Found problems: 85335
Let $H_k =\Sigma_{i=1}^k \frac{1}{i}$ for all positive integers $k$. Find an closed-form expression for $\Sigma_{i=1}^k H_i$ in terms of $n$ and $H_n$.
Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$
We say that a positive real number $d$ is $good$ if there exists an infinite squence $a_1,a_2,a_3,...\in (0,d)$ such that for each $n$, the points $a_1,a_2,...,a_n$ partition the interval $[0,d]$ into segments of length at most $\frac{1}{n}$ each . Find
$\text{sup}\{d| d \text{is good}\}$.
Let $\Omega$ be the inscribed circle of a triangle $\vartriangle ABC$. Let $D, E$ and $F$ be the tangency points of $\Omega$ and the sides $BC, CA$ and $AB$, respectively, and let $AD, BE$ and $CF$ intersect $\Omega$ at $K, L$ and $M$, respectively, such that $D, E, F, K, L$ and $M$ are all distinct. The tangent line of $\Omega$ at $K$ intersects $EF$ at $X$, the tangent line of $\Omega$ at $L$ intersects $DE$ at $Y$ , and the tangent line of $\Omega$ at M intersects $DF$ at $Z$. Prove that $X,Y$ and $Z$ are collinear.
Let $p$ prime and $m$ a positive integer. Determine all pairs $( p,m)$ satisfying the equation: $ p(p+m)+p=(m+1)^3$
Ivan is playing Lego with $4n^2$ $1 \times 2$ blocks. First, he places $2n^2$ $1 \times 2$ blocks to fit a $2n \times 2n$ square as the bottom layer. Then he builds the top layer on top of the bottom layer using the remaining $2n^2$ $1 \times 2$ blocks. Note that the blocks in the bottom layer are connected to the blocks above it in the top layer, just like real Lego blocks. He wants the whole two-layered building to be connected and not in seperate pieces.
Prove that if he can do so, then the four $1\times 2$ blocks connecting the four corners of the bottom layer, must be all placed horizontally or all vertically.
[i]Proposed by Ivan Chan Kai Chin[/i]
[b]Problem Section #4
a) There is a $6 * 6$ grid, each square filled with a grasshopper. After the bell rings, each grasshopper
jumps to an adjacent square (A square that shares a side). What is the maximum number of empty
squares possible?
A circle $\Gamma$ and a line $\ell$ which does not intersect $\Gamma$ are given. Suppose $P, Q,R, S$ are variable points on circle $\Gamma$ such that the points $A = PQ\cap RS$ and $B = PS \cap QR$ lie on $\ell$. Prove that the circle on $AB$ as a diameter passes through two fixed points.
Put the numbers $1,2,3,...,n$ on the circumference of a circle in such a way that the difference between neighboring numbers is at most $2$. Prove that there is just one solution (if regard is paid only to the order in which the numbers are arranged).
Let $ABCD$ be an isosceles trapezoid such that $AD = BC$, $AB = 3$, and $CD = 8$. Let $E$ be a point in the plane such that $BC = EC$ and $AE \perp EC$. Compute $AE$.
In triangle $ABC$ $\omega$ is incircle and $\omega_1$,$\omega_2$,$\omega_3$ is tangents to $\omega$ and two sides of $ABC$. $r, r_1, r_2, r_3$ is radius of $\omega, \omega_1, \omega_2, \omega_3$. Prove that $\sqrt{r_1 r_2}+\sqrt{r_2 r_3}+\sqrt{r_3 r_1}=r$
Consider all possible rectangles that can be drawn on a $8 \times 8$ chessboard, covering only whole cells. Calculate the sum of their areas.
What formula is obtained if “$8 \times 8$” is replaced with “$a \times b$”, where $a, b$ are positive integers?
How many solutions has the boundary-value problem
\[u_{xx}+\lambda u=\sin x,\;u(0)=u(\pi)=0\]
How many $5\times5$ grids are possible such that each element is either $0$ or $1$ and each row sum and column sum is $4$?
$\textbf{(A)}~64$
$\textbf{(B)}~32$
$\textbf{(C)}~120$
$\textbf{(D)}~96$
$$Problem 4 $$
Straight lines $k,l,m$ intersecting each other in three different points are drawn on a classboard. Bob remembers that in some coordinate system the lines$ k,l,m$ have the equations $y = ax, y = bx$ and $y = c +2\frac{ab}{a+b}x$ (where $ab(a + b)$ is non zero).
