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values | problem stringlengths 19 6.18k | generation stringlengths 345 68.1k |
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aops_forum | Find the largest $n$ such that the last nonzero digit of $n!$ is $1$.
| <think>
Okay, so I need to find the largest integer n such that the last nonzero digit of n! (n factorial) is 1. Hmm, interesting problem. Let me try to figure out how to approach this.
First, I know that factorials get really big really quickly, and the trailing zeros come from multiplying factors of 10, which are 2 ... |
olympiads | Example 3.3.2 Let \( S=\{1,2,3,4\}, n \) terms of the sequence \( a_{1}, a_{2}, \cdots, a_{n}\left(a_{i} \in S\right) \) have the following property: for any non-empty subset \( B \) of \( S \), there are adjacent \( |B| \) terms in the sequence that exactly form the set \( B \). Find the minimum value of \( n \). | <think>
Okay, so I have this problem where I need to find the minimum length n of a sequence composed of elements from the set S = {1, 2, 3, 4}. The sequence has to satisfy the condition that for any non-empty subset B of S, there are |B| adjacent terms in the sequence that exactly form the set B. Hmm, that sounds a bi... |
cn_contest | 1. The real number $m=2005^{3}-2005$. Among the following numbers, which one cannot divide $m$? ( ).
(A) 2006
(B) 2005
(C) 2004
(D) 2003 | <think>
Okay, let's see. The problem is about determining which of the given options cannot divide the number m, where m is equal to 2005 cubed minus 2005. The options are 2006, 2005, 2004, and 2003. So, I need to check each one to see if they can divide m, and figure out which one cannot.
First, let's write out what ... |
olympiads_ref | A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals. Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isosceles t... | <think>
Okay, so I need to solve this problem about a regular 2006-gon. The question is about finding the maximum number of isosceles triangles with two odd sides in a dissection of the polygon into triangles using 2003 non-intersecting diagonals. First, let me try to understand the problem step by step.
First, what i... |
olympiads | 5. Let $f(x)=a x^{5}+b x^{3}+c x+10$, and $f(3)$ $=3$. Then $f(-3)=$ | <think>
Okay, let me try to figure out this problem. So, we have the function f(x) = ax⁵ + bx³ + cx + 10. We know that when we plug in x = 3, the output is 3. And we need to find f(-3). Hmm, okay. Let me think.
First, maybe I should recall some properties of polynomial functions. The function here is a fifth-degree po... |
olympiads | $$
\begin{array}{l}
\text { 7. If } f(x)=x^{6}-2 \sqrt{2006} x^{5}-x^{4}+x^{3}-2 \sqrt{2007} x^{2}+2 x-\sqrt{2006} \text {, then } f(\sqrt{2006}+ \\
\sqrt{2007})= \\
\end{array}
$$ | <think>
Okay, let's see. The problem is to find the value of the function f at the point √2006 + √2007. The function f(x) is given as:
f(x) = x⁶ - 2√2006 x⁵ - x⁴ + x³ - 2√2007 x² + 2x - √2006.
Hmm. So I need to compute f(√2006 + √2007). That seems a bit complicated with all those square roots and high-degree terms. M... |
olympiads | \section*{Problem 1 - 271011}
How many ordered pairs of integers \((x, y)\) are there in total for which \(x \cdot y=1987\) holds? | <think>
Okay, so I need to find how many ordered pairs of integers (x, y) satisfy the equation x multiplied by y equals 1987. Hmm, let me think about this step by step.
First, I remember that if the product of two numbers is a particular value, then those numbers must be factors of that value. So, essentially, I need ... |
olympiads | 7.5. In triangle $A B C$, we construct $A D \perp B C$, where $D \in (B C)$. Points $E$ and $F$ are the midpoints of segments $A D$ and $C D$, respectively. Determine the measure in degrees of angle $B A C$, knowing that $B E \perp A F$. | <think>
Okay, so I need to figure out the measure of angle BAC in triangle ABC where AD is perpendicular to BC, and E and F are midpoints of AD and CD, respectively. Also, BE is perpendicular to AF. Hmm, let me start by drawing a diagram to visualize this.
First, triangle ABC. Let me label the vertices A, B, and C. T... |
olympiads | 1. Find the number of divisors of the number $a=2^{3} \cdot 3^{2} \cdot 5^{2}$, which are divisible by 3. Find the sum of such divisors. | <think>
Okay, let's see. I need to find the number of divisors of the number a = 2³ × 3² × 5² that are divisible by 3, and also find the sum of those divisors. Hmm, alright. Let me start by recalling how to find divisors of a number based on its prime factorization.
So, if a number is given in its prime factorization ... |
cn_contest | Initial 26. Find all natural numbers of the form $\overline{30 x 070 y 03}$ that are divisible by 37. | <think>
Okay, so I need to find all natural numbers of the form 30x070y03 that are divisible by 37. Hmm, let me break this down. The number is given as a 10-digit number: the first two digits are 3 and 0, then there's a digit x, followed by 0, 7, 0, then a digit y, and finally 0 and 3. So the structure is: 3 0 x 0 7 0 ... |
olympiads | We folded a rectangular piece of paper along one of its diagonals. After the fold, the four vertices ended up at the four vertices of a trapezoid, three sides of which are of equal length. What is the length of the shorter side of the original rectangle if the longer side is 12 cm? | <think>
Okay, so I have this problem where a rectangular piece of paper is folded along one of its diagonals, and after folding, the four vertices form a trapezoid with three sides of equal length. The longer side of the original rectangle is 12 cm, and I need to find the length of the shorter side. Hmm, let me try to ... |
olympiads | 2. (6 points) Calculate: $1+11+21+\cdots+1991+2001+2011=$ $\qquad$ | <think>
Okay, let me see. I need to calculate the sum of the series: 1 + 11 + 21 + ... + 1991 + 2001 + 2011. Hmm, this looks like an arithmetic sequence, right? Let me check the differences between the terms.
Starting with the first term, 1. Then the next term is 11. The difference is 11 - 1 = 10. Then 21 - 11 = 10. O... |
olympiads | 4. Given $a_{0}=-1, a_{1}=1, a_{n}=2 a_{n-1}+3 a_{n-2}+3^{n}(n \geqslant 3)$, find $a_{n}$. | <think>
Okay, so I need to find a closed-form expression for the sequence defined by the recurrence relation: \( a_0 = -1 \), \( a_1 = 1 \), and for \( n \geq 2 \), \( a_n = 2a_{n-1} + 3a_{n-2} + 3^n \). Hmm, let's see. I remember that for linear recurrence relations with constant coefficients, especially non-homogeneo... |
olympiads | 2. Solve the equation $\left(\frac{3 x}{2}\right)^{\log _{3}(8 x)}=\frac{x^{7}}{8}$. | <think>
Okay, so I need to solve this equation: \(\left(\frac{3x}{2}\right)^{\log_{3}(8x)} = \frac{x^{7}}{8}\). Hmm, logarithms and exponents can be tricky, but let me try to break it down step by step.
