help. But expanding (x + 1)^5 and (x - 1)^5 seems like a lot of work. Wait, but I remember that when you expand binomials w... |
olympiads | ## 107. Math Puzzle $4 / 74$
What is the distance of a $6 \mathrm{~cm}$ long chord from the center of a circle with a radius of $5 \mathrm{~cm}$? | <think>
Okay, so I need to find the distance from the center of a circle to a chord that's 6 cm long. The circle has a radius of 5 cm. Hmm, let's think about this. I remember there's some relationship between the radius, the chord length, and the distance from the center to the chord. Maybe it's related to right triang... |
olympiads | 1. Provide an example of a natural number that is divisible by 2019, and the sum of its digits is also divisible by 2019. Don't forget to show that your example meets the condition. | <think>
Okay, so I need to find a natural number that is divisible by 2019, and the sum of its digits is also divisible by 2019. Hmm, let's start by recalling some number theory concepts.
First, a number divisible by 2019 must be a multiple of 2019. Since 2019 is equal to 3 × 673, the number must be divisible by both... |
olympiads | Two workers, $A$ and $B$, completed a task assigned to them as follows. First, only $A$ worked for $\frac{2}{3}$ of the time it would take $B$ to complete the entire task alone. Then, $B$ took over from $A$ and finished the work. In this way, the work took 2 hours longer than if they had started working together and co... | <think>
Okay, so I need to solve this work rate problem. Let me try to parse all the information given here.
First, there are two workers, A and B. They completed a task in a specific way: first, only A worked for 2/3 of the time it would take B to complete the entire task alone. Then, B took over and finished the rem... |
olympiads | 10. (5 points) Write down the natural numbers from 1 to 2015 in sequence, to get a large number $123456789 \cdots 20142015$. When this large number is divided by 9, the remainder is | <think>
Okay, so I need to find the remainder when the number formed by writing all natural numbers from 1 to 2015 in sequence is divided by 9. Let me think about how to approach this.
First, I remember that there's a rule for divisibility by 9: if the sum of a number's digits is divisible by 9, then the number itself... |
olympiads | Problem 8. In 28 examination tickets, each includes two theoretical questions and one problem. The student has prepared 50 theoretical questions and can solve the problems in 22 tickets. What is the probability that, by randomly picking one ticket, the student will be able to answer all the questions in the ticket? | <think>
Okay, so I need to find the probability that a student can answer all the questions on a randomly picked ticket. Each ticket has two theoretical questions and one problem. The student prepared 50 theoretical questions out of... hmm, wait, how many theoretical questions are there in total? The problem doesn't sa... |
olympiads | 3.266. $\sin (x+2 \pi) \cos \left(2 x-\frac{7 \pi}{2}\right)+\sin \left(\frac{3 \pi}{2}-x\right) \sin \left(2 x-\frac{5 \pi}{2}\right)$. | <think>
Okay, let's see. I need to simplify this trigonometric expression: sin(x + 2π)cos(2x - 7π/2) + sin(3π/2 - x)sin(2x - 5π/2). Hmm, trigonometric identities might help here. Let me start by recalling some periodicity and shift identities.
First, sin(θ + 2π) is the same as sinθ because sine has a period of 2π. Sim... |
olympiads | 5. How many diagonals in a regular 32-sided polygon are not parallel to any of its sides | <think>
Alright, so I need to figure out how many diagonals in a regular 32-sided polygon are not parallel to any of its sides. Hmm, okay. Let's start by recalling some basics about polygons and diagonals.
First, in a regular n-sided polygon, the number of diagonals can be calculated using the formula n(n - 3)/2. For ... |
olympiads | A totó requires tipping 12 matches, in each case with the symbols $\gg 1 \ll, » 2 \ll$ or $\gg \mathrm{x} \ll$. What is the expected number of correct tips if we fill out the ticket randomly (e.g., by drawing lots) so that each of the three possibilities has an equal chance for every match? | <think>
Alright, let me try to figure out this problem. So, the question is about a totó, which I think is like a sports betting game where you predict the outcome of 12 matches. Each match can have three possible results: "1" (home win), "2" (away win), or "x" (draw). We need to find the expected number of correct tip... |
olympiads | ## Task 4 - 160814
Peter presents his friend Fritz with the following problem:
"Given a circle whose diameter is equal to the Earth's diameter, and a second concentric circle whose circumference is $1 \mathrm{~m}$ longer than the circumference of the first circle. Determine the distance between the two circle lines!"... | <think>
Okay, let's see. So Peter gives Fritz this problem where there's a circle with the Earth's diameter, and then another concentric circle (which means they share the same center) that has a circumference 1 meter longer than the first one. Fritz says the distance between the two circles is the same as if the first... |
olympiads | 1. (17 points) Three cyclists, Dima, Misha, and Petya, started a 1 km race simultaneously. At the moment Misha finished, Dima still had to cover one tenth of the distance, and at the moment Dima finished, Petya still had to cover one tenth of the distance. How far apart (in meters) were Petya and Misha when Misha finis... | <think>
Okay, let's see. So there's this problem about three cyclists: Dima, Misha, and Petya. They started a 1 km race at the same time. The question is asking how far apart Petya and Misha were when Misha finished. Hmm. Let me try to parse the information step by step.
First, when Misha finished the race, which is 1... |
olympiads | Problem 8.4. Given a square $A B C D$. Point $L$ is on side $C D$ and point $K$ is on the extension of side $D A$ beyond point $A$ such that $\angle K B L=90^{\circ}$. Find the length of segment $L D$, if $K D=19$ and $C L=6$.
. This process is carried out repeatedly until the... | <think>
Okay, so there's this problem where the numbers from 1 to 2002 are written in order, and then we start erasing numbers in a specific pattern. The first time, we erase every 3rd number starting from the first position, which would be the 1st, 4th, 7th, etc. Then, with the remaining numbers, we repeat the process... |
olympiads | 3.176. Given: $\operatorname{ctg} \alpha=4, \operatorname{ctg} \beta=\frac{5}{3}, 0<\alpha<\frac{\pi}{2}, 0<\beta<\frac{\pi}{2}$. Find $\alpha+\beta$. | <think>
Okay, so I need to find α + β given that ctg α is 4 and ctg β is 5/3, with both α and β in the first quadrant. Hmm, cotangent is the reciprocal of tangent, right? So maybe I should first find tan α and tan β by taking reciprocals of the given cotangent values. Let me write that down.