Misfortunately, both axes are erased. Also, Bob remembers that there is missing a line $n$ ($y = -ax + c$), but he has forgotten $a,b,c$. How can he reconstruct the line $n$?
Let $ABCD$ be a parallelogram and let $E$, $F$, $G$, and $H$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively. If $BH \cap AC = I$, $BD \cap EC = J$, $AC \cap DF = K$, and $AG \cap BD = L$, prove that the quadrilateral $IJKL$ is a parallelogram.
Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. What is the probability that Flora will eventually land at $10$?
$\textbf{(A) } \frac{5}{512} \qquad \textbf{(B) } \frac{45}{1024} \qquad \textbf{(C) } \frac{127}{1024} \qquad \textbf{(D) } \frac{511}{1024} \qquad \textbf{(E) } \frac{1}{2}$
Determine the largest constant $C$ such that
$$(x_1 + x_2 + \cdots + x_6)^2 \ge C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \cdots + x_6(x_1 + x_2))$$
holds for all real numbers $x_1, x_2, \cdots , x_6$.
For this $C$, determine all $x_1, x_2, \cdots x_6$ such that equality holds.
(Walther Janous)
Let $\triangle ABC$ have side lengths $AB=7$, $BC=8$, and $CA=9$. Let $D$ be the projection from $A$ to $\overline{BC}$ and $D'$ be the reflection of $D$ over the perpendicular bisector of $\overline{BC}$. Let $P$ and $Q$ be distinct points on the line through $D'$ parallel to $\overline{AC}$ such that $\angle APB = \angle AQB = 90^{\circ}$. The value of $AP+AQ$ can be written as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $b$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
[i]Proposed by i3435[/i]
Given $n \ (n \geq 3)$ points in space such that every three of them form a triangle with one angle greater than or equal to $120^\circ$, prove that these points can be denoted by $A_1,A_2, \ldots,A_n$ in such a way that for each $i, j, k, 1 \leq i < j < k \leq n$, angle $A_iA_jA_k$ is greater than or equal to $120^\circ . $
A square with side $2008$ is broken into regions that are all squares with side $1$. In every region, either $0$ or $1$ is written, and the number of $1$'s and $0$'s is the same. The border between two of the regions is removed, and the numbers in each of them are also removed, while in the new region, their arithmetic mean is recorded. After several of those operations, there is only one square left, which is the big square itself. Prove that it is possible to perform these operations in such a way, that the final number in the big square is less than $\frac{1}{2^{10^6}}$.
A round robin tournament is held with $2016$ participants. Each round, after seeing the results from the previous round, the tournament organizer chooses two players to play a game with each other that will result in a win for one of the players and a loss for the other. The tournament organizer wants each person to have a different total number of wins at the end of $k$ rounds. Find the minimum possible value of $k$ for which this can always be guaranteed.
[i]Proposed by Nathan Ramesh
Find all functions $f: \mathbb{Q}^{+}\to \mathbb{Q}^{+}$ such that for all $x,y \in \mathbb{Q}$: \[f \left( x+\frac{y}{x}\right) =f(x)+\frac{f(y)}{f(x)}+2y, \; x,y \in \mathbb{Q}^{+}.\]
Suppose there are $n\geq{5}$ different points arbitrarily arranged on a circle, the labels are $1, 2,\dots $, and $n$, and the permutation is $S$. For a permutation , a “descending chain” refers to several consecutive points on the circle , and its labels is a clockwise descending sequence (the length of sequence is at least $2$), and the descending chain cannot be extended to longer .The point with the largest label in the chain is called the "starting point of descent", and the other points in the chain are called the “non-starting point of descent” . For example: there are two descending chains $5, 2$and $4, 1$ in $5, 2, 4, 1, 3$ arranged in a clockwise direction, and $5$ and $4$ are their starting points of descent respectively, and $2, 1$ is the non-starting point of descent . Consider the following operations: in the first round, find all descending chains in the permutation $S$, delete all non-starting points of descent , and then repeat the first round of operations for the arrangement of the remaining points, until no more descending chains can be found. Let $G(S)$ be the number of all descending chains that permutation $S$ has appeared in the operations, $A(S)$ be the average value of $G(S)$of all possible n-point permutations $S$.
(1) Find $A(5)$.
(2)For $n\ge{6}$ , prove that $\frac{83}{120}n-\frac{1}{2} \le A(S) \le \frac{101}{120}n-\frac{1}{2}.$
Determine all functions $f : R \to R$ such that for all $x, y\in R$ holds $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$