Let $ABC$ be a triangle such that $\angle A=90^{\circ }$ and $\angle B<\angle C$. The tangent at $A$ to the circumcircle $\omega$ of triangle $ABC$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ in the line $BC$, let $X$ be the foot of the perpendicular from $A$ to $BE$, and let $Y$ be the midpoint of the segment $AX$. Let the line $BY$ intersect the circle $\omega$ again at $Z$. Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$. [hide="comment"] [i]Edited by Orl.[/i] [/hide]

$P(x)$ and $Q(x)$ are two polynomials with integer coefficients such that $P(x)|Q(x)^2+1$. [b]a)[/b] Prove that there exists polynomials $A(x)$ and $B(x)$ with rational coefficients and a rational number $c$ such that $P(x)=c(A(x)^2+B(x)^2)$. [b]b)[/b] If $P(x)$ is a monic polynomial with integer coefficients, Prove that there exists two polynomials $A(x)$ and $B(x)$ with integer coefficients such that $P(x)$ can be written in the form of $A(x)^2+B(x)^2$. [i]Proposed by Mohammad Gharakhani[/i]

On the $1$ km long ridge of Mount SPAM, there are $2006$ lemmings. In the beginning, each of them walks along the ridge in one of the two possible directions with speed $1$ m/s . When two lemmings meet, they both reverse the directions they walk but keep their walking speed. When some lemming reaches the end of the ridge, he falls down and dies. Find the least upper bound for the time it can take until all the lemmings are dead.

Consider ten concentric circles and ten rays as in the following figure. At the points where the inner circle is intersected by the rays write successively, in direction clockwise, the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. In the next circle we write the numbers $11, 12, 13, 14, 15, 16, 17, 18, 19,20$ successively, and so on successively until the last round were we write the numbers $91, 92, 93, 94, 95, 96, 97, 98, 99, 100$ successively. In this orde, the numbers $1, 11, 21, 31, 41, 51, 61, 71, 81, 91$ are in the same ray, and similarly for the other rays. In front of $50$ of those $100$ numbers, we use the sign ''$-$'' such as: a) in each of the ten rays, exist exactly $5$ signs ''$-$'' , and also b) in each of the ten concentric circles, to be exactly $5$ signs ''$-$''. Prove that the sum of the $100$ signed numbers that occur, equals zero. [img]https://cdn.artofproblemsolving.com/attachments/9/d/ffee6518fcd1b996c31cf06d0ce484a821b4ae.gif[/img]

Find all positive integers $n=p_1p_2 \cdots p_k$ which divide $(p_1+1)(p_2+1)\cdots (p_k+1)$ where $p_1 p_2 \cdots p_k$ is the factorization of $n$ into prime factors (not necessarily all distinct).

Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i]. Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.

In a triangle $ABC$, let the $B$-excircle touch $CA$ at $E$, $C$-excircle touch $AB$ at $F$. If $M$ is the midpoint of $BC$, then let the angle bisector of $\angle{BAC}$ meet $BC,EF,ME,MF$ at $D,P,E',F'$. Suppose that the circumcircles of $\triangle{EPE'}$ and $\triangle{FPF'}$ meet again at a point $Q$ and the circumcircle of $\triangle{DPQ}$ meets line $EF$ again at $X$. If $BC=10,CA=20,AB=18$, compute $\left|XE-XF\right|$. [i]2018 CCA Math Bonanza Individual Round #15[/i]

Given a square $ABCD$, point $E$ is the midpoint of $AD$. Let $F$ be the foot of the perpendicular drawn from point $B$ on $EC$. Point $K$ on $AB$ is such that $\angle DFK = 90^o$. The point $N$ on the $CE$ is such that $\angle NKB = 90^o$. Prove that the point $N$ lies on the segment $BD$. (Matvii Kurskyi) [img]https://cdn.artofproblemsolving.com/attachments/4/2/d42b8c8117ec1d5e5c5b981904779b156fce93.png[/img]

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

Let $n$ and $k$ be positive integers, and let be the sets $X=\{1,2,3,...,n\}$ and $Y=\{1,2,3,...,k\}$. Let $P$ be the set of all the subsets of the set $X$. Find the number of functions $ f: P \to Y$ that satisfy $f(A \cap B)=\min(f(A),f(B))$ for all $A,B \in P$.

Let $f:[0,1) \to \mathbb{R}$ be a monotonic function. Prove that the limits [center]$\lim_{x \nearrow 1} \int_0^x f(t) \mathrm{d}t$ and $\lim_{n \to \infty} \frac{1}{n} \left[ f(0) + f \left(\frac{1}{n}\right) + \ldots + f \left( \frac{n-1}{n} \right) \right]$[/center] exist and are equal.

Rational Man and Irrational Man both buy new cars, and they decide to drive around two racetracks from time $t=0$ to time $t=\infty$. Rational Man drives along the path parametrized by \begin{align*}x&=\cos(t)\\y&=\sin(t)\end{align*} and Irrational Man drives along the path parametrized by \begin{align*}x&=1+4\cos\frac{t}{\sqrt{2}}\\ y&=2\sin\frac{t}{\sqrt{2}}.\end{align*} Find the largest real number $d$ such that at any time $t$, the distance between Rational Man and Irrational Man is not less than $d$.

