question stringlengths 35 2.07k | thinking stringlengths 2.4k 2.9k | answer stringlengths 1 261 | id int64 5 93.7k | content stringlengths 2.41k 3.02k | id_ stringlengths 3 6 |
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Let's determine the value of the parameter $k$ such that the roots of the following equation, when listed in ascending order, have the same difference between each pair of consecutive roots.
$$
4 x^{4}-(4 k+13) x^{2}+(9 k+9)=0
$$ | <think>I. solution. Our equation is quadratic in terms of $x^{2}$, and it can have at most two solutions for $x^{2}$. If these solutions are real, distinct, and positive, then we will get four roots for $x$, which are equal in absolute value in pairs. (1) can be rewritten as:
$$
x^{4}-\left(\frac{13}{4}+k\right) x^{2}... | k_{1}=-\frac{3}{4},k_{2}=\frac{77}{4} | 83,095 | <think>I. solution. Our equation is quadratic in terms of $x^{2}$, and it can have at most two solutions for $x^{2}$. If these solutions are real, distinct, and positive, then we will get four roots for $x$, which are equal in absolute value in pairs. (1) can be rewritten as:
$$
x^{4}-\left(\frac{13}{4}+k\right) x^{2}... | 1_0 |
In a square $ABCD$ of diagonals $AC$ and $BD$, we call $O$ at the center of the square. A square $PQRS$ is constructed with sides parallel to those of $ABCD$ with $P$ in segment $AO, Q$ in segment $BO, R$ in segment $CO, S$ in segment $DO$. If area of $ABCD$ equals two times the area of $PQRS$, and $M$ is the midpoi... | <think>1. **Define the coordinates of the square \(ABCD\):**
- Let \(A(0,0)\), \(B(1,0)\), \(C(1,1)\), and \(D(0,1)\).
- The center \(O\) of the square \(ABCD\) is at \(\left(\frac{1}{2}, \frac{1}{2}\right)\).
2. **Define the coordinates of the square \(PQRS\):**
- Since \(PQRS\) is a smaller square with side... | \frac{45^\circ}{2} | 49,494 | <think>1. **Define the coordinates of the square \(ABCD\):**
- Let \(A(0,0)\), \(B(1,0)\), \(C(1,1)\), and \(D(0,1)\).
- The center \(O\) of the square \(ABCD\) is at \(\left(\frac{1}{2}, \frac{1}{2}\right)\).
2. **Define the coordinates of the square \(PQRS\):**
- Since \(PQRS\) is a smaller square with side... | 1_1 |
## Task 4 - 260924
For a right triangle $A B C$ with the right angle at $C$, it is required that this right angle be divided into four equal angles by the median of side $A B$, the angle bisector of $\angle A C B$, and the altitude perpendicular to side $A B$.
Investigate whether there exists a triangle $A B C$ that ... | <think>Let $W$ be the intersection of the angle bisector with $AB$, $H$ the foot of the perpendicular from $C$ to $AB$, and $M$ the midpoint of the segment $AB$.
Since the angle bisector divides the right angle at $C$ into the two angles $\angle A C W = \angle W C B = 45^{\circ}$, the altitude and the median must bise... | \angleBAC=22.5,\angleABC=67.5 | 52,034 | <think>Let $W$ be the intersection of the angle bisector with $AB$, $H$ the foot of the perpendicular from $C$ to $AB$, and $M$ the midpoint of the segment $AB$.
