Find the number of $ x $ which follows the following : $ x-\frac{1}{x}=[x]-[\frac{1}{x}] $ $ ( \frac{1}{100} \le x \le {100} ) $

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$. [i]Proposed by Aaron Lin[/i]

Does there exist a positive integer $n$ and a) complex numbers $a_0, a_1, \ldots, a_n;$ b) reals $a_0, a_1, \ldots, a_n, $ such that $P(x) Q(x)=x^{2024}+1$ where $P(x)=a_nx^n+\ldots +a_1x+a_0$ and $Q(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n?$

Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer

Calculate the following indefinite integrals. [1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$ [2] $\int x\log_{10} x dx$ [3] $\int \frac{x}{\sqrt{2x-1}}dx$ [4] $\int (x^2+1)\ln x dx$ [5] $\int e^x\cos x dx$

Consider solutions to the equation \[x^2-cx+1 = \dfrac{f(x)}{g(x)},\] where $f$ and $g$ are polynomials with nonnegative real coefficients. For each $c>0$, determine the minimum possible degree of $f$, or show that no such $f,g$ exist. [i]Proposed by Linus Hamilton and Calvin Deng[/i]

We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof. Prove that \[\sum_{k=1}^{n}[\alpha^{k}F_{k}+\frac{1}{2}]=F_{2n+1}\; \forall n>1.\]

The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to: ${{ \textbf{(A)}\ t^3 \qquad\textbf{(B)}\ t^2+t \qquad\textbf{(C)}\ |t^2+t| \qquad\textbf{(D)}\ t\sqrt{t^2+1} }\qquad\textbf{(E)}\ |t|\sqrt{1+t^2} } $

Given $n$ positive real numbers $x_1,x_2,x_3,...,x_n$ such that $$\left (1+\frac{1}{x_1}\right )\left(1+\frac{1}{x_2}\right)...\left(1+\frac{1}{x_n}\right)=(n+1)^n$$ Determine the minimum value of $x_1+x_2+x_3+...+x_n$. [i]Proposed by Loh Kwong Weng[/i]

Given the expression \[P_n(x) =\frac{1}{2^n}\left[(x +\sqrt{x^2 - 1})^n+(x-\sqrt{x^2 - 1})^n\right],\] prove: $(a) P_n(x)$ satisfies the identity \[P_n(x) - xP_{n-1}(x) + \frac{1}{4}P_{n-2}(x) \equiv 0.\] $(b) P_n(x)$ is a polynomial in $x$ of degree $n.$

Let $a, b, c$ and $d$ be real numbers with $a < b < c < d$. Sort the numbers $x = a \cdot b + c \cdot d, y = b \cdot c + a \cdot d$ and $z = c \cdot a + b \cdot d$ in ascending\order and prove the correctness of your result. (R. Henner, Vienna)

Solve in the real numbers the equation $ \arctan\sqrt{3^{1-2x}} +\arctan {3^x} =\frac{7\pi }{12} . $ [i]Ovidiu Țâțan[/i]

Let $ p$ be a prime number. Let $ h(x)$ be a polynomial with integer coefficients such that $ h(0),h(1),\dots, h(p^2\minus{}1)$ are distinct modulo $ p^2.$ Show that $ h(0),h(1),\dots, h(p^3\minus{}1)$ are distinct modulo $ p^3.$

Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation \[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\] holds.

Let $a,b,c,d$ be integers such that the number $a-b+c-d$ is odd and it divides the number $a^2-b^2+c^2-d^2$. Show that, for every positive integer $n$, $a-b+c-d$ divides $a^n-b^n+c^n-d^n$.

Together, the two positive integers $a$ and $b$ have $9$ digits and contain each of the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once. For which possible values of $a$ and $b$ is the fraction $a/b$ closest to $1$?

[b]6.1[/b] Which number is greater $$\underbrace{1000. . . 001}_{1965\, zeroes} / \underbrace{1000 . . . 001}_{1966\, zeroes} \,\,\, or \,\,\, \underbrace{1000. . . 001}_{1966\, zeroes} / \underbrace{1000 . . . 001}_{1967\, zeroes} \,\,?$$ [b]6.2[/b] $30$ teams participate in the football championship. Prove that at any moment there will be two teams that have played at this point the same number of matches. [b]6.3./ 7.1 [/b] All integers from $1$ to $1966$ are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board. [b]6.4 / 7.5[/b] Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two. [b]6.5[/b] In a chess tournament, there are more than three chess players, and each player plays each other the same number of times. There were $26$ rounds in the tournament. After the $13$th round, one of the participants discovered that he had an odd number points, and each of the other participants has an even number of points. How many chess players participated in the tournament? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].

Ana, Bruno and Carolina played table tennis with each other. In each game, only two of the friends played, with the third one resting. Every time one of the friends won a game, they rested during the next game. Ana played $12$ games, Bruno played $21$ games and Carolina rested for$ 8$ games. Who rested in the last game?

Let $a, b, c \in [-1, 1] $ such that $1 + 2abc \ge a^2 + b^2 + c^2$. Prove that $1 + 2a^2b^2c^2 \ge a^4 + b^4 + c^4$.

Prove that the ratio $$\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}}$$ is an odd integer.

Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $\frac{ab}{a^5+b^5+ab} +\frac{bc}{b^5+c^5+bc}+\frac{ac}{c^5+a^5+ac}\leq 1$

From point $A$ to point $B$, the car drove at a speed of $50$ km / h, and from $B$ to $A$ , at a speed of $30$ km / h. What was the average vehicle speed?

Let $\Gamma$ a fixed circunference. Find all finite sets $S$ of points in $\Gamma$ such that: For each point $P\in \Gamma$, there exists a partition of $S$ in sets $A$ and $B$ ($A\cup B=S$, $A\cap B=\phi$) such that $\sum_{X\in A}PX = \sum_{Y\in B}PY$.