Let \(\mathcal{S}=\left\{S_1,S_2,\ldots,S_n\right\}\) be a set of \(n\geq 2020\) distinct points on the Euclidean plane, no three of which are collinear. Andy the ant starts at some point \(S_{i_1}\) in \(\mathcal{S}\) and wishes to visit a series of 2020 points \(\left\{S_{i_1},S_{i_2},\ldots,S_{i_{2020}}\right\}\subseteq\mathcal{S}\) in order, such that \(i_j>i_k\) whenever \(j>k\). It is known that ants can only travel between points in \(\mathcal{S}\) in straight lines, and that an ant's path can never self-intersect. Find a positive integer \(n\) such that Andy can always fulfill his wish. (Lower n will be awarded more marks. Bounds for this problem may be used as a tie-breaker, should the need to do so arise.) [i]Proposed by the ICMC Problem Committee[/i]

In all rectangles with same diagonal $d$ find that one with bigger area .

Let $n$ be a positive integer. Jadzia has to write all integers from $1$ to $2n-1$ on a board, and she writes each integer in blue or red color. We say that pair of numbers $i,j\in \{1,2,3,...,2n-1\}$, where $i\leqslant j$, is $\textit{good}$ if and only if number of blue numbers among $i,i+1,...,j$ is odd. Determine, in terms of $n$, maximal number of good pairs.

In triangle $ABC$, points $P$ and $Q$ are projections of point $A$ onto the bisectors of angles $ABC$ and $ACB$, respectively. Prove that $PQ\parallel BC$.

Portia’s high school has $3$ times as many students as Lara’s high school. The two high schools have a total of $2600$ students. How many students does Portia’s high school have? $\textbf{(A) }600 \qquad \textbf{(B) }650 \qquad \textbf{(C) }1950 \qquad \textbf{(D) }2000 \qquad \textbf{(E) }2050$

Find all real numbers $a$ such that exists function $\mathbb {R} \rightarrow \mathbb {R} $ satisfying the following conditions: 1) $f(f(x)) =xf(x)-ax$ for all real $x$ 2) $f$ is not constant 3) $f$ takes the value $a$

There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect.

A postman has to deliver five letters to five different houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters?

Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$. Suppose $AD$ and $BC$ meet in $F$. Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively. If $\Gamma $ is the circumcircle of triangle $EGH$, prove that $FE$ is tangent to $\Gamma $.

Let $f(x)$ be a cubic polynomial with rational coefficients. If the graph of $f(x)$ is tangent to the $x$ axis, prove that the roots of $f(x)$ are all rational.

For a positive integer $n$, define $f(n)$ to be the number of sequences $(a_1,a_2,\dots,a_k)$ such that $a_1a_2\cdots a_k=n$ where $a_i\geq 2$ and $k\ge 0$ is arbitrary. Also we define $f(1)=1$. Now let $\alpha>1$ be the unique real number satisfying $\zeta(\alpha)=2$, i.e $ \sum_{n=1}^{\infty}\frac{1}{n^\alpha}=2 $ Prove that [list] (a) \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\alpha) \] (b) There is no real number $\beta<\alpha$ such that \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\beta) \] [/list]

$3n$ lines are drawn on the plane ($n > 1$), such that no two of them are parallel and no three of them are concurrent. Prove that, if $2n$ of the lines are coloured red and the other $n$ lines blue, there are at least two regions of the plane such that all of their borders are red. Note: for each region, all of its borders are contained in the original set of lines, and no line passes through the region.

What is the largest possible height of a right cylinder with radius $3$ that can fit in a cube with side length $12$?

Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant. [i]Victor Wang.[/i]

Solve the system $\begin{cases} x^2y^2+|xy|=\frac{4}{9}\\ xy+1=x+y^2\\ \end{cases}$

Let $x$, $y$, and $z$ be positive real numbers satisfying $x^2+y^2+z^2=1$. Prove that \[x^2yz+xy^2z+xyz^2\le\frac{1}{3}.\]

Decide whether there are infinitely many primes $p$ having a multiple in the form $n^2 + n + 1$ for some natural number $n$

On a faded piece of paper it is possible to read the following: \[(x^2 + x + a)(x^{15}- \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90.\] Some parts have got lost, partly the constant term of the first factor of the left side, partly the majority of the summands of the second factor. It would be possible to restore the polynomial forming the other factor, but we restrict ourselves to asking the following question: What is the value of the constant term $a$? We assume that all polynomials in the statement have only integer coefficients.

Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms? ${{ \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72}\qquad\textbf{(E)}\ 80} $

Let $O$ be the center of an equilateral triangle $ABC$ of area $1/\pi$. As shown in the diagram below, a circle centered at $O$ meets the triangle at points $D$, $E$, $F$, $G$, $H$, and $I$, which trisect each of the triangle's sides. Compute the total area of all six shaded regions. [asy] unitsize(90); pair A = dir(0); pair B = dir(120); pair C = dir(240); draw(A -- B -- C -- cycle); pair D = (2*A + B)/3; pair E = (A + 2*B)/3; pair F = (2*B + C)/3; pair G = (B + 2*C)/3; pair H = (2*C + A)/3; pair I = (C + 2*A)/3; draw(E -- F); draw(G -- H); draw(I -- D); draw(D -- G); draw(E -- H); draw(F -- I); pair O = (0, 0); real r = 1/sqrt(3); draw(circle(O, r)); fill(O -- D -- E -- cycle, gray); fill(O -- F -- G -- cycle, gray); fill(O -- H -- I -- cycle, gray); fill(arc(O, r, -30, 30) -- cycle, gray); fill(arc(0, r, 90, 150) -- cycle, gray); fill(arc(0, r, 210, 270) -- cycle, gray); label("$A$", A, A); label("$B$", B, B); label("$C$", C, C); label("$D$", D, unit(D)); label("$E$", E, unit(E)); label("$F$", F, unit(F)); label("$G$", G, unit(G)); label("$H$", H, unit(H)); label("$I$", I, unit(I)); label("$O$", O, C); [/asy]

Find the greatest prime divisor of $29! + 33!$.

Find the ratio of two numbers if the ratio of their arithmetic mean to their geometric mean is $25 : 24$

Let $c>0$ be a real number, and suppose that for every positive integer $n$, at least one percent of the numbers $1^c, 2^c, \cdots , n^c$ are integers. Prove that $c$ is an integer.

Let $(a, b, c, d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set $\{0,1,2,3\}$. For how many such quadruples is it true that $a\cdot d-b\cdot c$ is odd$?$ (For example, $(0, 3, 1, 1)$ is one such quadruple, because $0\cdot 1-3\cdot 1=-3$ is odd.) $\textbf{(A) } 48 \qquad \textbf{(B) } 64 \qquad \textbf{(C) } 96 \qquad \textbf{(D) } 128 \qquad \textbf{(E) } 192$

Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).