First, ctg α = 4 means tha... |
aops_forum | Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell... | <think>
Alright, let's tackle this geometry problem. So, we have triangle ABC with sides AB=26, BC=51, and CA=73. There's a point O inside the triangle, and three lines passing through O, each parallel to one of the sides. These lines intersect the sides of the triangle, forming a hexagon with area A. We need to find t... |
olympiads | 7. As shown in the figure, construct the inscribed circle $\odot O$ inside the square $A B C D$, tangent to $A B$ at point $Q$ and to $C D$ at point $P$. Connect $P A, P B$ to form isosceles $\triangle P A B$. Rotating these three figures (the square, the circle, and the isosceles triangle) around their common axis of ... | <think>
Okay, so I need to find the ratio of the volumes of a cylinder, a sphere, and a cone that are formed by rotating a square, a circle, and an isosceles triangle around their common axis of symmetry PQ. Let me start by visualizing the problem.
First, the square ABCD has an inscribed circle, which is tangent to A... |
aops_forum | Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$. | <think>
Okay, so I need to find a two-digit number, let's call it ab, such that when you take 5^n and 2^n for some positive integer n, both of these numbers start with the digits a and b in that order. The question is asking for the specific two-digit number ab that satisfies this condition for some n.
First, let me ... |
olympiads | Solve the following inequality on the set of real numbers:
$$
\sqrt{x+6}>x-6 .
$$ | <think>
Okay, so I need to solve the inequality √(x + 6) > x - 6. Hmm, let's see. First, I should figure out the domain where this inequality is defined. The square root is only defined when the expression inside is non-negative. So, x + 6 ≥ 0, which means x ≥ -6. That's the first thing.
Now, the right side of the ine... |
olympiads | 2. (10 points) A team of 8 people completed $\frac{1}{3}$ of a project in 30 days. Then, 4 more people were added to complete the remaining part of the project. Therefore, the total time to complete the project was $\qquad$ days.
The team of 8 people completed $\frac{1}{3}$ of a project in 30 days. Then, 4 more people... | <think>
Okay, let's see. So there's this problem where a team of 8 people completed 1/3 of a project in 30 days. Then they added 4 more people to finish the remaining 2/3. We need to find the total time taken to complete the project. Hmm, this seems like a work-rate problem. Let me recall how those work.
I think the ... |
olympiads | 5. The first four members of a sequence are: $2,0,1,8$. Each subsequent member of the sequence is the unit digit of the sum of the previous four members. (e.g., the fifth member is 1). Is the 2018th member of the sequence an even or odd number? Justify your answer! | <think>
Okay, so I have this problem here where the first four numbers in a sequence are 2, 0, 1, 8. Then each next number is the units digit of the sum of the previous four. They want to know if the 2018th term is even or odd. Hmm, let's see. Alright, let's start by understanding the problem step by step.
First, the ... |
aops_forum | Let $n$ be an integer bigger than $0$. Let $\mathbb{A}= ( a_1,a_2,...,a_n )$ be a set of real numbers. Find the number of functions $f:A \rightarrow A$ such that $f(f(x))-f(f(y)) \ge x-y$ for any $x,y \in \mathbb{A}$, with $x>y$. | <think>
Okay, let's try to figure out how to solve this problem. The question is about finding the number of functions f from a set A to itself, where A has n elements. The condition given is that for any x and y in A with x > y, the difference f(f(x)) - f(f(y)) must be at least x - y. Hmm, interesting. So the function... |
olympiads | (JBMO 2002 - Shorlist, Problem G3), treated in class
Let $A B C$ be an isosceles triangle at $A$, with $\widehat{C A B}=20^{\circ}$. Let $D$ be a point on the segment $[A C]$, such that $A D=B C$. Calculate the angle $\widehat{B D C}$. | <think>
Okay, let me try to figure out this geometry problem. So, we have an isosceles triangle ABC with AB equal to AC because it's isosceles at A. The angle at A, which is angle CAB, is 20 degrees. Then there's a point D on segment AC such that AD equals BC. We need to find the measure of angle BDC. Hmm, okay, let's ... |
olympiads | 6. (15 points) From a homogeneous straight rod, a piece of length $s=60 \mathrm{~cm}$ was cut. By how much did the center of gravity of the rod move as a result? | <think>
Okay, so I have this problem here where a piece of length 60 cm is cut from a homogeneous straight rod. The question is asking by how much the center of gravity (or center of mass, since it's homogeneous) of the rod moved as a result. Hmm, let me try to visualize this.
First, let me make sure I understand the... |
olympiads | Solve the following equation:
$$
2^{x+3}+4^{x+1}=320 .
$$ | <think>
Okay, so I need to solve the equation 2^{x+3} + 4^{x+1} = 320. Hmm, let me think about how to approach this. Both terms have exponents with base 2 and 4. I remember that 4 is a power of 2, right? 4 is 2 squared. Maybe I can rewrite everything in terms of base 2 to simplify the equation. That often helps with ex... |
olympiads | 9. If $\tan x+\tan y=5$ and $\tan (x+y)=10$, find $\cot ^{2} x+\cot ^{2} y$.
Answer: 96
Solution: We know that
$$
\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}=10
$$ | <think>
Okay, let's see here. The problem says that tan x plus tan y is 5, and tan(x + y) is 10. We need to find cot squared x plus cot squared y. Hmm, okay. So first, let me recall some trigonometric identities that might help here.
I remember that the tangent of a sum formula is tan(x + y) = (tan x + tan y)/(1 - tan... |
olympiads | 1. A snowdrift 468 cm high decreased in height by 6 cm in the first hour, by 12 cm in the second hour, ..., by $6 k$ cm in the -th hour. After some time $T$, the snowdrift melted completely. What fraction of the snowdrift's height melted in the time $\frac{T}{2}$? | <think>
Okay, so I need to figure out this snowdrift problem. Let me read it again. The snowdrift is 468 cm high. Each hour, it decreases by 6 cm in the first hour, 12 cm in the second hour, and so on, so each hour it's melting 6k cm where k is the hour number. After some time T, it's completely melted. The question is... |
olympiads | ## Task 18/77
Given is a cylindrical beaker of height $H$. The (homogeneous) mantle has the mass $M$, the mass of the bottom is negligible. The glass is filled with a (homogeneous) liquid of mass $m_{H}$ to the brim, but the liquid drips out through an opening in the bottom until it is completely empty.