Since the angle bisector divides the right angle at $C$ into the two angles $\angle A C W = \angle W C B = 45^{\circ}$, the altitude and the median must bise... | 1_2 |
B1. Determine the smallest possible value of the expression $\left|25^{m}-36^{n}\right|$, where $m$ and $n$ are natural numbers. | <think>B1. By considering $25=5^{2}$ and $36=6^{2}$, the expression can be factored as
$$
\left|25^{m}-36^{n}\right|=\left|5^{2 m}-6^{2 n}\right|=\left|\left(5^{m}\right)^{2}-\left(6^{n}\right)^{2}\right|=\left|\left(5^{m}+6^{n}\right)\left(5^{m}-6^{n}\right)\right|=\left(5^{m}+6^{n}\right)\left|\left(5^{m}-6^{n}\righ... | 11 | 41,038 | <think>B1. By considering $25=5^{2}$ and $36=6^{2}$, the expression can be factored as
$$
\left|25^{m}-36^{n}\right|=\left|5^{2 m}-6^{2 n}\right|=\left|\left(5^{m}\right)^{2}-\left(6^{n}\right)^{2}\right|=\left|\left(5^{m}+6^{n}\right)\left(5^{m}-6^{n}\right)\right|=\left(5^{m}+6^{n}\right)\left|\left(5^{m}-6^{n}\righ... | 1_3 |
In a room there are $144$ people. They are joined by $n$ other people who are each carrying $k$ coins. When these coins are shared among all $n + 144$ people, each person has $2$ of these coins. Find the minimum possible value of $2n + k$. | <think>1. Let the number of people initially in the room be \(144\).
2. Let \(n\) be the number of additional people who join the room.
3. Each of these \(n\) additional people is carrying \(k\) coins.
4. When these coins are shared among all \(n + 144\) people, each person has \(2\) coins.
We need to find the minimum... | 50 | 30,969 | <think>1. Let the number of people initially in the room be \(144\).
2. Let \(n\) be the number of additional people who join the room.
3. Each of these \(n\) additional people is carrying \(k\) coins.
4. When these coins are shared among all \(n + 144\) people, each person has \(2\) coins.
We need to find the minimum... | 1_4 |
11. (This sub-question is worth 20 points) It is known that any positive integer $n$ can be uniquely represented as
$$
n=a_{0}+a_{1} 9+a_{2} 9^{2}+\cdots+a_{m} 9^{m}
$$
where $m$ is a non-negative integer, $a_{j} \in\{0,1, \cdots, 8\}(j=0,1, \cdots, m-1), a_{m} \in\{1, \cdots, 8\}$. Find the sum of all positive intege... | <think>Let $A$ and $B$ denote the sets of all positive integers that form strictly increasing and decreasing sequences, respectively, in $(*)$. The symbol $S(M)$ represents the sum of all numbers in the set $M$, and the positive integers satisfying $(*)$ are denoted as:
$$
n=\overline{a_{m} a_{m-1} \cdots a_{1} a_{0}}
... | 984374748 | 59,590 | <think>Let $A$ and $B$ denote the sets of all positive integers that form strictly increasing and decreasing sequences, respectively, in $(*)$. The symbol $S(M)$ represents the sum of all numbers in the set $M$, and the positive integers satisfying $(*)$ are denoted as:
$$
n=\overline{a_{m} a_{m-1} \cdots a_{1} a_{0}}
... | 1_5 |
13.3.27 Given two concentric circles in a plane with radii $R$ and $r$ ($R > r$). Let $P$ be a fixed point on the smaller circle, $B$ be a moving point on the larger circle, and the line $BP$ intersects the larger circle at another point $C$. The line $l$ through point $P$ and perpendicular to $BP$ intersects the small... | <think>(1) As shown in the figure, establish a Cartesian coordinate system with the centers of the two circles as the origin, so that point $P$ is on the negative half-axis of the $x$-axis. Then the coordinates of point $P$ are $(-r, 0)$, and the equations of the two circles are: $x^{2}+y^{2}=R^{2}$, $x^{2}+y^{2}=r^{2}... | 6R^2+2r^2 | 93,351 | <think>(1) As shown in the figure, establish a Cartesian coordinate system with the centers of the two circles as the origin, so that point $P$ is on the negative half-axis of the $x$-axis. Then the coordinates of point $P$ are $(-r, 0)$, and the equations of the two circles are: $x^{2}+y^{2}=R^{2}$, $x^{2}+y^{2}=r^{2}... | 1_6 |
Let's check the following equalities, look for regularities in the table, and based on that, write down the next two rows of the table.
$$
\begin{aligned}
3^{2}+4^{2} & =5^{2} \\
10^{2}+11^{2}+12^{2} & =13^{2}+14^{2} \\
21^{2}+22^{2}+23^{2}+24^{2} & =25^{2}+26^{2}+27^{2} \\
36^{2}+37^{2}+38^{2}+39^{2}+40^{2} & =41^{2}... | <think>The equalities hold, the values on both sides in the subsequent rows are 25, 365, 2030, 7230, respectively.
