Let $ f:[0,1]\longrightarrow (0,\infty ) $ be a continuous function and $ \left( b_n \right)_{n\ge 1} $ be a sequence of numbers from the interval $ (0,1) $ that converge to $ 0. $ [b]a)[/b] Demonstrate that for any fixed $ n, $ the equation $ F(x)=b_nF(1)+\left( 1-b_n\right) F(0) $ has an unique solution, namely $ x_n, $ where $ F $ is a primitive of $ f. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{x_n}{b_n} . $

Let $ \mathcal{A}_n$ denote the set of all mappings $ f: \{1,2,\ldots ,n \} \rightarrow \{1,2,\ldots, n \}$ such that $ f^{-1}(i) :=\{ k \colon f(k)=i\ \} \neq \varnothing$ implies $ f^{-1}(j) \neq \varnothing, j \in \{1,2,\ldots, i \} .$ Prove \[ |\mathcal{A}_n| = \sum_{k=0}^{\infty} \frac{k^n}{2^{k+1}}.\] [i]L. Lovasz[/i]

There are 2003 points chosen randomly in the plane in such a way that no three of them lie on a straight line. Prove that there exists a circle which contains at least three of the given points on its circumference, and no other given points inside.

Let $a,b,d$ be integers such that $\left|a\right| \geqslant 2$, $d \geqslant 0$ and $b \geqslant \left( \left|a\right| + 1\right)^{d + 1}$. For a real coefficient polynomial $f$ of degree $d$ and integer $n$, let $r_n$ denote the residue of $\left[ f(n) \cdot a^n \right]$ mod $b$. If $\left \{ r_n \right \}$ is eventually periodic, prove that all the coefficients of $f$ are rational.

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

In $\triangle ABC$ with $AB=10$, $AC=13$, and $\measuredangle ABC = 30^\circ$, $M$ is the midpoint of $\overline{BC}$ and the circle with diameter $\overline{AM}$ meets $\overline{CB}$ and $\overline{CA}$ again at $D$ and $E$, respectively. The area of $\triangle DEM$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Compute $100m + n$. [i]Based on a proposal by Matthew Babbitt[/i]

For any positive integer $b\ge2$, we write the base-$b$ numbers as follows: \[(d_kd_{k-1}\dots d_0)_b=d_kb^k+d_{k-1}b^{k-1}+\dots+d_1b^1+d_0b^0,\]where each digit $d_i$ is a member of the set $S=\{0,1,2,\dots,b-1\}$ and either $d_k\not=0$ or $k=0$. There is a unique way to write any nonnegative integer in the above form. If we select the digits from a different set $S$ instead, we may obtain new representations of all positive integers or, in some cases, all integers. For example, if $b=3$ and the digits are selected from $S=\{-1,0,1\}$, we obtain a way to uniquely represent all integers, known as a $\emph{balanced ternary}$ representation. As further examples, the balanced ternary representation of numbers $5$, $-3$, and $25$ are: \[5=(1\ {-1}\ {-1})_3,\qquad{-3}=({-1}\ 0)_3,\qquad25=(1\ 0\ {-1}\ 1)_3.\]However, not all digit sets can represent all integers. If $b=3$ and $S=\{-2,0,2\}$, then no odd number can be represented. Also, if $b=3$ and $S=\{0,1,2\}$ as in the usual base-$3$ representation, then no negative number can be represented. Given a set $S$ of four integers, one of which is $0$, call $S$ a $\emph{4-basis}$ if every integer $n$ has at least one representation in the form \[n=(d_kd_{k-1}\dots d_0)_4=d_k4^k+d_{k-1}4^{k-1}+\dots+d_14^1+d_04^0,\]where $d_k,d_{k-1},\dots,d_0$ are all elements of $S$ and either $d_k\not=0$ or $k=0$. [list=a] [*]Show that there are infinitely many integers $a$ such that $\{-1,0,1,4a+2\}$ is not a $4$-basis. [*]Show that there are infinitely many integers $a$ such that $\{-1,0,1,4a+2\}$ is a $4$-basis.[/list]

Initially, on the lower left and right corner of a $2018\times 2018$ board, there're two horses, red and blue, respectively. $A$ and $B$ alternatively play their turn, $A$ start first. Each turn consist of moving their horse ($A$-red, and $B$-blue) by, simultaneously, $20$ cells respect to one coordinate, and $17$ cells respect to the other; while preserving the rule that the horse can't occupied the cell that ever occupied by any horses in the game. The player who can't make the move loss, who has the winning strategy?

Let $a>0$, $S_1 =\ln a$ and $S_n = \sum_{i=1 }^{n-1} \ln( a- S_i )$ for $n >1.$ Show that $$ \lim_{n \to \infty} S_n = a-1.$$

For which $n$ is $n^4+6n^3+11n^2+3n+31$ a perfect square?

