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48,329 | Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$.
[i]Proposed by Lewis Chen[/i] | 10004 |
48,334 | Let
\[f(x)=\cos(x^3-4x^2+5x-2).\]
If we let $f^{(k)}$ denote the $k$th derivative of $f$, compute $f^{(10)}(1)$. For the sake of this problem, note that $10!=3628800$. | 907200 |
48,340 | Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$, where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$, of the function. | (2, 1 + \sqrt{2}] |
48,373 | Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that
$$ m^{2}+f(n) \mid m f(m)+n $$
for all positive integers $m$ and $n$. (Malaysia) Answer. $f(n)=n$. | f(n)=n |
48,374 | Given positive real numbers $a, b, c, d$ that satisfy equalities
$$
a^{2}+d^{2}-a d=b^{2}+c^{2}+b c \quad \text{and} \quad a^{2}+b^{2}=c^{2}+d^{2}
$$
find all possible values of the expression $\frac{a b+c d}{a d+b c}$.
Answer: $\frac{\sqrt{3}}{2}$. | \frac{\sqrt{3}}{2} |
48,386 | 13. Let \( f(x, y) = \frac{a x^{2} + x y + y^{2}}{x^{2} + y^{2}} \), satisfying
\[
\max _{x^{2}+y^{2}+0} f(x, y) - \min _{x^{2}+y^{2}+0} f(x, y) = 2 \text{. }
\]
Find \( a \). | a=1 \pm \sqrt{3} |
48,400 | 10. (20 points) Draw any line through the fixed point $M(m, 0)$, intersecting the parabola $y^{2}=16 x$ at points $P$ and $Q$. If $\frac{1}{|P M|^{2}}+\frac{1}{|Q M|^{2}}$ is a constant, find all possible values of the real number $m$.
| 8 |
48,408 | Integers $x_1,x_2,\cdots,x_{100}$ satisfy \[ \frac {1}{\sqrt{x_1}} + \frac {1}{\sqrt{x_2}} + \cdots + \frac {1}{\sqrt{x_{100}}} = 20. \]Find $ \displaystyle\prod_{i \ne j} \left( x_i - x_j \right) $. | 0 |
48,410 | 1. Let the left and right vertices of the hyperbola $x^{2}-y^{2}=6$ be $A_{1}$ and $A_{2}$, respectively, and let $P$ be a point on the right branch of the hyperbola such that $\angle P A_{2} x=3 \angle P A_{1} x+10^{\circ}$. Then the degree measure of $\angle P A_{1} x$ is $\qquad$. | 20^{\circ} |
48,421 | 4. Equation
$$
\begin{array}{l}
\frac{x_{1} x_{2} \cdots x_{2010}}{x_{2011}}+\frac{x_{1} x_{2} \cdots x_{2009} x_{2011}}{x_{2010}}+\cdots+ \\
\frac{x_{2} x_{3} \cdots x_{2011}}{x_{1}}=2011
\end{array}
$$
The number of different ordered integer solutions $\left(x_{1}, x_{2}, \cdots, x_{2011}\right)$ is | 2^{2010} |
48,423 | Fix integers $n\ge k\ge 2$. We call a collection of integral valued coins $n-diverse$ if no value occurs in it more than $n$ times. Given such a collection, a number $S$ is $n-reachable$ if that collection contains $n$ coins whose sum of values equals $S$. Find the least positive integer $D$ such that for any $n$-diverse collection of $D$ coins there are at least $k$ numbers that are $n$-reachable.
[I]Proposed by Alexandar Ivanov, Bulgaria.[/i] | n+k-1 |
48,436 | Example 8 Quadratic Function
$$
f(x)=a x^{2}+b x+c(a, b \in \mathbf{R} \text {, and } a \neq 0)
$$
satisfies the conditions:
(1) For $x \in \mathbf{R}$, $f(x-4)=f(2-x)$, and
$$
f(x) \geqslant x \text {; }
$$
(2) For $x \in(0,2)$, $f(x) \leqslant\left(\frac{x+1}{2}\right)^{2}$;
(3) The minimum value of $f(x)$ on $\mathbf{R}$ is 0.
Find the largest $m(m>1)$, such that there exists $t \in \mathbf{R}$, for any $x \in[1, m]$, we have $f(x+t) \leqslant x$.
