Olympiad problems

2023 239 Open Mathematical Olympiad, 5

Tags: combinatorics , permutation

On a table, cards numbered $1, 2, \ldots , 200$ are laid out in a row in some order, and a line is drawn on the table between some two of them. It is allowed to swap two adjacent cards if the number on the left is greater than the number on the right. After a few such moves, the cards were arranged in ascending order. Prove we have swapped pairs of cards separated by the line no more than 1884 times.

2015 Math Prize for Girls Olympiad, 3

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Let $f$ be the cubic polynomial \[ f(x) = x^3 + bx^2 + cx + d, \] where $b$, $c$, and $d$ are real numbers. Let $x_1$, $x_2$, $\ldots\,$, $x_n$ be nonnegative numbers, and let $m$ be their average. Suppose that $m \ge - \dfrac{b}{2}\,$. Prove that \[ \sum_{i = 1}^n f(x_i) \ge n f(m). \]

2014 Contests, 1

Tags: geometry , circumcircle , perpendicular bisector

In the figure of [url]http://www.artofproblemsolving.com/Forum/download/file.php?id=50643&mode=view[/url] $\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$. The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$, and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$. If $DE \parallel O_1A$, prove that $DC \perp CO_2$.

2016 China Team Selection Test, 4

Tags: number theory

Let $a,b,b',c,m,q$ be positive integers, where $m>1,q>1,|b-b'|\ge a$. It is given that there exist a positive integer $M$ such that $$S_q(an+b)\equiv S_q(an+b')+c\pmod{m}$$ holds for all integers $n\ge M$. Prove that the above equation is true for all positive integers $n$. (Here $S_q(x)$ is the sum of digits of $x$ taken in base $q$).

2020 CHMMC Winter (2020-21), 1

Tags: geometry

A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$, and the distance from the vertex of the cone to any point on the circumference of the base is $3$, then the minimum possible volume of the cone can be written as $\frac{m\pi\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is squarefree. Find $m + n + p$.

2006 All-Russian Olympiad, 4

Tags: geometry , circumcircle

Given a triangle $ABC$. Let a circle $\omega$ touch the circumcircle of triangle $ABC$ at the point $A$, intersect the side $AB$ at a point $K$, and intersect the side $BC$. Let $CL$ be a tangent to the circle $\omega$, where the point $L$ lies on $\omega$ and the segment $KL$ intersects the side $BC$ at a point $T$. Show that the segment $BT$ has the same length as the tangent from the point $B$ to the circle $\omega$.

2003 AMC 10, 25

Tags: modular arithmetic

How many distinct four-digit numbers are divisible by $ 3$ and have $ 23$ as their last two digits? $ \textbf{(A)}\ 27 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 33 \qquad \textbf{(D)}\ 81 \qquad \textbf{(E)}\ 90$

1982 All Soviet Union Mathematical Olympiad, 336

Tags: broken line , combinatorial geometry , combinatorics

The closed broken line $M$ has odd number of vertices -- $A_1,A_2,..., A_{2n+1}$ in sequence. Let us denote with $S(M)$ a new closed broken line with vertices $B_1,B_2,...,B_{2n+1}$ -- the midpoints of the first line links: $B_1$ is the midpoint of $[A_1A_2], ... , B_{2n+1}$ -- of $[A_{2n+1}A_1]$. Prove that in a sequence $M_1=S(M), ... , M_k = S(M_{k-1}), ...$ there is a broken line, homothetic to the $M$.

2023 India IMO Training Camp, 2

Tags: combinatorics , graph theory

In a school, every pair of students are either friends or strangers. Friendship is mutual, and no student is friends with themselves. A sequence of (not necessarily distinct) students $A_1, A_2, \dots, A_{2023}$ is called [i]mischievous[/i] if $\bullet$ Total number of friends of $A_1$ is odd. $\bullet$ $A_i$ and $A_{i+1}$ are friends for $i=1, 2, \dots, 2022$. $\bullet$ Total number of friends of $A_{2023}$ is even. Prove that the total number of [i]mischievous[/i] sequences is even.

2000 National High School Mathematics League, 13

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Let $S_n=1+2+\cdots+n$ for $n\in\mathbb{N}$, find the maximum value of $f(n)=\frac{S_n}{(n+32)S_{n+1}}$.