Found problems: 4776
Let $ E$ be the set of all triangles whose only points with integer coordinates (in the Cartesian coordinate system in space), in its interior or on its sides, are its three vertices, and let $ f$ be the function of area of a triangle. Determine the set of values $ f(E)$ of $ f.$
The expression $16^n+4^n+1$ is equiavalent to the expression $(2^{p(n)}-1)/(2^{q(n)}-1)$ for all positive integers $n>1$ where $p(n)$ and $q(n)$ are functions and $\tfrac{p(n)}{q(n)}$ is constant. Find $p(2006)-q(2006)$.
There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?
Find all continuous functions $f:[1,8]\to\mathbb R$, such that
$$\int^2_1f(t^3)^2dt+2\int^2_1f(t^3)dt=\frac23\int^8_1f(t)dt-\int^2_1(t^2-1)^2dt.$$
Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$.
Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$.
Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.
Let $u$ and $v$ be real numbers such that
\[ (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. \]
Determine, with proof, which of the two numbers, $u$ or $v$, is larger.
Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ with the property that
$$ f\left( p(x)\right) =p\left( f(x)\right) ,\quad\forall x\in\mathbb{Q} , $$
for all integer polynomials $ p. $
Let $x_1, x_2, \ldots, x_n$ be real positive numbers such that $x_1\cdot x_2\cdots x_n=1$. Prove the inequality
$\frac1{x_1(x_1+1)}+\frac1{x_2(x_2+1)}+\cdots+\frac1{x_n(x_n+1)}\geq\frac n2$
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$,
\[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).\]
Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that
a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$;
b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$.
Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if
\[\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).\]
[i]Evan O'Dorney.[/i]
Find all real numbers $c$ such that there exists a function $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ which satisfies the following.
For all nonnegative reals $x, y$, $f(x+y^2) \ge cf(x)+y$.
Here $\mathbb{R}_{\ge 0}$ is the set of all nonnegative reals.
Find all real numbers $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=88$. The notation $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.
Given real numbers $a,c,d$ show that there exists at most one function $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies:
\[f(ax+c)+d\le x\le f(x+d)+c\quad\text{for any}\ x\in\mathbb{R}\]
Let $D$ be the set of real numbers excluding $-1$. Find all functions $f: D \to D$ such that for all $x,y \in D$ satisfying $x \neq 0$ and $y \neq -x$, the equality $$(f(f(x))+y)f \left(\frac{y}{x} \right)+f(f(y))=x$$ holds.
Find all real numbers $\alpha$ such that both values $\cot(\alpha)$ and $\cot(2\alpha)$ are integers.
Distinct points $A, B , C, D, E$ are given in this order on a semicircle with radius $1$. Prove that
\[AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.\]
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy
\[f(f(x)+y)=2x+f(f(y)-x)\quad\text{for all real}\ x,y. \]
$ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$. Find the perimeter of $\Delta PBQ$.
Find the functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy $$(f(x)-y)f(x+f(y))=f(x^2)-yf(y),$$ for all real numbers $x$ and $y.$
Prove that there does not exist a function $f : \mathbb R^+\to\mathbb R^+$ such that \[f(f(x)+y)=f(x)+3x+yf(y)\] for all positive reals $x,y$.
Let $\mathbb{R}^{+}$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^{+} \to \mathbb{R}^{+}$ satisfying for all $x, y \in \mathbb{R}^{+}$ the equality
\[f(xf(y))=f(xy)+x\]
Find the minimal value of the function
\[\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}\]
Evaluate the following definite integral.
\[\int_{-1}^{1}\frac{e^{2x}+1-(x+1)(e^{x}+e^{-x})}{x(e^{x}-1)}dx\]
Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$.
[i]Laurentiu Panaitopol[/i]
Construct a continuous function $ f(x)$, periodic with period $ 2 \pi$, such that the Fourier series of $ f(x)$ is divergent at $ x\equal{}0$, but the Fourier series of $ f^2(x)$ is uniformly convergent on $ [0,2 \pi].$
[i]P. Turan[/i]