Prove that for any positive integer $m$, the equation \[ \frac{n}{m}=\lfloor\sqrt[3]{n^2}\rfloor+\lfloor\sqrt{n}\rfloor+1\] has (at least) a positive integer solution $n_{m}$. Cezar Lupu & Dan Schwarz

The $ 2^N$ vertices of the $ N$-dimensional hypercube $ \{0,1\}^N$ are labelled with integers from $ 0$ to $ 2^N - 1$, by, for $ x = (x_1,x_2,\ldots ,x_N)\in \{0,1\}^N$, \[v(x) = \sum_{k = 1}^{N}x_k2^{k - 1}.\] For which values $ n$, $ 2\leq n \leq 2^n$ can the vertices with labels in the set $ \{v|0\leq v \leq n - 1\}$ be connected through a Hamiltonian circuit, using edges of the hypercube only? E. Bazavan & C. Talau

Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals. Dan Schwarz

Let $ P(x) \in \mathbb{Z}[x]$ be a polynomial of degree $ \text{deg} P = n > 1$. Determine the largest number of consecutive integers to be found in $ P(\mathbb{Z})$. B. Berceanu

Click for solution Consider the polynomial $ P(x+1)-P(x)-1$ it has degree $ \le 1$ so we get that we can not have more than $ n$ consecutive elements in the range of $ P$. This is reached when for example $ P(x)=x+\prod_{i=1}^n (x-i)$

Let $\sqrt{23}>\frac{m}{n}$ where $ m,n$ are positive integers. i) Prove that $ \sqrt{23}>\frac{m}{n}+\frac{3}{mn}.$ ii) Prove that $ \sqrt{23}<\frac{m}{n}+\frac{4}{mn}$ occurs infinitely often, and give at least three such examples. Dan Schwarz

Let $ k > 1$ be an integer, and consider the infinite array given by the integer lattice in the first quadrant of the plane, filled with real numbers. The array is said to be constant if all its elements are equal in value. The array is said to be $ k$-balanced if it is non-constant, and the sums of the elements of any $ k\times k$ sub-square have a constant value $ v_k$. An array which is both $ p$-balanced and $ q$-balanced will be said to be $ (p, q)$-balanced, or just doubly-balanced, if there is no confusion as to which $ p$ and $ q$ are meant. If $p, q$ are relatively prime, the array is said to be co-prime. We will call $ (M\times N)$-seed a $ M \times N$ array, anchored with its lower left corner in the origin of the plane, which extended through periodicity in both dimensions in the plane results into a $ (p, q)$-balanced array; more precisely, if we denote the numbers in the array by $ a_{ij}$ , where $ i, j$ are the coordinates of the lower left corner of the unit square they lie in, we have, for all non-negative integers $ i, j$ \[ a_{i + M,j} = a_{i,j} = a_{i,j + N}\] (a) Prove that $ q^2v_p = p^2v_q$ for a $ (p, q)$-balanced array. (b) Prove that more than two different values are used in a co-prime $ (p,q)$-balanced array. Show that this is no longer true if $ (p, q) > 1$. (c) Prove that any co-prime $ (p, q)$-balanced array originates from a seed. (d) Show there exist $ (p, q)$-balanced arrays (using only three different values) for arbitrary values $ p, q$. (e) Show that neither a $ k$-balanced array, nor a $ (p, q)$-balanced array if $ (p, q) > 1$, need originate from a seed. (f) Determine the minimal possible value $ T$ for a square $ (T\times T)$-seed resulting in a co-prime $ (p, q)$-balanced array, when $p,q$ are both prime. (g) Show that for any relatively prime $ p, q$ there must exist a co-prime $ (p, q)$-balanced array originating from a square $ (T\times T)$-seed, with no lesser $ (M\times N)$-seed available ($ M\leq T, N\leq T$ and $MN< T^2$). Dan Schwarz