Let $0<\alpha<\beta <\gamma\in \mathbb{R}$. Let $K=\{(x,y,z)\in \mathbb{R}^{3}\;|\; x,y,z\geq 0\; \text{and}\; x^{\beta}+y^{\beta}+z^{\beta}=1\}$. Define $f:K\rightarrow \mathbb{R},\; (x,y,z)\mapsto x^{\alpha}+y^{\beta}+z^{\gamma}$. At what points of $K$ does $f$ assume its minimal and maximal values?

$n$ vectors are on the plane. We can move each vector forward and backeard on the line that the vector is on it. If there are 2 vectors that their endpoints concide we can omit them and replace them with their sum (If their sum is nonzero). Suppose with these operations with 2 different method we reach to a vector. Prove that these vectors are on a common line

Let $AA_1,BB_1,CC_1$ be the altitudes of acute $\Delta ABC$. Let $O_a,O_b,O_c$ be the incentres of $\Delta AB_1C_1,\Delta BC_1A_1,\Delta CA_1B_1$ respectively. Also let $T_a,T_b,T_c$ be the points of tangency of the incircle of $\Delta ABC$ with $BC,CA,AB$ respectively. Prove that $T_aO_cT_bO_aT_cO_b$ is an equilateral hexagon.

We have a polyhedron all faces of which are triangle. Let $P$ be an arbitrary point on one of the edges of this polyhedron such that $P$ is not the midpoint or endpoint of this edge. Assume that $P_0 = P$. In each step, connect $P_i$ to the centroid of one of the faces containing it. This line meets the perimeter of this face again at point $P_{i+1}$. Continue this process with $P_{i+1}$ and the other face containing $P_{i+1}$. Prove that by continuing this process, we cannot pass through all the faces. (The centroid of a triangle is the point of intersection of its medians.) Proposed by Mahdi Etesamifard - Morteza Saghafian

Let $S$ be the set of all points $(x_1, x_2, x_3, \dots, x_{2017})$ in $\mathbb{R}^{2017}$ satisfying $|x_i|+|x_j|\leq 1$ for any $1\leq i< j\leq 2017$. The volume of $S$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$. [i]Proposed by Yannick Yao[/i]

Let $ ABCD $ be a convex quadrilateral with no pair of parallel sides, such that $ \angle ABC = \angle CDA $. Assume that the intersections of the pairs of neighbouring angle bisectors of $ ABCD $ form a convex quadrilateral $ EFGH $. Let $ K $ be the intersection of the diagonals of $ EFGH$. Prove that the lines $ AB $ and $ CD $ intersect on the circumcircle of the triangle $ BKD $.

Let $n, m$ be positive integers, and let $\alpha$ be an irrational number satisfying $1 < \alpha < n$. Define the set \[ X = \{a + b\alpha : 0 \le a \le n \text{ and } 0 \le b \le m \}. \] Let $x_0\le x_1\le \cdots \le x_{(n+1)(m+1)-1}$ be the elements of $X$. Show that for all $i+j\le (n+1)(m+1)-1$, we have that $x_{i+j} \le x_i + x_j$ .

In a regular nonagon, each vertex is colored either red or green. Three corners of the nonagon determine a triangle. Such a triangle is called [i]red [/i] or [i]green [/i] if all its vertices are red or green if all are green. Prove that for each such coloring of the nonagon there are at least two different ones , that are congruent triangles of the same color.

Let $\, k_1 < k_2 < k_3 < \cdots \,$ be positive integers, no two consecutive, and let $\, s_m = k_1 + k_2 + \cdots + k_m \,$ for $\, m = 1,2,3, \ldots \; \;$. Prove that, for each positive integer $\, n, \,$ the interval $\, [s_n, s_{n+1}) \,$ contains at least one perfect square.

For each $x$ with $|x|<1$, compute the sum of the series $$1+4x+9x^2+\ldots+n^2x^{n-1}+\ldots.$$

[list=a] [*]Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $g\colon\mathbb{R}\rightarrow\mathbb{R}$, $g(x)=f(x)+f(2x)$, and $h\colon\mathbb{R}\rightarrow\mathbb{R}$, $h(x)=f(x)+f(4x)$, are continuous functions. Prove that $f$ is also continuous. [*]Give an example of a discontinuous function $f\colon\mathbb{R} \rightarrow\mathbb{R}$, with the following property: there exists an interval $I\subset\mathbb{R}$, such that, for any $a$ in $I$, the function $g_{a} \colon\mathbb{R}\rightarrow\mathbb{R}$, $g_{a}(x)=f(x)+f(ax)$, is continuous.[/list]

In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.

Find all polynomials $P(x)$ with real coefficents such that \[P(P(x))=(x^2+x+1)\cdot P(x)\] where $x \in \mathbb{R}$