How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions? $\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 12 \qquad$

Does there exist an integer having the form $444...4443$ (all fours, and ending with a three) that is divisible by $13$? If so, give an integer having that form that is divisible by $13$, if not, prove that such an integer cannot exist.

Find the largest natural number $n$ for which $4^{995} +4^{1500} +4^n$ is a square.

Let $ABC$ be a triangle with $AB=7, BC=8, CA=9.$ Denote by $D$ and $G$ the foot from $A$ to $BC$ and the centroid of $\triangle ABC,$ respectively. Let $M$ be the midpoint of $BC,$ and $K$ be the other intersection of the reflection of $AM$ over the angle bisector of $\angle BAC$ with $(ABC).$ Let $E$ the intersection of $DG$ and $KM.$ Find the area of $ABCE.$ [i]Team #10[/i]

Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$? ${ \textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}}\ a+6\qquad\textbf{(E)}\ a+7$

Let $n$ be a positive integer, $p$ and $q$ be prime numbers such that \[ pq \mid n^p+2 \quad \text{and} \quad n+2 \mid n^p+q^p. \] Prove that there exists a positive integer $m$ satisfying $q \mid 4^m \cdot n +2$.

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]D1.[/b] A Giant Hopper is $200$ meters away from you. It can hop $50$ meters. How many hops would it take for it to reach you? [b]D2.[/b] A rope of length $6$ is used to form the edges of an equilateral triangle (a triangle with equal side lengths). What is the length of one of these edges? [b]D3 / Z1.[/b] Point $E$ is on side $AB$ of rectangle $ABCD$. Find the area of triangle $ECD$ divided by the area of rectangle $ABCD$. [b]D4 / Z2.[/b] Garb and Grunt have two rectangular pastures of area $30$. Garb notices that his has a side length of $3$, while Grunt’s has a side length of $5$. What’s the positive difference between the perimeters of their pastures? [b]D5.[/b] Let points $A$ and $B$ be on a circle with radius $6$ and center $O$. If $\angle AOB = 90^o$, find the area of triangle $AOB$. [b]D6 / Z3.[/b] A scalene triangle (the $3$ side lengths are all different) has integer angle measures (in degrees). What is the largest possible difference between two angles in the triangle? [b]D7.[/b] Square $ABCD$ has side length $6$. If triangle $ABE$ has area $9$, find the sum of all possible values of the distance from $E$ to line $CD$. [b]D8 / Z4.[/b] Let point $E$ be on side $\overline{AB}$ of square $ABCD$ with side length $2$. Given $DE = BC+BE$, find $BE$. [b]Z5.[/b] The two diagonals of rectangle $ABCD$ meet at point $E$. If $\angle AEB = 2\angle BEC$, and $BC = 1$, find the area of rectangle $ABCD$. [b]Z6.[/b] In $\vartriangle ABC$, let $D$ be the foot of the altitude from $A$ to $BC$. Additionally, let $X$ be the intersection of the angle bisector of $\angle ACB$ and $AD$. If $BD = AC = 2AX = 6$, find the area of $ABC$. [b]Z7.[/b] Let $\vartriangle ABC$ have $\angle ABC = 40^o$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{AC}$ respectively such that DE is parallel to $\overline{BC}$, and the circle passing through points $D$, $E$, and $C$ is tangent to $\overline{AB}$. If the center of the circle is $O$, find $\angle DOE$. [b]Z8.[/b] Consider $\vartriangle ABC$ with $AB = 3$, $BC = 4$, and $AC = 5$. Let $D$ be a point of $AC$ other than $A$ for which $BD = 3$, and $E$ be a point on $BC$ such that $\angle BDE = 90^o$. Find $EC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$a)$ Let $n$ a positive integer. Prove that $gcd(n, \lfloor n\sqrt{2} \rfloor)<\sqrt[4]{8}\sqrt{n}$. $b)$ Prove that there are infinitely many positive integers $n$ such that $gcd(n, \lfloor n\sqrt{2} \rfloor)>\sqrt[4]{7.99}\sqrt{n}$.

Let $n$ be a positive integer. There is a board of $(n + 1) \times (n + 1)$ whose squares are numbered in a diagonal pattern, as as the picture shows. Chepito starts from the lower left square, and moving only up or to the right until he reaches the upper right box. During his tour, Chepito writes down the number of each box on the which made a change of direction, and in the end calculates the sum of all the numbers entered. Determine the maximum value of this sum. [img]https://cdn.artofproblemsolving.com/attachments/e/d/f9dc43092a1407d6fe6f1b2c741af015079946.png[/img]

Define the [i]distance [/i] between two $5$-digit numbers $\overline{a_1a_2a_3a_4a_5}$ and $\overline{b_1b_2b_3b_4b_5}$ to be the largest integer $j$ such that $a_j \ne b_j$ . (Example: the distance between $16523$ and $16452$ is $5$.) Suppose all $5$-digit numbers are written in a line in some order. What is the minimal possible sum of the distances of adjacent numbers in that written order?

Prove the following theorem: If there are $n$ pairs of different points $P_i$, $i = 1, 2, ..., n$, $n > 2$ in three dimensions space, such that each of them is at a smaller distance from one and the same point $Q$ than any other $P_i$, then $n < 15$.

Let \begin{align*}X&=\begin{pmatrix}7&8&9\\8&-9&-7\\-7&-7&9\end{pmatrix}\\Y&=\begin{pmatrix}9&8&-9\\8&-7&7\\7&9&8\end{pmatrix}.\end{align*}Let $A=Y^{-1}X$ and let $B$ be the inverse of $X^{-1}+A^{-1}$. Find a matrix $M$ such that $M^2=XY-BY$ (you may assume that $A$ and $X^{-1}+A^{-1}$ are invertible).

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

Let $a, b, c$ be positive reals such that $abc=1$. Show that \[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]

For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: [list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, [*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list] Determine $N(n)$ for all $n\geq 2$. [i]Proposed by Dan Schwarz, Romania[/i]

Consider $x_1 > 0, y_1 > 0, x_2 < 0, y_2 > 0, x_3 < 0, y_3 < 0, x_4 > 0, y_4 < 0.$ Suppose that for each $i = 1, ... ,4$ we have $ (x_i -a)^2 +(y_i -b)^2 \le c^2$. Prove that $a^2 + b^2 < c^2$. Restate this fact in the form of geometric result in plane geometry.