Found problems: 85335
In how many ways can $2^{n}$ be expressed as the sum of four squares of natural numbers?
A regular pentagon is inscribed in a circle of radius $r$. $P$ is any point inside the pentagon. Perpendiculars are dropped from $P$ to the sides, or the sides produced, of the pentagon.
a) Prove that the sum of the lengths of these perpendiculars is constant.
b) Express this constant in terms of the radius $r$.
Answer the following questions:
(1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$
(2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$
35 points
Two persons are playing the following game on a $n\times m$ table, with drawn lines:
Person $\#1$ starts the game. Each person in their move, folds the table on one of its lines. The one that could not fold the table on their turn loses the game.
Who has a winning strategy?
Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of 10 mph. If he completes the first three laps at a constant speed of only 9 mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal?
Find the volume of the uion $ A\cup B\cup C$ of the three subsets $ A,\ B,\ C$ in $ xyz$ space such that:
\[ A\equal{}\{(x,\ y,\ z)\ |\ |x|\leq 1,\ y^2\plus{}z^2\leq 1\}\]
\[ B\equal{}\{(x,\ y,\ z)\ |\ |y|\leq 1,\ z^2\plus{}x^2\leq 1\}\]
\[ C\equal{}\{(x,\ y,\ z)\ |\ |z|\leq 1,\ x^2\plus{}y^2\leq 1\}\]
For how many $n$ in $\{1, 2, 3, ..., 100 \}$ is the tens digit of $n^2$ odd?
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 50 $
An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face?
Let $f:(0,\infty) \rightarrow (0,\infty)$ be a function such that
\[
10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90
\]
for every $x,y \in (0,\infty)$. What is $f(\frac 1{11})$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 21
\qquad\textbf{(D)}\ 31
\qquad\textbf{(E)}\ \text{There is more than one solution}
$
In $\triangle ABC$, points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that each of the quadrilaterals $AFDE$, $BDEF$, and $CEFD$ has an incircle. Prove that the inradius of $\triangle ABC$ is twice the inradius of $\triangle DEF$.
Find all real solutions $x$ to the equation $\lfloor x^2 - 2x \rfloor + 2\lfloor x \rfloor = \lfloor x \rfloor^2$.
Given two finite collections of pairs of real numbers
It turned out that for any three pairs $(a_1, b_1)$, $(a_2, b_2)$ and $(a_3, b_3)$ from the first collection there is a pair $(c, d)$ from the second collection, such that the following three inequalities hold:
\[
a_1c + b_1d \geq 0,a_2c + b_2c \geq 0 \text{ and } a_3c + b_3d \geq 0
\]
Prove that there is a pair $(\gamma, \delta)$ in the second collection, such that for any pair $(\alpha, \beta)$ from the first collection inequality $\alpha \gamma + \beta \delta \geq 0$ holds.
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations.
[i]Proposed by Warut Suksompong, Thailand[/i]
A sequence $x_0, x_1, x_2, . . .$ is given by $x_0 = 8$ and $x_{n+1} =\frac{1 + x_n}{1- x_n}$ for $n = 0, 1, 2, . . . .$ Determine the number $x_{2013}$.
Find the non-constant function $f(x)$ such that $f(x)=x^2-\int_0^1 (f(t)+x)^2dt.$
A triangle $ABC$ has the orthocenter $H$ different from the vertexes and the circumcenter $O$. Let $M, N$ and $P$ be the circumcenters of triangles $HBC, HCA$ and $HAB$. Prove that the lines $AM, BN, CP$ and $OH$ are concurrent.
[b]Problem Section #4
b) Let $A$ be a unit square. What is the largest area of a triangle whose vertices lie on the perimeter of
$A$? Justify your answer.
Let $p = 2027$ be the smallest prime greater than $2018$, and let $P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2$. Let $\mathrm{GF}(p)$ be the integers modulo $p$, and let $\mathrm{GF}(p)(X)$ be the set of rational functions with coefficients in $\mathrm{GF}(p)$ (so that all coefficients are taken modulo $p$). That is, $\mathrm{GF}(p)(X)$ is the set of fractions $\frac{P(X)}{Q(X)}$ of polynomials with coefficients in $\mathrm{GF}(p)$, where $Q(X)$ is not the zero polynomial. Let $D\colon \mathrm{GF}(p)(X)\to \mathrm{GF}(p)(X)$ be a function satisfying \[
D\left(\frac fg\right) = \frac{D(f)\cdot g - f\cdot D(g)}{g^2}
\]for any $f,g\in \mathrm{GF}(p)(X)$ with $g\neq 0$, and such that for any nonconstant polynomial $f$, $D(f)$ is a polynomial with degree less than that of $f$. If the number of possible values of $D(P(X))$ can be written as $a^b$, where $a$, $b$ are positive integers with $a$ minimized, compute $ab$.
[i]Proposed by Brandon Wang[/i]
Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
\begin{align*}
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\end{align*}
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.
How many $6$-digit positive integers whose digits are different from $0$ are there such that each number generated by rearranging the digits of the original number is always divisible by $7$?
$
\textbf{(A)}\ 11
\qquad\textbf{(B)}\ 77
\qquad\textbf{(C)}\ 133
\qquad\textbf{(D)}\ 166
\qquad\textbf{(E)}\ 255
$
$\ddot{x}=3x-x^3-1$. In which of the potential wells is the period of oscillation greater (in the shallower or the deeper) with equal values of the total energy?
a) Prove that there doesn't exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: gcd(a_i+j,a_j+i)=1$
b) Let $p$ be an odd prime number. Prove that there exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: p \not | gcd(a_i+j,a_j+i)$
Find the largest prime $p$ less than $210$ such that the number $210 - p$ is composite.
Prove that there exist $ 2004 $ pairwise distinct numbers $ n_1,n_2,\ldots ,n_{2004} , $ all greater than $ 1, $ satisfying:
$$ \binom{n_1}{2} +\binom{n_2}{2} +\cdots +\binom{n_{2003}}{2} =\binom{n_{2004}}{2} . $$
If $\overrightarrow{u_1},\overrightarrow{u_2}, ...,\overrightarrow{u_n}$ be vectors in the plane such that the sum of their lengths is at least $1$, then between them we find vectors whose sum is a vector of length at least $\sqrt2/8$. Prove it.