Let $\triangle{ABC}$ exist such that $AB=6, BC=8, AC=10.$ Let $P$ lie on the circumcircle of $ABC,$ $\omega,$ such that $P$ lies strictly on the arc between $B$ and $C$ (i.e. $P \neq B, C$). Drop altitudes from $P$ to $BC, AC$ at points $J$ and $Q$ respectively. Let $l$ be a line through $B$ such that it intersects $AC$ at a point $K.$ Let $M$ be the midpoint of $BQ.$ Let $CM$ intersect line $l$ at a point $I.$ Let $AI$ intersect $JQ$ at a point $U.$ Now, $B, J, U, M$ are cyclic. Now, let $\angle{QJC}=\theta.$ If we set $y=\sin(\theta), x=\cos(\theta),$ they satisfy the equation $$768(xy)=(16-8x^2+6xy)(x^2y^2(8x-6y)^2+(8x-8xy^2+6y^3)^2)$$ The numerical values of $x,y$ are approximately: $$x=0.72951, y=0.68400$$ Let $BK$ intersect the circumcircle of $ABC,$ $\omega,$ at a point $L.$ Find the value of $BL.$ We will only look up to two decimal places for correctness.

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$. An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to: a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition. b) Check that if $K$ is a commutative field, the elements of$ M$ that have inverse they are characterized by the condition $ad - bc \ne 0$. c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.

Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]

If five boys and three girls are randomly divided into two four-person teams, what is the probability that all three girls will end up on the same team? $\text{(A) }\frac{1}{7}\qquad\text{(B) }\frac{2}{7}\qquad\text{(C) }\frac{1}{10}\qquad\text{(D) }\frac{1}{14}\qquad\text{(E) }\frac{1}{28}$

The non-parallel sides of a trapezium serve as the diameters of two circles. Prove that all four tangents to the circles drawn from the point of intersection of the diagonals are equal (if this point lies outside the circles). (S Markelov)

Rectangle $ABCD$ has perimeter $178$ and area $1848$. What is the length of the diagonal of the rectangle? [i]2016 CCA Math Bonanza Individual Round #2[/i]

Let be a natural number $ n\ge 3. $ Solve the equation $ \lfloor x/n \rfloor =\lfloor x-n \rfloor $ in $ \mathbb{R} . $ [i]Constantin Nicolau[/i]

Consider $ABC$ an acute-angled triangle with $AB \ne AC$. Denote by $M$ the midpoint of $BC$, by $D, E$ the feet of the altitudes from $B, C$ respectively and let $P$ be the intersection point of the lines $DE$ and $BC$. The perpendicular from $M$ to $AC$ meets the perpendicular from $C$ to $BC$ at point $R$. Prove that lines $PR$ and $AM$ are perpendicular.

Let $ n$ be a natural number such that $ n \equal{} a^2 \plus{} b^2 \plus{}c^2$ for some natural numbers $ a,b,c$. Prove that \[ 9n \equal{} (p_1a\plus{}q_1b\plus{}r_1c)^2 \plus{} (p_2a\plus{}q_2b\plus{}r_2c)^2 \plus{} (p_3a\plus{}q_3b\plus{}r_3c)^2\] where $ p_j$'s , $ q_j$'s , $ r_j$'s are all [b]nonzero[/b] integers. Further, if $ 3$ does [b]not[/b] divide at least one of $ a,b,c,$ prove that $ 9n$ can be expressed in the form $ x^2\plus{}y^2\plus{}z^2$, where $ x,y,z$ are natural numbers [b]none[/b] of which is divisible by $ 3$.

The incircle of $ABC$ touches $AB$ at $M$. $N$ is any point on the segment $BC$. Show that the incircles of $AMN, BMN, ACN$ have a common tangent.

