A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?

find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have $$\{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\}$$.

Let $C$ be the curve $y^2 = x^3$ (where $x$ takes all non-negative real values). Let $O$ be the origin, and $A$ be the point where the gradient is $1.$ Find the length of the curve from $O$ to $A.$

Consider these two geoboard quadrilaterals. Which of the following statements is true? [asy] for (int a = 0; a < 5; ++a) { for (int b = 0; b < 5; ++b) { dot((a,b)); } } draw((0,3)--(0,4)--(1,3)--(1,2)--cycle); draw((2,1)--(4,2)--(3,1)--(3,0)--cycle); label("I",(0.4,3),E); label("II",(2.9,1),W); [/asy] $\text{(A)}\ \text{The area of quadrilateral I is more than the area of quadrilateral II.}$ $\text{(B)}\ \text{The area of quadrilateral I is less than the area of quadrilateral II.}$ $\text{(C)}\ \text{The quadrilaterals have the same area and the same perimeter.}$ $\text{(D)}\ \text{The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II.}$ $\text{(E)}\ \text{The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II.}$

Prove that the number $\overline{40...0}9$ (with at least one zero) is not a perfect square. (VA Senderov)

Let $|a|<\frac{\pi}{2}$. Evaluate \[\int_0^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^2}\]

Through a point $M$ inside an angle $a$ line is drawn. It cuts off this angle a triangle of the least possible area. Prove that $M$ is the midpoint of the segment on this line that the angle intercepts.

In a $5 \times 9$ checkered rectangle, the middle row and middle column are colored gray. You leave the corner cell and move to the cell adjacent to the side with each move. For each transition from a gray cell to a gray one you need to pay a ruble. What is the smallest number of rubles you need to pay to go around all the squares of the board exactly once (it is not necessary to return to the starting square)?

Fill in the circles to the right with the numbers 1 through 16 so that each number is used once (the number 1 has been filled in already). The number in any non-circular region is equal to the greatest difference between any two numbers in the circles on that region's vertices. You do not need to prove that your configuration is the only one possible; you merely need to find a valid conguration. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justication acceptable.) [asy] size(190); defaultpen(linewidth(0.8)); int i,j; path p; for(i=0;i<=3;++i){ draw((i,0)--(i,3)); draw((0,i)--(3,i)); } draw((0,3)--(1,2)^^(0,1)--(2,3)^^(1,0)--(3,2)^^(3,0)--(2,1)); for(i=0;i<=3;++i){ for(j=0;j<=3;++j){ p=circle((i,j),1/4); unfill(p); draw(p); } } label("$1$",(0,3)); label("$7$",(1/3,2+1/3)); label("$8$",(2/3,2+2/3)); label("$2$",(1+1/3,2+2/3)); label("$2$",(1/3,1+2/3)); label("$2$",(2+2/3,1+1/3)); label("$8$",(1+2/3,1/3)); label("$5$",(2+1/3,1/3)); label("$4$",(2+2/3,2/3)); label("$4$",(1/2,1/2)); label("$10$",(3/2,3/2)); label("$11$",(5/2,5/2)); [/asy]

In rectangle $ABCD$, $AB=12$ and $BC=10$. Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE=9$, $DF=8$, $\overline{BE} \parallel \overline{DF}$, $\overline{EF} \parallel \overline{AB}$, and line $BE$ intersects segment $\overline{AD}$. The length $EF$ can be expressed in the form $m\sqrt{n}-p$, where $m,n,$ and $p$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n+p$.

Determine $w^2+x^2+y^2+z^2$ if \[ \begin{array}{l} \displaystyle \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 \\ \displaystyle \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 \\ \displaystyle \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 \\ \displaystyle \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 \\ \end{array} \]

A dissection of a convex polygon into finitely many triangles by segments is called a [i]trilateration[/i] if no three vertices of the created triangles lie on a single line (vertices of some triangles might lie inside the polygon). We say that a trilateration is [i]good[/i] if its segments can be replaced with one-way arrows in such a way that the arrows along every triangle of the trilateration form a cycle and the arrows along the whole convex polygon also form a cycle. Find all $n\ge 3$ such that the regular $n$-gon has a good trilateration.