Required is a... | <think>
Okay, let's see. I have to find the position of the center of mass as a function of the fill height h for a cylindrical beaker. The beaker has a height H, its mantle (which I think means the side part) has mass M, and the bottom's mass is negligible. The liquid, which originally fills the beaker to the brim (so... |
olympiads | 1. A necklace consists of 80 beads of red, blue, and green colors. It is known that on any segment of the necklace between two blue beads, there is at least one red bead, and on any segment of the necklace between two red beads, there is at least one green bead. What is the minimum number of green beads that can be in ... | <think>
Okay, let's try to figure out this necklace problem. So, we have 80 beads in total, colored red, blue, or green. The conditions are:
1. Between any two blue beads, there's at least one red bead.
2. Between any two red beads, there's at least one green bead.
We need to find the minimum number of green beads po... |
olympiads | $3.420 \sin 10^{\circ} \cdot \sin 20^{\circ} \cdot \sin 30^{\circ} \cdot \sin 40^{\circ} \cdot \sin 50^{\circ} \cdot \sin 60^{\circ} \cdot \sin 70^{\circ} \cdot \sin 80^{\circ}=\frac{3}{256} \cdot$ | <think>
Okay, let me try to figure out how to solve this problem. The question is to verify whether the product of these sines multiplied by 3.420 equals 3/256. The expression given is:
3.420 × sin10° × sin20° × sin30° × sin40° × sin50° × sin60° × sin70° × sin80° = 3/256
First, I need to check if this equality holds.... |
olympiads | 34. Let $S=\{1,2,3, \ldots, 20\}$ be the set of all positive integers from 1 to 20 . Suppose that $N$ is the smallest positive integer such that exactly eighteen numbers from $S$ are factors of $N$, and the only two numbers from $S$ that are not factors of $N$ are consecutive integers. Find the sum of the digits of $N$... | <think>
Okay, so I need to find the smallest positive integer N such that exactly eighteen numbers from the set S = {1,2,3,...,20} are factors of N. The two numbers that aren't factors of N are consecutive integers. Then, I need to find the sum of the digits of this smallest N. Hmm, let's break this down step by step.
... |
olympiads_ref | 4. A4 (BUL 2) Determine all real values of the parameter $a$ for which the equation
$$ 16 x^{4}-a x^{3}+(2 a+17) x^{2}-a x+16=0 $$
has exactly four distinct real roots that form a geometric progression. | <think>
Okay, let's try to figure out this problem. So, we have the equation:
$$16 x^{4} - a x^{3} + (2 a + 17) x^{2} - a x + 16 = 0$$
And we need to find all real values of the parameter \(a\) such that this equation has exactly four distinct real roots that form a geometric progression. Hmm, geometric progression. ... |
olympiads | 11.1. At first, there were natural numbers, from 1 to 2021. And they were all white. Then the underachiever Borya painted every third number blue. Then the top student Vova came and painted every fifth number red. How many numbers remained white? (7 points)
# | <think>
Okay, let's tackle this problem step by step. So, the question is about natural numbers from 1 to 2021. Initially, all are white. Then Borya paints every third number blue, and after that, Vova paints every fifth number red. We need to find how many numbers remained white after both painted. Hmm.
First, let me... |
olympiads | 1. (6 points) Calculate: $4.165 \times 4.8 + 4.165 \times 6.7 - 4.165 \div \frac{2}{3}=$ | <think>
Okay, let's see. I need to calculate this expression: 4.165 times 4.8 plus 4.165 times 6.7 minus 4.165 divided by (2/3). Hmm, that looks a bit complicated, but maybe there's a way to simplify it.
First, let me write down the problem so I can visualize it better:
4.165 × 4.8 + 4.165 × 6.7 - 4.165 ÷ (2/3)
Hmm... |
olympiads | B3. In triangle ABC, the angle $\alpha=58^{\circ}$ and the angle $\beta=84^{\circ}$. What is the measure of angle $x$ between the angle bisector of $\gamma$ and the altitude to side $c$? | <think>
Okay, so I need to find the measure of angle x between the angle bisector of γ and the altitude to side c in triangle ABC. The given angles are α=58° and β=84°. First, let me visualize the triangle. In triangle ABC, angle α is at vertex A, angle β at vertex B, and angle γ at vertex C. Since the sum of angles in... |
olympiads | 7.5. One hundred non-zero integers are written in a circle such that each number is greater than the product of the two numbers following it in a clockwise direction. What is the maximum number of positive numbers that can be among these 100 written numbers? | <think>
Alright, so I have this problem here: 100 non-zero integers are written in a circle, and each number is greater than the product of the two numbers following it in a clockwise direction. The question is asking for the maximum number of positive numbers that can be among these 100 numbers. Hmm, okay, let me try ... |
olympiads | 14. (2 marks) In $\triangle A B C, \angle A C B=3 \angle B A C, B C=5, A B=11$. Find $A C$.
(2 分) 在 $\triangle A B C$ 中 $\angle A C B=3 \angle B A C, B C=5, A B=11$ 。求 $A C$ 。 | <think>
嗯，我现在要解这个三角形的问题。题目是：在△ABC中，∠ACB = 3∠BAC，BC = 5，AB = 11，求AC的长度。好的，首先我需要理清楚已知条件和所求的量。
首先，已知的是BC = 5，AB = 11，角ACB是角BAC的三倍。要求的是AC的长度。我需要画一个三角形，标出各个边和角，然后找出适用的定理来解决这个问题。
首先，我应该给各个角命名，方便之后讨论。假设角BAC为α，那么根据题意，角ACB就是3α。接下来，剩下的角是角ABC，也就是角B，根据三角形内角和为180度，我们可以得到：
α + 角B + 3α = 180°
所以，角B = 180° - 4α
现在，已知AB = 11，BC = 5... |
olympiads | Problem 3. Martin has three containers, two of which hold exactly $3 l$ and $5 l$, while the third contains $10 \mathrm{l}$ of milk. He needs to measure out exactly $4 l$ of milk using the three containers.
Can Martin do this, and if so, how? | <think>
Okay, let's see. Martin has three containers: 3 liters, 5 liters, and a 10-liter container that's already full. He needs to measure out exactly 4 liters. Hmm, how can he do this using these containers? I need to think step by step.
First, let's list the containers he has: 3L, 5L, and 10L. The 10L is full. The... |
olympiads | A vertical-axis, rotational paraboloid-shaped tall container has its focus 0.05 meters above the axis point. We pour a little water into the container. At what angular velocity $\omega$ should we rotate the container around its axis so that the water flows out at the top of the container? | <think>
Okay, so I have this problem about a vertical-axis paraboloid-shaped container. The focus is 0.05 meters above the axis point. We pour a little water into it, and we need to find the angular velocity ω such that when we rotate the container, the water just starts to spill out from the top. Hmm, let me think.
