Determine all positive integers $n$ for which the square $n \times n$ can be cut into squares $2\times 2$ and $3\times3$ (with the sides parallel to the sides of the big square).

Problem. The sum of two consecutive squares can be a square; for instance $3^2 + 4^2 = 5^2$. (a) Prove that the sum of $m$ consecutive squares cannot be a square for $m \in \{3, 4, 5, 6\}$. (b) Find an example of eleven consecutive squares whose sum is a square. Can anyone help me with this? Thanks.

Let $ABC$ be a triangle, $\Gamma$ its circumcircle and $l$ the tangent to $\Gamma$ through $A$. The altitudes from $B$ and $C$ are extended and meet $l$ at $D$ and $E$, respectively. The lines $DC$ and $EB$ meet $\Gamma$ again at $P$ and $Q$, respectively. Show that the triangle $APQ$ is isosceles.

Sixteen wooden Cs are placed in a $4$-by-$4$ grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four two-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjacent square in the subgrid that is $90$ degrees away in the clockwise direction, without changing the orientation of the C. Given that two colorings are the considered same if and only if one can be obtained from the other by a series of quadrant operations, determine the number of distinct colorings of the Cs. [img]https://cdn.artofproblemsolving.com/attachments/a/9/1e59dce4d33374960953c0c99343eef807a5d2.png[/img]

Find the area of the region bounded by $C: y=-x^4+8x^3-18x^2+11$ and the tangent line which touches $C$ at distinct two points.

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed. The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$. Find the length of $AB$.

Two different points $X$ and $Y$ are chosen in the plane. Find all the points $Z$ in this plane for which the triangle $XYZ$ is isosceles.

The altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at point $H$. Let $F$ be the intersection of the line $AB$ and the line that is parallel to the side BC and goes through the circumcenter of $ABC$. Let $M$ be the midpoint of the segment $AH$. Prove that $\angle CMF = 90^o$

A planar country has an odd number of cities separated by pairwise distinct distances. Some of these cities are connected by direct two-way flights. Each city is directly connected to exactly two ther cities, and the latter are located farthest from it. Prove that, using these flights, one may go from any city to any other city

Let $n\in\mathbb{N}^*$ and $v_1,v_2,\ldots ,v_n$ be vectors in the plane with lengths less than or equal to $1$. Prove that there exists $\xi_1,\xi_2,\ldots ,\xi_n\in\{-1,1\}$ such that \[ | \xi_1v_1+\xi_2v_2+\ldots +\xi_nv_n|\le\sqrt{2}\]

Let $a, b, c,$ be nonnegative reals with $ a+b+c=3 $, find the largest positive real $ k $ so that for all $a,b,c,$ we have $$ a^2+b^2+c^2+k(abc-1)\ge 3 $$

Show that, exist infinite triples $(a, b, c)$ where $a, b, c$ are natural numbers, such that: $2a^2 + 3b^2 - 5c^2 = 1997$

$ A_1 , A_2 , \cdots , A_n $ are given subsets. Let $ S = \left\{ 1, 2, \cdots , n \right\} $. For any $ X \subset S $, let \[ N(X)= \left\{ i \in S-X \ | \ \forall j \in X, \ A_i \cap A_j \ne \emptyset \right\} \] Let $ m $ be an integer such that $ 3 \le m \le n-2 $. Prove that there exist $ X \subset S $ such that $ |X|=m $ and $ |N(X)| \ne 1 $.

Let $n \geq 2$ be a fixed positive integer and let $a_{0},a_{1},...,a_{n-1}$ be real numbers. Assume that all of the roots of the polynomial $P(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{1}x+a_{0}$ are strictly positive real numbers. Determine the smallest possible value of $\frac{a_{n-1}^{2}}{a_{n-2}}$ over all such polynomials. [i]Proposed by Nikola Velov[/i]

Let $\alpha(n)$ be the number of digits equal to one in the dyadic representation of a positive integer $n$. Prove that [list=a] [*] the inequality $\alpha(n^2 ) \le \frac{1}{2} \alpha(n) (1+\alpha(n))$ holds, [*] equality is attained for infinitely $n\in\mathbb{N}$, [*] there exists a sequence $\{n_i\}$ such that $\lim_{i \to \infty} \frac{ \alpha({n_{i}}^2 )}{ \alpha(n_{i}) } = 0$.[/list]

(My problem. :D) Call the number of times that the digits of a number change from increasing to decreasing, or vice versa, from the left to right while ignoring consecutive digits that are equal the [i]flux[/i] of the number. For example, the flux of 123 is 0 (since the digits are always increasing from left to right) and the flux of 12333332 is 1, while the flux of 9182736450 is 8. What is the average value of the flux of the positive integers from 1 to 999, inclusive?

The expression $ (81)^{ \minus{} 2^{ \minus{} 2}}$ has the same value as: $ \textbf{(A)}\ \frac {1}{81} \qquad \textbf{(B)}\ \frac {1}{3} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 81 \qquad \textbf{(E)}\ 81^4$

Point $C$ is chosen on the arc of a semicircle with diameter $AB$. The two circles with diameters of $AC$ and $BC$ intersect again at point $D$. If $DA=20$ and $DB=16$, compute the length of $DC$.

A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon.

Prove that for every integer $n > 10000$ there exists an integer $m$ such that it can be written as the sum of two squares, and $0<m-n<3\sqrt[4]n$.

A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th runner to finish behind Ammar. There were twice as many runners who had finished before Julia compared to the number who finished behind her. How many runners were there in the race?

For a given integer $ n\geq 2,$ determine the necessary and sufficient conditions that real numbers $ a_{1},a_{2},\cdots, a_{n},$ not all zero satisfy such that there exist integers $ 0<x_{1}<x_{2}<\cdots<x_{n},$ satisfying $ a_{1}x_{1}\plus{}a_{2}x_{2}\plus{}\cdots\plus{}a_{n}x_{n}\geq 0.$

[b]T[/b]wo ordered pairs $(a,b)$ and $(c,d)$, where $a,b,c,d$ are real numbers, form a basis of the coordinate plane if $ad \neq bc$. Determine the number of ordered quadruples $(a,b,c)$ of integers between $1$ and $3$ inclusive for which $(a,b)$ and $(c,d)$ form a basis for the coordinate plane.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(0) \in \mathbb Q$ and \[f(x+f(y)^2 ) = {f(x+y)}^2.\] (25 points)