(2002, National High School Mathematics Competition) | 9 |
48,459 | 5. (Dongtai City) In the tetrahedron $S-ABC$, $SA$ is perpendicular to the base $ABC$, and the lateral faces $SAB$ and $SBC$ form a right dihedral angle. If $\angle BSC = 45^{\circ}, SB = a$, find the volume of the circumscribed sphere of this tetrahedron. | \frac{\sqrt{2}}{3} \pi a^{3} |
48,464 | A gambler plays the following coin-tossing game. He can bet an arbitrary positive amount of money. Then a fair coin is tossed, and the gambler wins or loses the amount he bet depending on the outcome. Our gambler, who starts playing with $ x$ forints, where $ 0<x<2C$, uses the following strategy: if at a given time his capital is $ y<C$, he risks all of it; and if he has $ y>C$, he only bets $ 2C\minus{}y$. If he has exactly $ 2C$ forints, he stops playing. Let $ f(x)$ be the probability that he reaches $ 2C$ (before going bankrupt). Determine the value of $ f(x)$. | \frac{x}{2C} |
48,476 | Let $\{n,p\}\in\mathbb{N}\cup \{0\}$ such that $2p\le n$. Prove that $\frac{(n-p)!}{p!}\le \left(\frac{n+1}{2}\right)^{n-2p}$. Determine all conditions under which equality holds. | \frac{(n-p)!}{p!} \le \left( \frac{n+1}{2} \right)^{n-2p} |
48,494 | 4. A semicircle with diameter $A B=2$, a perpendicular line to the plane of the circle is drawn through point $A$, and a point $S$ is taken on this perpendicular line such that $A S=A B$. $C$ is a moving point on the semicircle, and $M, N$ are the projections of point $A$ on $S B, S C$ respectively. When the volume of the pyramid $S-A M N$ is maximized, the sine of the angle formed by $S C$ and the plane $A B C$ is $\qquad$ | \frac{\sqrt{3}}{2} |
48,535 | Let $S$ be the set of $2\times2$-matrices over $\mathbb{F}_{p}$ with trace $1$ and determinant $0$. Determine $|S|$. | p(p + 1) |
48,558 | A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of
\[
\sum_{i = 1}^{2012} | a_i - i |,
\]
then compute the sum of the prime factors of $S$.
[i]Proposed by Aaron Lin[/i] | 2083 |
48,576 | Let $\omega_1$ be a circle of radius $1$ that is internally tangent to a circle $\omega_2$ of radius $2$ at point $A$. Suppose $\overline{AB}$ is a chord of $\omega_2$ with length $2\sqrt3$ that intersects $\omega_1$ at point $C\ne A$. If the tangent line of $\omega_1$ at $C$ intersects $\omega_2$ at points $D$ and $E$, find $CD^4 + CE^4$. | 63 |
48,600 | Find all continuous functions $f : R \rightarrow R$ such, that $f(xy)= f\left(\frac{x^2+y^2}{2}\right)+(x-y)^2$ for any real numbers $x$ and $y$ | f(x) = c - 2x |
48,629 | Let $r_1$, $r_2$, $\ldots$, $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$. If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form $\tfrac mn$ where $m$ and $n$ are positive coprime integers, find $m+n$. | 240 |
48,630 | Four. (20 points) Let the line $y=\sqrt{3} x+b$ intersect the parabola $y^{2}=2 p x(p>0)$ at points $A$ and $B$. The circle passing through $A$ and $B$ intersects the parabola $y^{2}=2 p x(p>0)$ at two other distinct points $C$ and $D$. Find the size of the angle between the lines $AB$ and $CD$.
| 60^{\circ} |
48,635 | Find all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that
\[ 3f(f(f(n))) \plus{} 2f(f(n)) \plus{} f(n) \equal{} 6n, \quad \forall n\in \mathbb{N}.\] | f(n) = n |
48,657 | One of Euler's conjectures was disproved in then 1960s by three American mathematicians when they showed there was a positive integer $ n$ such that \[133^5 \plus{} 110^5 \plus{} 84^5 \plus{} 27^5 \equal{} n^5.\] Find the value of $ n$. | 144 |
48,659 | Example 5 Let $a, b, c$ be the lengths of the three sides of a right-angled triangle, and $a \leqslant b \leqslant c$. Find the maximum constant $k$, such that
$$
\begin{array}{l}
a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b) \\
\geqslant \text { kabc }
\end{array}
$$
holds for all right-angled triangles, and determine when equality occurs. | 2+3\sqrt{2} |
48,688 | Let $n$ be positive integer. Define a sequence $\{a_k\}$ by
\[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\]
(1) Find $a_2$ and $a_3$.
