In $\triangle ABC$, let $D$ be the point on side $BC$ such that $AB+BD=DC+CA.$ The line $AD$ intersects the circumcircle of $\triangle ABC$ again at point $X \neq A$. Prove that one of the common tangents of the circumcircles of $\triangle BDX$ and $\triangle CDX$ is parallel to $BC$.

Given that $\sqrt{10} \approx 3.16227766$, find the largest integer $n$ such that $n^2 \leq 10,000,000$. [i]Proposed by Jacob Stavrianos[/i]

Let $ n\ge12$ be an integer, and let $ P_1,P_2,...P_n, Q$ be distinct points in a plane. Prove that for some $ i$, at least $ \frac{n}{6}\minus{}1$ of the distances $ P_1P_i,P_2P_i,...P_{i\minus{}1}P_i,P_{i\plus{}1}P_i,...P_nP_i$ are less than $ P_iQ$.

Let $\vartriangle ABC$ be such that $\angle BAC$ is acute. The line perpendicular on side $AB$ from $C$ and the line perpendicular on $AC$ from $B$ intersect the circumscribed circle of $\vartriangle ABC$ at $D$ and $E$ respectively. If $DE = BC$ , calculate $\angle BAC$.

Sixteen points are placed in the centers of a $4 \times 4$ chess table in the following way: • • • • • • • • • • • • • • • • (a) Prove that one may choose $6$ points such that no isoceles triangle can be drawn with the vertices at these points. (b) Prove that one cannot choose $7$ points with the above property.

In a room with $9$ students, there are $n$ clubs with $4$ participants in each club. For any pairs of clubs no more than $2$ students belong to both clubs. Prove that $n \le 18$ [i]Proposed by Manuel Aguilera, Valle[/i]

In $\Delta ABC$ with $AC=10$ and $BC=15$ the points $G$ and $I$ are its centroid and the center of its inscribed circle respectively. Find $AB$, if $\angle GIC=90^\circ$.

Bob draws the graph of $y = x^3 - 13x^2 + 40x + 25$ and is dismayed to find out that it only has one root. Alice comes to the rescue, translating (without rotating or dilating) the axes so that the origin is at the point that used to be $(-20, 16)$. This new graph has three $x$-intercepts; compute their sum.

It is given regular polygon with $2n$ sides and center $S$. Consider every quadrilateral with vertices as vertices of polygon. Let $u$ be number of such quadrilaterals which contain point $S$ inside and $v$ number of remaining quadrilaterals. Find $u-v$

A right-angled triangle made of paper is folded along a straight line so that the vertex at the right angle coincides with one of the other vertices of the triangle and a quadrilateral is obtained . (a) What is the ratio into which the diagonals of this quadrilateral divide each other? (b) This quadrilateral is cut along its longest diagonal. Find the area of the smallest piece of paper thus obtained if the area of the original triangle is $1$ . (A Shapovalov)

Let $\omega_1, \omega_2$ be two circles with different radii, and let $H$ be the exsimilicenter of the two circles. A point X outside both circles is given. The tangents from $X$ to $\omega_1$ touch $\omega_1$ at $P, Q$, and the tangents from $X$ to $\omega_2$ touch $\omega_2$ at $R, S$. If $PR$ passes through $H$ and is not a common tangent line of $\omega_1, \omega_2$, prove that $QS$ also passes through $H$.

Each vertex of a regular $17$-gon is colored red, blue, or green in such a way that no two adjacent vertices have the same color. Call a triangle “multicolored” if its vertices are colored red, blue, and green, in some order. Prove that the $17$-gon can be cut along nonintersecting diagonals to form at least two multicolored triangles. (A diagonal of a polygon is a a line segment connecting two nonadjacent vertices. Diagonals are called nonintersecting if each pair of them either intersect in a vertex or do not intersect at all.)

Let $ABC$ be a triangle with $AB = 4$, $BC = 5$, $CA = 6$. Triangles $APB$ and $CQA$ are erected outside $ABC$ such that $AP=PB$, $\overline{AP}\perp \overline{PB}$ and $CQ=QA$, $\overline{CQ}\perp \overline{QA}$. Pick a point $X$ uniformly at random from segment $\overline{BC}$. What is the expected value of the area of triangle $PXQ$?

