For non-negative real numbers $x_1, x_2, \ldots, x_n$ which satisfy $x_1 + x_2 + \cdots + x_n = 1$, find the largest possible value of $\sum_{j = 1}^{n} (x_j^{4} - x_j^{5})$.

Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.

Let $S = \lbrace 1, 2, \ldots, 15 \rbrace$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions: I. $|A_i| = 7, i = 1, 2, \ldots, n$; II. $|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n$ III. For any 3-element subset $M$ of $S$, there exists $A_k$ such that $M \subset A_k$. Find the smallest possible value of $n$.

A circle is tangential to sides $AB$ and $AD$ of convex quadrilateral $ABCD$ at $G$ and $H$ respectively, and cuts diagonal $AC$ at $E$ and $F$. What are the necessary and sufficient conditions such that there exists another circle which passes through $E$ and $F$, and is tangential to $DA$ and $DC$ extended?

Click for solution orl wrote: A circle is tangential to sides $AB$ and $AC$ of convex quadrilateral $ABCD$ at $G$ and $H$ respectively, and cuts diagonal $AC$ at $E$ and $F$. What are the necessary and sufficient conditions such that there exists another circle which passes through $E$ and $F$, and is tangential to $DA$ and $DC$ extended? Assume that when correcting the typo, the problem would begin "A circle is tangential to sides $AB$ and $BC$..." The 1st circle $(P)$ tangent to the sides $AB$ and $AC$ is centered on the internal bisector of the angle $\angle ABC$. The points $E, F$ lie both inside of the segment $AC$. The 2nd circle $(Q)$ tangent to the lines $DA, DC$ has to be centered the internal bisector of the angle $\angle CDA$. In order to pass through the points $E, F$, it also has to be centered on the perpendicular bisector of the segment $EF$. Let $Q$ be the intersection of the the internal bisector of the angle $\angle ABC$ with the perpendicular bisector of the segment $Q$. Let $(Q)$ be a circle centered at this point and passing through the points $E, F$. The quadrilateral diagonal $AC \equiv EF$ is the radical axis of the circles $(P), (Q)$. Iff the circle $(Q)$ is tangent to the lines $DA, DC$ at points $K, L$, the tangent lengths $AG = AK$ to the circles $(P), (Q)$ from the vertex $A$ are equal and the tangent lengths $CH = CL$ to the circles $(P), (Q)$ from the vertex $C$ are also equal, because these 2 vertices are on the radical axis of the circles $(P), (Q)$. The tangent lengths $BG = BH$ to the circle $(P)$ from the vertex $B$ are equal the tangent lengths $DK = DL$ to the circle $(Q)$ from the vertex $D$ are also equal. Consequently, $AB + CD = (AG + BG) + (CL + DL) = AK + BH + CH + DK =$ $= (BH + CH) + (DK + AK) = BC + DA$ which is equivalent to the quadrilateral $ABCD$ being tangential. Since the circles $(P), (Q)$ are supposed to be different, the given circle $(P)$ must not be identical with the quadrilateral incircle $(I)$.

For a fixed natural number $m \geq 2$, prove that a.) There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\] b.) For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.

For every permutation $ \tau$ of $ 1, 2, \ldots, 10$, $ \tau = (x_1, x_2, \ldots, x_{10})$, define $ S(\tau) = \sum_{k = 1}^{10} |2x_k - 3x_{k - 1}|$. Let $ x_{11} = x_1$. Find I. The maximum and minimum values of $ S(\tau)$. II. The number of $ \tau$ which lets $ S(\tau)$ attain its maximum. III. The number of $ \tau$ which lets $ S(\tau)$ attain its minimum.