(2) Find the general term $a_k$.
(3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$.
50 points | \ln 2 |
48,715 | Let $ f:[0,1]\longrightarrow [0,1] $ be a nondecreasing function. Prove that the sequence
$$ \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} $$
is convergent and calculate its limit. | 1 |
48,717 | Point $D$ lies on side $\overline{BC}$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC.$ The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F,$ respectively. Given that $AB=4,BC=5,$ and $CA=6,$ the area of $\triangle AEF$ can be written as $\tfrac{m\sqrt{n}}p,$ where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p.$ | 36 |
48,728 | 4 Let $X=\{00,01, \cdots, 98,99\}$ be the set of 100 two-digit numbers, and $A$ be a subset of $X$ such that: in any infinite sequence of digits from 0 to 9, there are two adjacent digits that form a two-digit number in $A$. Find the minimum value of $|A|$. (52nd Moscow Mathematical Olympiad) | 55 |
48,741 | Find the number of integer solutions of the equation
$x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$ | 0 |
48,755 | Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\in\mathbb N$ such that $n$ and $n+1$ are both in $S$. | 111 |
48,771 | Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$. She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$. Mary wants to prevent Pat from winning the game.
Determine, with proof, the smallest initial integer Mary could choose in order to achieve this. | 2022 |
48,799 | Given is a $n \times n$ chessboard. With the same probability, we put six pawns on its six cells. Let $p_n$ denotes the probability that there exists a row or a column containing at least two pawns. Find $\lim_{n \to \infty} np_n$. | 30 |
48,801 | Four. (20 points) Given points $A$ and $B$ are the upper and lower vertices of the ellipse $\frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1(a>b>0)$, and $P$ is a point on the hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$ in the first quadrant. The lines $PA$ and $PB$ intersect the ellipse at points $C$ and $D$, respectively, and $D$ is exactly the midpoint of $PB$.
(1) Find the slope of the line $CD$;
(2) If $CD$ passes through the upper focus of the ellipse, find the eccentricity of the hyperbola. | \frac{\sqrt{7}}{2} |
48,834 | 29. In the right-angled $\triangle A B C$, find the largest positive real number $k$ such that the inequality $a^{3}+b^{3}+c^{3} \geqslant k(a+$ $b+c)^{3}$ holds. (2006 Iran Mathematical Olympiad) | \frac{1}{\sqrt{2}(1+\sqrt{2})^{2}} |
48,840 | A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane, such that
(i) no three points in $P$ lie on a line and
(ii) no two points in $P$ lie on a line through the origin.
A triangle with vertices in $P$ is fat, if $O$ is strictly inside the triangle. Find the maximum number of fat triangles.
(Austria, Veronika Schreitter)
Answer: $2021 \cdot 505 \cdot 337$ | 2021 \cdot 505 \cdot 337 |
48,864 |
1. Let $A B C$ be an acute angled triangle. The circle $\Gamma$ with $B C$ as diameter intersects $A B$ and $A C$ again at $P$ and $Q$, respectively. Determine $\angle B A C$ given that the orthocenter of triangle $A P Q$ lies on $\Gamma$.
| 45 |
48,871 | Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$. | \frac{a^2 + b^2 + c^2}{3} |
48,946 | For $p \in \mathbb{R}$, let $(a_n)_{n \ge 1}$ be the sequence defined by
\[ a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx. \]
Determine all possible values of $p$ for which the series $\sum_{n=1}^\infty a_n$ converges.