[u]Round 5[/u] [b]p13.[/b] Five different schools are competing in a tournament where each pair of teams plays at most once. Four pairs of teams are randomly selected and play against each other. After these four matches, what is the probability that Chad's and Jordan's respective schools have played against each other, assuming that Chad and Jordan come from different schools? [b]p14.[/b] A square of side length $1$ and a regular hexagon are both circumscribed by the same circle. What is the side length of the hexagon? [b]p15.[/b] From the list of integers $1,2, 3,...,30$ Jordan can pick at least one pair of distinct numbers such that none of the $28$ other numbers are equal to the sum or the difference of this pair. Of all possible such pairs, Jordan chooses the pair with the least sum. Which two numbers does Jordan pick? [u]Round 6[/u] [b]p16.[/b] What is the sum of all two-digit integers with no digit greater than four whose squares also have no digit greater than four? [b]p17.[/b] Chad marks off ten points on a circle. Then, Jordan draws five chords under the following constraints: $\bullet$ Each of the ten points is on exactly one chord. $\bullet$ No two chords intersect. $\bullet$ There do not exist (potentially non-consecutive) points $A, B,C,D,E$, and $F$, in that order around the circle, for which $AB$, $CD$, and $EF$ are all drawn chords. In how many ways can Jordan draw these chords? [b]p18.[/b] Chad is thirsty. He has $109$ cubic centimeters of silicon and a 3D printer with which he can print a cup to drink water in. He wants a silicon cup whose exterior is cubical, with five square faces and an open top, that can hold exactly $234$ cubic centimeters of water when filled to the rim in a rectangular-box-shaped cavity. Using all of his silicon, he prints a such cup whose thickness is the same on the five faces. What is this thickness, in centimeters? [u]Round 7[/u] [b]p19.[/b] Jordan wants to create an equiangular octagon whose side lengths are exactly the first $8$ positive integers, so that each side has a different length. How many such octagons can Jordan create? [b]p20.[/b] There are two positive integers on the blackboard. Chad computes the sum of these two numbers and tells it to Jordan. Jordan then calculates the sum of the greatest common divisor and the least common multiple of the two numbers, and discovers that her result is exactly $3$ times as large as the number Chad told her. What is the smallest possible sum that Chad could have said? [b]p21.[/b] Chad uses yater to measure distances, and knows the conversion factor from yaters to meters precisely. When Jordan asks Chad to convert yaters into meters, Chad only gives Jordan the result rounded to the nearest integer meters. At Jordan's request, Chad converts $5$ yaters into $8$ meters and $7$ yaters into $12$ meters. Given this information, how many possible numbers of meters could Jordan receive from Chad when requesting to convert $2014$ yaters into meters? [u]Round 8[/u] [b]p22.[/b] Jordan places a rectangle inside a triangle with side lengths $13$, $14$, and $15$ so that the vertices of the rectangle all lie on sides of the triangle. What is the maximum possible area of Jordan's rectangle? [b]p23.[/b] Hoping to join Chad and Jordan in the Exeter Space Station, there are $2014$ prospective astronauts of various nationalities. It is given that $1006$ of the astronaut applicants are American and that there are a total of $64$ countries represented among the applicants. The applicants are to group into $1007$ pairs with no pair consisting of two applicants of the same nationality. Over all possible distributions of nationalities, what is the maximum number of possible ways to make the $1007$ pairs of applicants? Express your answer in the form $a \cdot b!$, where $a$ and $b$ are positive integers and $a$ is not divisible by $b + 1$. Note: The expression $k!$ denotes the product $k \cdot (k - 1) \cdot ... \cdot 2 \cdot 1$. [b]p24.[/b] We say a polynomial $P$ in $x$ and $y$ is $n$-[i]good [/i] if $P(x, y) = 0$ for all integers $x$ and $y$, with $x \ne y$, between $1$ and $n$, inclusive. We also define the complexity of a polynomial to be the maximum sum of exponents of $x$ and $y$ across its terms with nonzero coeffcients. What is the minimal complexity of a nonzero $4$-good polynomial? In addition, give an example of a $4$-good polynomial attaining this minimal complexity. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2915803p26040550]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all the triples $(a, b, c)$ of positive real numbers such that the system \[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\] is compatible in the set of real numbers, and then find all its real solutions.

Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$, Anne and Carl in $5$. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?

$ 25$ boys and some girls came to the party and discovered an interesting property of their company. Take an arbitrary group of $ \geq 10$ boys and all the girls which are acquainted with at least one of them. Then in the joint group, the number of girls is by one greater than the number of boys. Prove that there exists a girl who is acquainted with at least $ 16$ boys.

We define a sequence $a_n$ so that $a_0=1$ and \[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases} \] for all postive integers $n$. Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.

Find all real parameters $q$ for which there is a $p\in[0,1]$ such that the equation $$x^4+2px^3+(2p^2-p)x^2+(p-1)p^2x+q=0$$has four real roots.

An integer $m\ge 3$ and an infinite sequence of positive integers $(a_n)_{n\ge 1}$ satisfies the equation \[a_{n+2} = 2\sqrt[m]{a_{n+1}^{m-1} + a_n^{m-1}} - a_{n+1}. \] for all $n\ge 1$. Prove that $a_1 < 2^m$.

Real numbers $a_{1},a_{2},\dots,a_{n}$ satisfy $a_{i}\geq\frac{1}{i}$, for all $i=\overline{1,n}$. Prove the inequality: \[\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).\]

Let be two real numbers $ a<b, $ a nonempty and non-maximal subset $ K $ of the interval $ (a,b) $ and three functions $$ f:(a,b)\longrightarrow\mathbb{R}, g,h:\mathbb{R}\longrightarrow\mathbb{R} $$ satisfying the following relations. $ \text{(i)} g $ and $ h $ are primitivable. $ \text{(ii)} g-h $ hasn't any root in $ (a,b). $ $ \text{(iii)} $ The restrictions of $ f $ at $ K $ and $ (a,b)\setminus K $ are equal to $ g,h, $ respectively. Prove that $ f $ is not primitivable.

In a 100-dimensional hypercube, each edge has length $ 1$. The box contains $2^{100} + 1$ hyperspheres with the same radius $ r$. The center of one hypersphere is the center of the hypercube, and it touches all the other spheres. Each of the other hyperspheres is tangent to $100$ faces of the hypercube. Thus, the hyperspheres are tightly packed in the hypercube. Find $ r$.

In rectangular coordinate system, if $m(x^2+y^2+2y+1)=(x-2y+3)^2$ refers to an ellipse, then the range value of $m$ is $\text{(A)}(0,1)\qquad\text{(B)}(1,+\infty)\qquad\text{(C)}(0,5)\qquad\text{(D)}(5,+\infty)$

A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?