Felix picks four points uniformly at random inside a unit circle $\mathcal{C}$. He then draws the four possible triangles which can be formed using these points as vertices. Finally, he randomly chooses of the six possible pairs of the triangles he just drew. What is the probability that the center of the circle $\mathcal{C}$ is contained in the union of the interiors of the two triangles that Felix chose?

Three circles with radii $1$, $2$, and $3$ are pairwise tangent to each other. Find the radius of the circle that is externally tangent to all three of these circles. [i]Proposed by Tamar Turashvili, Georgia [/i]

Let $a_1, a_2, \cdots, a_n$ be a sequence of real numbers with $a_1+a_2+\cdots+a_n=0$. Define the score $S(\sigma)$ of a permutation $\sigma=(b_1, \cdots b_n)$ of $(a_1, \cdots a_n)$ to be the minima of the sum $$(x_1-b_1)^2+\cdots+(x_n-b_n)^2$$ over all real numbers $x_1\le \cdots \le x_n$. Prove that $S(\sigma)$ attains the maxima over all permutations $\sigma$, if and only if for all $1\le k\le n$, $$b_1+b_2+\cdots+b_k\ge 0.$$ [i]Proposed by Anzo Teh Zhao Yang[/i]

Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$

Let $a < b < c < d < e$ be positive integers. Prove that $$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$ where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?

Suppose that $P(x)$ is a polynomial with the property that there exists another polynomial $Q(x)$ to satisfy $P(x)Q(x)=P(x^2)$. $P(x)$ and $Q(x)$ may have complex coefficients. If $P(x)$ is a quintic with distinct complex roots $r_1,\dots,r_5$, find all possible values of $|r_1|+\dots+|r_5|$.

Facundo and Luca have been given a cake that is shaped like the quadrilateral in the figure. [img]https://cdn.artofproblemsolving.com/attachments/3/2/630286edc1935e1a8dd9e704ed4c813c900381.png[/img] They are going to make two straight cuts on the cake, thus obtaining $4$ portions in the shape of a quadrilateral. Then Facundo will be left with two portions that do not share any side, the other two will be for Luca. Show how they can cut the cuts so that both children get the same amount of cake. Justify why cutting in this way achieves the objective.

Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?

Let $ABC$ be a triangle with $AB<AC<BC$ with circumcircle $\Gamma_1$. The circle $\Gamma_2$ has center $D$ lying on $\Gamma_1$ and touches $BC$ at $E$ and the extension of $AB$ at $F$. Let $\Gamma_1$ and $\Gamma_2$ meet at $K, G$ and the line $KG$ meets $EF$ and $CD$ at $M, N$. Show that $BCNM$ is cyclic.

A tortoise is given an $80$-second head start in a race. When Achilles catches up to where the tortoise was when he (Achilles) began running, he finds that while he is now $40$ meters ahead of the starting line, the tortoise is now $5$ meters ahead of him. At this point, how long will it be, in seconds, before Achilles passes the tortoise? [i]2015 CCA Math Bonanza Team Round #3[/i]

A square is divided into $16$ equal squares, obtaining the set of $25$ different vertices. What is the least number of vertices one must remove from this set, so that no $4$ points of the remaining set are the vertices of any square with sides parallel to the sides of the initial square?

Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

Find all primes $p$ such that $\dfrac{2^{p+1}-4}{p}$ is a perfect square.

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

Positive integers $a, b$ satisfy equality $b^2 = a^2 + ab + b$. Prove that $b$ is a square of a positive integer. (Patrik Bak)

A cube stands on one of the squares of an infinite chessboard. On each face of the cube there is an arrow pointing in one of the four directions parallel to the sides of the face. Anton looks on the cube from above and rolls it over an edge in the direction pointed by the arrow on the top face. Prove that the cube cannot cover any $5\times 5$ square.

Find all natural solutions of $3^{x}= 2^{x}y+1.$

6) $G$ group, $G_{m}$ and $G_{n}$ commutative subgroups being the $m$ and $n$ th powers of the elements in $G$. Prove $G_{gcd(m,n)}$ is commutative.

Let $C (O)$ be a circle (with center $O$ ) and $A, B$ points on the circle with $\angle AOB = 90^o$. Circles $C_1 (O_1)$ and $C_2 (O_2)$ are tangent internally with circle $C$ at $A$ and $B$, respectively, and, also, are tangent to each other. Consider another circle $C_3 (O_3)$ tangent externally to the circles $C_1, C_2$ and tangent internally to circle $C$, located inside angle $\angle AOB$. Show that the points $O, O_1, O_2, O_3$ are the vertices of a rectangle.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Let $ \alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $ n.$ Prove that: (a) the inequality $ \alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds; (b) the above inequality is an equality for infinitely many positive integers, and (c) there exists a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i }$ goes to zero as $ i$ goes to $ \infty.$ [i]Alternative problem:[/i] Prove that there exists a sequence a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i )}$ (d) $ \infty;$ (e) an arbitrary real number $ \gamma \in (0,1)$; (f) an arbitrary real number $ \gamma \geq 0$; as $ i$ goes to $ \infty.$

A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is $\textbf{(A)}\ \text{500 thousand} \qquad \textbf{(B)}\ \text{5 million} \qquad \textbf{(C)}\ \text{50 million} \qquad \textbf{(D)}\ \text{500 million} \qquad \textbf{(E)}\ \text{5 billion}$

In triangle $ABC$ , $AA_0,BB_0,CC_0$ are the angle bisectors , $A_0,B_0,C_0$are on sides $BC,CA,AB,$ draw $A_0A_1//BB_0,A_0A_2//CC_0$ ,$A_1$ lies on $AC$ ,$A_2$ lies on $AB$ , $A_1A_2$ intersects $BC$ at $A_3$.$B_3$ ,$C_3$ are constructed similarly.Prove that : $A_3,B_3,C_3$ are collinear.

[u]Set 5[/u] [b]p13.[/b] An equiangular $12$-gon has side lengths that alternate between $2$ and $\sqrt3$. Find the area of the circumscribed circle of this $12$-gon. [b]p14.[/b] For positive integers $n$, let $\sigma(n)$ denote the number of positive integer factors of $n$. Then $\sigma(17280) = \sigma (2^7 \cdot 3^3 \cdot 5)= 64$. Let $S$ be the set of factors $k$ of $17280$ such that $\sigma(k) = 32$. If $p$ is the product of the elements of $S$, find $\sigma(p)$. [b]p15.[/b] How many odd $3$-digit numbers have exactly four $1$’s in their binary (base $2$) representation? For example, $225_{10} = 11100001_2$ would be valid. PS. You should use hide for answers.

The triangles $BCD$ and $ACE$ are externally constructed to sides $BC$ and $CA$ of a triangle $ABC$ such that $AE = BD$ and $\angle BDC+\angle AEC = 180^\circ$. Let $F$ be a point on segment $AB$ such that ${AF\over FB}={CD\over CE}$. Prove that ${DE\over CD+CE}={EF\over BC}={FD\over AC}$.