| p > \frac{3}{2} |
48,951 | Find the maximum value of
$$\int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx$$over all continuously differentiable functions $f:[0,1]\to\mathbb R$ with $f(0)=0$ and
$$\int^1_0|f'(x)|^2dx\le1.$$ | \frac{2}{3} |
48,964 | 10. For the sequence $\left\{a_{n}\right\}$, if there exists a sequence $\left\{b_{n}\right\}$, such that for any $n \in \mathbf{Z}_{+}$, we have $a_{n} \geqslant b_{n}$, then $\left\{b_{n}\right\}$ is called a "weak sequence" of $\left\{a_{n}\right\}$. Given
$$
\begin{array}{l}
a_{n}=n^{3}-n^{2}-2 t n+t^{2}\left(n \in \mathbf{Z}_{+}\right), \\
b_{n}=n^{3}-2 n^{2}-n+\frac{5}{4}\left(n \in \mathbf{Z}_{+}\right),
\end{array}
$$
and $\left\{b_{n}\right\}$ is a weak sequence of $\left\{a_{n}\right\}$. Then the range of the real number $t$ is
$\qquad$ | \left(-\infty, \frac{1}{2}\right] \cup\left[\frac{3}{2},+\infty\right) |
48,998 | 1. Given that the circumradius of $\triangle A B C$ is $R$, and
$$
2 R\left(\sin ^{2} A-\sin ^{2} C\right)=(\sqrt{2} a-b) \sin B,
$$
where $a$ and $b$ are the sides opposite to $\angle A$ and $\angle B$ respectively. Then the size of $\angle C$ is | 45^{\circ} |
49,037 | Let $E(n)$ denote the largest integer $k$ such that $5^k$ divides $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$ Calculate
$$\lim_{n\to \infty} \frac{E(n)}{n^2 }.$$ | \frac{1}{8} |
49,040 | Let $n$ be a fixed integer, $n \geqslant 2$.
a) Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$
holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n} \geqslant 0$;
b) For this constant $c$, determine the necessary and sufficient conditions for equality to hold.
This article provides a simple solution.
The notation $\sum_{1 \leq i<j \leq \leqslant} f\left(x_{i}, x_{j}\right)$ represents the sum of all terms $f\left(x_{i}, x_{j}\right)$ for which the indices satisfy $1 \leqslant i<j \leqslant n$, and in the following text, it is simply denoted as $\sum f\left(x_{i}, x_{j}\right)$. | \frac{1}{8} |
49,051 | 3. Given $a, b, c \in \mathbf{R}$, and
$$
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c} \text {, }
$$
then there exists an integer $k$, such that the following equations hold for
$\qquad$ number of them.
(1) $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^{2 k+1}=\frac{1}{a^{2 k+1}}+\frac{1}{b^{2 k+1}}+\frac{1}{c^{2 k+1}}$;
(2) $\frac{1}{a^{2 k+1}}+\frac{1}{b^{2 k+1}}+\frac{1}{c^{2 k+1}}=\frac{1}{a^{2 k+1}+b^{2 k+1}+c^{2 k+1}}$;
(3) $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^{2 k}=\frac{1}{a^{2 k}}+\frac{1}{b^{2 k}}+\frac{1}{c^{2 k}}$;
(4) $\frac{1}{a^{2 k}}+\frac{1}{b^{2 k}}+\frac{1}{c^{2 k}}=\frac{1}{a^{2 k}+b^{2 k}+c^{2 k}}$. | 2 |
49,053 | Let $f_{0}(x)=x$, and for each $n\geq 0$, let $f_{n+1}(x)=f_{n}(x^{2}(3-2x))$. Find the smallest real number that is at least as large as
\[ \sum_{n=0}^{2017} f_{n}(a) + \sum_{n=0}^{2017} f_{n}(1-a)\]
for all $a \in [0,1]$. | 2018 |
49,059 | Let $P(n)$ be the number of permutations $\left(a_{1}, \ldots, a_{n}\right)$ of the numbers $(1,2, \ldots, n)$ for which $k a_{k}$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P(n)$ is a multiple of 2010. | 4489 |
49,060 | Let \(Q\) be a set of permutations of \(1,2,...,100\) such that for all \(1\leq a,b \leq 100\), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Find the largest possible number of elements in \(Q\). | 100 |
49,062 | 14. (16 points) As shown in Figure 4, $A$ and $B$ are the common vertices of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $(a>b>0)$ and the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. $P$ and $Q$ are moving points on the hyperbola and the ellipse, respectively, different from $A$ and $B$, and satisfy
$$
\overrightarrow{A P}+\overrightarrow{B P}=\lambda(\overrightarrow{A Q}+\overrightarrow{B Q})(\lambda \in \mathbf{R},|\lambda|>1) \text {. }
$$
Let $k_{1} 、 k_{2} 、 k_{3} 、 k_{4}$, then $k_{1}+k_{2}+k_{3}+k_{4}$ is a constant. | 0 |
49,080 | Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$. | \frac{1}{4} |
49,124 | For each polynomial $P(x)$, define $$P_1(x)=P(x), \forall x \in \mathbb{R},$$ $$P_2(x)=P(P_1(x)), \forall x \in \mathbb{R},$$ $$...$$ $$P_{2024}(x)=P(P_{2023}(x)), \forall x \in \mathbb{R}.$$ Let $a>2$ be a real number. Is there a polynomial $P$ with real coefficients such that for all $t \in (-a, a)$, the equation $P_{2024}(x)=t$ has $2^{2024}$ distinct real roots? | \text{Yes} |
49,169 | 12. Let the function $f(x)$ be a differentiable function defined on the interval $(-\infty, 0)$, with its derivative being $f^{\prime}(x)$, and $2 f(x) + x f^{\prime}(x) > x^{2}$. Then
$$
(x+2017)^{2} f(x+2017)-f(-1)>0
$$
The solution set is $\qquad$ | (-\infty,-2018) |
49,203 | Let $ p > 2$ be a prime number. Find the least positive number $ a$ which can be represented as
\[ a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X),
\]
where $ f(X)$ and $ g(X)$ are integer polynomials.