F... |
olympiads | 5th Chinese 1990 Problem B1 Given a positive integer n, for what real u, v do the simultaneous equations a 2 + b 2 + c 2 = 169n 2 , a 2 (ua 2 + vb 2 ) + b 2 (ub 2 + vc 2 ) + c 2 (uc 2 + va 2 ) = (2u + v)(13n) 4 /4 have a solution in positive integers a, b, c? Solution | <think>
Alright, let's try to tackle this problem. So, we have a positive integer n, and we need to find real numbers u and v such that the given simultaneous equations have a solution in positive integers a, b, c. The equations are:
1. \( a^2 + b^2 + c^2 = 169n^2 \)
2. \( a^2(ua^2 + vb^2) + b^2(ub^2 + vc^2) + c^2(uc^... |
olympiads_ref | A permutation of the set $\{1, \ldots, 2021\}$ is a sequence $\sigma=\left(\sigma_{1}, \ldots, \sigma_{2021}\right)$ such that each element of the set $\{1, \ldots, 2021\}$ is equal to exactly one term $\sigma_{i}$. We define the weight of such a permutation $\sigma$ as the sum
$$
\sum_{i=1}^{2020}\left|\sigma_{i+1}-\... | <think>
Okay, let's see. The problem is about finding the greatest possible weight of a permutation of the set {1, ..., 2021}. The weight is defined as the sum of the absolute differences between consecutive elements in the permutation. So, if we have a permutation σ = (σ₁, σ₂, ..., σ₂₀₂₁), the weight is Σ|σ_{i+1} - σ_... |
olympiads | 6. Let three non-collinear unit vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ in the plane satisfy $\boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=\mathbf{0}$. If $0 \leq t \leq 1$, then the range of $|-2 \boldsymbol{a}+t \boldsymbol{b}+(1-t) \boldsymbol{c}|$ is $\qquad$. | <think>
Okay, so I need to find the range of the expression \(|-2 \boldsymbol{a} + t \boldsymbol{b} + (1 - t) \boldsymbol{c}|\) where \(0 \leq t \leq 1\), given that \(\boldsymbol{a}\), \(\boldsymbol{b}\), and \(\boldsymbol{c}\) are non-collinear unit vectors in the plane and they satisfy \(\boldsymbol{a} + \boldsymbol... |
olympiads | Alice wanders in the Forest of Forgetfulness where she cannot remember the day of the week. She meets the lion and the unicorn. The lion lies on Monday, Tuesday, and Wednesday and tells the truth the other days, while the unicorn lies only on Thursday, Friday, and Saturday.
"Yesterday was a day when I lied," says the ... | <think>
Okay, let's try to figure out what day it is based on the statements from the lion and the unicorn. Both of them say, "Yesterday was a day when I lied." Hmm, so we need to check for each day of the week whether both statements make sense given their lying schedules.
First, let me list out the days of the week ... |
cn_contest | 1. Given that $\alpha, \beta$ are the roots of the equation $x^{2}-7 x+8=0$, and $\alpha$ $>\beta$. Without solving the equation, use the relationship between roots and coefficients to find the value of $\frac{2}{\alpha}+3 \beta^{2}$.
(8th "Zu Chongzhi Cup" Mathematics Invitational Competition) | <think>
Okay, let me try to figure out how to solve this problem. So, the question is: given that α and β are the roots of the equation x² -7x +8 =0, with α > β. We need to find the value of (2/α) + 3β² without solving the equation. Instead, we should use the relationships between the roots and the coefficients. Alrigh... |
olympiads | 17. Let $f:(0,1) \rightarrow(0,1)$ be a differentiable function with a continuous derivative such that for every positive integer $n$ and odd positive integer $a<2^{n}$, there exists an odd positive integer $b<2^{n}$ such that $f\left(\frac{a}{2^{n}}\right)=\frac{b}{2^{n}}$. Determine the set of possible values of $f^{... | <think>
Okay, let's try to tackle this problem. So, we have a function f that maps the interval (0,1) to itself. It's differentiable with a continuous derivative. The key condition here is that for every positive integer n and every odd positive integer a < 2^n, there exists an odd positive integer b < 2^n such that f(... |
olympiads | 2. For numbers $a$ and $b$ such that $a + 10b = 1$, what is the maximum possible value of the product $a \cdot b$? Determine the values of $a$ and $b$ for which this value is achieved. | <think>
Okay, so I need to find the maximum product of two numbers, a and b, given that their combination a + 10b equals 1. Hmm, this seems like an optimization problem with a constraint. Let me think about how to approach this.
First, the problem gives me the equation a + 10b = 1. I need to maximize the product P = a... |
olympiads | 76. Teacher Wang bought 132 pencils, 75 exercise books, and 37 pencil sharpeners, and divided them into several identical prize packages, with the same number of pencils, exercise books, and pencil sharpeners remaining. How many prize packages did Teacher Wang make at most?
| <think>
Okay, let's try to figure out this problem. So, Teacher Wang bought 132 pencils, 75 exercise books, and 37 pencil sharpeners. She wants to divide them into several identical prize packages. The key here is that each package must have the same number of pencils, exercise books, and pencil sharpeners. Also, it me... |
olympiads | Solve the following system of equations without using logarithms:
$$
a^{7 x} \cdot a^{15 y}=\sqrt{a^{19}}, \quad \sqrt[3]{a^{25 y}}: \sqrt{a^{13 x}}=\sqrt[12]{a}
$$ | <think>
Okay, let's see. I need to solve this system of equations without using logarithms. The equations are:
1. \( a^{7x} \cdot a^{15y} = \sqrt{a^{19}} \)
2. \( \sqrt[3]{a^{25y}} : \sqrt{a^{13x}} = \sqrt[12]{a} \)
Hmm, okay. Both equations involve exponents with the same base \( a \). Since the base is the same, I ... |
olympiads | 21. (1999 National Team Selection Test) For non-negative real numbers $x_{1}, x_{2}$, $\cdots, x_{n}$ satisfying the condition $x_{1}+x_{2}+\cdots+x_{n}=1$, find the maximum value of $\sum_{j=1}^{n}\left(x_{j}^{4}-x_{j}^{5}\right)$. | <think>
Okay, so I have this problem here where I need to find the maximum value of the sum of (x_j^4 - x_j^5) for j from 1 to n, given that all x_j are non-negative real numbers and their sum is 1. Hmm, interesting. Let me think about how to approach this.
First, since all the variables are non-negative and add up to... |
olympiads | 4. Given that $x$ and $y$ are real numbers, $x^{2}-x y+y^{2}=1$. Then the maximum value of $x+2 y$ is $\qquad$ . | <think>
Okay, so I have this problem here: given that x and y are real numbers satisfying the equation x² - xy + y² = 1, I need to find the maximum value of x + 2y. Hmm, let's see. I need to maximize the linear expression x + 2y under the quadratic constraint x² - xy + y² = 1.
First, I recall that when dealing with m... |
inequalities | 5. 137 Find all polynomials $P(x)$ satisfying $P(0)=0$ and
$$P(x) \equiv \frac{1}{2}(P(x+1)+P(x-1)), x \in R$$ | <think>
Okay, so I need to find all polynomials P(x) such that P(0) = 0 and the equation P(x) = 1/2 [P(x+1) + P(x-1)] holds for all real numbers x. Let me try to break this down step by step.