First, let me recall some logarithm and exponent rules. Maybe taking the logarithm of both sides could help simplify... |
olympiads | What is the least possible number of cells that can be marked on an $n \times n$ board such that for each $m>\frac{n}{2}$ both diagonals of any $m \times m$ sub-board contain a marked cell?
## Answer: $n$. | <think>
Okay, so I have this problem here: I need to find the least number of cells to mark on an n×n board so that for every m×m sub-board where m is greater than n/2, both of its diagonals contain at least one marked cell. The answer is supposed to be n, but I need to figure out why. Let me try to work through this s... |
olympiads | 34. Find the number of 20 -tuples of integers $x_{1}, \ldots, x_{10}, y_{1}, \ldots, y_{10}$ with the following properties:
- $1 \leq x_{i} \leq 10$ and $1 \leq y_{i} \leq 10$ for each $i$;
- $x_{i} \leq x_{i+1}$ for $i=1, \ldots, 9$
- if $x_{i}=x_{i+1}$, then $y_{i} \leq y_{i+1}$. | <think>
Okay, so I need to find the number of 20-tuples (x₁, ..., x₁₀, y₁, ..., y₁₀) that satisfy the given conditions. Let me parse the problem again to make sure I understand all the constraints.
First, each x_i and y_i must be an integer between 1 and 10 inclusive. So, each x_i ∈ {1, 2, ..., 10} and same for each y... |
aops_forum | There exist two positive numbers $ x$ such that $ \sin(\arccos(\tan(\arcsin x)))\equal{}x$. Find the product of the two possible $ x$. | <think>
Alright, let me try to work through this problem step by step. So, we need to find all positive numbers x such that sin(arccos(tan(arcsin x))) equals x. Then, we have to find the product of these two possible x values. Hmm, that's quite a mouthful. Let me break it down piece by piece.
First, let's parse the eq... |
olympiads | Folklore
In an acute-angled triangle $A B C$, the bisector $A N$, the altitude $B H$, and the line perpendicular to side $A B$ and passing through its midpoint intersect at one point. Find the angle $BAC$.
# | <think>
Okay, so I need to solve this geometry problem where in an acute-angled triangle ABC, the bisector AN, the altitude BH, and the line perpendicular to side AB passing through its midpoint intersect at one point. The goal is to find the angle BAC. Hmm, let's start by understanding the problem step by step.
First... |
olympiads | 8 The line $\sqrt{3} x-y-\sqrt{3}=0$ intersects the parabola $y^{2}=4 x$ at points $A$ and $B$ ($A$ is above the x-axis), and intersects the x-axis at point $F$. If $\overrightarrow{O F}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}$, then $\mu^{2}-$ $\lambda^{2}=$ $\qquad$ . | <think>
Okay, let's see. I need to solve this problem where a line intersects a parabola and the x-axis, and then find μ² - λ² based on some vector equation. Let me try to break it down step by step.
First, the given line is √3 x - y - √3 = 0. Let me rewrite that in slope-intercept form to maybe visualize it better. S... |
olympiads | 37th Putnam 1976 Problem B5 Find ∑ 0 n (-1) i nCi ( x - i ) n , where nCi is the binomial coefficient. Solution | <think>
Okay, let's see. I need to find the sum from i=0 to n of (-1)^i times the binomial coefficient (n choose i) times (x - i)^n. Hmm, that seems a bit complicated, but maybe I can break it down step by step.
First, I remember that binomial coefficients with alternating signs often show up in inclusion-exclusion pr... |
olympiads | 5. Given a grid board $3 \times 2021$ (3 cells vertically and 2021 cells horizontally) and an unlimited supply of cardboard strips $1 \times 3$. Petya and Vasya play the following game. They take turns placing the strips on the board (cell by cell) without overlapping, one strip per turn. Petya places his strips horizo... | <think>
Okay, so we have this game on a 3x2021 grid. Petya goes first, placing horizontal 1x3 strips, and Vasya places vertical 1x3 strips. They alternate turns, and the first one who can't move loses. The question is, who can force a win regardless of the opponent's moves?
Hmm, let's start by understanding the game m... |
olympiads | 3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x, y)$ such that there exists a pair of real numbers $a, b$ for which the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
(x-a)^{2}+(y-b)^{2} \leqslant 8 \\
a^{2}+b^{2} \leqslant \min (-4 a+4 b ; 8)
\end{array}\right.
$$... | <think>
Alright, let's try to tackle this problem step by step. So, we need to find the area of the figure M, which consists of all points (x, y) such that there exist real numbers a and b satisfying two conditions:
1. The point (x, y) lies inside or on the circle centered at (a, b) with radius √8.
2. The pair (a, b) ... |
olympiads | 4. (3 points) On the side $B C$ of triangle $A B C$ as a diameter, a circle with a radius of 20 cm is constructed. This circle intersects sides $A B$ and $A C$ at points $X$ and $Y$ respectively. Find $B X \cdot A B + C Y \cdot A C$. | <think>
Okay, let me try to figure out this problem. So we have triangle ABC, and on side BC as a diameter, they've constructed a circle with a radius of 20 cm. That means the diameter BC is 40 cm because the radius is 20. Got that. Then this circle intersects sides AB and AC at points X and Y respectively. We need to ... |
olympiads | [ Arithmetic. Mental arithmetic, etc. ] $[$ Word problems (other). ]
Snow White cut out a large square from batiste and put it in a chest. The First Dwarf came, took out the square, cut it into four squares, and put all four back into the chest. Then the Second Dwarf came, took out one of the squares, cut it into four... | <think>
Okay, let me try to figure out this problem step by step. So, Snow White starts with a large square and puts it in a chest. Then each dwarf comes in, takes out one square, cuts it into four smaller squares, and puts those four back. We need to find out how many squares are there after the seventh dwarf has done... |
olympiads | For example, the rules of a "level-passing game" stipulate: on the $n$-th level, a die must be rolled $n$ times. If the sum of the points from these $n$ rolls is greater than $2^{n}$, the player passes the level. Questions:
( I ) What is the maximum number of levels a person can pass in this game?
( II ) What is the pr... | <think>
Okay, so I have this problem about a level-passing game. Let me try to understand it step by step.
The problem says that on the n-th level, a die must be rolled n times. If the sum of the points from these n rolls is greater than 2^n, the player passes the level. There are two questions here:
I. What is the... |
olympiads | How many five-digit numbers are there that end in six and are divisible by three? | <think>
Okay, so I need to figure out how many five-digit numbers end with six and are divisible by three. Let me think about this step by step.