In each row, the squares of consecutive natural numbers are listed, the number of terms in the subsequent rows is 3, 5, 7, 9, and on the left side, there is always one more term than on the right side. Ac... | \begin{aligned}55^{2}+56^{2}+57^{2}+58^{2}+59^{2}+60^{2}&=61^{2}+62^{2}+63^{2}+64^{2}+65^{2}\\78^{2}+79^{2}+80^{2}+81^{2}+8 | 9,243 | <think>The equalities hold, the values on both sides in the subsequent rows are 25, 365, 2030, 7230, respectively.
In each row, the squares of consecutive natural numbers are listed, the number of terms in the subsequent rows is 3, 5, 7, 9, and on the left side, there is always one more term than on the right side. Ac... | 1_7 |
Two candidates participated in an election with $p+q$ voters. Candidate $A$ received $p$ votes and candidate $B$ received $q$ votes, with $p>q$. During the counting, only one vote is recorded at a time on a board. Let $r$ be the probability that the number associated with candidate $A$ on the board is always greater th... | <think>Solution
a) We can make lists with the letters $A$ and $B$ representing the possible orders of votes counted. For example, the list $A A B A B$ indicates that the first two and the fourth votes counted were for candidate $A$, the third and fifth for candidate $B$. There are exactly 10 lists with 3 letters $A$ a... | \frac{1}{2019} | 82,167 | <think>Solution
a) We can make lists with the letters $A$ and $B$ representing the possible orders of votes counted. For example, the list $A A B A B$ indicates that the first two and the fourth votes counted were for candidate $A$, the third and fifth for candidate $B$. There are exactly 10 lists with 3 letters $A$ a... | 1_8 |
2. Determine the largest possible number of non-empty, pairwise disjoint sets with the same sum of elements, into which the set can be divided:
a) $\{1,2, \ldots, 2017\}$,
b) $\{1,2, \ldots, 2018\}$.
If a set consists of a single number, we consider it as the sum of its elements. | <think>SOLUTION. Let's first look at the sets with the smallest possible number of elements; we can probably have at most one one-element set - otherwise, those two one-element sets would not have the same sum of elements. The other sets are therefore at least two-element.
In part a), let's take the one-element set to... | 1009 | 24,894 | <think>SOLUTION. Let's first look at the sets with the smallest possible number of elements; we can probably have at most one one-element set - otherwise, those two one-element sets would not have the same sum of elements. The other sets are therefore at least two-element.
In part a), let's take the one-element set to... | 1_9 |
Let be given a triangle $ABC$ and points $D,M,N$ belong to $BC,AB,AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. If $S_{\vartriangle BMD} = 9$ cm $^2, S_{\vartriangle DNC} = 25$ cm$^2$, compute $S_{\vartriangle AMN}$?
| <think>1. **Assign coordinates to points:**
Let \( A(0,0) \), \( B(b,0) \), and \( C(a,c) \).
Choose \( M(\lambda,0) \) on \( AB \), then \( D \) on \( BC \) and \( N \) on \( AC \).
2. **Determine coordinates of \( D \) and \( N \):**
Since \( MD \parallel AC \), the coordinates of \( D \) can be expressed ... | 15 \, \text{cm}^2 | 33,696 | <think>1. **Assign coordinates to points:**
Let \( A(0,0) \), \( B(b,0) \), and \( C(a,c) \).
Choose \( M(\lambda,0) \) on \( AB \), then \( D \) on \( BC \) and \( N \) on \( AC \).
2. **Determine coordinates of \( D \) and \( N \):**
Since \( MD \parallel AC \), the coordinates of \( D \) can be expressed ... | 1_10 |
8. $n$ chess players participate in a chess tournament, with each pair of players competing in one match. The rules are: the winner gets 1 point, the loser gets 0 points, and in the case of a draw, both players get 0.5 points. If it is found after the tournament that among any $m$ players, there is one player who has w... | <think>8. First, prove two lemmas.
Lemma 1 When $n \geqslant m$, if the match results of $n$ players have property $P(m)$, then there must be a player who wins all others.