First, let's restate the problem to make sure I understand it. We have a polynomial P(x) with real coefficients, and it satisfi... |
olympiads | Example 15 (2003 Beijing Competition Question) Given that $x, y$ are positive real numbers, and satisfy $x y+x+y=71, x^{2} y+x y^{2}=880$, then $x^{2}+y^{2}=$ $\qquad$ . | <think>
Okay, let me try to solve this problem step by step. So, we have two equations involving x and y, both positive real numbers. The equations are:
1. xy + x + y = 71
2. x²y + xy² = 880
And we need to find x² + y². Hmm, okay. Let me think about how to approach this.
First, I notice that both equations involve x... |
olympiads | Example 8 (I) Given two identical equilateral triangular pieces of paper, it is required to cut and assemble one into a regular tetrahedron model and the other into a regular triangular prism model, such that their total surface areas are equal to the area of the original triangle. Please design a cutting and assembly ... | <think>
Okay, let's see. I need to solve this problem where I have two identical equilateral triangular pieces of paper. For each, I have to cut and assemble one into a regular tetrahedron and the other into a regular triangular prism. The total surface areas of both models should equal the area of the original triangl... |
olympiads | 14. (6 points) If the sum of the "day" numbers in three consecutive days is 18, then these three "day" numbers are 5, 6, 7. If the sum of the "day" numbers in three consecutive days is 33, then these three "day" numbers are $\qquad$ | <think>
Okay, let's see. The problem says that if the sum of the day numbers in three consecutive days is 18, then those days are 5, 6, and 7. Now, they're asking what the three consecutive days would be if the sum is 33. Hmm, okay.
First, I need to understand what "day numbers" mean. I think they're referring to the ... |
olympiads | [ $\left.\frac{\text { Pairings and Groups; Bijections }}{\text { Regular Polygons }}\right]$
[ Examples and Counterexamples. Constructions ]
A circus arena is illuminated by $n$ different spotlights. Each spotlight illuminates a certain convex shape. It is known that if one arbitrary spotlight is turned off, the aren... | <think>
Okay, let me try to work through this problem step by step. So, we have a circus arena with n different spotlights. Each spotlight illuminates a convex shape. The conditions are:
1. If any one spotlight is turned off, the arena is still fully illuminated.
2. If any two spotlights are turned off, the arena is n... |
aops_forum | The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations?
[i]Proposed by Michael Kural[/i] | <think>
Okay, let's try to figure out this problem. So, we start with the numbers 0 and 1 written on a chalkboard. Every minute, Denys inserts the sum of each pair of consecutive numbers between them. We need to determine how many odd numbers will be on the board after 10 such operations. Hmm, interesting. Let me think... |
olympiads | 1. The price of one pencil is a whole number of eurocents. The total price of 9 pencils is greater than 11, but less than 12 euros, while the total price of 13 pencils is greater than 15, but less than 16 euros. How much does one pencil cost? | <think>
Okay, let's try to figure out how much one pencil costs. The problem says the price is a whole number of eurocents, so we're dealing with integers here. Let me break down the information given.
First, the total price of 9 pencils is greater than 11 euros but less than 12 euros. Let's convert those euros to cen... |
olympiads | 8.4. It is known that $a b c=1$. Calculate the sum
$$
\frac{1}{1+a+a b}+\frac{1}{1+b+b c}+\frac{1}{1+c+c a}
$$ | <think>
Okay, so I have this problem here where I need to calculate the sum of three fractions: 1 divided by (1 + a + ab), plus 1 divided by (1 + b + bc), plus 1 divided by (1 + c + ca). The given condition is that abc = 1. Hmm, let's see how I can approach this.
First, I remember that when dealing with variables mul... |
aops_forum | What is the maximum number of subsets of $S = {1, 2, . . . , 2n}$ such that no one is contained in another and no two cover whole $S$?
[i]Proposed by Fedor Petrov[/i] | <think>
Okay, so I need to figure out the maximum number of subsets of the set S = {1, 2, ..., 2n} such that two conditions are satisfied: first, no subset is contained within another, and second, no two subsets cover the entire set S. Hmm, let me start by understanding the problem better.
The problem is asking for th... |
olympiads | Given $x, y, z \geq 0, x+y+z=4$, find the maximum value of $x^{3} y+y^{3} z+z^{3} x$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | <think>
Okay, so I have this problem here: Given that x, y, z are non-negative real numbers and their sum is 4, I need to find the maximum value of the expression x³y + y³z + z³x. Hmm, let's see how to approach this.
First, since we're dealing with an optimization problem under a constraint (x + y + z = 4), maybe I ca... |
olympiads | $\begin{array}{l}\text { 1. In } \triangle A B C, A B=4, B C=7, C A=5, \\ \text { let } \angle B A C=\alpha. \text { Find } \sin ^{6} \frac{\alpha}{2}+\cos ^{6} \frac{\alpha}{2} \text {. }\end{array}$ | <think>
Okay, so I need to find the value of \(\sin^6 \frac{\alpha}{2} + \cos^6 \frac{\alpha}{2}\) where \(\alpha\) is the angle at vertex A in triangle ABC with sides AB=4, BC=7, and CA=5. Hmm, let's start by recalling the formula for \(\sin^6 x + \cos^6 x\). I remember there are some identities that can simplify this... |
olympiads_ref | Given a positive integer $n$. Determine all positive real numbers $x$ such that
$$
n x^{2}+\frac{2^{2}}{x+1}+\frac{3^{2}}{x+2}+\ldots+\frac{(n+1)^{2}}{x+n}=n x+\frac{n(n+3)}{2} .
$$ | <think>
Okay, let's see. I have this equation to solve for positive real numbers x, given a positive integer n. The equation is:
n x² + (2²)/(x + 1) + (3²)/(x + 2) + ... + ((n + 1)²)/(x + n) = n x + [n(n + 3)]/2.
Hmm, that's quite a complex equation. Let me try to parse it step by step.
First, on the left-hand side ... |
olympiads | At most how many numbers can we select from the first 1983 positive integers so that the product of any two selected numbers does not appear among the selected numbers? | <think>
Okay, so I need to figure out the maximum number of numbers I can select from the first 1983 positive integers such that the product of any two selected numbers isn't also in the selected set. Hmm, interesting problem. Let me start by understanding the problem better.
