Found problems: 85335
Find $$\sum_{i=-2}^3 \sum_{j=1}^4 (j-3){i^{2j-1}}.$$
[i]Lightning 3.2[/i]
Let $n$ be a positive integer. Prove that there is no positive integer solution to thxe equation $(x + 2)^n - x^n = 1 + 7^n$.
Let $ \lambda_1 \leq \lambda_2 \leq...$ be a positive sequence and let $ K$ be a constant such that \[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\] Prove that there exists a constant $ K'$ such that \[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]
[i]L. Leindler[/i]
The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC.$ Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ.$
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
Let $ ABCD$ be a cyclic quadrilateral and $ r$ and $ s$ the lines obtained reflecting $ AB$ with respect to the internal bisectors of $ \angle CAD$ and $ \angle CBD$, respectively. If $ P$ is the intersection of $ r$ and $ s$ and $ O$ is the center of the circumscribed circle of $ ABCD$, prove that $ OP$ is perpendicular to $ CD$.
Let $x$, $y$, and $z$ be positive real numbers satisfying the system of equations
\begin{align*}
\sqrt{2x - xy} + \sqrt{2y - xy} & = 1\\
\sqrt{2y - yz} + \hspace{0.1em} \sqrt{2z - yz} & = \sqrt{2}\\
\sqrt{2z - zx\vphantom{y}} + \sqrt{2x - zx\vphantom{y}} & = \sqrt{3}.
\end{align*}Then $\big[ (1-x)(1-y)(1-z) \big] ^2$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
King Midas spent $\frac{100}{x}\%$ of his gold deposit yesterday. He is set to earn gold today. What percentage
of the amount of gold King Midas currently has would he need to earn today to end up with as much
gold as he started?
Felix is playing a card-flipping game. $n$ face-down cards are randomly colored, each with equal probability of being black or red. Felix starts at the $1$st card. When Felix is at the $k$th card, he guesses its color and then flips it over. For $k < n$, if he guesses correctly, he moves onto the $(k + 1)$-th card. If he guesses incorrectly, he gains $k$ penalty points, the cards are replaced with newly randomized face-down cards, and he moves back to card $1$ to continue guessing. If Felix guesses the $n$th card correctly, the game ends.
What is the expected number of penalty points Felix earns by the end of the game?
In the Euclidean plane, let be a point $ O $ and a finite set $ \mathcal{M} $ of points having at least two points.
Prove that there exists a proper subset of $ \mathcal{M}, $ namely $ \mathcal{M}_0, $ such that the following inequality is true:
$$ \sum_{P\in \mathcal{M}_0} OP\ge \frac{1}{4}\sum_{Q\in\mathcal{M}} OQ $$
Find all prime numbers $p,q,r,s$ such that their sum is a prime number and $p^2+qs$ and $p^2 +qr$ are squares of integers.
If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.
A square is given. Lines divide it into $n$ polygons.
What is he the largest possible sum of the internal angles of all polygons?
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
For positive integer $n$ let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999,488$ and $a_7 = 9,999,999,488$. Find the value of $n$ so that an is divisible by the highest power of $2$.
Let \(ABCDEF\) be a convex cyclic hexagon such that quadrilateral \(ABDF\) is a square, and the incenter of \(\triangle ACE\) lines on \(\overline{BF}\). Diagonal \(CE\) intersects diagonals \(BD\) and \(DF\) at points \(P\) and \(Q\), respectively. Prove that the circumcircle of \(\triangle DPQ\) is tangent to \(\overline{BF}\).
[i]Proposed by Elliott Liu[/i]
Prove that, for every set $X=\{x_{1},x_{2},\dots,x_{n}\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that
\[\left|m+\sum_{s\in S}s\right|\le\frac1{n+1}\]
How many functions $f:\{1,2,3,4\}\rightarrow \{1,2,3\}$ are surjective?