First, let's recall that a five-digit number ranges from 10000 to 99999. That part is straightforward. Now, the number has to end with six, so the last digit is fixed as 6. T... |
olympiads | 1. Find all values of $p$, for each of which the numbers $-p-12, 2 \cdot \sqrt{p}$, and $p-5$ are respectively the first, second, and third terms of some geometric progression. | <think>
Okay, so I need to find all values of p where -p -12, 2 times the square root of p, and p -5 form a geometric progression. Hmm, let me recall what a geometric progression is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio, r. So... |
olympiads | Write in ascending order the numbers
$$
\sqrt{121}, \quad \sqrt[3]{729} \quad \text { and } \quad \sqrt[4]{38416}
$$ | <think>
Okay, so I need to figure out the order of these three numbers from smallest to largest: √121, the cube root of 729, and the fourth root of 38416. Let me start by calculating each one individually.
First, √121. That's straightforward because 121 is a perfect square. I remember that 11 times 11 is 121. So, √12... |
olympiads | 6. Given three points $A, B, C$ on a plane satisfying $|\overrightarrow{A B}|=3,|\overrightarrow{B C}|=4,|\overrightarrow{C A}|=5$, then $\overrightarrow{A B} \cdot \overrightarrow{B C}+\overrightarrow{B C} \cdot \overrightarrow{C A}+\overrightarrow{C A} \cdot \overrightarrow{A B}$ is equal to | <think>
Okay, let me try to figure out this problem. So we have three points A, B, C on a plane. The lengths of the vectors between them are given: AB is 3 units, BC is 4 units, and CA is 5 units. We need to find the sum of the dot products: AB · BC + BC · CA + CA · AB. Hmm, dot products. Alright, let's recall that the... |
olympiads | 5. Let integer $n \geqslant 2$,
$$
A_{n}=\sum_{k=1}^{n} \frac{3 k}{1+k^{2}+k^{4}}, B_{n}=\prod_{k=2}^{n} \frac{k^{3}+1}{k^{3}-1} \text {. }
$$
Then the size relationship between $A_{n}$ and $B_{n}$ is | <think>
Alright, so I have this problem here where I need to figure out the relationship between two expressions, A_n and B_n, for integers n ≥ 2. Let me write them down again to make sure I have them right:
A_n is the sum from k=1 to n of (3k)/(1 + k² + k⁴), and B_n is the product from k=2 to n of (k³ + 1)/(k³ - 1). ... |
olympiads | 9. Find the 12-combination number of the multiset $S=\{4 \cdot a, 3 \cdot b, 4 \cdot c, 5 \cdot d\}$. | <think>
Okay, so I need to find the 12-combination number of the multiset S = {4·a, 3·b, 4·c, 5·d}. Hmm, let me recall what a combination in a multiset means. From what I remember, when dealing with combinations in multisets, we're essentially looking for the number of ways to select 12 elements from S, considering the... |
olympiads |
A2 Find the maximum value of $z+x$, if $(x, y, z, t)$ satisfies the conditions:
$$
\left\{\begin{array}{l}
x^{2}+y^{2}=4 \\
z^{2}+t^{2}=9 \\
x t+y z \geq 6
\end{array}\right.
$$
| <think>
Okay, so I need to find the maximum value of z + x given these three conditions. Let me see... First, let me write down the problem again to make sure I have it right.
We have variables x, y, z, t. The conditions are:
1. x² + y² = 4
2. z² + t² = 9
3. xt + yz ≥ 6
And we need to maximize z + x. Hmm. Alright.
... |
olympiads | $7 \cdot 48$ In an acute triangle $\triangle ABC$, the three sides are $a, b, c$, and the distances from its circumcenter to the three sides are $m, n, p$. Then, $m: n: p$ equals
(A) $\frac{1}{a}: \frac{1}{b}: \frac{1}{c}$.
(B) $a: b: c$.
(C) $\cos A: \cos B: \cos C$.
(D) $\sin A: \sin B: \sin C$.
(China Junior High Sc... | <think>
Okay, so I need to solve this problem about an acute triangle ABC. The question is asking about the ratio of the distances from the circumcenter to the three sides, which are labeled as m, n, p. The options given are different ratios, and I need to figure out which one is correct. The answer choices are A throu... |
aops_forum | A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$. | <think>
Okay, let's see. I need to solve this problem where a positive number m/n is equal to the ratio of two expressions: one is 7 plus the number's reciprocal, and the other is 65 minus the number's reciprocal. Then, since m and n are coprime, I have to find 2m + n. Hmm, let me break this down step by step.
First, ... |
olympiads | 9. Solution. Suppose Olga Pavlovna has \( x \) liters of jam left, and Maria Petrovna has \( y \) liters of jam left. The numbers \( x \) and \( y \) are randomly and independently chosen from the interval from 0 to 1. We will consider that a random point with coordinates \((x; y)\) is selected from the unit square \( ... | <think>
Okay, so the problem is about Olga Pavlovna and Maria Petrovna each having some amount of jam left, represented by x and y liters respectively. Both x and y are randomly chosen between 0 and 1, so we can think of this as picking a random point (x, y) in the unit square. The event A is that together they have at... |
olympiads | 3. Among the four functions $y=\sin |x|, y=\cos |x|, y=|\cot x|, y=\lg |\sin x|$, the even function that is periodic with $\pi$ and monotonically increasing in $\left(0, \frac{\pi}{2}\right)$ is
A. $y=\sin |x|$
B. $y=\cos |x|$
C. $y=|\cot x|$
D. $y=\lg |\sin x|$ | <think>
Okay, let's try to solve this problem step by step. The question is asking which of the four given functions is an even function, periodic with period π, and monotonically increasing on the interval (0, π/2). The options are A through D, each with a different function. Let me break down the problem into parts a... |
olympiads | 23. Given that $n, k$ are natural numbers, and $1<k<n$. If $\frac{1+2+3+\cdots \cdots+(n-2)+(n-1)+n-k}{n-1}=10$, then $n+k=$ $\qquad$ | <think>
Okay, let's try to figure out this problem. So we have two natural numbers, n and k, where 1 < k < n. The equation given is [1 + 2 + 3 + ... + (n-2) + (n-1) + n - k] divided by (n - 1) equals 10. We need to find n + k. Alright, let's break this down step by step.
First, the numerator of the left side is the su... |
olympiads | 8. Let $A=\left\{a_{1}, a_{2}, \cdots, a_{7}\right\}$. Here $a_{i} \in \mathbf{Z}^{\prime}$, and let $n_{A}$ denote the number of triples $(x, y, z)$ such that: $x<y$, $x+y=z, x, y, z \in A$. Then the maximum possible value of $n_{A}$ is $\qquad$. | <think>
Okay, let me try to figure out this problem. So, we have a set A with 7 elements, all positive integers. We need to find the maximum number of triples (x, y, z) such that x < y and x + y = z, where x, y, z are in A. The goal is to maximize the number of such triples. Hmm, interesting.