The question is asking for the largest pos... |
olympiads | 9. (24th American Mathematics Competition) There are 1990 piles of stones, with the number of stones in each pile being $1, 2, \cdots, 1990$. Perform the following operation: each time, you can choose any number of piles and remove the same number of stones from each chosen pile. How many operations are required at min... | <think>
Okay, let me try to figure out this problem. So, there are 1990 piles of stones, each pile having from 1 up to 1990 stones, right? Each operation allows me to choose any number of piles and remove the same number of stones from each chosen pile. The question is, what's the minimum number of operations needed to... |
olympiads | 2. Hudson labels each of the four vertices of a triangular pyramid with a different integer chosen from 1 to 15 . For each of the four triangular faces, he then calculates the mean of the three numbers at the vertices of the face.
Given that the means calculated by Hudson are all integers, how many different sets of fo... | <think>
Okay, let's tackle this problem step by step. The question is about Hudson labeling the four vertices of a triangular pyramid (which is basically a tetrahedron) with different integers from 1 to 15. For each of the four triangular faces, he calculates the mean of the three numbers at the vertices, and all these... |
olympiads | 5. On the website of the football club "Rostov," a poll is being conducted to determine which of the $m$ football players the website visitors consider the best at the end of the season. Each visitor votes once for one player. The website displays the rating of each player, which is the percentage of votes cast for the... | <think>
Okay, let's try to figure out this problem step by step. The question is about a poll on a football club's website where visitors vote for one of m players. The rating of each player is the percentage of votes they've received, rounded to the nearest whole number. After some votes have been cast, the total rati... |
olympiads | 12.287. The edges of a rectangular parallelepiped are in the ratio $3: 4: 12$. A diagonal section is made through the larger edge. Find the sine of the angle between the plane of this section and a diagonal of the parallelepiped not lying in it. | <think>
Okay, so I have this problem here: a rectangular parallelepiped (which is just a fancy way of saying a box shape with all sides at right angles) has edges in the ratio 3:4:12. Then, someone makes a diagonal section through the larger edge, and I need to find the sine of the angle between the plane of this secti... |
olympiads | How many squares with sides along the grid lines can be drawn on an $8 \times 8$ board?
# | <think>
Okay, so I need to figure out how many squares can be drawn on an 8x8 chessboard with sides along the grid lines. Hmm, let's start by visualizing an 8x8 grid. It's like a regular chessboard, right? So there are 8 rows and 8 columns, making 64 small squares. But the question isn't just about the 1x1 squares. It'... |
olympiads | Problem 1. A train consists of 20 cars, numbered from 1 to 20, starting from the beginning of the train. Some of the cars are postal cars. It is known that
- the total number of postal cars is even;
- the number of the nearest postal car to the beginning of the train is equal to the total number of postal cars;
- the ... | <think>
Alright, let's try to tackle this problem step by step. Okay, so we have a train with 20 cars numbered from 1 to 20. Some of these are postal cars, and we need to figure out which ones based on the given conditions. Let me break down each condition and see how they can help us.
First, the total number of posta... |
olympiads | Example 3 Find the largest positive integer $n$, such that in three-dimensional space there exist $n$ points $P_{1}, P_{2}, \cdots, P_{n}$, where no three points are collinear, and for any $1 \leqslant i<j<k \leqslant n, \triangle P_{i} P_{j} P_{k}$ is not an obtuse triangle. | <think>
Alright, let's tackle this problem. So, we need to find the largest positive integer n such that in three-dimensional space, there exist n points where no three are collinear, and every triangle formed by any three points is not obtuse. Hmm, okay. Let me start by understanding the problem thoroughly.
First, le... |
olympiads | 31. In one month, three Wednesdays fell on even dates. What date will the second Sunday of the month be? | <think>
Okay, let's see. The problem says that in one month, three Wednesdays fell on even dates. We need to find out what date the second Sunday of the month will be. Hmm, alright. Let me break this down step by step.
First, let me visualize a calendar. A month can have 28 to 31 days. Since there are three Wednesdays... |
olympiads | 10. A number lies between 2013 and 2156, and when divided by $5$, $11$, and $13$, the remainders are the same. The largest possible remainder is $\qquad$ _. | <think>
Okay, so I need to find a number between 2013 and 2156. When this number is divided by 5, 11, or 13, it leaves the same remainder each time. And I need to find the largest possible remainder. Hmm, let me think.
First, let me paraphrase the problem. There's a number N such that 2013 < N < 2156. When N is divide... |
olympiads | 9.5. We will call a palindrome a natural number whose decimal representation reads the same from left to right as from right to left (the decimal representation cannot start with zero; for example, the number 1221 is a palindrome, while the numbers 1231, 1212, and 1010 are not). Which palindromes among the numbers from... | <think>
Okay, so I need to figure out whether there are more palindromic numbers between 10,000 and 999,999 with an odd sum of digits or an even sum, and by what factor. Hmm, let's start by understanding the problem better.
First, the range given is from 10,000 to 999,999. That means we're dealing with 5-digit and 6-d... |
olympiads | Question 231, Positive integer $n \geq 2$, given pairwise distinct real numbers $a_{1}, a_{2}, \ldots, a_{n}$ forming set $S$, define $k(S)$ as the number of all distinct numbers of the form $a_{i}+2^{j}(i, j=1, 2, \ldots, n)$, for all sets $S$, find the minimum possible value of $k(S)$.
For all sets $S$, find the min... | <think>
Okay, let's try to tackle this problem. So, we have a positive integer n ≥ 2, and we need to consider a set S of n distinct real numbers a₁, a₂, ..., aₙ. The task is to define k(S) as the number of all distinct numbers of the form a_i + 2^j, where i and j range from 1 to n. Then, for all possible sets S, we nee... |
olympiads | For what value of $k$ is $k \%$ of 25 equal to $20 \%$ of 30 ? | <think>
Okay, let's see. I have to find the value of k such that k% of 25 is equal to 20% of 30. Hmm. Alright, so first, let me remember what percentage means. Percentage means per hundred, right? So k% of 25 would be (k/100) multiplied by 25. Similarly, 20% of 30 is (20/100)*30. The problem says these two expressions ... |
olympiads_ref | Determine all four digit numbers $\overline{a b c d}$ such that
$$
a(a+b+c+d)\left(a^{2}+b^{2}+c^{2}+d^{2}\right)\left(a^{6}+2 b^{6}+3 c^{6}+4 d^{6}\right)=\overline{a b c d}
$$ | <think>
Alright, so I need to find all four-digit numbers \(\overline{a b c d}\) that satisfy the equation:
\[
a(a + b + c + d)(a^2 + b^2 + c^2 + d^2)(a^6 + 2b^6 + 3c^6 + 4d^6) = \overline{a b c d}
\]
First, since \(\overline{a b c d}\) is a four-digit number, \(a\) must be between 1 and 9, and \(b\), \(c\), \(d\) ar... |
olympiads | 5. Let the cost of a pound of rice be x coins, a pound of brown sugar be y coins, and a pound of refined sugar be z coins. We obtain the system:
$$
\left\{\begin{array}{l}
4 x+\frac{9}{2} y+12 z=6 ; \\
12 x+6 y+6 z=8
\end{array}\right. \text { Subtract the first equation from twice the second equation and }
$$
expres... | <think>
Okay, so I need to figure out how they arrived at the conclusion that 4x + 3y + 6z equals 4 coins. Let me start by going through the problem step by step.