Proof of Lemma 1: When $n=m$, the proposition is obviously true.
Assume the proposition holds for $n$.
Then for $n+1$ players, choose any $n$ playe... | 2m-3 | 90,971 | <think>8. First, prove two lemmas.
Lemma 1 When $n \geqslant m$, if the match results of $n$ players have property $P(m)$, then there must be a player who wins all others.
Proof of Lemma 1: When $n=m$, the proposition is obviously true.
Assume the proposition holds for $n$.
Then for $n+1$ players, choose any $n$ playe... | 1_11 |
Example 6. Solve the system of differential equations
$$
\left\{\begin{array}{l}
x^{\prime}=3 x+4 y+2 z \\
y^{\prime}=x+4 y+z \\
z^{\prime}=4 x+6 y+5 z
\end{array}\right.
$$ | <think>Solution. The characteristic equation of the matrix $A=\left(\begin{array}{lll}3 & 4 & 2 \\ 1 & 4 & 1 \\ 4 & 6 & 5\end{array}\right)$ is: $\left|\begin{array}{ccc}3-\lambda & 4 & 2 \\ 1 & 4-\lambda & 1 \\ 4 & 6 & 5-\lambda\end{array}\right|=0$, or $(3-\lambda)(4-\lambda) \times$ $\times(5-\lambda)+12+16-6(3-\lam... | {\begin{pmatrix}C_{1}e^{}&+7C_{3}e^{9}\\&C_{2}e^{2}+4C_{3}e^{9}\\-C_{1}e^{}-2C_{2}e^{2}+13C_{3}e^{9}\end{pmatrix}.} | 76,054 | <think>Solution. The characteristic equation of the matrix $A=\left(\begin{array}{lll}3 & 4 & 2 \\ 1 & 4 & 1 \\ 4 & 6 & 5\end{array}\right)$ is: $\left|\begin{array}{ccc}3-\lambda & 4 & 2 \\ 1 & 4-\lambda & 1 \\ 4 & 6 & 5-\lambda\end{array}\right|=0$, or $(3-\lambda)(4-\lambda) \times$ $\times(5-\lambda)+12+16-6(3-\lam... | 1_12 |
Find the smallest real number $p$ such that the inequality $\sqrt{1^2+1}+\sqrt{2^2+1}+...+\sqrt{n^2+1} \le \frac{1}{2}n(n+p)$ holds for all natural numbers $n$. | <think>1. We start with the given inequality:
\[
\sum_{x=1}^n \sqrt{x^2 + 1} \leq \frac{n(n + p)}{2}
\]
for all natural numbers \( n \).
2. To find the smallest \( p \), we first transform the inequality:
\[
\sum_{x=1}^n (\sqrt{x^2 + 1} - x) \leq \frac{n(n + p)}{2} - \sum_{x=1}^n x
\]
Simplifyi... | 2\sqrt{2} - 1 | 19,024 | <think>1. We start with the given inequality:
\[
\sum_{x=1}^n \sqrt{x^2 + 1} \leq \frac{n(n + p)}{2}
\]
for all natural numbers \( n \).