[i]Proposed by Nathan Ramesh
Let $f, g$ be bijections on $\{1, 2, 3, \dots, 2016\}$. Determine the value of $$\sum_{i=1}^{2016}\sum_{j=1}^{2016}[f(i)-g(j)]^{2559}.$$
The sides of a triangle are in the ratio $ 6: 8: 9$. Then:
$ \textbf{(A)}\ \text{the triangle is obtuse} \qquad\textbf{(B)}\ \text{the angles are in the ratio } 6: 8: 9$
$ \textbf{(C)}\ \text{the triangle is acute}$
$ \textbf{(D)}\ \text{the angle opposite the largest side is double the angle opposite the smallest side}$
$ \textbf{(E)}\ \text{none of these}$
Lines determined by sides $AB$ and $CD$ of the convex quadrilateral $ABCD$ intersect at point $P$. Prove that $\alpha +\gamma =\beta +\delta$ if and only if $PA\cdot PB=PC\cdot PD$, where $\alpha ,\beta ,\gamma ,\delta$ are the measures of the internal angles of vertices $A, B, C, D$ respectively.
If rationals $x, y$ satisfy $x^5 + y^5 = 2 x^2 y^2$, show that $1-x y$ is the square of a rational.
In the triangle $ABC$, $\angle C$ is obtuse and $D$ is a fixed point on the side $BC$, different from $B$ and $C$. For any point $M$ on the side $BC$, different from $D$, the ray $AM$ intersects the circumcircle $S$ of $ABC$ at $N$. The circle through $M, D$ and $N$ meets $S$ again at $P$, different from $N$. Find the location of the point $M$ which minimises $MP$.
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \text{ cm}$ and $6 \text{ cm}$. Into each cone is dropped a spherical marble of radius $1 \text{ cm}$, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
$\textbf{(A) }1:1 \qquad \textbf{(B) }47:43 \qquad \textbf{(C) }2:1 \qquad \textbf{(D) }40:13 \qquad \textbf{(E) }4:1$
[asy]
size(350);
defaultpen(linewidth(0.8));
real h1 = 10, r = 3.1, s=0.75;
pair P = (r,h1), Q = (-r,h1), Pp = s * P, Qp = s * Q;
path e = ellipse((0,h1),r,0.9), ep = ellipse((0,h1*s),r*s,0.9);
draw(ellipse(origin,r*(s-0.1),0.8));
fill(ep,gray(0.8));
fill(origin--Pp--Qp--cycle,gray(0.8));
draw((-r,h1)--(0,0)--(r,h1)^^e);
draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4"));
draw(subpath(ep,reltime(ep,0.5),reltime(ep,1)));
draw(Qp--(0,Qp.y),Arrows(size=8));
draw(origin--(0,12),linetype("4 4"));
draw(origin--(r*(s-0.1),0));
label("$3$",(-0.9,h1*s),N,fontsize(10));
real h2 = 7.5, r = 6, s=0.6, d = 14;
pair P = (d+r-0.05,h2-0.15), Q = (d-r+0.05,h2-0.15), Pp = s * P + (1-s)*(d,0), Qp = s * Q + (1-s)*(d,0);
path e = ellipse((d,h2),r,1), ep = ellipse((d,h2*s+0.09),r*s,1);
draw(ellipse((d,0),r*(s-0.1),0.8));
fill(ep,gray(0.8));
fill((d,0)--Pp--Qp--cycle,gray(0.8));
draw(P--(d,0)--Q^^e);
draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4"));
draw(subpath(ep,reltime(ep,0.5),reltime(ep,1)));
draw(Qp--(d,Qp.y),Arrows(size=8));
draw((d,0)--(d,10),linetype("4 4"));
draw((d,0)--(d+r*(s-0.1),0));
label("$6$",(d-r/4,h2*s-0.06),N,fontsize(10));
[/asy]
$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$, for all $ n \in N$. Let $ r_n$ be the average of $ g_n(x)$'s roots. If $ r_{19}\equal{}99$, find $ r_{99}$.
In triangle $ABC$, let $O$ be the circumcenter and let $G$ be the centroid. The line perpendicular to $OG$ at $O $ intersects $BC$ at $M$ such that $M$, $G$, and $A$ are collinear and $OM = 3$. Compute the area of $ABC$, given that $OG = 1$.