Let $ABC$ be a triangle, ($BC < AB$). The line $l$ passing trough the vertices $C$ and orthogonal to the angle bisector $BE$ of $\angle B$, meets $BE$ and the median $BD$ of the side $AC$ at points $F$ and $G$, respectively. Prove that segment $DF$ bisects the segment $EG$.

Find the minimum value of $\int_0^{\pi} (x-y)^2 (\sin x)|\cos x|dx$.

Let $ABC$ be a triangle with area $K$. Points $A^*$, $B^*$, and $C^*$ are chosen on $AB$, $BC$, and $CA$ respectively such that $\triangle{A^*B^*C^*}$ has area $J$. Suppose that \[\frac{AA^*}{AB}=\frac{BB^*}{BC}=\frac{CC^*}{CA}=\frac{J}{K}=x\] for some $0<x<1$. What is $x$? [i]2019 CCA Math Bonanza Lightning Round #4.3[/i]

For all valid values of $a$ and $b$, simplify the expression $$\frac{\sqrt{4b-a^2+2ab+4}+a}{\sqrt{4ab-10b^2-8}+b}.$$

A quadrilateral $ ABCD $ is inscribed in a circle. The lines $AB$ and $CD$ intersect at point $E$, and the lines $AD$ and $BC$ intersect at point $F$. The bisector of the angle $ AEC $ intersects the side $ BC $ at the point $ M $ and the side $ AD $ at the point $ N $; and the bisector of the angle $ BFD $ intersects the side $ AB $ at the point $ P $ and the side $ CD $ at the point $ Q $. Prove that the quadrilateral $MPNQ$ is a rhombus.

In a country, there are $100$ towns. Some pairs of towns are joined by roads. The roads do not intersect one another except meeting at towns. It is possible to go from any town to any other town by road. Prove that it is possible to pave some of the roads so that the number of paved roads at each town is odd.

Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then \[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \] [i]Proposed by Greece.[/i]

How many of the numbers $2,6,12,20,\ldots,14520$ are divisible by $120$? $\text{(A) }2\qquad\text{(B) }8\qquad\text{(C) }12\qquad\text{(D) }24\qquad\text{(E) }32$

Triangle $ABC$ is a triangle with side lengths $13$, $14$, and $15$. A point $Q$ is chosen uniformly at random in the interior of $\triangle{ABC}$. Choose a random ray (of uniformly random direction) with endpoint $Q$ and let it intersect the perimeter of $\triangle{ABC}$ at $P$. What is the expected value of $QP^2$? [i]2018 CCA Math Bonanza Tiebreaker Round #4[/i]

Let $ABC$ be an isosceles triangle with $CA = CB$. Let $D$ be the foot of the alttiude from $C$, and $\ell$ be the external angle bisector at $C$. Take a point $N$ on $\ell$ so that $AN> AC$ , on the same side as $A$ wrt $CD$. The bisector of the angle $NAC$ cuts $\ell$'at $F$. Show that $\angle NCD + \angle BAF> 180^o.$

Let $a_0=1,a_1=2,$ and $a_n=4a_{n-1}-a_{n-2}$ for $n\ge 2.$ Find an odd prime factor of $a_{2015}.$

The area of the ring between two concentric circles is $12\tfrac12\pi$ square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is: $\textbf{(A) }\dfrac5{\sqrt2}\qquad \textbf{(B) }5\qquad \textbf{(C) }5\sqrt2\qquad \textbf{(D) }10\qquad \textbf{(E) }10\sqrt2$

Given a board of $3 \times 3$ you want to write the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and a number in their boxes positive integer $M$, not necessarily different from the above. The goal is that the sum of the three numbers in each row be the same $a)$ Find all the values of $M$ for which this is possible. $b)$ For which of the values of $M$ found in $a)$ is it possible to arrange the numbers so that no only the three rows add the same but also the three columns add the same?

Graph $G$ is defined in 3d space. It has $e$ edges and every vertex are connected if the distance between them is $1.$ Given that there exists the Hamilton cycle, prove that for $e>1,$ we have $$\min d(v)\le 1+2\left(\frac{e}{2}\right)^{0.4}.$$

Let $RICE$ be a quadrilateral with an inscribed circle $O$ such that every side of $RICE$ is tangent to $O$. Given taht $RI=3$, $CE=8$, and $ER=7$, compute $IC$.

Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ? [i]Proposed by Hungary.[/i] [hide="Remark"]I'm not sure I'm posting this in a right Forum.[/hide]

Given a regular heptagon $A_1...A_7$. Prove that $$\frac{1}{|A_1A_5|} + \frac{1}{|A_1A_3| }= \frac{1}{|A_1A_7|}$$.

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.

Let $ K$ be the circumscribed circle of the trapezoid $ ABCD$ . In this trapezoid the diagonals $ AC$ and $ BD$ are perpendicular. The parallel sides $ AB\equal{}a$ and $ CD\equal{}c$ are diameters of the circles $ K_{a}$ and $ K_{b}$ respectively. Find the perimeter and the area of the part inside the circle $ K$, that is outside circles $ K_{a}$ and $ K_{b}$.

Suppose $a$, $b$ and $c$ are nonzero real numberss satisfying $abc=2$. Prove that among the three numbers $2a-\frac{1}{b}$, $2b-\frac{1}{c}$ and $2c-\frac{1}{a}$, at most two of them are greater than $2$.

In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. For what is the smallest amount of rubles you can rearrange chips in reverse order?

Let $\mathcal E$ be an ellipse with foci $F_1$ and $F_2$. Parabola $\mathcal P$, having vertex $F_1$ and focus $F_2$, intersects $\mathcal E$ at two points $X$ and $Y$. Suppose the tangents to $\mathcal E$ at $X$ and $Y$ intersect on the directrix of $\mathcal P$. Compute the eccentricity of $\mathcal E$. (A [i]parabola[/i] $\mathcal P$ is the set of points which are equidistant from a point, called the [i]focus[/i] of $\mathcal P$, and a line, called the [i]directrix[/i] of $\mathcal P$. An [i]ellipse[/i] $\mathcal E$ is the set of points $P$ such that the sum $PF_1 + PF_2$ is some constant $d$, where $F_1$ and $F_2$ are the [i]foci[/i] of $\mathcal E$. The [i]eccentricity[/i] of $\mathcal E$ is defined to be the ratio $F_1F_2/d$.)

Two circles with radii $R$ and $r$ intersect at $C$ and $D$ and are tangent to a line $\ell$ at $A$ and $B$. Prove that the circumradius of triangle $ABC$ does not depend on the length of segment $AB$.

Determine the largest integer $n$ such that there exist monic quadratic polynomials $p_1(x)$, $p_2(x)$, $p_3(x)$ with integer coefficients so that for all integers $ i \in [1, n]$ there exists some $j \in [1, 3]$ and $m \in Z$ such that $p_j (m) = i$.

Let $a,b,c$ be real numbers satisfying: \[ab-a=b+119\] \[bc-b=c+59\] \[ca-c=a+71\] Determine all possible values of $a+b+c$.