[i]Mircea Becheanu[/i]. | p |
49,215 | Let the matrices of order 2 with the real elements $A$ and $B$ so that $AB={{A}^{2}}{{B}^{2}}-{{\left( AB \right)}^{2}}$ and $\det \left( B \right)=2$.
a) Prove that the matrix $A$ is not invertible.
b) Calculate $\det \left( A+2B \right)-\det \left( B+2A \right)$. | 6 |
49,217 | Let $ n$ be a positive integer. Consider
\[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \}
\]
as a set of $ (n \plus{} 1)^{3} \minus{} 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $ S$ but does not include $ (0,0,0)$.
[i]Author: Gerhard Wöginger, Netherlands [/i] | 3n |
49,251 | $\square$ Example 11 Positive integers $n \geqslant 3, x_{1}, x_{2}, \cdots, x_{n}$ are positive real numbers, $x_{n+j}=x_{j}(1 \leqslant$ $j \leqslant n-1)$, find the minimum value of $\sum_{i=1}^{n} \frac{x_{j}}{x_{j+1}+2 x_{j+2}+\cdots+(n-1) x_{j+n-1}}$. (1995 Chinese National Mathematical Olympiad Training Team Question) | \frac{2}{n-1} |
49,254 | Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \][i]Proposed by Michael Kural[/i] | 164 |
49,293 | Determine all functions $ f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy:
$ f(x\plus{}f(y))\equal{}f(x)\plus{}y$ for all $ x,y \in \mathbb{N}$. | f(x) = x |
49,300 | 10. (20 points) In the sequence $\left\{a_{n}\right\}$, let $S_{n}=\sum_{i=1}^{n} a_{i}$ $\left(n \in \mathbf{Z}_{+}\right)$, with the convention: $S_{0}=0$. It is known that
$$
a_{k}=\left\{\begin{array}{ll}
k, & S_{k-1}<k ; \\
-k, & S_{k-1} \geqslant k
\end{array}\left(1 \leqslant k \leqslant n, k 、 n \in \mathbf{Z}_{+}\right)\right. \text {. }
$$
Find the largest positive integer $n$ not exceeding 2019 such that
$$
S_{n}=0 .