Let $D$ the touchpoint of the inscribed circle of triangle $ABC$ be with side $AB$ . From $A$ the perpendicular lines on the angle bisectors of vertices $B$ and $C$ intersect them at points $A_1$ and $A_2$ respectively . Prove that $A_1A_2 = AD$.

Consider a $7\times7$ chessboard that starts out with all the squares white. We start painting squares black, one at a time, according to the rule that after painting the first square, each newly painted square must be adjacent along a side to only the square just previously painted. The final figure painted will be a connected “snake” of squares. (a) Show that it is possible to paint $31$ squares. (b) Show that it is possible to paint $32$ squares. (c) Show that it is possible to paint $33$ squares.

Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$, compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \][i]Proposed by Evan Chen[/i]

Let $n$ and $k$ be positive integers. Consider $n$ infinite arithmetic progressions of nonnegative integers with the property that among any $k$ consecutive nonnegative integers, at least one of $k$ integers belongs to one of the $n$ arithmetic progressions. Let $d_1,d_2,\ldots,d_n$ denote the differences of the arithmetic progressions, and let $d=\min\{d_1,d_2,\ldots,d_n\}$. In terms of $n$ and $k$, what is the maximum possible value of $d$?

A fair coin is repeatedly flipped until $2019$ consecutive coin flips are the same. Compute the probability that the first and last flips of the coin come up differently.

Seven cookies of radius $1$ inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie? [asy] draw(circle((0,0),3)); draw(circle((0,0),1)); draw(circle((1,sqrt(3)),1)); draw(circle((-1,sqrt(3)),1)); draw(circle((-1,-sqrt(3)),1)); draw(circle((1,-sqrt(3)),1)); draw(circle((2,0),1)); draw(circle((-2,0),1)); [/asy] $\textbf{(A) } \sqrt{2} \qquad \textbf{(B) } 1.5 \qquad \textbf{(C) } \sqrt{\pi} \qquad \textbf{(D) } \sqrt{2\pi} \qquad \textbf{(E) } \pi$

Prove that the line segment, joining the orthocenter and the intersection point of the medians of the acute-angled triangle $ABC$ is parallel to the side $AB$ iff $\tan \angle A \cdot \tan \angle B = 3$.

Mr Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each or Mr Green's steps is two feet long. Mr Green expect half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr Green expect from his garden? $ \textbf{(A) }600\qquad\textbf{(B) }800\qquad\textbf{(C) }1000\qquad\textbf{(D) }1200\qquad\textbf{(E) }1400 $

Let $S$ be a set of $n$ points in the plane such that any two points of $S$ are at least $1$ unit apart. Prove there is a subset $T$ of $S$ with at least $\frac{n}{7}$ points such that any two points of $T$ are at least $\sqrt{3}$ units apart.

In a triangle $ABC$ it is given that $2AB=AC+BC$. Prove that the incentre of $\triangle ABC$, the circumcentre of $\triangle ABC$, and the midpoints of $AC$ and $BC$ are concyclic.

Let $n \ge 1$. Prove that it is possible to select some of the integers $1,2,...,2^n$ so that for each $p = 0,1,...,n - 1$ the sum of the $p$-th powers of the selected numbers is equal to the sum of the $p$-th powers of the remaining numbers.

Let $ABCD$ be a rhombus and suppose $E$ and $F$ are the midpoints of $\overline{AD}$ and $\overline{EF}$ are the midpoints of $\overline{AD}$ and $\overline{BC},$ respectively. If $G$ is the intersection of $\overline{AC}$ and $\overline{EF},$ find the ratio of the area of $AEG$ to the area of $AGFB.$

How many distinct integral solutions of the form $(x, y)$ exist to the equation$ 21x + 22y = 43$ such that $1 < x < 11$ and $y < 22$?

Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$. [asy] pair A,B,C,D,E,F,G; B=origin; A=5*dir(60); C=(5,0); E=0.6*A+0.4*B; F=0.6*A+0.4*C; G=rotate(240,F)*A; D=extension(E,F,B,dir(90)); draw(D--G--A,grey); draw(B--0.5*A+rotate(60,B)*A*0.5,grey); draw(A--B--C--cycle,linewidth(1.5)); dot(A^^B^^C^^D^^E^^F^^G); label("$A$",A,dir(90)); label("$B$",B,dir(225)); label("$C$",C,dir(-45)); label("$D$",D,dir(180)); label("$E$",E,dir(-45)); label("$F$",F,dir(225)); label("$G$",G,dir(0)); label("$\ell$",midpoint(E--F),dir(90)); [/asy]

Let $n$ be a positive integer. In how many ways can we mark cells on an $n\times n$ board such that no two rows and no two columns have the same number of marked cells? [i]Selim Bahadir, Turkey[/i]