Found problems: 85335
Let $$h_n := \sum_{k=0}^n \binom{n}{k} \frac{2^{k+1}}{(k+1)}.$$ Find $$\sum_{n=0}^\infty \frac{h_n}{n!}.$$
Let $a$, $b$, $c$, and $d$ be positive integers such that the following two inequalities hold: $a < 10^{20} \cdot c$ and $b > 10^{23} \cdot d$.
Determine the minimum possible value of the total number of positive integer pairs $(n, m)$ for which $n \cdot m = 2^{2023}$ and
$$ \frac {ab}{n} + \frac{cd}{m} < \frac{(a + c)(b + d)}{n + m}$$
[i]Proposed by Stijn Cambie, Belgium[/i]
Prove that if $c$ is a positive number distinct from $1$ and $n$ a positive integer, then
\[n^{2}\leq \frac{c^{n}+c^{-n}-2}{c+c^{-1}-2}. \]
Suppose that $a, b, c$ are real numbers such that $a + b + c> 0$, and so the equation $ax^2 + bx + c = 0$ has no real solutions. Show that $c> 0$.
[b]10.[/b] Given a straight line $g$ in the plane and a point $O$ on $g$. Construct, without making use of the Parallel Axiom, the half-line perpendicular to $g$ at the point $O$ and lying in one of the half-planes defined by $g$, under the following restrictions: The construction must be effected by use of a ruler and of a length standard (i.e. an etalon-segment) only; moreover, all lines and points of the construction must lie in the chosen half-plane. [b](G. 20)[/b]
The game of Greed starts with an initial configuration of one or more piles of stones.
Player $1$ and Player $2$ take turns to remove stones, beginning with Player $1$. At each turn, a player has two choices:
• take one stone from any one of the piles (a simple move);
• take one stone from each of the remaining piles (a greedy move).
The player who takes the last stone wins.
Consider the following two initial configurations:
(a) There are $2018$ piles, with either $20$ or $18$ stones in each pile.
(b) There are four piles, with $17, 18, 19$, and $20$ stones, respectively.
In each case, find an appropriate strategy that guarantees victory to one of the players.
Let $a_0,a_1,a_2,\dots$ be a periodic sequence of real numbers(that is, there is a fixed positive integer $k$ such that $a_n=a_{n+k}$ for every integer $n\geq 0$). The following equality is true, for all $n\geq 0$:
$a_{n+2}=\frac{1}{n+2} (a_n - \frac{n+1}{a_{n+1}})$
if $a_0=2020$, determine the value of $a_1$.
Find all triplets of positive real numbers $(x,y,z)$ that satisfy the following system of equations:
$$ \begin{cases}
x+y^2+z^3=3\\
y+z^2+x^3=3\\
z+x^2+y^3=3.
\end{cases}$$
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.
Find the integer part of $ \frac{1}{\sqrt1}+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}+...+\frac{1}{\sqrt{1000}} $
$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.
A convex quadrilateral of area $1$ is given. Prove that there exist four points in the interior or on the sides of the quadrilateral such that each triangle with the vertices in three of these four points has an area greater than or equal to $1/4$.
Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation
$k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.
An incircle of triangle $ABC$ touches $AB$, $BC$, $AC$ at points $C_1$, $A_1$,$ B_1$ respectively. Let $A'$ be the reflection of $A_1$ about $B_1C_1$, point $C'$ is defined similarly. Lines $A'C_1$ and $C'A_1$ meet at point $D$. Prove that $BD \parallel AC$.
Consider the system of equations $\begin{cases} x^2 - 3y + p = z, \\
y^2 - 3z + p = x, \\
z^2 - 3x + p = y \end{cases}$ with real parameter $p$.
a) For $p \ge 4$, solve the considered system in the field of real numbers.
b) Prove that for $p \in (1, 4)$ every real solution of the system satisfies $x = y = z$.
(Jaroslav Svrcek)
Find the minimum of $x^2 - 2x$ over all real numbers $x.$
Let $ABC$ be a triangle with $\angle BAC = 60^o$. Let $E$ be the point on the side $BC$ , such that $2 \angle BAE = \angle ACB$ . Let $D$ be the second intersection of $AB$ and the circumcircle of the triangle $AEC$ and $P$ be the second intersection of $CD$ and the circumcircle of the triangle $DBE$. Calculate the angle $\angle BAP$.
Given prime number $p$. $a_1,a_2 \cdots a_k$ ($k \geq 3$) are integers not divible by $p$ and have different residuals when divided by $p$. Let
\[ S_n= \{ n \mid 1 \leq n \leq p-1, (na_1)_p < \cdots < (na_k)_p \} \]
Here $(b)_p$ denotes the residual when integer $b$ is divided by $p$. Prove that $|S|< \frac{2p}{k+1}$.
Let $B$ be a $1 \times 2 \times 4$ box (rectangular parallelepiped). Let $R$ be the set of points that are within distance 3 of some point in $B$. (Note that $R$ contains $B$.) What is the volume of $R$?
Let $f_1(x)=x,f_n(x)=x+\frac{1}{14}\int_0^\pi xf_{n-1}(t)\cos ^ 3 t\ dt\ (n\geq 2)$.
Find $\lim_{n\to\infty} f_n(x)$
Brandon wants to split his orchestra of $20$ violins, $15$ violas, $10$ cellos, and $5$ basses into three distinguishable groups, where all of the players of each instrument are indistinguishable. He wants each group to have at least one of each instrument and for each group to have more violins than violas, more violas than cellos, and more cellos than basses. How many ways are there for Brandon to split his orchestra following these conditions?
Let $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that
$$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$
is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial.
[i]Proposed by Vlad Matei, Romania[/i]
World Cup in America introduced a new point system. For a victory $3$ points are given, for a draw $1$ point and for defeat $0$ points. In the preliminary games, the teams are divided into groups of $4$ teams. In groups, teams play with each other, once, then in accordance with the points scored $a,b,c$ and $d$ ($a>b>c>d$) teams take the first, second, third and fourth place in their groups. Give all possible options for the distribution points $a,b,c$ and $d$
If real numbers $a>b>1$ satisfy the inequality$$(ab+1)^2+(a+b)^2\leq 2(a+b)(a^2-ab+b^2+1)$$what is the minimum possible value of $\dfrac{\sqrt{a-b}}{b-1}$
Al is bored of Rock Paper Scissors, and wants to invent a new game: $Z-Y-X-W-V.$ Two players, each choose to play either $Z, Y, X, W,$ or $V.$ If they play the same thing, the result is a tie. However, Al must come up with a ’pecking order’, that is, he must decide which plays beat which. For each of the $10$ pairs of distinct plays that the two players can make, Al randomly decides a winner. For example, he could decide that $W$ beats $Y$ and that $Z$ beats $X,$ etc. What is the probability that after Al makes all of these $10$ choices, the game is balanced, that is, playing each letter results in an equal probability of winning?