$$ | 1092 |
49,302 | 18. C6 (FRA 2) Let \( O \) be a point of three-dimensional space and let \( l_{1}, l_{2}, l_{3} \) be mutually perpendicular straight lines passing through \( O \). Let \( S \) denote the sphere with center \( O \) and radius \( R \), and for every point \( M \) of \( S \), let \( S_{M} \) denote the sphere with center \( M \) and radius \( R \). We denote by \( P_{1}, P_{2}, P_{3} \) the intersection of \( S_{M} \) with the straight lines \( l_{1}, l_{2}, l_{3} \), respectively, where we put \( P_{i} \neq O \) if \( l_{i} \) meets \( S_{M} \) at two distinct points and \( P_{i}=O \) otherwise ( \( i=1,2,3 \) ). What is the set of centers of gravity of the (possibly degenerate) triangles \( P_{1} P_{2} P_{3} \) as \( M \) runs through the points of \( S \) ? | \frac{2R}{3} |
49,316 | Find the greatest positive real number $M$ such that for all positive real sequence $(a_n)$ and for all real number $m < M$, it is possible to find some index $n \ge 1$ that satisfies the inequality $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$. | M = 4 |
49,320 | 3. Let $A, B, C$ be the three interior angles of $\triangle ABC$, then the imaginary part of the complex number
$$
\frac{(1+\cos 2B+i \sin 2 B)(1+\cos 2 C+i \sin 2 C)}{1+\cos 2 A-i \sin 2 A}
$$
is . $\qquad$ | 0 |
49,326 | Given a set of points in space, a [i]jump[/i] consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in $10$-dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
[i]Author: Anderson Wang[/i] | 1025 |
49,339 | Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of
\[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\]
Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$ | n |
49,342 | Five. (15 points) Given a function $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any real numbers $x, y, z$ we have
$$
\begin{array}{l}
\frac{1}{2} f(x y)+\frac{1}{2} f(x z)-f(x) f(y z) \geqslant \frac{1}{4} . \\
\text { Find }[1 \times f(1)]+[2 f(2)]+\cdots+[2011 f(2011)]
\end{array}
$$
where $[a]$ denotes the greatest integer not exceeding the real number $a$. | 1011030 |
49,356 |
A1. Determine all functions $f$ from the set of non-negative integers to itself such that
$$
f(a+b)=f(a)+f(b)+f(c)+f(d)
$$
whenever $a, b, c, d$, are non-negative integers satisfying $2 a b=c^{2}+d^{2}$.
| f(n)=kn^{2} |
49,367 | Given triangle $ ABC$ with sidelengths $ a,b,c$. Tangents to incircle of $ ABC$ that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of $ ABC$). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle $ ABC$ is equal to
$ \frac{\pi (a^{2}\plus{}b^{2}\plus{}c^{2})(b\plus{}c\minus{}a)(c\plus{}a\minus{}b)(a\plus{}b\minus{}c)}{(a\plus{}b\plus{}c)^{3}}$
(hmm,, looks familiar, isn't it? :wink: ) | \frac{\pi (a^{2} + b^{2} + c^{2})(b + c - a)(c + a - b)(a + b - c)}{(a + b + c)^{3}} |
49,386 | For all positive integers $n$, let
\[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute $f(2019) - f(2018)$. Here $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$. | 11431 |
49,395 | We denote by $\mathbb{N}^{*}$ the set of strictly positive integers. Find all $^{2}$ functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that
$$
m^{2}+f(n) \mid m f(m)+n
$$
for all strictly positive integers $m$ and $n$. | f(n)=n |
49,404 | 9. The number of prime pairs $(a, b)$ that satisfy the equation
$$
a^{b} b^{a}=(2 a+b+1)(2 b+a+1)
$$
is $\qquad$. | 2 |
49,424 | Let all possible $2023$-degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$,
where $P(0)+P(1)=0$, and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$. What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$ | 2^{-2023} |
49,431 | Let $\{\omega_1,\omega_2,\cdots,\omega_{100}\}$ be the roots of $\frac{x^{101}-1}{x-1}$ (in some order). Consider the set $$S=\{\omega_1^1,\omega_2^2,\omega_3^3,\cdots,\omega_{100}^{100}\}.$$ Let $M$ be the maximum possible number of unique values in $S,$ and let $N$ be the minimum possible number of unique values in $S.$ Find $M-N.$ | 99 |
49,438 | On the table there are $k \ge 3$ heaps of $1, 2, \dots , k$ stones. In the first step, we choose any three of the heaps, merge them into a single new heap, and remove $1$ stone from this new heap. Thereafter, in the $i$-th step ($i \ge 2$) we merge some three heaps containing more than $i$ stones in total and remove $i$ stones from the new heap. Assume that after a number of steps a single heap of $p$ stones remains on the table. Show that the number $p$ is a perfect square if and only if so are both $2k + 2$ and $3k + 1$. Find the least $k$ with this property. | 161 |
49,454 | The positive integers are colored with black and white such that:
- There exists a bijection from the black numbers to the white numbers,
- The sum of three black numbers is a black number, and
- The sum of three white numbers is a white number.