2. To find the smallest \( p \), we first transform the inequality:
\[
\sum_{x=1}^n (\sqrt{x^2 + 1} - x) \leq \frac{n(n + p)}{2} - \sum_{x=1}^n x
\]
Simplifyi... | 1_13 |
Let $c$ be the constant number such that $c>1.$Find the least area of the figure surrounded by the line passing through the point $(1,\ c)$ and the palabola $y=x^{2}$ on $x-y$ plane. | <think>1. **Equation of the Line:**
The equation of the line \( L \) passing through the point \((1, c)\) with slope \( m \) is given by:
\[
y = m(x - 1) + c
\]
2. **Intersection Points:**
Let \((a, a^2)\) and \((b, b^2)\) be the points of intersection of the line \( L \) and the parabola \( y = x^2 \).... | \frac{4}{3} (c - 1)^{3/2} | 32,369 | <think>1. **Equation of the Line:**
The equation of the line \( L \) passing through the point \((1, c)\) with slope \( m \) is given by:
\[
y = m(x - 1) + c
\]
2. **Intersection Points:**
Let \((a, a^2)\) and \((b, b^2)\) be the points of intersection of the line \( L \) and the parabola \( y = x^2 \).... | 1_14 |
Seven people of seven different ages are attending a meeting. The seven people leave the meeting one at a
time in random order. Given that the youngest person leaves the meeting sometime before the oldest
person leaves the meeting, the probability that the third, fourth, and fifth people to leave the meeting do so in ord... | <think>1. **Determine the total number of possible orders:**
The total number of ways the seven people can leave the meeting is \(7!\). Since the youngest person must leave before the oldest person, we divide by 2 (due to symmetry), giving us:
\[
\frac{7!}{2} = \frac{5040}{2} = 2520
\]
2. **Define the peop... | 25 | 13,960 | <think>1. **Determine the total number of possible orders:**
The total number of ways the seven people can leave the meeting is \(7!\). Since the youngest person must leave before the oldest person, we divide by 2 (due to symmetry), giving us:
\[
\frac{7!}{2} = \frac{5040}{2} = 2520
\]
2. **Define the peop... | 1_15 |
1・185 On the blackboard, there are natural numbers $1,2,3, \cdots, n$, where $n \geqslant 3$. Each time, it is allowed to erase any two numbers $p$ and $q$, and replace them with $p+q$ and $|p-q|$. After such several rewrites, all the numbers on the blackboard become $k$. What values can $k$ possibly be? | <think>[Solution] Let $s$ be any natural number satisfying the inequality $2^{s} \geqslant n$.
Obviously, after each step, non-negative integers will be written on the blackboard. If the sum and difference of two non-negative integers can both be divided by an odd number $d$, then these two numbers themselves can also ... | 2^{} | 33,330 | <think>[Solution] Let $s$ be any natural number satisfying the inequality $2^{s} \geqslant n$.
Obviously, after each step, non-negative integers will be written on the blackboard. If the sum and difference of two non-negative integers can both be divided by an odd number $d$, then these two numbers themselves can also ... | 1_16 |
4. Let M be a set of six distinct positive integers whose sum is 60. We will write all of them on the faces of a cube, with exactly one on each face. In one step, we choose any three faces of the cube that share a common vertex and increase each of the numbers on these three faces by 1. Determine the number of all such... | <think>4. Let the faces of the cube be denoted by $S_{1}, S_{2}, \ldots, S_{6}$ such that face $S_{1}$ is opposite to face $S_{6}$, face $S_{2}$ is opposite to $S_{5}$, and $S_{3}$ is opposite to $S_{4}$. Let the number on face $S_{i}$ be denoted by $c_{i}$. Clearly, any vertex of the cube belongs to exactly one pair o... | 84 | 33,580 | <think>4. Let the faces of the cube be denoted by $S_{1}, S_{2}, \ldots, S_{6}$ such that face $S_{1}$ is opposite to face $S_{6}$, face $S_{2}$ is opposite to $S_{5}$, and $S_{3}$ is opposite to $S_{4}$. Let the number on face $S_{i}$ be denoted by $c_{i}$. Clearly, any vertex of the cube belongs to exactly one pair o... | 1_17 |
Solve the following system of equations:
$$
\begin{aligned}
& \left|a_{1}-a_{2}\right| \cdot x_{2}+\left|a_{1}-a_{3}\right| \cdot x_{3}+\left|a_{1}-a_{4}\right| \cdot x_{4}=1 \\
& \left|a_{2}-a_{1}\right| \cdot x_{1} \quad+\left|a_{2}-a_{3}\right| \cdot x_{3}+\left|a_{2}-a_{4}\right| \cdot x_{4}=1 \\
& \left|a_{3}-a_{... | <think>) First, we solve the system for the case of the parameter size relationship $a_{1}<a_{2}<a_{3}<a_{4}$. We will reduce other cases to this one. (I) is thus structured:
$$
\begin{array}{ccc}
\quad\left(a_{2}-a_{1}\right) x_{2} & +\left(a_{3}-a_{1}\right) x_{3}+\left(a_{4}-a_{1}\right) x_{4} & =1 \\
\left(a_{2}-a... | x_{}=x_{k}=\frac{1}{a_{k}-a_{}},\quadx_{i}=x_{j}=0 | 58,318 | <think>) First, we solve the system for the case of the parameter size relationship $a_{1}<a_{2}<a_{3}<a_{4}$. We will reduce other cases to this one. (I) is thus structured:
$$
\begin{array}{ccc}
\quad\left(a_{2}-a_{1}\right) x_{2} & +\left(a_{3}-a_{1}\right) x_{3}+\left(a_{4}-a_{1}\right) x_{4} & =1 \\
\left(a_{2}-a... | 1_18 |
Suppose three direction on the plane . We draw $ 11$ lines in each direction . Find maximum number of the points on the plane which are on three lines . | <think>1. **Affine Transformation and Line Directions**:
By an affine transformation, we can assume that the three directions of the lines are:
- Parallel to the \(x\)-axis.
