Every positive integer can be represented as a sum of one or more consecutive positive integers. For each $n$ , find the number of such represententation of $n$.

A convex quadrilateral $ABCD$ is such that $\angle B = \angle D$ and are both acute angles. $E$ is on $AB$ such that $CB = CE$ and $F$ is on $AD$ such that $CF = CD$. If the circumcenter of $CEF$ is $O_1$ and the circumcenter of $ABD$ is $O_2$. Prove that $C,O_1,O_2$ are collinear. [i]Proposed by Kapil Pause[/i]

Left focal point and right focal point of a hyperbola are $F_1,F_2$, left focal point and right focal point of a hyperbola are $M,N$. If $P$ is a point on the hyperbola, then the tangent point of inscribed circle of $\triangle PF_1F_2$ on $F_1F_2$ is $\text{(A)}$a point on segment $MN$ $\text{(B)}$a point on segment $F_1M$ or $F_2N$ $\text{(C)}$point $M$ or $N$ $\text{(D)}$not sure

Let $P, Q$ be points taken on the side $BC$ of a triangle $ABC$, in the order $B, P, Q, C$. Let the circumcircles of $\vartriangle PAB$, $\vartriangle QAC$ intersect at $M$ ($\ne A$) and those of $\vartriangle PAC, \vartriangle QAB$ at N. Prove that $A, M, N$ are collinear if and only if $P$ and $Q$ are symmetric in the midpoint $A' $ of $BC$.

Given real numbers $0=x_1<x_2<\ldots<x_{2n}<x_{2n+1}=1$ such that $x_{i+1}-x_i\le h$ for $1\le i\le2n$, show that $$\frac{1-h}2<\sum_{i=1}^nx_{2i}(x_{2i+1}-x_{2i-1})<\frac{1+h}2.$$

A circle is divided into $n$ sectors ($n \geq 3$). Each sector can be filled in with either $1$ or $0$. Choose any sector $\mathcal{C}$ occupied by $0$, change it into a $1$ and simultaneously change the symbols $x, y$ in the two sectors adjacent to $\mathcal{C}$ to their complements $1-x$, $1-y$. We repeat this process as long as there exists a zero in some sector. In the initial configuration there is a $0$ in one sector and $1$s elsewhere. For which values of $n$ can we end this process?

Let $0 \leq k < n$ be integers and $A=\{a \: : \: a \equiv k \pmod n \}.$ Find the smallest value of $n$ for which the expression \[ \frac{a^m+3^m}{a^2-3a+1} \] does not take any integer values for $(a,m) \in A \times \mathbb{Z^+}.$

When simplified, $ (x^{ \minus{} 1} \plus{} y^{ \minus{} 1})^{ \minus{} 1}$ is equal to: $ \textbf{(A)}\ x \plus{} y \qquad\textbf{(B)}\ \frac {xy}{x \plus{} y} \qquad\textbf{(C)}\ xy \qquad\textbf{(D)}\ \frac {1}{xy} \qquad\textbf{(E)}\ \frac {x \plus{} y}{xy}$

Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.

A circle moves so that it is continually in the contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle.

In a $m\times n$ square grid, with top-left corner is $A$, there is route along the edges of the grid starting from $A$ and visits all lattice points (called "nodes") exactly once and ending also at $A$. a. Prove that this route exists if and only if at least one of $m,\ n$ is odd. b. If such a route exists, then what is the least possible of turning points? *A turning point is a node that is different from $A$ and if two edges on the route intersect at the node are perpendicular.

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.

