Let $f(0), f(1), f(2), \dots$ be a sequence satisfying \[ f(0) = 0 \quad \text{and} \quad f(n) = n - f(f(n-1)) \] for $n=1,2,3,\dots$. Give a formula for $f(n)$ such that its value can be immediately computed using $n$ without having to compute the previous terms.

Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$). It is known that the circumcircles of triangles $AEM$ and $CDM$ are tangent. Find the angle $\angle BMC.$

Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.

Let $f$ be a function on nonnegative integers such that $f(0)=0$ and $$f(3n+2)=f(3n+1)=f(3n)+1=3f(n)+1$$ for all integers $n \ge 0.$ Compute the sum of all nonnegative integers $m$ such that $f(m)=13.$

On the board is written a number with nine non-zero and distinct digits. Prove that we can delete at most seven digits so that the number formed by the digits left to be a perfect square.

Cities $P_1,...,P_{1025}$ are connected to each other by airlines $A_1,...,A_{10}$ so that for any two distinct cities $P_k$ and $P_m$ there is an airline offering a direct flight between them. Prove that one of the airlines can offer a round trip with an odd number of flights.

For which $n\ge 2$ is it possible to find $n$ pairwise non-similar triangles $A_1, A_2,\ldots , A_n$ such that each of them can be divided into $n$ pairwise non-similar triangles, each of them similar to one of $A_1,A_2 ,\ldots ,A_n$?

On the $1$ km long ridge of Mount SPAM, there are $2006$ lemmings. In the beginning, each of them walks along the ridge in one of the two possible directions with speed $1$ m/s . When two lemmings meet, they both reverse the directions they walk but keep their walking speed. When some lemming reaches the end of the ridge, he falls down and dies. Find the least upper bound for the time it can take until all the lemmings are dead.

In an All-Area track meet, $216$ sprinters enter a $100-$meter dash competition. The track has $6$ lanes, so only $6$ sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter? $\textbf{(A)}\mbox{ }36\qquad\textbf{(B)}\mbox{ }42\qquad\textbf{(C)}\mbox{ }43\qquad\textbf{(D)}\mbox{ }60\qquad\textbf{(E)}\mbox{ }72$

Let $H \subset P(X)$ be a system of subsets of $X$ and $\kappa > 0$ be a cardinal number such that every $x \in X$ is contained in less than $\kappa$ members of $H$. Prove that there exists an $f \colon X \rightarrow \kappa$ coloring, such that every nonempty $A \in H$ has a “unique” point, that is, an element $x \in A$ such that $f(x) \neq f(y)$ for all $x \neq y \in A$. (translated by Miklós Maróti)

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and incenter $I$. The midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$ is denoted $S$. Points $E, F$ were chosen on line $OI$ for which $BE$ and $CF$ are both perpendicular to $OI$. Point $X$ was chosen so that $XE\perp AC$ and $XF\perp AB$. Point $Y$ was chosen so that $YE\perp SC$ and $YF\perp SB$. $D$ was chosen on $BC$ so that $DI\perp BC$. Prove that $X$, $Y$, and $D$ are collinear.

Suppose that $ \frac{2}{3}$ of $ 10$ bananas are worth as much as $ 8$ oranges. How many oranges are worth as much is $ \frac{1}{2}$ of $ 5$ bananas? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ \frac{5}{2} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{7}{2} \qquad \textbf{(E)}\ 4$

$B$ is the center of unit circle. $A,C$ are points on the circle (the order of $A,B,C$ is clockwise), and $\angle ABC=2\alpha(0<\alpha<\frac{\pi}{3})$. Then we will rotate $\triangle ABC$ anticlockwise. In the first rotation, $A$ is the center of rotation, the result is that $B$ is on the circle. In the second rotation, $B$ is the center of rotation, the result is that $C$ is on the circle. In the third rotation, $C$ is the center of rotation, the result is that $A$ is on the circle. ... After we rotate for $100$ times, the distance $A$ travelled is $\text{(A)}22\pi(1+\sin\alpha)-66\alpha\qquad\text{(B)}\frac{67}{3}\pi\qquad\text{(C)}22\pi+\frac{68}{3}\pi\sin\alpha-66\alpha\qquad\text{(D)}33\pi-66\alpha$

Let $ABCDEF$ be a regular octahedron with unit side length, such that $ABCD$ is a square. Points $G, H$ are on segments $BE, DF$ respectively. The planes $AGD$ and $BCH$ divide the octahedron into three pieces, each with equal volume. Compute $BG$.

A real function has been assigned to every cell of an $n \times n$ table. Prove that a function can be assigned to each row and each column of this table such that the function assigned to each cell is equivalent to the combination of functions assigned to the row and the column containing it.

The sequence $a_0, a_1, a_2, ..$ is defined by $a_0 = a, a_{n+1} = a_n + L(a_n)$, where $L(m)$ is the last digit of $m$ (eg $L(14) = 4$). Suppose that the sequence is strictly increasing. Show that infinitely many terms must be divisible by $d = 3$. For what other d is this true?

Let $N$ be the fifth largest number that can be created by combining $2021$ $1$'s using addition, multiplication, and exponentiation, in any order (parentheses are allowed). If $f(x)=\log_2(x)$, and $k$ is the least positive integer such that $f^k(N)$ is not a power of $2$, what is the value of $f^k(N)$? (Note: $f^k(N)=f(f(\cdots(f(N))))$, where $f$ is applied $k$ times.) [i]Proposed by Adam Bertelli[/i]

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $\text{8:00 AM}$, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50\%$ of a house, quitting at $\text{4:00 PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $\text{2:12 PM}$. On Wednesday Paula worked by herself and finished the house by working until $\text{7:12 PM}$. How long, in minutes, was each day's lunch break? $ \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 42 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 60 $

Let $ \vartriangle ABC $ be a triangle with $ \angle ACB = 60º $. Let $ E $ be a point inside $ \overline {AC} $ such that $ CE <BC $. Let $ D $ over $ \overline {BC} $ such that $$ \frac {AE} {BD} = \frac {BC} {CE} -1 .$$ Let us call $ P $ the intersection of $ \overline {AD} $ with $ \overline {BE} $ and $ Q $ the other point of intersection of the circumcircles of the triangles $ AEP $ and $ BDP $. Prove that $QE \parallel BC $.

The continuous function $f(x)$ satisfies $c^2f(x + y) = f(x)f(y)$ for all real numbers $x$ and $y,$ where $c > 0$ is a constant. If $f(1) = c$, find $f(x)$ (with proof).

Let $ a \in \mathbb{R}, 0 < a < 1,$ and $ f$ a continuous function on $ [0, 1]$ satisfying $ f(0) \equal{} 0, f(1) \equal{} 1,$ and \[ f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.\] Determine $ f \left( \frac{1}{7} \right).$

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

What is the largest possible number of edges in a graph on $2n$ nodes, if there exists exactly one way to split its nodes into $n$ pairs so that the nodes from each pair are connected by an edge? [i]Proposed by Anton Trygub[/i]

Prove that a pyramid $A_1A_2 \ldots A_{2k+1}S$ with equal lateral edges and equal space angles between adjacent lateral walls is regular.

In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m \cdot n$ and $n \cdot m$ as the same product.