- Parallel to the \(y\)-axis.
- Parallel to the line \(x = y\).
2. **Formation of the Grid**:
Fix the lines of the first two group... | 91 | 88,240 | <think>1. **Affine Transformation and Line Directions**:
By an affine transformation, we can assume that the three directions of the lines are:
- Parallel to the \(x\)-axis.
- Parallel to the \(y\)-axis.
- Parallel to the line \(x = y\).
2. **Formation of the Grid**:
Fix the lines of the first two group... | 1_19 |
Consider the polynomial $P(x)=x^3+x^2-x+2$. Determine all real numbers $r$ for which there exists a complex number $z$ not in the reals such that $P(z)=r$. | <think>1. Given the polynomial \( P(x) = x^3 + x^2 - x + 2 \), we need to determine all real numbers \( r \) for which there exists a complex number \( z \) (not in the reals) such that \( P(z) = r \).
2. If \( P(z) = r \), then we have:
\[
z^3 + z^2 - z + 2 = r \implies z^3 + z^2 - z + 2 - r = 0
\]
Let \(... | r \in (-\infty, \frac{49}{27}) \cup (3, \infty) | 81,992 | <think>1. Given the polynomial \( P(x) = x^3 + x^2 - x + 2 \), we need to determine all real numbers \( r \) for which there exists a complex number \( z \) (not in the reals) such that \( P(z) = r \).
2. If \( P(z) = r \), then we have:
\[
z^3 + z^2 - z + 2 = r \implies z^3 + z^2 - z + 2 - r = 0
\]
Let \(... | 1_20 |
[b]Q14.[/b] Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$. Suppose that $IH$ is perpendicular to $BC$ ($H$ belongs to $BC$). If $HB=5 \text{cm}, \; HC=8 \text{cm}$, compute the area of $\triangle ABC$. | <think>1. Given a right triangle \( \triangle ABC \) with \( \angle A = 90^\circ \), the incenter \( I \) is the intersection of the angle bisectors of \( \angle B \) and \( \angle C \). The perpendicular from \( I \) to \( BC \) meets \( BC \) at \( H \), with \( HB = 5 \) cm and \( HC = 8 \) cm.
2. The lengths \( HB... | 40 | 74,803 | <think>1. Given a right triangle \( \triangle ABC \) with \( \angle A = 90^\circ \), the incenter \( I \) is the intersection of the angle bisectors of \( \angle B \) and \( \angle C \). The perpendicular from \( I \) to \( BC \) meets \( BC \) at \( H \), with \( HB = 5 \) cm and \( HC = 8 \) cm.
2. The lengths \( HB... | 1_21 |
Call a pair of integers $(a,b)$ [i]primitive[/i] if there exists a positive integer $\ell$ such that $(a+bi)^\ell$ is real. Find the smallest positive integer $n$ such that less than $1\%$ of the pairs $(a, b)$ with $0 \le a, b \le n$ are primitive.
[i]Proposed by Mehtaab Sawhney[/i] | <think>1. **Understanding the Problem:**
We need to find the smallest positive integer \( n \) such that less than \( 1\% \) of the pairs \((a, b)\) with \( 0 \le a, b \le n \) are primitive. A pair \((a, b)\) is called primitive if there exists a positive integer \(\ell\) such that \((a + bi)^\ell\) is real.