First, I need to understa... |
olympiads | 13.2.5* Consider all line segments of length 4, one end of which lies on the line $y=x$, and the other end on the line $y=2x$. Find: the equation of the locus of the midpoints of these line segments. | <think>
Okay, so I need to find the equation of the locus of the midpoints of all line segments of length 4, where one end is on the line y = x and the other end is on y = 2x. Hmm, let me think about how to approach this.
First, let me visualize the problem. There are two lines: y = x is a straight line passing throug... |
olympiads | 10. (10 points) Among the following $2 n$ numbers, what is the maximum number of numbers that can be selected such that the ratio of any two selected numbers is not 2 or $\frac{1}{2}$?
$3, 3 \times 2, 3 \times 2^{2}, 3 \times 2^{3}, 3 \times 2^{4}, 3 \times 2^{5}, \cdots, 3 \times 2^{2 n-1}$. | <think>
Okay, so the problem here is about selecting the maximum number of numbers from a given list such that the ratio of any two selected numbers isn't 2 or 1/2. The list provided is 3, 3×2, 3×2², 3×2³, up to 3×2²ⁿ⁻¹. So there are 2n numbers in total. My goal is to figure out the largest possible subset where no two... |
olympiads | 8. In an acute-angled triangle $A B C$, a point $Q$ is chosen on side $A C$ such that $A Q: Q C=1: 2$. From point $Q$, perpendiculars $Q M$ and $Q K$ are dropped to sides $A B$ and $B C$ respectively. It is given that $B M: M A=4: 1, B K=K C$. Find $M K: A C$. | <think>
Okay, let's try to tackle this geometry problem step by step. It's an acute-angled triangle ABC, so all angles are less than 90 degrees. A point Q is on side AC such that AQ:QC is 1:2. From Q, we drop perpendiculars QM and QK to sides AB and BC respectively. We know that BM:MA is 4:1, and BK = KC. We need to fi... |
olympiads | Find the remainder of the Euclidean division of $2018^{2019^{2020}}$ by 11. | <think>
Okay, so I need to find the remainder when 2018^{2019^{2020}} is divided by 11. That sounds like a modular arithmetic problem. Maybe I can use Euler's theorem or Fermat's little theorem here? Let me recall: Fermat's little theorem says that if p is a prime number and a is not divisible by p, then a^{p-1} ≡ 1 mo... |
olympiads | 11. Given $S_{1}=1, S_{2}=1-2, S_{3}=1-2+3, S_{4}=1-2+3-4, S_{5}=1-2+3-4+5, \cdots$, then $S_{1}+S_{2}+S_{3}+\cdots+S_{299}=$ $\qquad$. | <think>
Alright, let's try to tackle this problem. So, we have these sequences S₁, S₂, S₃, and so on, where each Sₙ is the alternating sum of numbers from 1 to n. For example, S₁ is just 1, S₂ is 1 - 2, S₃ is 1 - 2 + 3, etc. Then we need to find the sum of all these Sₖ from k = 1 to 299. That is, compute S₁ + S₂ + S₃ +... |
olympiads | 11. The maximum value of the function $y=2 x+\sqrt{1-2 x}$ is | <think>
Okay, so I need to find the maximum value of the function y = 2x + sqrt(1 - 2x). Hmm, let me think. Alright, first, since there's a square root involved, I should probably figure out the domain of the function. The expression inside the square root has to be non-negative, right? So 1 - 2x must be greater than o... |
olympiads | 5. On the cells of a chessboard of size $8 \times 8$, 5 identical pieces are randomly placed. Find the probability that four of them will be located either on the same row, or on the same column, or on one of the two main diagonals. | <think>
Okay, so I have this probability problem here about a chessboard and placing 5 identical pieces on it. The question is asking for the probability that four of them are located either on the same row, column, or one of the two main diagonals. Hmm, let's break this down step by step.
First, let's understand the ... |
olympiads | [ Area of a triangle (using two sides and the angle between them).] Law of Cosines
Two sides of a triangle are equal to $2 \sqrt{2}$ and 3, and the area of the triangle is 3. Find the third side. | <think>
Okay, let me try to figure out how to find the third side of this triangle. The problem says two sides are 2√2 and 3, and the area is 3. Hmm, I need to find the third side. Alright, let's start by recalling some formulas related to the area of a triangle.
I remember that the area of a triangle can be calculat... |
olympiads | 8.5. Eleven of the best football teams each played one match against each other. It turned out that each team scored 1 goal in their first match, 2 goals in their second match, ..., 10 goals in their tenth match. What is the maximum number of matches that could have ended in a draw? | <think>
Okay, let me try to figure out this problem. So, we have eleven football teams, each playing ten matches against the other teams. Each team scores 1 goal in their first match, 2 goals in the second, and so on up to 10 goals in their tenth match. The question is asking for the maximum number of matches that coul... |
cn_contest | Example 2: Two people, A and B, play the following game: They take turns writing a natural number on the blackboard, with the requirement that the newly written number cannot be expressed as a non-negative integer linear combination of the numbers already on the blackboard. That is, after writing numbers \(a_{1}, a_{2}... | <think>
Okay, let me try to figure out this problem step by step. So, we have two players, A and B, taking turns writing natural numbers on a blackboard. The rule is that you can't write a number that's a non-negative integer linear combination of the numbers already present. The person who is forced to write 1 loses. ... |
olympiads_ref | The vertices $A$ and $B$ of an equilateral $\triangle A B C$ lie on a circle $k$ of radius 1, and the vertex $C$ is inside $k$. The point $D \neq B$ lies on $k, A D=A B$ and the line $D C$ intersects $k$ for the second time in point $E$. Find the length of the segment $C E$. | <think>
Okay, let me try to figure out this geometry problem. So, we have an equilateral triangle ABC, with vertices A and B on a circle k of radius 1, and vertex C inside the circle. Then there's a point D (not B) on the circle such that AD = AB. The line DC intersects the circle again at E, and we need to find the le... |
olympiads | 12th Swedish 1972 Problem 1 Find the largest real number a such that x - 4y = 1, ax + 3y = 1 has an integer solution. | <think>
Okay, let me try to figure out this problem. So, we need to find the largest real number 'a' such that the system of equations:
x - 4y = 1,
a x + 3y = 1
has an integer solution. Hmm, integer solution means that both x and y have to be integers, right? So, we need to find 'a' such that there exist integers x ... |
olympiads | ## Task A-3.1.