Let ABC be a triangle and $M$ be an interior point. Prove that \[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]

A unit hexagon is composed of a regular haxagon of side length 1 and its equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon? [asy] defaultpen(linewidth(0.7)); draw(polygon(3)); pair D=origin+1*dir(270), E=origin+1*dir(150), F=1*dir(30); draw(D--E--F--cycle);[/asy] $\textbf{(A)}\: 1:1\qquad \textbf{(B)}\: 6:5\qquad \textbf{(C)}\: 3:2\qquad \textbf{(D)}\: 2:1\qquad \textbf{(E)}\: 3:1\qquad $

In equilateral triangle $ABC$ with side length $2$, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_1$ and $B_2$, respectively. Finally, let the parabola with focus $C$ and directrix $AB$ intersect sides $CA$ and $C_B$ at $C_1$ and $C_2$, respectively. Find the perimeter of the triangle formed by lines $A_1A_2$, $B_1B_2$, $C_1C_2$.

Find all sets of natural numbers $(a, b, c)$ such that $$a+1|b^2+c^2\,\, , b+1|c^2+a^2\,\,, c+1|a^2+b^2.$$

Points $A_1,B_1,C_1$ are selected on the sides $BC$,$CA$,$AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$, $A_1BC_1$, $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.

For $k=1,2,\ldots ,2011$ we denote $S_k=\frac{1}{k}+\frac{1}{k+1}+\cdots +\frac{1}{2011}$. Compute the sum $S_1+S_1^2+S_2^2+\cdots +S_{2011}^2$.

Let $O$ be the circumcenter of triangle $ABC$: Points $D$ and $E$ are chosen at sides $AB$ and $AC$ respectively such that $\angle ADO = \angle AEO = 60^o$ and $BDEC$ is inscribed quadrangle. Prove or disprove that $ABC$ is isosceles triangle.

Given a real number $p>1$, a continuous function $h\colon [0,\infty)\to [0,\infty)$, and a smooth vector field $Y\colon \mathbb{R}^n \to \mathbb{R}^n$ with $\mathrm{div}~Y=0$, prove the following inequality \[\int_{\mathbb{R}^n}h(|x|)|x|^{p}\leq \int_{\mathbb{R}^{n}}h(|x|)|x+Y(x)|^{p}.\]

A sequence $P_1, \dots, P_n$ of points in the plane (not necessarily diferent) is [i]carioca[/i] if there exists a permutation $a_1, \dots, a_n$ of the numbers $1, \dots, n$ for which the segments $$P_{a_1}P_{a_2}, P_{a_2}P_{a_3}, \dots, P_{a_n}P_{a_1}$$ are all of the same length. Determine the greatest number $k$ such that for any sequence of $k$ points in the plane, $2023-k$ points can be added so that the sequence of $2023$ points is [i]carioca[/i].

Let $ ABC$ be a triangle with $ AB \equal{} AC$. A ray $ Ax$ is constructed in space such that the three planar angles of the trihedral angle $ ABCx$ at its vertex $ A$ are equal. If a point $ S$ moves on $ Ax$, find the locus of the incenter of triangle $ SBC$.

Let $f : Z_{>0} \to Z_{>0}$ be a function which satisfies $k|f^k(x)-x$ for all $k, x \in Z_{>0}$ and $f(x)-x \le 2023$. If $f(1) = 2000$, what can $f$ be? [i]Remark[/i]: Here, $f^k (x)$ denotes the $k$-fold application of $f$ to $x$.

In triangle $ABC$, the altitude from $B$ is tangent to the circumcircle of $ABC$. Prove that the largest angle of the triangle is between $90^o$ and $135^o$. If the altitudes from both $B$ and from $C$ are tangent to the circumcircle, then what are the angles of the triangle?

Find all polynomials $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients.

IF $ a$, $ b$ and $ c$ are positive reals such that $ a^{2}\plus{}b^{2}\plus{}c^{2}\equal{}1$ prove the inequality: \[ \frac{a^{5}\plus{}b^{5}}{ab(a\plus{}b)}\plus{} \frac {b^{5}\plus{}c^{5}}{bc(b\plus{}c)}\plus{}\frac {c^{5}\plus{}a^{5}}{ca(a\plus{}b)}\geq 3(ab\plus{}bc\plus{}ca)\minus{}2.\]