A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane. [i]P.A.Kozhevnikov[/i]

Determine which of the following numbers is greater $$10^{10^{10^{10}}}, (10^{10})!$$

a) There are $2n$ rays marked in a plane, with $n$ being a natural number. Given that no two marked rays have the same direction and no two marked rays have a common initial point, prove that there exists a line that passes through none of the initial points of the marked rays and intersects with exactly $n$ marked rays. (b) Would the claim still hold if the assumption that no two marked rays have a common initial point was dropped?

We call a path Valid if i. It only comprises of the following kind of steps: A. $(x, y) \rightarrow (x + 1, y + 1)$ B. $(x, y) \rightarrow (x + 1, y - 1)$ ii. It never goes below the x-axis. Let $M(n)$ = set of all valid paths from $(0,0) $, to $(2n,0)$, where $n$ is a natural number. Consider a Valid path $T \in M(n)$. Denote $\phi(T) = \prod_{i=1}^{2n} \mu_i$, where $\mu_i$= a) $1$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y + 1)$ b) $y$, if the $i^{th} $ step is $(x, y) \rightarrow (x + 1, y - 1)$ Now Let $f(n) =\sum _{T \in M(n)} \phi(T)$. Evaluate the number of zeroes at the end in the decimal expansion of $f(2021)$

Suppose an integer-valued function $f$ satisfies $$\sum_{k=1}^{2n+1}f(k)=\ln|2n+1|-4\ln|2n-1|\enspace\text{and}\enspace\sum_{k=0}^{2n}f(k)=4e^n-e^{n-1}$$ for all non-negative integers $n$. Determine $\sum_{n=0}^\infty\frac{f(n)}{2^n}$.

In triangle $ABC,AB>AC,I$ is the incenter, $AM$ is the midline. The line crosses $I$ and is perpendicular to $BC $ intersect $AM$ at point $L$, and the symmetry of $I$ with respect to point $A$ is $J$ Prove that: $\angle ABJ= \angle LBI$.

A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$. Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular.

We have a white table with $2$ rows and $5$ columns , and would like to colour all cells of the table according to the following rules: $\bullet$ We must colour the cell in the bottom left corner first. $\bullet$ After that, we can only colour a cell if some adjacent cell has already been coloured. (Two cells are adjacent if they share an edge.) How many different orders are there for colouring all $10$ squares (following these rules)?

Let $ a(k) $ be the largest odd number by which $ k $ is divisible. Prove that $$ \sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).$$

In an acute triangle $ABC$ the heights $AD, BE$ and $CF$ intersecting at $H{}$. Let $O{}$ be the circumcenter of the triangle $ABC$. The tangents to the circle $(ABC)$ drawn at $B{}$ and $C{}$ intersect at $T{}$. Let $K{}$ and $L{}$ be symmetric to $O{}$ with respect to $AB$ and $AC$ respectively. The circles $(DFK)$ and $(DEL)$ intersect at a point $P{}$ different from $D{}$. Prove that $P, D$ and $T{}$ lie on the same line. [i]Proposed by Don Luu (Vietnam)[/i]

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Take five good haikus Scramble their lines randomly What are the chances That you end up with Five completely good haikus (With five, seven, five)? Your answer will be m over n where m,n Are numbers such that m,n positive Integers where gcd Of m,n is 1. Take this answer and Add the numerator and Denominator. [i]Proposed by Jeff Lin[/i]

The quadrilateral $ABCD$ is circumscribed around a circle $k$ with center $I$ and $DA\cap CB=E$, $AB\cap DC=F$. In $\Delta EAF$ and $\Delta ECF$ are inscribed circles $k_1 (I_1,r_1)$ and $k_2 (I_2,r_2)$ respectively. Prove that the middle point $M$ of $AC$ lies on the radical axis of $k_1$ and $k_2$.

The incircle of the triangle $ABC$, with the center $I$ , touches the sides $AB, AC$ and $BC$ in the points $K, L$ and $M$ respectively. Points $P$ and $Q$ are taken on the sides $AC$ and $BC$ respectively, such that $|AP| = |CL|$ and $|BQ| = |CM|$. Prove that the difference of areas of the figures $APIQB$ and $CPIQ$ is equal to the area of the quadrangle $CLIM$