Found problems: 85335
Let $\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f \colon \mathbb{R}^+ \to \mathbb{R}^+$ such that, for all $x,y \in \mathbb{R}^+$,
$$f(xy+f(x))=xf(y)+2.$$
Calculate $ \lim_{n\to\infty }\left(\lim_{x\to 0} \left( -\frac{n}{x}+1+\frac{1}{x}\sum_{r=2}^{n+1}\sqrt[r!]{1+\sin rx}\right)\right) . $
[i]Constantin Rusu[/i]
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Let $A$ be a subset of the edges of an $n\times n $ table. Let $V(A)$ be the set of vertices from the table which are connected to at least on edge from $A$ and $j(A)$ be the number of the connected components of graph $G$ which it's vertices are the set $V(A)$ and it's edges are the set $A$. Prove that for every natural number $l$:
$$\frac{l}{2}\leq min_{|A|\geq l}(|V(A)|-j(A)) \leq \frac{l}{2}+\sqrt{\frac{l}{2}}+1$$
The closed line segment $I$ is covered by finitely many closed line segments.
Show that one can choose a subfamily $S$ of the family of line segments having the properties:
(1) the chosen line segments are disjoint,
(2) the sum of the lengths of the line segments of S is more than half of the length of $I.$
Show that the claim does not hold any more if the line segment $I$ is replaced by a circle and other occurences of the compound word ''line segment" by the word ''circular arc".
Let $x$ and $y$ be complex numbers such that $x+y=\sqrt{20}$ and $x^2+y^2=15$. Compute $|x-y|$.
In triangle $ABC$, point $I$ is the center of the inscribed circle points, points $I_A$ and $I_C$ are the centers of the excircles, tangent to sides $BC$ and $AB$, respectively. Point $O$ is the center of the circumscribed circle of triangle $II_AI_C$. Prove that $OI \perp AC$
Determine all the functions $f : R \to R$ such that: $$x + f(xf(y)) = f(y) + yf(x)$$ for all $x, y \in R$.
Let $a = (\sqrt2 +\sqrt3 +\sqrt6)(\sqrt2 +\sqrt3 -\sqrt6)(\sqrt3 +\sqrt6 -\sqrt2)(\sqrt6 +\sqrt2 -\sqrt3)$
$b = (\sqrt2 +\sqrt3 +\sqrt5)(\sqrt2 +\sqrt3 -\sqrt5)(\sqrt3 +\sqrt5 -\sqrt2)(\sqrt5 +\sqrt2 -\sqrt3)$
The difference $a - b$ belongs to the set:
A. $(-\infty,-4)$ B. $[-4,0)$ C.$\{0\}$ D. $(0,4]$ E. $(4,\infty)$
Find all positive integers $n$ such that there exist non-constant polynomials with integer coefficients $f_1(x),...,f_n(x)$ (not necessarily distinct) and $g(x)$ such that $$1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)$$
Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer
Compute $\sum_{k=0}^{2n} (-1)^k a_k^2$ where $a_k$ are the coefficients in the expansion
\[(1- \sqrt 2 x +x^2)^n =\sum_{k=0}^{2n} a_k x^k.\]
Let $a$ and $b$ be rational numbers such that $a+b=a^2+b^2$ . Suppose the common value $s=a+b=a^2+b^2$ is not an integer, and let's write it as an irreducible fraction: $s=\frac{m}{n}$. Let $p$ be the smallest prime divisor of $n$. Find the minimum value of $p$.
Consider the polynomials $P\left(x\right)=16x^4+40x^3+41x^2+20x+16$ and $Q\left(x\right)=4x^2+5x+2$. If $a$ is a real number, what is the smallest possible value of $\frac{P\left(a\right)}{Q\left(a\right)}$?
[i]2016 CCA Math Bonanza Team #6[/i]
Find all primes $q_1, q_2, q_3, q_4, q_5$ such that $q_1^4+q_2^4+q_3^4+q_4^4+q_5^4$ is the product of two consecutive even integers.
The incircle of a triangle $ABC$ touches the sides $BC$ and $AC$ at points $D$ and $E$, respectively. Suppose $P$ is the point on the shorter arc $DE$ of the incircle such that $\angle APE = \angle DPB$. The segments $AP$ and $BP$ meet the segment $DE$ at points $K$ and $L$, respectively. Prove that $2KL = DE$.
[i]Dušan Djukić[/i]
Let $n\in\mathbb{N}^*$. Prove that \[ \lim_{x\to 0}\frac{ \displaystyle (1+x^2)^{n+1}-\prod_{k=1}^n\cos kx}{ \displaystyle x\sum_{k=1}^n\sin kx}=\frac{2n^2+n+12}{6n}. \]
Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back?
$\textbf{(A)}\ \dfrac{1}{12} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{6} \qquad \textbf{(D)}\ \dfrac{1}{3} \qquad \textbf{(E)}\ \dfrac{2}{3}$
Let $ABCD$ be a parallelogram. The tangent to the circumcircle of triangle $BCD$ at $C$ intersects $AB$ at $P$ and intersects $AD$ at $Q$. The tangents to the circumcircle of triangle $APQ$ at $P$ and $Q$ meet at $R$. Show that points $A$, $C$, and $R$ are collinear.
Circumcenter and incentre of $\triangle ABC$ are $O,I$. $AD$ is the height on side $BC$. If $I$ is on line $OC$, prove that the radius of circumcircle and escribed circle (in \angle BAC) are equal.
Let $X$ be the set of all lattice points in the plane (points $(x, y)$ with $x, y \in \mathbb{Z}$). A path of length $n$ is a chain $(P_0, P_1, ... , P_n)$ of points in $X$ such that $P_{i-1}P_i = 1$ for $i = 1, ... , n$. Let $F(n)$ be the number of distinct paths beginning in $P_0=(0,0)$ and ending in any point $P_n$ on line $y = 0$. Prove that $F(n) = \binom{2n}{n}$
Let $I$ be the center of a circle inscribed in triangle $ABC$, in which $\angle BAC = 60 ^o$ and $AB \ne AC$. The points $D$ and $E$ were marked on the rays $BA$ and $CA$ so that $BD = CE = BC$. Prove that the line $DE$ passes through the point $I$.
The square was cut into acute -angled triangles. Prove that there are at least eight of them.
We say that a sequence $a_1,a_2,\cdots$ is [i]expansive[/i] if for all positive integers $j,\; i<j$ implies $|a_i-a_j|\ge \tfrac 1j$. Find all positive real numbers $C$ for which one can find an expansive sequence in the interval $[0,C]$.
Captain Jack and his pirate men plundered six chests of treasure $(A_1,A_2,A_3,A_4,A_5,A_6)$. Every chest $A_i$ contains $a_i$ coins of gold, and all $a_i$s are pairwise different $(i=1,2,\cdots ,6)$. They place all chests according to a layout (see the attachment) and start to alternately take out one chest a time between the captain and a pirate who serves as the delegate of the captain’s men. A rule must be complied with during the game: only those chests that are not adjacent to other two or more chests are allowed to be taken out. The captain will win the game if the coins of gold he obtains are not less than those of his men in the end. Let the captain be granted to take chest firstly, is there a certain strategy for him to secure his victory?