Olympiad problems

2021 Alibaba Global Math Competition, 12

Let $A=(a_{ij})$ be a $5 \times 5$ matrix with $a_{ij}=\min\{i,j\}$. Suppose $f:\mathbb{R}^5 \to \mathbb{R}^5$ is a smooth map such that $f(\Sigma) \subset \Sigma$, where $\Sigma=\{x \in \mathbb{R}^5: xAx^T=1\}$. Denote by $f^{(n)}$ te $n$-th iterate of $f$. Prove that there does not exist $N \ge 1$ such that \[\inf_{x \in \Sigma} \| f^{(n)}(x)-x\|>0, \forall n \ge N.\]

2018 Purple Comet Problems, 4

Tags: combinatorics

The following diagram shows a grid of $36$ cells. Find the number of rectangles pictured in the diagram that contain at least three cells of the grid. [img]https://cdn.artofproblemsolving.com/attachments/a/4/e9ba3a35204ec68c17a364ebf92cc107eb4d7a.png[/img]

2022 BMT, 1

Tags: algebra , easy

Define an operation $\Diamond$ as $ a \Diamond b = 12a - 10b.$ Compute the value of $((((20 \Diamond 22) \Diamond 22) \Diamond 22) \Diamond22).$

2020 CHMMC Winter (2020-21), 3

Tags: nt

A [i]Beaver-number[/i] is a positive 5 digit integer whose digit sum is divisible by 17. Call a pair of [i]Beaver-numbers[/i] differing by exactly $1$ a [i]Beaver-pair[/i]. The smaller number in a [i]Beaver-pair[/i] is called an [i]MIT Beaver[/i], while the larger number is called a [i]CIT Beaver[/i]. Find the positive difference between the largest and smallest [i]CIT Beavers[/i] (over all [i]Beaver-pairs[/i]).

2007 F = Ma, 4

Tags:

An object is released from rest and falls a distance $h$ during the first second of time. How far will it fall during the next second of time? $ \textbf{(A)}\ h\qquad\textbf{(B)}\ 2h \qquad\textbf{(C)}\ 3h \qquad\textbf{(D)}\ 4h\qquad\textbf{(E)}\ h^2 $

2016 ASDAN Math Tournament, 27

Tags:

Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\tfrac{n}{100}$ probability of $1$ meter forward, to meter $n+1$, and a $\tfrac{100-n}{100}$ of going $1$ meter backward, to meter $n-1$. What is the expected value of the number of steps it takes for you to step off the bridge (i.e., to get to meter $0$ or $100$)? Let $C$ be the actual answer and $A$ be the answer you will submit. Your score will be given by $\max\{0,\lceil25-25|\log_6(\tfrac{A-C/2}{C/2})|^{0.8}\rceil\}$.

2010 China Western Mathematical Olympiad, 1

Tags: modular arithmetic , number theory

Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that [b](a)[/b] $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$; [b](b)[/b] $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$.

2017 Sharygin Geometry Olympiad, 8

Tags: geometry , homothety

Let $AK$ and $BL$ be the altitudes of an acute-angled triangle $ABC$, and let $\omega$ be the excircle of $ABC$ touching side $AB$. The common internal tangents to circles $CKL$ and $\omega$ meet $AB$ at points $P$ and $Q$. Prove that $AP =BQ$. [i]Proposed by I.Frolov[/i]

2021 Indonesia TST, A

Tags: inequality , inequalities , algebra

A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying $$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$ then the following inequality holds: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$ (a) Prove that $M=20-\frac{1}{20}$ is not $strong$. (b) Prove that $M=20-\frac{1}{21}$ is $strong$.

2004 Thailand Mathematical Olympiad, 3

Tags: combinatorics

$18$ students with pairwise distinct heights line up. Ideally, the teacher wants the students to be ordered by height so that the tallest student is in the back of the line. However, it turns out that this is not the case, so when the teacher sees two consecutive students where the taller of the two is in front, the two students are swapped. It turns out that $150$ swaps must be made before the students are lined up in the correct order. How many possible starting orders are there?

2024 Pan-African, 4

Tags: combinatorics

Consider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\{1, 2, \ldots, 2024\}$, and no two segments have the same length. No segment is entirely contained in another segment, but two segments may partially overlap each other. What is the maximum value of $m$?

2011 Iran MO (2nd Round), 1

Tags: algebra

find the smallest natural number $n$ such that there exists $n$ real numbers in the interval $(-1,1)$ such that their sum equals zero and the sum of their squares equals $20$.

