Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$. Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined.

Let $f(x)=\frac{P(x)}{Q(x)}$, where $P(x), Q(x)$ are two non-constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ for infinitely many values of $x$. a) Show that $\text{deg}(P)<\text{deg}(Q)$. b) Show that $P'(1)=2Q'(1)-\text{deg}(Q)\cdot Q(1)$. Here, $P'(x)$ denotes the derivative of $P(x)$ as usual.

Let $a$ be a fixed positive integer and $(e_n)$ the sequence, which is defined by $e_0=1$ and $$ e_n=a + \prod_{k=0}^{n-1} e_k$$ for $n \geq 1$. Prove that (a) There exist infinitely many prime numbers that divide one element of the sequence. (b) There exists one prime number that does not divide an element of the sequence. (Theresia Eisenkölbl)

Solve the quasi-homogeneous equation \[\frac{dy}{dx}=x+\frac{x^3}{y}\]

Write either $1$ or $-1$ in each of the cells of a $(2n) \times (2n)$-table, in such a way that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$. Determine the largest possible value of $M$.

Let $a$ and $b$ be real numbers with $a<b,$ and let $f$ and $g$ be continuous functions from $[a,b]$ to $(0,\infty)$ such that $\int_a^b f(x)\,dx=\int_a^b g(x)\,dx$ but $f\ne g.$ For every positive integer $n,$ define \[I_n=\int_a^b\frac{(f(x))^{n+1}}{(g(x))^n}\,dx.\] Show that $I_1,I_2,I_3,\dots$ is an increasing sequence with $\displaystyle\lim_{n\to\infty}I_n=\infty.$

In a circle we enscribe a regular $2001$-gon and inside it a regular $667$-gon with shared vertices. Prove that the surface in the $2001$-gon but not in the $667$-gon is of the form $k.sin^3\left(\frac{\pi}{2001}\right).cos^3\left(\frac{\pi}{2001}\right)$ with $k$ a positive integer. Find $k$.

Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]

Let $C_1$ be a circle inside the circle $C_2$ and let $P$ in the interior of $C_1$, $Q$ in the exterior of $C_2$. One draws variable lines $l_i$ through $P$, not passing through $Q$. Let $l_i$ intersect $C_1$ in $A_i,B_i$, and let the circumcircle of $QA_iB_i$ intersect $C_2$ in $M_i,N_i$. Show that all lines $M_i,N_i$ are concurrent.

Let $ABC$ be a triangle with incenter $I$. Denote the midpoints of $AI$, $AC$ and $CI$ by $L$, $M$ and $N$ respectively. Point $D$ lies on segment $AM$ such that $BC= BD$. Let the incircle of triangle $ABD$ be tangent to $AD$ and $BD$ at $E$ and $F$ respectively. Denote the circumcenter of triangle $AIC$ by $J$, and the circumcircle of triangle $JMD$ by $\omega$. Lines $MN$ and $JL$ meet $\omega$ again at $P$ and $Q$ respectively. Prove that $PQ$, $LN$ and $EF$ are concurrent.

Andy flips a strange coin for which the probability of flipping heads is $\frac{1}{2^k+1}$, where $k$ is the number of heads that appeared previously. If Andy flips the coin repeatedly until he gets heads 10 times, what is the expected number of total flips he performs? [i]Proposed by Harry Kim[/i]

Let two chords $AC$ and $BD$ of a circle $k$ meet at the point $K$, and let $O$ be the center of $k$. Let $M$ and $N$ be the circumcenters of triangles $AKB$ and $CKD$. Show that the quadrilateral $OMKN$ is a parallelogram.

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?

Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital $\textit{every}$ month, and having half the women work while only $2$ of the $7$ men work on some months means that some of the women risk getting burned out. After all, nearly all the members of the volunteer group have other jobs. Hannah comes up with a plan that the committee likes. Beginning in October, the comittee of five volunteer teachers will consist of any five members of the volunteer group, so long as there is at least one woman and at least one man teaching each month. Under this new plan, what is the least number of months that $\textit{must}$ go by (including October when the first set of five teachers is selected, but not September) such that some five-member comittee $\textit{must have}$ taught together twice (all five members are the same during two different months)?

$2n$ clubs want to play a league. Each club must play every other club exactly once. Each club is only allowed to play one game per day. Prove that the competition can be completed in $2n - 1$ days.

The longest diagonal of a convex hexagon is $2$. Is there necessarily a side or diagonal in this hexagon whose length does not exceed $1$?

Pete puts 100 stones in a row: black one, white one, black one, white one, ..., black one, white one. In a single move either Pete chooses two black stones with only white stones between them, and repaints all these white stones in black, or Pete chooses two white stones with only black stones between them, and repaints all these black stones in white. Can Pete with a sequence of moves described above obtain a row of 50 black stones followed by 50 white stones? Egor Bakaev

The rectangle $ABCD$ lies inside a circle. The rays $BA$ and $DA$ meet this circle at points $A_1$ and $A_2$. Let $A_0$ be the midpoint of $A_1A_2$. Points $B_0$, $C_0, D_0$ are defined similarly. Prove that $A_0C_0 = B_0D_0$.

Find all integers $n$ such that $1<n<10^6$ and $n^3-1$ is divisible by $10^6 n-1$.

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.

Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.

Let $[n]$ denote the set of integers $0, 1, \ldots, n-1.$ Let $\omega_n=e^{2\pi i/n}.$ Let $$f(n) = \prod_{\overset{i \in [n]}{\gcd(i,n)=1}} \prod_{\overset{j \in [n]}{\gcd(j,n)=1}} (\omega_n^i - \omega_n^j).$$ Then, $f(2024)=2^{e_1} \cdot 11^{e_2} \cdot 23^{e_3}$ for positive integers $e_1, e_2, e_3.$ Find $e_1+e_2+e_3.$

If $M$ is $30 \%$ of $Q$, $Q$ is $20 \%$ of $P$, and $N$ is $50 \%$ of $P$, then $\frac{M}{N} =$ $\textbf{(A)}\ \frac{3}{250} \qquad \textbf{(B)}\ \frac{3}{25} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ \displaystyle \frac{6}{5} \qquad \textbf{(E)}\ \displaystyle \frac{4}{3}$

$$Problem 1$$ ;Find all composite numbers $n$ with the following property: For every proper divisor $d$ of $n$ (i.e. $1 < d < n$), it holds that $n-12 \geq d \geq n-20$.

The value of $ x \minus{} y^{x \minus{} y}$ when $ x \equal{} 2$ and $ y \equal{} \minus{}2$ is: $ \textbf{(A)}\ \minus{}18 \qquad \textbf{(B)}\ \minus{}14\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 256$