Determine all quadruples $(a, b, c, d)$ of natural numbers for which
$$
a+b=c d, \quad a b=c+d
$$ | <think>
Okay, let's see. I need to find all quadruples of natural numbers (a, b, c, d) such that two equations are satisfied: a + b = c*d and a*b = c + d. Natural numbers usually start from 1, right? So a, b, c, d are all positive integers.
First, maybe I can express some variables in terms of others. Let me look at ... |
olympiads | 35. On the coordinate plane, there is a line $l, O \perp l$. Consider a vector $\overrightarrow{O P}=\vec{p}$, where $P \in l$; then $p=|\vec{p}|$ is the distance from the line to the origin. Denote by $\varphi$ the angle that $\vec{e}=\frac{\vec{p}}{p}$, the unit vector of $\vec{p}$, forms with the unit vector $\vec{i... | <think>
Okay, let's try to figure out how to derive the vector equation of the line l and then get its normal equation in coordinates. The problem mentions that line l is perpendicular to vector OP, which is given as vector p. Point P is on line l, and the length of vector p is the distance from the origin to the line.... |
olympiads | Problem 5.3. The figure shows a plan of the road system of a certain city. In this city, there are 8 straight streets, and 11 intersections are named with Latin letters $A, B, C, \ldots, J, K$.
Three police officers need to be placed at some intersections so that at least one police officer is on each of the 8 streets... | <think>
Okay, so I need to figure out where to place three police officers at the intersections of this city's road system such that every one of the eight streets has at least one officer on it. Let me start by understanding the layout of the streets based on the description provided.
First, the horizontal streets ar... |
olympiads | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{\sin 5 x}{\tan 3 x}$ | <think>
Okay, so I need to find the limit of sin(5x)/tan(3x) as x approaches π. Hmm, let's start by recalling some trigonometric identities and limit properties. First, I know that tan(3x) is the same as sin(3x)/cos(3x), so maybe I can rewrite the original expression to make it easier to handle. Let me try that.
So, r... |
olympiads | [ Case Analysis ] [ Proof by Contradiction ]
In the cells of a $3 \times 3$ table, numbers are arranged such that the sum of the numbers in each column and each row is zero. What is the smallest number of non-zero numbers that can be in this table, given that this number is odd?
# | <think>
Okay, so I have this problem here about a 3x3 table where the sum of each row and each column is zero. I need to find the smallest number of non-zero numbers that can be in the table, and that number has to be odd. Hmm, let's see.
First, let me understand the problem better. We have a 3x3 grid, each cell has ... |
olympiads | 10. Given the parabola $y^{2}=4 p(x+p)$ (where $p>0$), draw any two mutually perpendicular chords $A B$ and $C D$ through the origin $O$. Find the minimum value of $|A B|+|C D|$. | <think>
Alright, let's tackle this problem. So, we have the parabola \( y^2 = 4p(x + p) \) where \( p > 0 \). We need to draw two mutually perpendicular chords AB and CD through the origin O, and find the minimum value of \( |AB| + |CD| \). Hmm, okay. Let me break this down step by step.
First, let's recall that a cho... |
aops_forum | $d(n)$ shows the number of positive integer divisors of positive integer $n$. For which positive integers $n$ one cannot find a positive integer $k$ such that $\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots )$ is a perfect square. | <think>
Okay, so the problem is about finding positive integers n for which you can't find a positive integer k such that applying the divisor function d (which counts the number of positive divisors) k times results in a perfect square. Hmm, interesting. Let me break this down step by step.
First, I need to recall th... |
olympiads | 7. Given the function $f(x)=x^{3}-2 x^{2}-3 x+4$, if $f(a)=f(b)=f(c)$, where $a<b<c$, then $a^{2}+b^{2}+c^{2}=$ | <think>
Okay, let's see. I have this problem: Given the function f(x) = x³ - 2x² - 3x + 4, and it says that f(a) = f(b) = f(c) with a < b < c. I need to find a² + b² + c². Hmm, interesting. So, first, I need to understand what this means.
The function is a cubic polynomial. If three different points a, b, c have the ... |
olympiads | 4.1. (12 points) In rectangle $A B C D$, point $E$ is marked on the extension of side $C D$ beyond point $D$. The bisector of angle $A B C$ intersects side $A D$ at point $K$, and the bisector of angle $A D E$ intersects the extension of side $A B$ at point $M$. Find $B C$, if $M K=8$ and $A B=3$. | <think>
Alright, let me try to figure out this geometry problem. Okay, so we have a rectangle ABCD. Let me visualize that. In a rectangle, opposite sides are equal and all angles are right angles. Let me label the rectangle: A is the bottom-left corner, B is the bottom-right, C is the top-right, and D is the top-left. ... |
cn_contest | 5. In a certain month, there are more Mondays than Tuesdays, and more Sundays than Saturdays. What day of the week is the 5th of this month? Could this month be December? | <think>
Okay, so I have this problem here: In a certain month, there are more Mondays than Tuesdays, and more Sundays than Saturdays. What day of the week is the 5th of this month? Could this month be December?
Hmm, let's try to figure this out step by step. First, I need to recall that months can have 28, 29, 30, or ... |
cn_contest | Example 1. Find $P_{n}=\prod_{k=1}^{n}\left(2 \cos 2^{k-1} \theta-1\right)$. | <think>
Okay, so I have to find the product P_n, which is the product from k=1 to n of (2 cos(2^{k-1} θ) - 1). Hmm, let's see. Products involving trigonometric functions often have some telescoping properties or use identities. Maybe I can use some trigonometric identities to simplify each term or find a pattern when m... |
cn_contest | 4. Given that $A_{1}, B_{1}, C_{1}, D_{1}$ are points on the four sides of square $A B C D$. The following are two propositions about it:
(1) If $A_{1} B_{1} C_{1} D_{1}$ is a rectangle with unequal adjacent sides, then the rectangle $A_{1} B_{1} C_{1} D_{1}$ must have a side parallel to one of the diagonals of the squ... | <think>
Okay, let's try to figure out this problem step by step. We have a square ABCD, and points A₁, B₁, C₁, D₁ on its four sides. There are two propositions about the shapes formed by connecting these points, and we need to determine if each is true or false. The options are whether both are true, both false, or one... |
olympiads | 6. One mole of an ideal gas was expanded so that during the process, the pressure of the gas turned out to be directly proportional to its volume. In this process, the gas heated up by $\Delta T=100{ }^{\circ} \mathrm{C}$. Determine the work done by the gas in this process. The gas constant $R=8.31$ J/mol$\cdot$K. (15 ... | <think>
Okay, so I have this problem here where one mole of an ideal gas is expanded, and during the process, the pressure of the gas is directly proportional to its volume. The gas heats up by 100 degrees Celsius, and I need to find the work done by the gas. The gas constant R is given as 8.31 J/mol·K. Hmm, let's see ... |
olympiads | Problem 7.6. Points $D$ and $E$ are marked on sides $A C$ and $B C$ of triangle $A B C$ respectively. It is known that $A B=B D, \angle A B D=46^{\circ}, \angle D E C=90^{\circ}$. Find $\angle B D E$, if it is known that $2 D E=A D$. | <think>
Okay, let's see. I need to find angle BDE in triangle ABC where points D and E are on sides AC and BC respectively. The given information is that AB equals BD, angle ABD is 46 degrees, angle DEC is 90 degrees, and 2 DE equals AD. Hmm, that seems a bit complex, but let's break it down step by step.
