Prove that, if every three consecutive vertices of a convex $n{}$-gon, $n\geqslant 4$, span a triangle of area at least 1, then the area of the $n{}$-gon is (strictly) greater than $(n\log_2 n)/4-1/2.$ [i]Radu Bumbăcea & Călin Popescu[/i]

[u]Round 1[/u] [b]p1.[/b] Ten children arrive at a birthday party and leave their shoes by the door. All the children have different shoe sizes. Later, as they leave one at a time, each child randomly grabs a pair of shoes their size or larger. After some kids have left, all of the remaining shoes are too small for any of the remaining children. What is the greatest number of shoes that might remain by the door? [b]p2.[/b] Turans, the king of Saturn, invented a new language for his people. The alphabet has only $6$ letters: A, N, R, S, T, U; however, the alphabetic order is different than in English. A word is any sequence of $6$ different letters. In the dictionary for this language, the first word is SATURN. Which word follows immediately after TURANS? [b]p3.[/b] Benji chooses five integers. For each pair of these numbers, he writes down the pair's sum. Can all ten sums end with different digits? [b]p4.[/b] Nine dwarves live in a house with nine rooms arranged in a $3\times3$ square. On Monday morning, each dwarf rubs noses with the dwarves in the adjacent rooms that share a wall. On Monday night, all the dwarves switch rooms. On Tuesday morning, they again rub noses with their adjacent neighbors. On Tuesday night, they move again. On Wednesday morning, they rub noses for the last time. Show that there are still two dwarves who haven't rubbed noses with one another. [b]p5.[/b] Anna and Bobby take turns placing rooks in any empty square of a pyramid-shaped board with $100$ rows and $200$ columns. If a player places a rook in a square that can be attacked by a previously placed rook, he or she loses. Anna goes first. Can Bobby win no matter how well Anna plays? [img]https://cdn.artofproblemsolving.com/attachments/7/5/b253b655b6740b1e1310037da07a0df4dc9914.png[/img] [u]Round 2[/u] [b]p6.[/b] Some boys and girls, all of different ages, had a snowball fight. Each girl threw one snowball at every kid who was older than her. Each boy threw one snowball at every kid who was younger than him. Three friends were hit by the same number of snowballs, and everyone else took fewer hits than they did. Prove that at least one of the three is a girl. [b]p7.[/b] Last year, jugglers from around the world travelled to Jakarta to participate in the Jubilant Juggling Jamboree. The festival lasted $32$ days, with six solo performances scheduled each day. The organizers noticed that for any two days, there was exactly one juggler scheduled to perform on both days. No juggler performed more than once on a single day. Prove there was a juggler who performed every day. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Quadrilateral $ABCD$ is inscribed in a circle, $E$ is an arbitrary point of this circle. It is known that distances from point $E$ to lines $AB, AC, BD$ and $CD$ are equal to $a, b, c$ and $d$ respectively. Prove that $ad= bc$.

Let $P_1,P_2,\ldots ,P_n$, be infinite arithmetic progressions of positive integers, of differences $d_1,d_2,\ldots ,d_n$, respectively. Prove that if every positive integer appears in at least one of the $n$ progressions then one of the differences $d_i$ divides the least common multiple of the remaining $n-1$ differences. Note: $P_i=\left \{ a_i,a_i+d_i,a_i+2d_i,a_i+3d_i,a_i+4d_i,\cdots \right \}$ with $ a_i$ and $d_i$ positive integers.

Using one straight cut we partition a rectangular piece of paper into two pieces. We call this one "operation". Next, we cut one of the two pieces so obtained once again, to partition [i]this piece[/i] into two smaller pieces (i.e. we perform the operation on any [i]one[/i] of the pieces obtained). We continue this process, and so, after each operation we increase the number of pieces of paper by $1$. What is the minimum number of operations needed to get $47$ pieces of $46$-sided polygons? [obviously there will be other pieces too, but we will have at least $47$ (not necessarily [i]regular[/i]) $46$-gons.]

Determine the value of the sum \[\left|\sum_{1\leq i<j\leq 50}ij(-1)^{i+j}\right|.\]

Let $A_1A_2...A_{101}$ be a regular $101$-gon, and colour every vertex red or blue. Let $N$ be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the $101$-gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour. $(1)$ Find the largest possible value of $N$. $(2)$ Find the number of ways to colour the vertices such that maximum $N$ is acheived. (Two colourings a different if for some $A_i$ the colours are different on the two colouring schemes).

Let $\mathcal{P}$ be the set of all polynomials with coefficients in $\{0, 1\}$. Suppose $a, b$ are non-zero integers such that for every $f \in \mathcal{P}$ with $f(a)\neq 0$, we have $f(a) \mid f(b)$. Prove that $a=b$. [i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]

Given $n$ positive real numbers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n \ge 0$ and $x_1^2+x_2^2+\cdots+x_n^2=1$, prove that \[\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.\]

Let $(a_n)$ be the Lucas sequence: $a_0=2,a_1=1, a_{n+1}=a_n+a_{n-1}$ for $n\geq 1$. Show that $a_{59}$ divides $(a_{30})^{59}-1$.

