Let $O$ be the center and $R$ be the radius of circumcircle $\omega$ of triangle $ABC$. Circle $\omega_1$ with center $O_1$ and radius $R$ pass through points $A, O$ and intersects the side $AC$ at point $K$. Let $AF$ be the diameter of circle $\omega$ and points $F, K, O_1$ are collinear. Determine $\angle ABC$:

Let $ f(x)\equal{}\sum_{k\equal{}1}^n a_k x^k$ and $ g(x)\equal{}\sum_{k\equal{}1}^n \frac{a_k x^k}{2^k \minus{}1}$ be two polynomials with real coefficients. Let g(x) have $ 0,2^{n\plus{}1}$ as two of its roots. Prove That $ f(x)$ has a positive root less than $ 2^n$.

Point $I{}$ is the center of the circle inscribed in the quadrilateral $ABCD$. Prove that there is a point $K{}$ on the ray $CI$ such that $\angle KBI=\angle KDI=\angle BAI$.

The sum of several positive numbers is equal to $10$, and the sum of their squares is greater than $20$. Prove that the sum of the cubes of these numbers is greater than $40$. [i](5 points)[/i]

The function $f(x) = ax + b$ satisfies the following equalities: \begin{align*} f(f(f(1))) &= 2023, \\ f(f(f(0))) &= 1996. \end{align*} Find the value of $a$.

This problem is about pairs of consecutive whole numbers satisfying the property that one of the numbers is a perfect square and the other one is the double of a perfect square. (a) The smallest such pairs are $(0,1)$ and $(8,9)$, Indeed $0=2 \cdot 0^2$ and $1=1^2$; $8=2 \cdot 2^2$ and $9=3^2$. Show that there are infinitely many pairs of the form $(2a^2,b^2)$ where the smaller number is the double of a perfect square satisfying the given property. (b) Find a pair of integers satisfying the property that is not in the form given in the first part, that is, find a pair of integers such that the smaller one is a perfect square and the larger one is the double of a perfect square.

Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?

In the collection of all right circular cylinders of fixed volume $c$, what is the ratio $\frac hr$ of the cylinder which has the least total surface area?

[asy] draw(unitsquare);draw((0,0)--(.25,sqrt(3)/4)--(.5,0)); label("Z",(0,1),NW);label("Y",(1,1),NE);label("A",(0,0),SW);label("X",(1,0),SE);label("B",(.5,0),S);label("P",(.25,sqrt(3)/4),N); //Credit to Zimbalono for the diagram[/asy] Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. The length of the path in inches traversed by vertex $P$ is equal to $\textbf{(A) }20\pi/3\qquad\textbf{(B) }32\pi/3\qquad\textbf{(C) }12\pi\qquad\textbf{(D) }40\pi/3\qquad \textbf{(E) }15\pi$

Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process, the contents of the two bags are the same? $ \textbf{(A) } \frac 1{10} \qquad \textbf{(B) } \frac 16 \qquad \textbf{(C) } \frac 15 \qquad \textbf{(D) } \frac 13 \qquad \textbf{(E) } \frac 12$

$3n$ lines are drawn on the plane ($n > 1$), such that no two of them are parallel and no three of them are concurrent. Prove that, if $2n$ of the lines are coloured red and the other $n$ lines blue, there are at least two regions of the plane such that all of their borders are red. Note: for each region, all of its borders are contained in the original set of lines, and no line passes through the region.

The polynomial $A(x) = x^2 + ax + b$ with integer coefficients has the following property: for each prime $p$ there is an integer $k$ such that $A(k)$ and $A(k + 1)$ are both divisible by $p$. Proof that there is an integer $m$ such that $A(m) = A(m + 1) = 0$.

Find the 3-digit number whose ratio with the sum of its digits it's minimal.

For given positive integers $r, v, n$ let $S(r, v, n)$ denote the number of $n$-tuples of non-negative integers $(x_1, \cdots, x_n)$ satisfying the equation $x_1 +\cdots+ x_n = r$ and such that $x_i \leq v$ for $i = 1, \cdots , n$. Prove that \[S(r, v, n)=\sum_{k=0}^{m} (-1)^k \binom nk \binom{r - (v + 1)k + n - 1}{n-1}\] Where $m=\left\{n,\left[\frac{r}{v+1}\right]\right\}.$

What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[\text{lcm}(m,n) = 3m \times \gcd(m,n)?\]

Determine the smallest integer $n \ge 4$ for which one can choose four different numbers $a, b, c, $ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$ .

For a non-constant arithmetic progression $(a_n)$ there exists a natural $n$ such that $a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}$ . Prove that there are no zero terms in this progression.