2005 National High School Mathematics League, 5

Tags: ellipse , hyperbola , conic

Which kind of curve does the equation $\frac{x^2}{\sin\sqrt2-\sin\sqrt3}+\frac{y^2}{\cos\sqrt2-\cos\sqrt3}=1$ refer to? $\text{(A)}$ An ellipse, whose focal points are on $x$-axis. $\text{(B)}$ A hyperbola, whose focal points are on $x$-axis. $\text{(C)}$ An ellipse, whose focal points are on $y$-axis. $\text{(D)}$ A hyperbola, whose focal points are on $y$-axis.

2006 Stanford Mathematics Tournament, 15

Tags: limit , geometric series

Let $c_i$ denote the $i$th composite integer so that $\{c_i\}=4,6,8,9,...$ Compute \[\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}\] (Hint: $\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$)

1952 Miklós Schweitzer, 4

Tags: superior algebra , algebra , polynomial

Let $ K$ be a finite field of $ p$ elements, where $ p$ is a prime. For every polynomial $ f(x)\equal{}\sum_{i\equal{}0}^na_ix^i$ ($ \in K[x]$) put $ \overline{f(x)}\equal{}\sum_{i\equal{}0}^n a_ix^{p^i}$. Prove that for any pair of polynomials $ f(x),g(x)\in K[x]$, $ \overline{f(x)}|\overline{g(x)}$ if and only if $ f(x)|g(x)$.

2010 Indonesia MO, 8

Tags: circumcircle , ratio , geometry

Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram. [i]Raja Oktovin, Pekanbaru[/i]

2006 Romania National Olympiad, 1

Tags: geometry , 3d geometry , prism

We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.

2008 IMAR Test, 2

Tags: analytic geometry , prime number , number theory

A point $ P$ of integer coordinates in the Cartesian plane is said [i]visible[/i] if the segment $ OP$ does not contain any other points with integer coordinates (except its ends). Prove that for any $ n\in\mathbb{N}^*$ there exists a visible point $ P_{n}$, at distance larger than $ n$ from any other visible point. [b]Dan Schwarz[/b]

2000 Greece Junior Math Olympiad, 3

Tags:

On a past Mathematical Olympiad the maximum possible score on a problem was 5. The average score of boys was 4, the average score of girls was 3.25, and the overall average score was 3.60. Find the total number of participants, knowing that it was in the range from 31 to 50.

2021 Sharygin Geometry Olympiad, 24

Tags: geometry , 3d geometry , pyramid

A truncated trigonal pyramid is circumscribed around a sphere touching its bases at points $T_1, T_2$. Let $h$ be the altitude of the pyramid, $R_1, R_2$ be the circumradii of its bases, and $O_1, O_2$ be the circumcenters of the bases. Prove that $$R_1R_2h^2 = (R_1^2-O_1T_1^2)(R_2^2-O_2T_2^2).$$

2015 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory

Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?

2011 Iran MO (2nd Round), 2

Tags: linear algebra , matrix , trigonometry , angle bisector , geometry

In triangle $ABC$, we have $\angle ABC=60$. The line through $B$ perpendicular to side $AB$ intersects angle bisector of $\angle BAC$ in $D$ and the line through $C$ perpendicular $BC$ intersects angle bisector of $\angle ABC$ in $E$. prove that $\angle BED\le 30$.

2014-2015 SDML (High School), 8

Tags: analytic geometry

A penny is placed in the coordinate plane $\left(0,0\right)$. The penny can be moved $1$ unit to the right, $1$ unit up, or diagonally $1$ unit to the right and $1$ unit up. How many different ways are there for the penny to get to the point $\left(5,5\right)$? $\text{(A) }8\qquad\text{(B) }25\qquad\text{(C) }99\qquad\text{(D) }260\qquad\text{(E) }351$

V Soros Olympiad 1998 - 99 (Russia), 11.1

Tags: algebra

Solve the equation $$x^5 + (x + 1)^5 + (x + 2)^5 + ... + (x + 1998)^5 = 0.$$

2023 Caucasus Mathematical Olympiad, 1

Tags: number theory

Let $n{}$ and $m$ be positive integers, $n>m>1$. Let $n{}$ divided by $m$ have partial quotient $q$ and remainder $r$ (so that $n = qm + r$, where $r\in\{0,1,...,m-1\}$). Let $n-1$ divided by $m$ have partial quotient $q^{'}$ and remainder $r^{'}$. a) It appears that $q+q^{'} =r +r^{'} = 99$. Find all possible values of $n{}$. b) Prove that if $q+q^{'} =r +r^{'}$, then $2n$ is a perfect square.