First, let m... |
olympiads | 13. Given that $b$ and $c$ are positive integers, and the three quadratic equations $x^{2}+b x+c=k$ (for $k=0,1,2$) all have real roots but no integer roots, find the minimum value of $b+c$. | <think>
Okay, let's see. The problem is asking for the minimum value of \( b + c \) where \( b \) and \( c \) are positive integers. The condition is that each of the three quadratic equations \( x^2 + b x + c = k \) for \( k = 0, 1, 2 \) all have real roots but no integer roots. Hmm, that sounds a bit tricky, but let'... |
olympiads | [Helper area. The area helps solve the task] Pythagorean Theorem (direct and inverse) ]
Find the area of an isosceles triangle if the height dropped to the base is 10, and the height dropped to the side is 12. | <think>
Okay, let's see. I need to find the area of an isosceles triangle where the height dropped to the base is 10, and the height dropped to the side is 12. Hmm, isosceles triangle... So two sides are equal. The height to the base is 10, which is the usual height we consider, right? And then there's another height, ... |
olympiads | 1.29 Calculate the sum of the cubes of two numbers if their sum and product are 11 and 21, respectively. | <think>
Okay, so I need to find the sum of the cubes of two numbers, given that their sum is 11 and their product is 21. Hmm, let's think. I remember there's a formula for the sum of cubes that relates to the sum and product of the numbers. Let me recall... Ah, yes! The formula for \(a^3 + b^3\) is \((a + b)^3 - 3ab(a ... |
olympiads | (7) The sum of all positive integers $n$ that satisfy $\frac{1}{4}<\sin \frac{\pi}{n}<\frac{1}{3}$ is $\qquad$ | <think>
Okay, so I have this problem here: I need to find the sum of all positive integers n such that 1/4 is less than sin(π/n) and that's also less than 1/3. Hmm, let's try to figure this out step by step.
First, I know that the sine function oscillates between -1 and 1, but since we're dealing with π/n and n is a ... |
olympiads | 18.4.11 * Find the integer solution of the equation $\left(1+\frac{1}{m}\right)^{m+1}=\left(1+\frac{1}{1988}\right)^{1988}$. | <think>
Okay, so I need to solve the equation \(\left(1+\frac{1}{m}\right)^{m+1} = \left(1+\frac{1}{1988}\right)^{1988}\) for integer values of \(m\). Hmm, let's see. I remember that expressions like \(\left(1+\frac{1}{n}\right)^n\) are related to the number \(e\), which is approximately 2.71828. As \(n\) increases, \(... |
olympiads | Example 3 As shown in Figure 3, in $\triangle A B C$, $\angle B=46^{\circ}$, point $D$ is on side $B C$, and satisfies $\angle B A D=21^{\circ}$. If $A B$ $=C D$, find the degree measure of $\angle C A D$. ${ }^{[3]}$ | <think>
Okay, let's see. So we have triangle ABC with angle B equal to 46 degrees. There's a point D on side BC such that angle BAD is 21 degrees. Also, we know that AB equals CD. We need to find the measure of angle CAD. Hmm, okay. Let me try to visualize this.
First, I'll draw triangle ABC. Let me label the vertice... |
olympiads | $11 、$ Satisfy all 7 conditions below, the five different natural numbers $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C} 、 \mathrm{D} 、 \mathrm{E}$ are $\mathrm{A}=(\quad), \mathrm{B}=(\quad)$, $\mathrm{C}=(\quad), \mathrm{D}=(\quad), \mathrm{E}=(\quad)$ respectively. What are they?
（1）These numbers are all less than 10;
(2) A... | <think>
Okay, let's see. I need to find five different natural numbers A, B, C, D, E that satisfy seven conditions. All numbers are less than 10, and they have to be different. Let me start by listing the conditions again to make sure I have them all straight:
1. All numbers A, B, C, D, E are different natural numbers... |
olympiads | 1. The polynomial $P(x)$ with integer coefficients satisfies the condition $P(29)=P(37)=2022$. Find the smallest possible value of $P(0)>0$ under these conditions. | <think>
Okay, let's see. The problem is about a polynomial P(x) with integer coefficients. It tells me that P(29) = P(37) = 2022, and I need to find the smallest possible value of P(0) that's positive. Hmm, alright. Let me try to break this down step by step.
First, since P(x) has integer coefficients, that makes me t... |
olympiads | Problem 8.1. In a box, there are oranges, pears, and apples, a total of 60 fruits. It is known that there are 3 times more apples than non-apples, and there are 5 times fewer pears than non-pears. How many oranges are in the box? | <think>
Okay, let's see. I need to figure out how many oranges are in the box. The total number of fruits is 60, which includes oranges, pears, and apples. Hmm. Let me break down the information given.
First, it says there are 3 times more apples than non-apples. Hmm. Non-apples would be oranges and pears combined, ri... |
olympiads | 12. There are two buckets, a large one and a small one, both containing some water. It is known that the weight of the water in the large bucket is half that in the small bucket. If 30 kilograms of water are added to the large bucket, the weight of the water in the large bucket becomes three times that in the small buc... | <think>
Okay, let's see. I need to figure out how much water was originally in the large bucket. The problem says there are two buckets, a large one and a small one, both with some water. The weight of the water in the large bucket is half that in the small bucket initially. Then, when 30 kilograms are added to the lar... |
olympiads | 10-8. In trapezoid $A B C D$, the bases $A D$ and $B C$ are 8 and 18, respectively. It is known that the circumcircle of triangle $A B D$ is tangent to the lines $B C$ and $C D$. Find the perimeter of the trapezoid. | <think>
Okay, let's see. I have this trapezoid ABCD with bases AD and BC. AD is 8 units and BC is 18 units. The problem says that the circumcircle of triangle ABD is tangent to the lines BC and CD. I need to find the perimeter of the trapezoid. Hmm, okay.