2. **... | 299 | 35,548 | <think>1. **Understanding the Problem:**
We need to find the smallest positive integer \( n \) such that less than \( 1\% \) of the pairs \((a, b)\) with \( 0 \le a, b \le n \) are primitive. A pair \((a, b)\) is called primitive if there exists a positive integer \(\ell\) such that \((a + bi)^\ell\) is real.
2. **... | 1_22 |
Let's determine the base $x$ of the number system if the following equation holds:
$$
2016_{x}=x^{3}+2 x+342
$$ | <think>I. solution. $x$ is a positive integer greater than 6, because the number written in base $x$ contains the digit 6. Since $2016_{x}=2 x^{3}+x+6$, we can write the following equation:
$$
\begin{aligned}
2 x^{3}+x+6 & =x^{3}+2 x+342 \\
x^{3}-x-336 & =0
\end{aligned}
$$
Let's examine the solutions of the equation... | 7 | 11,200 | <think>I. solution. $x$ is a positive integer greater than 6, because the number written in base $x$ contains the digit 6. Since $2016_{x}=2 x^{3}+x+6$, we can write the following equation:
$$
\begin{aligned}
2 x^{3}+x+6 & =x^{3}+2 x+342 \\
x^{3}-x-336 & =0
\end{aligned}
$$
Let's examine the solutions of the equation... | 1_23 |
Example. Find the work of the force
\[
\vec{F}=(x-y) \vec{i}+\vec{j}
\]
when moving along the curve \( L \)
\[
x^{2}+y^{2}=4 \quad(y \geq 0)
\]
from point \( M(2,0) \) to point \( N(-2,0) \). | <think>Solution.
1. The work $A$ of a force field is equal to the line integral of the second kind along the curve $L$:
$$
A=\int_{L}(\vec{F}, d \vec{r})=\int_{L}(x-y) d x+d y
$$
2. We compute the line integral. For this:
a) since $L$ is the upper semicircle, its parametric equations are written as
$$
\left\{\begi... | 2\pi | 51,447 | <think>Solution.
1. The work $A$ of a force field is equal to the line integral of the second kind along the curve $L$:
$$
A=\int_{L}(\vec{F}, d \vec{r})=\int_{L}(x-y) d x+d y
$$
2. We compute the line integral. For this:
a) since $L$ is the upper semicircle, its parametric equations are written as
$$
\left\{\begi... | 1_24 |
$1 \cdot 60$ Find the smallest positive integer $n>1$, such that the arithmetic mean of $1^{2}, 2^{2}, 3^{2}, \cdots, n^{2}$ is a perfect square. | <think>【Solution】Let
$$
\frac{1^{2}+2^{2}+\cdots+n^{2}}{n}=k^{2}
$$
where $k \in \mathbb{N}$. That is,
$$
\frac{1}{6}(n+1)(2 n+1)=k^{2}
$$
Since $n \geqslant 2$, we have
$$
\frac{1}{6}(n+1)(2 n+1) \geqslant \frac{5}{2},
$$
thus
$$
\begin{array}{l}
k^{2} \geqslant \frac{5}{2}, \\
k \geqslant 2.
\end{array}
$$
Since ... | 337 | 79,532 | <think>【Solution】Let
$$
\frac{1^{2}+2^{2}+\cdots+n^{2}}{n}=k^{2}
$$
where $k \in \mathbb{N}$. That is,
$$
\frac{1}{6}(n+1)(2 n+1)=k^{2}
$$
Since $n \geqslant 2$, we have
$$
\frac{1}{6}(n+1)(2 n+1) \geqslant \frac{5}{2},
$$
thus
$$
\begin{array}{l}
k^{2} \geqslant \frac{5}{2}, \\
k \geqslant 2.