In a triangle $ABC$, a point $D$ is on the segment $BC$, Let $X$ and $Y$ be the incentres of triangles $ACD$ and $ABD$ respectively. The lines $BY$ and $CX$ intersect the circumcircle of triangle $AXY$ at $P\ne Y$ and $Q\ne X$, respectively. Let $K$ be the point of intersection of lines $PX$ and $QY$. Suppose $K$ is also the reflection of $I$ in $BC$ where $I$ is the incentre of triangle $ABC$. Prove that $\angle BAC=\angle ADC=90^{\circ}$.

For what positive integer values of $x$ is $x^4 + 6x^3 + 11x^2 + 3x + 31$ a perfect square?

A sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is called "Devin" if for any $f\in C[0,1]$ $$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(x_i)=\int_0^1 f(x)\,dx $$ Prove that a sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is "Devin" if and only if for any non-negative integer $k$ it holds $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i^k=\frac{1}{k+1}.$$ [b]Remark[/b]. I left intact the text as it was proposed. Devin is a Bulgarian city and SPA resort, where this competition took place.

Given is a polynomial $P(x)$ of degree $n>1$ with real coefficients. The equation $P(P(P(x)))=P(x)$ has $n^3$ distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.

Circles $ C_1,C_2$ intersect at $ P$. A line $ \Delta$ is drawn arbitrarily from $ P$ and intersects with $ C_1,C_2$ at $ B,C$. What is locus of $ A$ such that the median of $ AM$ of triangle $ ABC$ has fixed length $ k$.

A triangle has a perimeter of $4$ [i]yards[/i] and an area of $6$ square [i]feet[/i]. If one of the angles of the triangle is right, what is the length of the largest side of the triangle, in feet? [i]2016 CCA Math Bonanza Lightning #1.4[/i]

Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]

What is the smallest possible value of $x^2 + y^2 - x -y - xy$?

Let $\theta_{i}\in(0,\frac{\pi}{2})(i=1,2,\cdots,n)$. Prove: $$(\sum_{i=1}^n\tan\theta_{i})(\sum_{i=1}^n\cot\theta_{i})\geq(\sum_{i=1}^n\sin\theta_{i})^2+(\sum_{i=1}^n\cos\theta_{i})^2.$$

Let $ \mathbb{Q}$ and $ \mathbb{R}$ be the set of rational numbers and the set of real numbers, respectively, and let $ f : \mathbb{Q} \rightarrow \mathbb{R}$ be a function with the following property. For every $ h \in \mathbb{Q} , \;x_0 \in \mathbb{R}$, \[ f(x\plus{}h)\minus{}f(x) \rightarrow 0\] as $ x \in \mathbb{Q}$ tends to $ x_0$. Does it follow that $ f$ is bounded on some interval? [i]M. Laczkovich[/i]

For a word $w$ consisting of $n$ lowercase letters, an [i]edit[/i] is specified by a pair $(i,c)$ where $i\in \{1,\ldots, n\}$ and $c$ is a lowercase letter, and transforms $w$ by replacing its $i^{\text{th}}$ letter with $c$. It is possible that $c$ is the same as the letter it replaced. How many sequences of six edits transform $\verb|frog|$ into $\verb|goat|$? Note that on the word $\verb|abcd|$, the edits $(1,\verb|a|)$ and $(2,\verb|b|)$ are considered distinct, even though both result in the word $\verb|abcd|$.

We say that a positive integer $n$ is nice if $\frac{4}{n}$ cannot be written as $\frac{1}{x}+\frac{1}{xy}+\frac{1}{z}$ for any positive integers $x, y, z$. Let us denote by $ a_n$ the number of nice numbers smaller than $n$. Prove that the sequence $\frac{n}{a_n}$ is not bounded.

A point $O$ inside a convex $n$-gon $A_1A_2 . . .A_n$ is connected with segments to its vertices. The sides of this $n$-gon are numbered $1$ to $n$ (distinct sides have distinct numbers). The segments $OA_1,OA_2, . . . ,OA_n$ are similarly numbered. a) For $n = 9$ find a numeration such that the sum of the sides’ numbers is the same for all triangles $A_1OA_2, A_2OA_3, . . . , A_nOA_1$. b) Prove that for $n = 10$ there is no such numeration.

A rectangular box $ P$ is inscribed in a sphere of radius $ r$. The surface area of $ P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $ r$? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

Given $2N$ real numbers. It is known that if they are arbitrarily divided into two groups of $N$ numbers each then the products of the numbers of each group differ by $2$ at most. Is it necessarily true that if we arbitrarily place these numbers along a circle then there are two neighboring numbers that differ by $2$ at most, for a) $N=50$; (3 marks) b) $N=25$? (5 marks)