Find the point $ p $ in the first quadrant on the line $ y = 2x $ such that the distance between $ p $ and $ p' $, the point reflected across the line $ y = x $, is equal to $ \sqrt{32} $.

For a positive integer $k\ge 2$ define $\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}$ to be a collection of $k^2$ lattice points on the cartesian coordinate plane. Let $d_1(k)>d_2(k)>\cdots$ be the decreasing sequence of the distinct distances between any two points in $T_k$. Suppose $S_i(k)$ be the number of distances equal to $d_i(k)$. Prove that for any three positive integers $m>n>i$ we have $S_i(m)=S_i(n)$.

Given integer $n > 1$ and real number $t \geq 1$. $P$ is a parallelogram with four vertices $(0, 0), (0, t), (tF_{2n+1}, tF_{2n}), (tF_{2n+1}, tF_{2n} + t)$. Here, ${F_n}$ is the $n$-th term of Fibonacci sequence defined by $F_0 = 0, F_1 = 1$ and $F_{m+1} = F_m + F_{m-1}$. Let $L$ be the number of integral points (whose coordinates are integers) interior to $P$, and $M$ be the area of $P$, which is $t^2F_{2n+1}.$ [b][i]i)[/i][/b] Prove that for any integral point $(a, b)$, there exists a unique pair of integers $(j, k)$ such that$ j(F_{n+1}, F_n) + k(F_n, F_{n-1}) = (a, b)$, that is,$ jF_{n+1} + kF_n = a$ and $jF_n + kF_{n-1} = b.$ [i][b]ii)[/b][/i] Using [i][b]i)[/b][/i] or not, prove that $|\sqrt L-\sqrt M| \leq \sqrt 2.$

Let $P$ be the parabola in the plane determined by the equation $y = x^2$ . Suppose a circle $C$ in the plane intersects $P$ at four distinct points. If three of these points are $(-28, 784)$,$(-2, 4)$, and $(13, 169)$, find the sum of the distances from the focus of $P$ to all four of the intersection points

Let $n$ be a positive integer and $\mathcal S$ be the set of points $(x, y)$ with $x, y \in \{1, 2, \ldots , n\}$. Let $\mathcal T$ be the set of all squares with vertices in the set $\mathcal S$. We denote by $a_k$ ($k \geq 0$) the number of (unordered) pairs of points for which there are exactly $k$ squares in $\mathcal T$ having these two points as vertices. Prove that $a_0 = a_2 + 2a_3$. [i]Yugoslavia[/i]

For real numbers, if $ |x| \plus{} |y| \equal{} 13$, then $ x^2 \plus{} 7x \minus{} 3y \plus{} y^2$ cannot be $\textbf{(A)}\ 208 \qquad\textbf{(B)}\ 15\sqrt {2} \qquad\textbf{(C)}\ \frac {35}{2} \qquad\textbf{(D)}\ 37 \qquad\textbf{(E)}\ \text{None}$

Let $ABCD$ be a square, and let $E$ and $F$ be points in $AB$ and $BC$ respectively such that $BE=BF$. In the triangle $EBC$, let N be the foot of the altitude relative to $EC$. Let $G$ be the intersection between $AD$ and the extension of the previously mentioned altitude. $FG$ and $EC$ intersect at point $P$, and the lines $NF$ and $DC$ intersect at point $T$. Prove that the line $DP$ is perpendicular to the line $BT$.

Find all polynomials $p$ with real coefficients that if for a real $a$,$p(a)$ is integer then $a$ is integer.

A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object? $ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $

For each positive integer $n$ determine the maximum number of points in space creating the set $A$ which has the following properties: $1)$ the coordinates of every point from the set $A$ are integers from the range $[0, n]$ $2)$ for each pair of different points $(x_1,x_2,x_3), (y_1,y_2,y_3)$ belonging to the set $A$ it is satisfied at least one of the following inequalities $x_1< y_1, x_2<y_2, x_3<y_3$ and at least one of the following inequalities $x_1>y_1, x_2>y_2,x_3>y_3$.

Let $(x_j,y_j)$, $1\le j\le 2n$, be $2n$ points on the half-circle in the upper half-plane. Suppose $\sum_{j=1}^{2n}x_j$ is an odd integer. Prove that $\displaystyle{\sum_{j=1}^{2n}y_j \ge 1}$.

Prove that no three points with integer coordinates can be the vertices of an equilateral triangle.

Let $P$ be a point on the plane. Three nonoverlapping equilateral triangles $PA_1A_2$, $PA_3A_4$, $PA_5A_6$ are constructed in a clockwise manner. The midpoints of $A_2A_3$, $A_4A_5$, $A_6A_1$ are $L$, $M$, $N$, respectively. Prove that triangle $LMN$ is equilateral.

Solve the following equation in integers with gcd (x, y) = 1 $x^2 + y^2 = 2 z^2$

In Cartesian space, let $\Omega = \{(a, b, c) : a, b, c$ are integers between $1$ and $30\}$. A point of $\Omega$ is said to be [i]visible [/i] from the origin if the segment that joins said point with the origin does not contain any other elements of $\Omega$. Find the number of points of $\Omega$ that are [i]visible [/i] from the origin.