First, we have two equations with three variables: x, y, and z, which represent the cost per pound of rice, brown sugar, and refined sugar, respectively. Th... |
olympiads | 11. The sequence $a_{0}, a_{1}, a_{2}, \cdots, a_{n}, \cdots$, satisfies the relation $\left(3-a_{n+1}\right)\left(6+a_{n}\right)=18$ and $a_{0}=3$, then $\sum_{i=0}^{n} \frac{1}{a_{i}}=$ $\qquad$ . | <think>
Okay, so I need to solve this problem where there's a sequence defined by the relation (3 - a_{n+1})(6 + a_n) = 18, and the initial term a_0 is 3. Then I have to find the sum of 1/a_i from i=0 to n. Hmm, let's start by understanding the recurrence relation better.
First, the given equation is (3 - a_{n+1})(6 +... |
olympiads | Fibonacci numbers Euclidean algorithm
For each natural $n$, provide an example of a rectangle that would be cut into exactly $n$ squares, among which there should be no more than two identical ones. | <think>
Alright, let's tackle this problem step by step. The question is asking for an example of a rectangle that can be cut into exactly n squares, with no more than two identical ones. Hmm, okay. So for each natural number n, we need to find such a rectangle.
First, I remember that tiling rectangles with squares i... |
olympiads | Two circles of radius $r$ touch each other. In addition, each of them touches externally a third circle of radius $R$ at points $A$ and $B$ respectively.
Find the radius $r$, if $A B=12, R=8$. | <think>
Okay, let's see. I need to find the radius r of two circles that touch each other and each touches externally a third, larger circle with radius R=8. The distance between the points where they touch the larger circle, AB, is 12. Hmm, okay.
First, I should visualize this. There's a big circle with radius R=8. T... |
olympiads | 6. The sequence $a_{n}$ is defined as follows:
$a_{1}=1, a_{n+1}=a_{n}+\frac{2 a_{n}}{n}$, for $n \geq 1$. Find $a_{999}$. | <think>
Okay, so I need to find the 999th term of this sequence where the first term is 1, and each subsequent term is defined by the previous term plus twice the previous term divided by the current index. Let me write that down again to make sure I have it right.
The problem says: a₁ = 1, and then for each n ≥ 1, aₙ... |
olympiads_ref | Queenie and Horst play a game on a \(20 \times 20\) chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when som... | <think>
Okay, so Queenie and Horst are playing this game on a 20x20 chessboard. The rules are that Horst places a black knight each turn, making sure it doesn't attack any previous knights. Then Queenie places a white queen anywhere she wants. The game ends when someone can't move, and we need to find the maximum K suc... |
olympiads | 14th Chinese 1999 Problem B3 How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it? Solution | <think>
Okay, so I need to figure out how many ways there are to color 16 unit cubes red in a 4x4x4 cube such that each 1x1x4, 1x4x1, and 4x1x1 cuboid contains exactly one red cube. Hmm, this seems like a problem related to permutations or maybe Latin cubes? Let me think step by step.
First, let me visualize the cube.... |
olympiads | 10. Given a sequence $\left\{a_{n}\right\}$ whose terms are all non-negative real numbers, and satisfies: for any integer $n \geqslant 2$, we have $a_{n+1}=$ $a_{n}-a_{n-1}+n$. If $a_{2} a_{2022}=1$, find the maximum possible value of $a_{1}$. | <think>
Alright, let's tackle this problem step by step. We have a sequence {a_n} where all terms are non-negative real numbers. The recurrence relation given is for n ≥ 2: a_{n+1} = a_n - a_{n-1} + n. Also, we know that a_2 * a_{2022} = 1, and we need to find the maximum possible value of a_1. Hmm, okay. Let's try to ... |
olympiads | Question 47, Find the maximum value of the function $f(x)=\frac{\sqrt{2} \sin x+\cos x}{\sin x+\sqrt{1-\sin x}}(0 \leq x \leq \pi)$. | <think>
Okay, so I need to find the maximum value of the function f(x) = [√2 sinx + cosx] / [sinx + √(1 - sinx)] for x between 0 and π. Hmm, let's start by understanding the function and how to approach this.
First, since the function involves sinx and cosx, which are trigonometric functions, maybe I can use trigonom... |
olympiads | 1373. Integrate the equation
$$
x d y=(x+y) d x
$$
and find the particular solution that satisfies the initial conditions $y=2$ when $x=-1$. | <think>
Okay, so I have this differential equation to solve: x dy = (x + y) dx. And then I need to find a particular solution that passes through the point (-1, 2). Hmm, let me think about how to approach this.
First, let me recall what kind of equation this is. The equation is x dy = (x + y) dx. Maybe I should rewri... |
olympiads | 11.1. Arkady, Boris, and Vasili decided to run the same distance, consisting of several laps. Arkady ran each lap 2 minutes faster than Boris, and Boris ran 3 minutes faster than Vasili, and all of them ran at a constant speed. When Arkady finished the distance, Boris had one lap left to run, and Vasili had two laps le... | <think>
Okay, let's try to figure out this problem step by step. So, we have three runners: Arkady, Boris, and Vasili. They all ran the same distance, which is several laps. The key information here is about their speeds and how much they had left when Arkady finished.
First, let's parse the problem again to make sur... |
olympiads | 33 (1275). Find all prime numbers $p$ and $q$ for which $p^{2} - 2 q^{2}=1$. | <think>
Okay, let's see. I need to find all prime numbers p and q such that p squared minus twice q squared equals 1. So the equation is p² - 2q² = 1. Hmm, primes are numbers greater than 1 that have no divisors other than 1 and themselves. So p and q are primes here. Let me think how to approach this.
First, maybe I ... |
amc_aime | A circle and a square have the same perimeter. Then:
$\text{(A) their areas are equal}\qquad\\ \text{(B) the area of the circle is the greater} \qquad\\ \text{(C) the area of the square is the greater} \qquad\\ \text{(D) the area of the circle is } \pi \text{ times the area of the square}\qquad\\ \text{(E) none of the... | <think>
Okay, so I have this problem here where a circle and a square have the same perimeter. I need to figure out which one has a larger area or if they're equal, and then choose the correct option from A to E. Let me start by recalling some formulas for perimeters and areas of circles and squares.