Find the number of possible colorings that satisfies the above conditions. | 2 |
49,551 | Example 10 Let $a, b, c$ be positive real numbers, satisfying
$$a+b+c+3 \sqrt[3]{a b c} \geqslant k(\sqrt{a b}+\sqrt{b c}+\sqrt{c a}),$$
Find the maximum value of $k$. | 2 |
49,597 | Example 4 Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for all real numbers $x_{1}, x_{2}, \cdots, x_{2000}$, we have
$$
\begin{array}{l}
\sum_{i=1}^{2005} f\left(x_{i}+x_{i+1}\right)+f\left(\sum_{i=1}^{2006} x_{i}\right) \\
\leqslant \sum_{i=1}^{2006} f\left(2 x_{i}\right) .
\end{array}
$$ | f(x)=c |
49,609 | Suppose $z_1, z_2 , \cdots z_n$ are $n$ complex numbers such that $min_{j \not= k} | z_{j} - z_{k} | \geq max_{1 \leq j \leq n} |z_j|$. Find the maximum possible value of $n$. Further characterise all such maximal configurations. | n = 7 |
49,681 | If $f(n)$ denotes the number of divisors of $2024^{2024}$ that are either less than $n$ or share at least one prime factor with $n$, find the remainder when $$\sum^{2024^{2024}}_{n=1} f(n)$$ is divided by $1000$. | 224 |
49,685 | Let $n$ be a positive integer and let $P$ be the set of monic polynomials of degree $n$ with complex coefficients. Find the value of
\[ \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \} \] | 1 |
49,692 | 7. In the tetrahedron $P-ABC$,
$$
AB=\sqrt{3}, AC=1, PB=PC=\sqrt{2} \text{.}
$$
Then the range of $S_{\triangle ABC}^{2}+S_{\triangle PBC}^{2}$ is | \left(\frac{1}{4}, \frac{7}{4}\right) |
49,694 | Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property:
$$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$
[b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal.
[b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $ | \ln 2 |
49,699 | The polynomial of seven variables
$$
Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2)
$$
is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients:
$$
Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2.
$$
Find all possible values of $P_1(1,1,\ldots,1)$.
[i](A. Yuran)[/i] | 3 |
49,710 | $3 \cdot 28$ Try to express $\sum_{k=0}^{n} \frac{(-1)^{k} C_{n}^{k}}{k^{3}+9 k^{2}+26 k+24}$ in the form $\frac{p(n)}{q(n)}$, where $p(n)$ and $q(n)$ are two polynomials with integer coefficients. | \frac{1}{2(n+3)(n+4)} |
49,738 | Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation
\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\] | f(x) = C |
49,740 | 15. A1 (POL) ${ }^{\mathrm{IMO} 2}$ Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such that the inequality
$$ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} $$
holds for every $x_{1}, \ldots, x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality. | \frac{1}{8} |
49,805 | Let $\mathbb{R}$ be the set of [real numbers](https://artofproblemsolving.com/wiki/index.php/Real_number). Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
$f(x^2 - y^2) = xf(x) - yf(y)$
for all pairs of real numbers $x$ and $y$. | f(x)=kx |
49,825 | 9. As shown in Figure 1, in $\triangle A B C$,
$$
\begin{array}{l}
\cos \frac{C}{2}=\frac{2 \sqrt{5}}{5}, \\
\overrightarrow{A H} \cdot \overrightarrow{B C}=0, \\
\overrightarrow{A B} \cdot(\overrightarrow{C A}+\overrightarrow{C B})=0 .
\end{array}
$$
Then the eccentricity of the hyperbola passing through point $C$ and having $A$ and $H$ as its foci is . $\qquad$ | 2 |
49,864 | Determine the smallest positive real number $r$ such that there exist differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ satisfying
(a) $f(0)>0$,
(b) $g(0)=0$,
(c) $\left|f^{\prime}(x)\right| \leq|g(x)|$ for all $x$,
(d) $\left|g^{\prime}(x)\right| \leq|f(x)|$ for all $x$, and
(e) $f(r)=0$. | \frac{\pi}{2} |
49,881 | Example 9 Let $n=1990$. Then
$$
\frac{1}{2^{n}}\left(1-3 \mathrm{C}_{n}^{2}+3^{2} \mathrm{C}_{n}^{4}-\cdots+3^{99} \mathrm{C}_{n}^{108}-3^{90} \mathrm{C}_{n}^{900}\right) \text {. }
$$
(1990, National High School Mathematics Competition) | -\frac{1}{2} |
49,885 | Determine continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that $\left( {{a}^{2}}+ab+{{b}^{2}} \right)\int\limits_{a}^{b}{f\left( x \right)dx=3\int\limits_{a}^{b}{{{x}^{2}}f\left( x \right)dx,}}$ for every $a,b\in \mathbb{R}$ . | f(x) = C |
49,928 | Example 12 (1995 National High School League Question) Given the family of curves $2(2 \sin \theta-\cos \theta+3) x^{2}-(8 \sin \theta+$ $\cos \theta+1) y=0$ ( $\theta$ is a parameter). Find the maximum value of the length of the chord intercepted by the family of curves on the line $y=2 x$.