Let $O$, $A$, $B$, and $C$ be points in space such that $\angle AOB=60^{\circ}$, $\angle BOC=90^{\circ}$, and $\angle COA=120^{\circ}$. Let $\theta$ be the acute angle between planes $AOB$ and $AOC$. Given that $\cos^2\theta=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Michael Ren[/i]

Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the following? [asy]import math; unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((0,0)--(4,0)--(4,3)--(0,3)--(0,0)--(4,3)); fill((0,0)--(4,0)--(4,3)--cycle,mediumgray); label(rotate(aTan(3.0/4.0))*"Diagonal",(2,1.5),NW); label(rotate(90)*"Height",(4,1.5),E); label("Length",(2,0),S);[/asy]$ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 20.5 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 21.5 \qquad \textbf{(E)}\ 22$

Let $X_1,X_2,\ldots$ be a sequence of independent random variables distributed exponentially with mean $1$. Suppose $\mathbb N$ is a random variable independent of $X_i$'s that has a Poisson distribution with mean $\lambda>0$. What is the expected value of $X_1+X_2+\ldots+X_N$? $\textbf{(A)}~N^2$ $\textbf{(B)}~\lambda+\lambda^2$ $\textbf{(C)}~\lambda^2$ $\textbf{(D)}~\lambda$

In a triangle $ABC ~(\overline{AB} < \overline{AC})$, points $D (\neq A, B)$ and $E (\neq A, C)$ lies on side $AB$ and $AC$ respectively. Point $P$ satisfies $\overline{PB}=\overline{PD}, \overline{PC}=\overline{PE}$. $X (\neq A, C)$ is on the arc $AC$ of the circumcircle of triangle $ABC$ not including $B$. Let $Y (\neq A)$ be the intersection of circumcircle of triangle $ADE$ and line $XA$. Prove that $\overline{PX} = \overline{PY}$.

$f(x)$ and $g(x)$ are linear functions such that for all $x$, $f(g(x)) = g(f(x)) = x$. If $f(0) = 4$ and $g(5) = 17$, compute $f(2006)$.

Two digits one are written at the ends of a segment. In the middle, their sum is written, the number 2. Then, in the middle between each two neighboring numbers written, their sum is written again, and so on, 1973 times. How many times will the number 1973 be written? [i]Proposed by G. Halperin[/i]

Suppose that $f$ is a function with two continuous derivatives 2and $f(0) = 0.$ Prove that the function $g,$ defined by $g(0) = f '(0), g(x) = f(x) / x$ for $x \ne 0, $ has a continuous derivative.

In $\triangle ABC$ we consider the points $A',B',C'$ in sides $BC,AC,AB$ such that $$3BA'=CA',~3CB'=AB',~3AC'=BA'$$$\triangle DEF$ is defined by the intersections of $AA',BB',CC'$. If the are of $\triangle ABC$ is $2020$ find the area of $\triangle DEF$. [i]Proposed by Alejandro Madrid, Valle[/i]

In one of the countries, there are $n \geq 5$ cities operated by two airline companies. Every two cities are operated in both directions by at most one of the companies. The government introduced a restriction that all round trips that a company can offer should have atleast six cities. Prove that there are no more than $\lfloor \tfrac{n^2}{3} \rfloor$ flights offered by these companies.

Let $n{}$ be a non-negative integer and consider the standard power expansion of the following polynomial \[\sum_{k=0}^n\binom{n}{k}^2(X+1)^{2k}(X-1)^{2(n-k)}=\sum_{k=0}^{2n}a_kX^k.\]The coefficients $a_{2k+1}$ all vanish since the polynomial is invariant under the change $X\mapsto -X.$ Prove that the coefficients $a_{2k}$ are all positive.

Let $\omega$ be a semicircle and $AB$ its diameter. $\omega_1$ and $\omega_2$ are two different circles, both tangent to $\omega$ and to $AB$, and $\omega_1$ is also tangent to $\omega_2$. Let $P,Q$ be the tangent points of $\omega_1$ and $\omega_2$ to $AB$ respectively, and $P$ is between $A$ and $Q$. Let $C$ be the tangent point of $\omega_1$ and $\omega$. Find $\tan\angle ACQ$.

The matrix \[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\] satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that \[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]

Given three cocentric circles $\omega_1$,$\omega_2$,$\omega_3$ with radius $r_1,r_2,r_3$ such that $r_1+r_3\geq {2r_2}$.Constrat a line that intersects $\omega_1$,$\omega_2$,$\omega_3$ at $A,B,C$ respectively such that $AB=BC$.