Given a $\triangle ABC$, determine which is the circle with the smallest area that contains it.

Real numbers $w$, $x$, $y$, and $z$ satisfy $w+x+y = 3$, $x+y+z = 4,$ and $w+x+y+z = 5$. What is the value of $x+y$? $\textbf{(A) }-\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }\frac{3}{2}\qquad\textbf{(D) }2\qquad\textbf{(E) }3$

Let $\tau(n)$ be the number of positive divisors of $n$, let $f(n) = \sum_{d \mid n} \tau(d)$, and let $g(n) = \sum_{d \mid n} f(d)$. Let $P_n$ be the product of the first $n$ prime numbers, and let $M = P_1 P_2 \cdots P_{2021}$. Then $\sum_{d \mid M} \frac{1}{g(d)} = \frac{a}{b}$, where $a, b$ are relatively prime positive integers. What is the remainder when $\tau(ab)$ is divided by $2017$? (Here, $\sum_{d \mid n}$ means a sum over the positive divisors of $n$.)

The [i]runs[/i] of a decimal number are its increasing or decreasing blocks of digits. Thus $024379$ has three [i]runs[/i] : $024, 43$, and $379$. Determine the average number of runs for a decimal number in the set $\{d_1d_2 \cdots d_n | d_k \neq d_{k+1}, k = 1, 2,\cdots, n - 1\}$, where $n \geq 2.$

From a $8\times 7$ rectangle divided into unit squares, we cut the corner, which consists of the first row and the first column. (that is, the corner has $14$ unit squares). For the following, when we say corner we reffer to the above definition, along with rotations and symmetry. Consider an infinite lattice of unit squares. We will color the squares with $k$ colors, such that for any corner, the squares in that corner are coloured differently (that means that there are no squares coloured with the same colour). Find out the minimum of $k$.

The three digit number $n=CCA$ (in base $10$), where $C\neq A$, is divisible by $14$. How many possible values for $n$ are there? [i]2016 CCA Math Bonanza Individual #4[/i]

Given a set of $4n$ positive numbers such that any distinct choice of ordered foursomes of these numbers constitutes a geometric progression. Prove that at least $4$ numbers of the set are identical.

a) Year 1872 Texas 3 gold miners found a peice of gold. They have a coin that with possibility of $\frac 12$ it will come each side, and they want to give the piece of gold to one of themselves depending on how the coin will come. Design a fair method (It means that each of the 3 miners will win the piece of gold with possibility of $\frac 13$) for the miners. b) Year 2005, faculty of Mathematics, Sharif university of Technolgy Suppose $0<\alpha<1$ and we want to find a way for people name $A$ and $B$ that the possibity of winning of $A$ is $\alpha$. Is it possible to find this way? c) Year 2005 Ahvaz, Takhti Stadium Two soccer teams have a contest. And we want to choose each player's side with the coin, But we don't know that our coin is fair or not. Find a way to find that coin is fair or not? d) Year 2005,summer In the National mathematical Oympiad in Iran. Each student has a coin and must find a way that the possibility of coin being TAIL is $\alpha$ or no. Find a way for the student.

In triangle $ ABC$, $ \overline{CA} \equal{} \overline{CB}$. On $ CB$ square $ BCDE$ is constructed away from the triangle. If $ x$ is the number of degrees in angle $ DAB$, then $ \textbf{(A)}\ x\text{ depends upon triangle }ABC \qquad\textbf{(B)}\ x\text{ is independent of the triangle}$ $ \textbf{(C)}\ x\text{ may equal angle }CAD \qquad\textbf{(D)}\ x\text{ can never equal angle }CAB$ $ \textbf{(E)}\ x\text{ is greater than }45^{\circ}\text{ but less than }90^{\circ}$

A rabbit, a skunk and a turtle are running a race. The skunk finishes the race in 6 minutes. The rabbit runs 3 times as quickly as the skunk. The rabbit runs 5 times as quickly as the turtle. How long does the turtle take to finish the race?

Compute the remainder when $\binom{205}{101}$ is divded by $101 \times 103$. [i]Proposed by Brandon Xu[/i]

Let $a,b,c$ and $d$ be positive real numbers such that $a^2+b^2+c^2+d^2 \ge 4$. Prove that $$(a+b)^3+(c+d)^3+2(a^2+b^2+c^2+d^2) \ge 4(ab+bc+cd+da+ac+bd)$$

Consider a circle $(O)$ and two fixed points $B,C$ on $(O)$ such that $BC$ is not the diameter of $(O)$. $A$ is an arbitrary point on $(O)$, distinct from $B,C$. Let $D,J,K$ be the midpoints of $BC,CA,AB$, respectively, $E,M,N$ be the feet of perpendiculars from $A$ to $BC$, $B$ to $DJ$, $C$ to $DK$, respectively. The two tangents at $M,N$ to the circumcircle of triangle $EMN$ meet at $T$. Prove that $T$ is a fixed point (as $A$ moves on $(O)$).

Let $P$ be a point inside or outside (but not on) of a triangle $ABC$. Prove that $PA +PB +PC$ is greater than half of the perimeter of the triangle

Find the median of the positive divisors of $6^4-1$.

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.