\end{array}
$$
Since ... | 1_25 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the conditions
\[ f(1+x y)-f(x+y)=f(x) f(y) \quad \text{for all } x, y \in \mathbb{R} \]
and \( f(-1) \neq 0 \). | <think>The only solution is the function $f(x)=x-1, x \in \mathbb{R}$. We set $g(x)=f(x)+1$ and show that $g(x)=x$ for all real $x$. The conditions take the form
$$ g(1+x y)-g(x+y)=(g(x)-1)(g(y)-1) \quad \text { for all } x, y \in \mathbb{R} \text { and } g(-1) \neq 1 $$
Denote $C=g(-1)-1 \neq 0$. Setting $y=-1$ in (... | f(x)=x-1, x \in \mathbb{R} | 58,611 | <think>The only solution is the function $f(x)=x-1, x \in \mathbb{R}$. We set $g(x)=f(x)+1$ and show that $g(x)=x$ for all real $x$. The conditions take the form
$$ g(1+x y)-g(x+y)=(g(x)-1)(g(y)-1) \quad \text { for all } x, y \in \mathbb{R} \text { and } g(-1) \neq 1 $$
Denote $C=g(-1)-1 \neq 0$. Setting $y=-1$ in (... | 1_26 |
We define the weight $W$ of a positive integer as follows: $W(1) = 0$, $W(2) = 1$, $W(p) = 1 + W(p + 1)$ for every odd prime $p$, $W(c) = 1 + W(d)$ for every composite $c$, where $d$ is the greatest proper factor of $c$. Compute the greatest possible weight of a positive integer less than 100. | <think>To solve the problem, we need to compute the weight \( W(n) \) for positive integers \( n \) and determine the greatest possible weight for \( n < 100 \). The weight function \( W \) is defined recursively as follows:
- \( W(1) = 0 \)
- \( W(2) = 1 \)
- \( W(p) = 1 + W(p + 1) \) for every odd prime \( p \)
- \( ... | 12 | 34,028 | <think>To solve the problem, we need to compute the weight \( W(n) \) for positive integers \( n \) and determine the greatest possible weight for \( n < 100 \). The weight function \( W \) is defined recursively as follows:
- \( W(1) = 0 \)
- \( W(2) = 1 \)
- \( W(p) = 1 + W(p + 1) \) for every odd prime \( p \)
- \( ... | 1_27 |
Suppose for independent events $A_{2}, A_{3}, \ldots, A_{n}$,
$$
P\left(A_{i}\right)=\frac{1}{2 i^{2}}
$$
What is the probability that an odd number of the events $A_{2}, A_{3}, \ldots, A_{n}$ occur? | <think>I. solution. Let $p_{n}$ be the probability asked for in the problem. Among the events $A_{2}, A_{3}, \ldots, A_{n+1}$, an odd number will occur either if an even number of the events $A_{2}, A_{3}, \ldots, A_{n}$ occur and $A_{n+1}$ also occurs, or if an odd number of the former occur and $A_{n+1}$ does not. Si... | \frac{n-1}{4n} | 53,869 | <think>I. solution. Let $p_{n}$ be the probability asked for in the problem. Among the events $A_{2}, A_{3}, \ldots, A_{n+1}$, an odd number will occur either if an even number of the events $A_{2}, A_{3}, \ldots, A_{n}$ occur and $A_{n+1}$ also occurs, or if an odd number of the former occur and $A_{n+1}$ does not. Si... | 1_28 |
Example 6 In isosceles $\triangle A B C$, $\angle B=\angle C=40^{\circ}$, extend $A B$ to point $D$, such that $A D=B C$. Find the degree measure of $\angle B C D$. | <think>Solution 1: As shown in Figure 7, construct an equilateral $\triangle ABE$ outside $\triangle ABC$ with $AB$ as a side, and connect $CE$. It is easy to see that
$$
AB=AC, \angle BAC=100^{\circ}.
$$
Since $\triangle ABE$ is an equilateral triangle, then
$$
\begin{array}{l}
AB=BE=AE, \\
\angle BAE=60^{\circ}, \\
... | 10^{\circ} | 16,387 | <think>Solution 1: As shown in Figure 7, construct an equilateral $\triangle ABE$ outside $\triangle ABC$ with $AB$ as a side, and connect $CE$. It is easy to see that
$$
AB=AC, \angle BAC=100^{\circ}.
$$
Since $\triangle ABE$ is an equilateral triangle, then
$$
\begin{array}{l}
AB=BE=AE, \\
\angle BAE=60^{\circ}, \\
... | 1_29 |
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