First, let me draw a rough sketch of the trapezoid to visualize... |
olympiads | B1. If the polynomial $p$ is divided by the polynomial $2-x$, the quotient is $2 x^{2}-x+3$. Determine the remainder of this division, given that the product of all the zeros of the polynomial $p$ is equal to $\frac{11}{2}$. | <think>
Okay, let's see. So the problem is about polynomial division. We have a polynomial p(x) which, when divided by (2 - x), gives a quotient of 2x² - x + 3. We need to find the remainder of this division. Additionally, we're told that the product of all the zeros of p(x) is 11/2. Hmm, let's break this down step by ... |
olympiads | 7. Let
$$
f(x)=x^{4}-6 x^{3}+26 x^{2}-46 x+65 \text {. }
$$
Let the roots of $f(x)$ be $a_{k}+i b_{k}$ for $k=1,2,3,4$. Given that the $a_{k}, b_{k}$ are all integers, find $\left|b_{1}\right|+\left|b_{2}\right|+\left|b_{3}\right|+\left|b_{4}\right|$. | <think>
Okay, let's see. I have this polynomial: f(x) = x⁴ - 6x³ + 26x² - 46x + 65. The problem says that all the roots are of the form a_k + i b_k, where a_k and b_k are integers. I need to find the sum of the absolute values of the imaginary parts of all roots. So, |b₁| + |b₂| + |b₃| + |b₄|.
First, since the coeffi... |
olympiads | In a spatial rectangular coordinate system, a sphere centered at the origin has a radius of 3 units. How many lattice points lie on the surface of the sphere? | <think>
Okay, so I have this problem here: In a spatial rectangular coordinate system, a sphere centered at the origin has a radius of 3 units. How many lattice points lie on the surface of the sphere? Hmm, lattice points are points where all the coordinates are integers, right? So I need to find all integer solutions ... |
olympiads | 7.4. Anton from the village was given several zucchini, and he decided to give them to his friends. He gave half of the received zucchini to Arina, and a third (also from the received amount) to Vera. It turned out that after this, the number of zucchini Arina had became a square of some natural number, and the number ... | <think>
Okay, let's see. So Anton gets some zucchinis, right? And he gives half of them to Arina and a third to Vera. Then after that, Arina's total is a perfect square and Vera's is a perfect cube. We need to find the smallest number of zucchinis Anton could have received. Hmm, let me break this down step by step.
Fi... |
olympiads | [Ratio of areas of triangles with a common angle] Auxiliary similar triangles $\quad]$
In triangle $A B C$, angle $C$ is $30^{\circ}$, and angle $A$ is acute. A line perpendicular to side $B C$ is drawn, cutting off triangle $C N M$ from triangle $A B C$ (point $N$ lies between vertices $B$ and $C$). The areas of tria... | <think>
Okay, let's try to tackle this geometry problem step by step. First, let me understand what's given and what we need to find.
We have triangle ABC with angle C being 30 degrees, and angle A is acute. So, angle A is less than 90 degrees. Then, a line perpendicular to side BC is drawn, cutting off triangle CNM f... |
olympiads | Some unit-radius circles are drawn on the plane, and we color the center of each circle blue. We mark some points on the circumference of the circles in red such that exactly 2 red points lie on each circumference. What is the maximum number of blue points if there are a total of 25 colored points? | <think>
Okay, so here's the problem: We have some unit-radius circles on a plane. The centers of these circles are colored blue, and on each circle's circumference, we mark exactly two red points. The total number of colored points (both blue and red) is 25. We need to find the maximum number of blue points possible. H... |
olympiads | 4. (15 points) A ball was thrown vertically upwards from a balcony. It is known that it hit the ground after 6 seconds. Given that the initial speed of the ball is 20 m/s, determine the height of the balcony. The acceleration due to gravity is 10 m/s². | <think>
Okay, so I need to figure out the height of the balcony from which the ball was thrown upwards. The problem says the ball was thrown vertically upwards with an initial speed of 20 m/s and hit the ground after 6 seconds. Gravity is 10 m/s². Alright, let's start by recalling some kinematic equations.
First, whe... |
olympiads | 3. Solve the equation
$$
2 x+2+\operatorname{arctg} x \cdot \sqrt{x^{2}+1}+\operatorname{arctg}(x+2) \cdot \sqrt{x^{2}+4 x+5}=0
$$ | <think>
Okay, so I have this equation to solve:
2x + 2 + arctg(x) * sqrt(x² + 1) + arctg(x + 2) * sqrt(x² + 4x + 5) = 0
Hmm, arctg is the inverse tangent function, right? So arctg(x) is the angle whose tangent is x. The equation looks pretty complicated with those square roots and arctg terms multiplied together. Le... |
olympiads | ## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{\sqrt{2}} \frac{x^{4} \cdot d x}{\left(4-x^{2}\right)^{3 / 2}}
$$ | <think>
Okay, so I need to calculate the definite integral of x^4 over (4 - x^2) raised to the 3/2 power, from 0 to sqrt(2). Hmm, let's see. The integral is ∫₀^√2 [x⁴ / (4 - x²)^(3/2)] dx. Alright, first, maybe I should simplify the integrand or find a substitution that can make this easier.
Looking at the denominator... |
olympiads | 11.4. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge length a. The ends of the segment intersecting edge $C_{1} D_{1}$ lie on the lines $A A_{1}$ and $B C$. What is the minimum length that this segment can have | <think>
Okay, so I have this cube labeled ABCD A₁B₁C₁D₁, and each edge has length a. The problem is asking for the minimum length of a segment that intersects the edge C₁D₁, with its endpoints lying on the lines AA₁ and BC. Hmm, let me try to visualize this cube first.
First, let me recall that in a cube, all edges ar... |
olympiads_ref | Find the largest integer $k(k \geq 2)$, for which there exists an integer $n(n \geq k)$ such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:
1. each chosen number is not divisible by 6, by 7 and by 8;
2. the positive difference of ... | <think>
Okay, so I need to find the largest integer k (where k is at least 2) such that there exists an integer n (n is at least k) where, from any collection of n consecutive positive integers, I can always choose k numbers that meet two conditions:
1. Each chosen number is not divisible by 6, 7, or 8.
2. The positiv... |
olympiads | 8. (10 points) In the inscribed quadrilateral $A B C D$, the degree measures of the angles are in the ratio $\angle A: \angle B: \angle C=2: 3: 4$. Find the length of $A C$, if $C D=21, B C=14 \sqrt{3}-10.5$. | <think>
Okay, so I need to find the length of AC in an inscribed quadrilateral ABCD. The angles at A, B, and C are in the ratio 2:3:4. They also gave me CD = 21 and BC = 14√3 - 10.5. Hmm, inscribed quadrilateral... that means it's cyclic, right? So all vertices lie on a circle. I remember that in cyclic quadrilaterals,... |
olympiads | 12. The sequence $\left\{a_{n}\right\}$ is defined as follows: $a_{0}=1, a_{1}=2, a_{n+2}=a_{n}+a_{n+1}^{2}$, then $a_{2006} \equiv$ $\qquad$ $(\bmod 7)$ | <think>
Okay, so I need to find the value of \( a_{2006} \mod 7 \) where the sequence is defined by \( a_0 = 1 \), \( a_1 = 2 \), and \( a_{n+2} = a_n + a_{n+1}^2 \). Hmm, modular arithmetic problem with a recursive sequence. Let's see. Since we're dealing with mod 7, maybe the sequence repeats after some period? That'... |
olympiads | 11.006. The diagonal of a rectangular parallelepiped is 13 cm, and the diagonals of its lateral faces are $4 \sqrt{10}$ and $3 \sqrt{17}$ cm. Determine the volume of the parallelepiped. | <think>
Okay, so I need to find the volume of a rectangular parallelepiped. The problem gives me the space diagonal, which is 13 cm, and the diagonals of two of its lateral faces: 4√10 cm and 3√17 cm. Hmm, let me recall what a rectangular parallelepiped is. It's basically a 3D shape with all faces being rectangles, so ... |
olympiads | 3. The base of the pyramid $P A B C D$ is a trapezoid $A B C D$, the base $A D$ of which is twice the length of the base $B C$. The segment $M N$ is the midline of the triangle $A B P$, parallel to the side $A B$. Find the ratio of the volumes of the solids into which the plane $D M N$ divides this pyramid. | <think>
Okay, let's try to tackle this problem step by step. First, let me understand the problem statement clearly. We have a pyramid PABCD with a trapezoid base ABCD. The base AD of the trapezoid is twice the length of BC. Then, there's a segment MN which is the midline of triangle ABP, and it's parallel to side AB. ... |
olympiads | Example 3.21. Find the points of discontinuity of the function
$$
z=\frac{x y+1}{x^{2}-y}
$$ | <think>
Okay, so I need to find the points of discontinuity for the function z = (xy + 1)/(x² - y). Hmm, let me think. Discontinuities in functions usually occur where the function isn't defined, right? So for a rational function like this, the denominator being zero would make it undefined. That makes sense. So first,... |
olympiads | Problem 8.2.1. Given trapezoid $A B C D(A D \| B C)$. It turns out that $\angle A B D=\angle B C D$. Find the length of segment $B D$, if $B C=36$ and $A D=64$.
 | <think>
Okay, so I have this trapezoid ABCD where AD is parallel to BC. The problem says that angle ABD is equal to angle BCD, and we need to find the length of segment BD. Given are BC = 36 and AD = 64. Hmm, let me try to visualize this first.
Since it's a trapezoid with AD parallel to BC, the bases are AD and BC. Th... |
olympiads | ## Task 3 - 330813
Sebastian is considering a three-digit natural number and notices:
(1) If you place the digit 5 in front of this three-digit number, the resulting four-digit number is a perfect square.
(2) If you append the digit 1 to the (original three-digit) number, the resulting four-digit number is also a pe... | <think>
Okay, let's see. Sebastian has a three-digit number, and when you put a 5 in front of it, making a four-digit number, that's a perfect square. Also, if you append a 1 to the original number, that four-digit number is also a perfect square. We need to find this three-digit number. Hmm, interesting.
First, let ... |
olympiads | 2.1. (5 points) The ratio of the sides of a rectangle is $3: 4$, and its diagonal is 9. Find the smaller side of the rectangle. | <think>
Okay, so I have this problem here about a rectangle with a side ratio of 3:4 and a diagonal of 9. I need to find the smaller side. Let me see how to approach this.
First, the ratio of the sides is 3 to 4. That means if the shorter side is 3 parts, the longer side would be 4 parts. But I need actual lengths, no... |
olympiads | 2. Diana wrote a two-digit number, and appended to it a two-digit number that was a permutation of the digits of the first number. It turned out that the difference between the first and second numbers is equal to the sum of the digits of the first number. What four-digit number was written? | <think>
Okay, let's see. So Diana wrote a two-digit number first. Let's call that number AB, where A is the tens digit and B is the units digit. Then, she appended a two-digit number that was a permutation of the digits of the first number. So appending means she wrote them next to each other, right? So if the first nu... |
olympiads_ref | Find all triples $(x, n, p)$ of positive integers $x$ and $n$ and prime numbers $p$ for which
$$
x^{3}+3 x+14=2 \cdot p^{n} \text {. }
$$ | <think>
Okay, so I need to find all triples of positive integers (x, n) and prime numbers p such that x³ + 3x + 14 = 2·pⁿ. Hmm, let me try to break this down step by step.
First, let me understand the equation: x³ + 3x + 14 equals twice a prime power. So, the right-hand side is 2 multiplied by p raised to the power of... |
olympiads | Problem 9.8. On the side $CD$ of trapezoid $ABCD (AD \| BC)$, a point $M$ is marked. A perpendicular $AH$ is dropped from vertex $A$ to segment $BM$. It turns out that $AD = HD$. Find the length of segment $AD$, given that $BC = 16$, $CM = 8$, and $MD = 9$.
 Point $M$ lies on the side of a regular hexagon with side length 10. Find the sum of the distances from point $M$ to the lines containing the other sides of the hexagon. | <think>
Okay, so I have this problem here: there's a regular hexagon with side length 10, and a point M lies on one of its sides. I need to find the sum of the distances from point M to the lines that contain the other sides of the hexagon. Hmm, let's try to visualize this first.
A regular hexagon has six sides, all e... |
olympiads | ## Task 3 - 150523
When a pioneer group reported on their trips taken over the past few years, the following was revealed:
(1) Exactly 13 members of this group have been to the Baltic Sea.
(2) Exactly 15 pioneers have been to the Harz.
(3) Exactly 6 pioneers have been to both the Baltic Sea and the Harz.
(4) Exact... | <think>
Okay, let's see. So, there's this problem about pioneers who went on trips to the Baltic Sea and the Harz. I need to find out the total number of pioneers in the group. The information given is:
1. Exactly 13 members have been to the Baltic Sea.
2. Exactly 15 have been to the Harz.
3. Exactly 6 have been to bo... |
olympiads | ## 9. Megacube
From identical small cubes, one large cube is assembled. The length of the edge of the large cube is a hundred times larger than the length of the edge of each small cube. The large cube is placed on a table, and all its sides, except the one that touches the table, are painted red. How many small cubes... | <think>
Okay, so I need to figure out how many small cubes have exactly two red sides after the large cube is painted. Let me start by visualizing the problem. The large cube is made up of smaller cubes, each with edge length 1 (let's say), and the large cube has an edge length of 100, so it's 100x100x100 small cubes. ... |
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