| 8\sqrt{5} |
49,946 | Find all prime numbers of the form $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$, where $n$ is a natural number. | 101 |
49,986 | For positive integers $n$, let $M_n$ be the $2n+1$ by $2n+1$ skew-symmetric matrix for which each entry in the first $n$ subdiagonals below the main diagonal is 1 and each of the remaining entries below the main diagonal is -1. Find, with proof, the rank of $M_n$. (According to one definition, the rank of a matrix is the largest $k$ such that there is a $k \times k$ submatrix with nonzero determinant.)
One may note that
\begin{align*}
M_1 &= \left( \begin{array}{ccc} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0
\end{array}\right) \\
M_2 &= \left( \begin{array}{ccccc} 0 & -1 & -1 & 1
& 1 \\ 1 & 0 & -1 & -1 & 1 \\ 1 & 1 & 0 & -1 & -1 \\ -1 & 1 & 1 & 0 &
-1 \\ -1 & -1 & 1 & 1 & 0 \end{array} \right).
\end{align*} | 2n |
49,999 | 8. Let $p$ be a prime number, and $A$ be a set of positive integers, satisfying the following conditions:
(1) The set of prime factors of the elements in $A$ contains $p-1$ elements;
(2) For any non-empty subset of $A$, the product of its elements is not a $p$-th power of an integer.
Find the maximum number of elements in $A$.
(Iran provided) | (p-1)^2 |
50,017 | Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions :
i) $a_0=1$, $a_1=3$
ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$
to be true that
$$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$. | M = 2 |
50,131 | Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \]
Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $. | \frac{49}{3} |
50,175 | For any integer $n\ge 2$ and two $n\times n$ matrices with real
entries $A,\; B$ that satisfy the equation
$$A^{-1}+B^{-1}=(A+B)^{-1}\;$$
prove that $\det (A)=\det(B)$.
Does the same conclusion follow for matrices with complex entries?
(Proposed by Zbigniew Skoczylas, Wroclaw University of Technology)
| \det(A) = \det(B) |
50,182 | Compute the number of ordered quadruples of complex numbers $(a,b,c,d)$ such that
\[ (ax+by)^3 + (cx+dy)^3 = x^3 + y^3 \]
holds for all complex numbers $x, y$.
[i]Proposed by Evan Chen[/i] | 18 |
50,199 | Suppose that $x$, $y$, and $z$ are complex numbers of equal magnitude that satisfy
\[x+y+z = -\frac{\sqrt{3}}{2}-i\sqrt{5}\]
and
\[xyz=\sqrt{3} + i\sqrt{5}.\]
If $x=x_1+ix_2, y=y_1+iy_2,$ and $z=z_1+iz_2$ for real $x_1,x_2,y_1,y_2,z_1$ and $z_2$ then
\[(x_1x_2+y_1y_2+z_1z_2)^2\]
can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b.$ | 1516 |
50,200 | Example 1. Let $n$ be an integer, calculate the following expression:
$$
\begin{array}{l}
{\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{2^{2}}\right]+\left[\frac{n+2^{2}}{2^{3}}\right]} \\
+\cdots .
\end{array}
$$
where the symbol $[x]$ denotes the greatest integer not exceeding $x$.
(IMO-10)
Analysis: In the expression of the problem, the denominators are powers of 2. Since $2^{\mathrm{k}}=(100 \cdots 0)_{2}$, in the binary number system, dividing by 2 is simply shifting the decimal point to the left by $k$ positions. Also, because the binary representation of a decimal integer is necessarily a binary integer, and the binary representation of a decimal pure fraction is necessarily a binary pure fraction, and vice versa, the operation of "taking the integer part" of a positive number in binary is the same as in decimal, which is to simply remove the fractional part. Thus, using binary number operations to solve this problem is very concise. | n |
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