First, the perime... |
olympiads | 8,9 [ relationships of linear elements of similar triangles
The height of a right triangle, dropped to the hypotenuse, divides this triangle into two triangles. The distance between the centers of the inscribed circles of these triangles is 1. Find the radius of the inscribed circle of the original triangle.
# | <think>
Okay, let's see. I need to find the radius of the inscribed circle of the original right triangle, given that the distance between the centers of the inscribed circles of the two smaller triangles formed by the height to the hypotenuse is 1. Hmm, right triangle, inscribed circles, similar triangles maybe? Let m... |
olympiads | 10. The teacher took out 13 cards, with the numbers on the cards ranging from 1 to 13. After keeping one of the cards, the teacher distributed the remaining 12 cards to three students, Jia, Yi, and Bing, with each receiving 4 cards. It is known that the sum of the numbers on the cards in Jia's hand and the sum of the n... | <think>
Alright, let's try to figure out this card problem. So, the teacher has 13 cards numbered from 1 to 13. They kept one card and gave the remaining 12 cards to three students: Jia, Yi, and Bing, each getting 4 cards. The problem states that the ratio of the sum of Jia's cards to Yi's cards is 2:7, and Bing's sum ... |
olympiads | [ equations in integers ] [ Decimal number system ]
a) In a three-digit number, the first digit from the left was erased, then the resulting two-digit number was multiplied by 7 and the original three-digit number was obtained. Find such a number.
b) In a three-digit number, the middle digit was erased and the result... | <think>
Alright, so I have these two problems here, part a and part b. Let me start with part a first. Let me read it again: "In a three-digit number, the first digit from the left was erased, then the resulting two-digit number was multiplied by 7 and the original three-digit number was obtained. Find such a number."
... |
olympiads | Example 2.1. $I=\int_{0}^{2} x^{3} d x$ | <think>
Okay, so I need to compute the integral of x cubed from 0 to 2. Hmm, let me start by remembering how to do definite integrals. I think the process is to first find the antiderivative of the function and then evaluate it at the upper and lower limits, subtracting the lower limit value from the upper one.
Alrig... |
olympiads | 9. (6 points) A hollow cone and a hollow cylinder, with base inner radii and heights of $12 \mathrm{~cm}, 20 \mathrm{~cm}$ respectively, are combined to form a vertically placed container as shown in the figure. Some fine sand is poured into this container, filling the cone and part of the cylinder. It is measured that... | <think>
Okay, let me try to figure out this problem. So, we have a container made by combining a hollow cone and a hollow cylinder. The cone and the cylinder both have the same base radius and height: radius 12 cm and height 20 cm. They are combined vertically, so probably the cone is on top of the cylinder or vice ver... |
olympiads | 8. Let $\mathcal{R}$ be the finite region in the plane bounded by the $x$-axis and the graph of the curve given by the equation $2 x^{2}+5 y=10$. Given three points in $\mathcal{R}$, which of the following statements is always true?
(A) at least two of the three points have a distance $\geq \frac{\sqrt{5}}{2}$
(B) at... | <think>
Okay, so I need to solve this problem about the region R bounded by the x-axis and the graph of 2x² + 5y = 10. Then, given three points in R, determine which of the statements (A) through (E) is always true. Let me try to break this down step by step.
First, let me visualize the region R. The equation given is... |
olympiads | 4. The numbers from 1 to 8 are arranged at the vertices of a cube such that the sum of the numbers in any three vertices lying on the same face is at least 10. What is the smallest possible sum of the numbers at the vertices of one face? | <think>
Okay, so I need to figure out the smallest possible sum of numbers on one face of a cube, where each vertex (there are 8 vertices) is labeled with numbers 1 through 8 without repetition. The condition is that the sum of the numbers on any three vertices of the same face must be at least 10. Hmm, let me try to b... |
olympiads_ref | 9. (HUN 3) The incenter of the triangle $A B C$ is $K$. The midpoint of $A B$ is $C_{1}$ and that of $A C$ is $B_{1}$. The lines $C_{1} K$ and $A C$ meet at $B_{2}$, the lines $B_{1} K$ and $A B$ at $C_{2}$. If the areas of the triangles $A B_{2} C_{2}$ and $A B C$ are equal, what is the measure of angle $\angle C A B$... | <think>
Okay, so I need to solve this geometry problem. Let me read it again to make sure I understand all the parts.
We have triangle ABC with incenter K. The midpoint of AB is C1, and the midpoint of AC is B1. The lines C1K and AC intersect at B2, and the lines B1K and AB intersect at C2. The areas of triangles AB2C... |
olympiads | At the "Lukomorye" station, they sell cards for one, five, and twenty rides. All cards cost a whole number of gold coins. Five cards for one ride are more expensive than one card for five rides, and four cards for five rides are more expensive than one card for twenty rides. It turned out that the cheapest way for 33 b... | <think>
Alright, let's try to figure out how much a card for five rides costs. Hmm, the problem is about finding the cheapest way for 33 bogatyrs to travel, which ended up being buying cards for 35 rides and spending 33 gold coins. We need to find the cost of a five-ride card. Let's break down the information step by s... |
olympiads | 1. [3 points] Find the number of eight-digit numbers, the product of the digits of each of which is equal to 3375. The answer should be presented as an integer. | <think>
Okay, so I need to find the number of eight-digit numbers where the product of the digits is 3375. Hmm, let's start by breaking down the problem. First, an eight-digit number can't start with zero, so the first digit has to be from 1 to 9, and the rest can be from 0 to 9. But since the product of the digits inc... |
olympiads | (a) Verify the identity
$$
(a+b+c)^{3}=a^{3}+b^{3}+c^{3}+3(a+b)(b+c)(c+a)
$$
(b) Solve the system
$$
\left\{\begin{array}{l}
x+y+z=1 \\
x^{2}+y^{2}+z^{2}=1 \\
x^{3}+y^{3}+z^{3}=1
\end{array}\right.
$$ | <think>
Okay, let me try to tackle these two problems. Starting with part (a): verifying the identity \((a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)\). Hmm, I need to check if this equation holds true for all values of a, b, and c.
First, I remember that expanding \((a + b + c)^3\) usually results in seve... |
olympiads | ## Task A-4.3.
A die is rolled three times in a row. Determine the probability that each subsequent roll (after the first) results in a number that is not less than the previous one. | <think>
Okay, so I have this problem where a die is rolled three times in a row, and I need to find the probability that each subsequent roll is not less than the previous one. That means, if the first roll is, say, a 2, the second roll can be 2, 3, 4, 5, or 6, and then the third roll has to be at least whatever the se... |
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