Olympiad problems

2022 LMT Fall, 1

Tags: number theory

Let $x$ be the positive integer satisfying $5^2 +28^2 +39^2 = 24^2 +35^2 + x^2$. Find $x$.

2017 Iran Team Selection Test, 6

Tags: algebra , number theory , Iran , Iranian TST

Let $k>1$ be an integer. The sequence $a_1,a_2, \cdots$ is defined as: $a_1=1, a_2=k$ and for all $n>1$ we have: $a_{n+1}-(k+1)a_n+a_{n-1}=0$ Find all positive integers $n$ such that $a_n$ is a power of $k$. [i]Proposed by Amirhossein Pooya[/i]

2005 Federal Math Competition of S&M, Problem 3

Tags: functional equation , fe , algebra , polynomial

Determine all polynomials $p$ with real coefficients for which $p(0)=0$ and $$f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,$$where $f(n)=\lfloor p(n)\rfloor$.

1940 Moscow Mathematical Olympiad, 059

Tags: Digit , number theory , Integer sequence

Consider all positive integers written in a row: $123456789101112131415...$ Find the $206788$-th digit from the left.

1984 Bundeswettbewerb Mathematik, 4

Tags: geometry , 3D geometry , sphere , geometry solved

A sphere is touched by all the four sides of a (space) quadrilateral. Prove that all the four touching points are in the same plane.

2021 JHMT HS, 5

Tags: general , 2021

Terry decides to practice his arithmetic by adding the numbers between $10$ and $99$ inclusive. However, he accidentally swaps the digits of one of the numbers, and thus gets the incorrect sum of $4941.$ What is the largest possible number whose digits Terry could have swapped in the summation?

2008 China Team Selection Test, 2

Tags: induction , number theory , algebra , Sequence , recurrence relation

The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.

2012 HMNT, 3

Tags: geometry

$ABCD$ is a rectangle with $AB = 20$ and $BC = 3$. A circle with radius $5$, centered at the midpoint of $DC$, meets the rectangle at four points: $W, X, Y$ , and $Z$. Find the area of quadrilateral $WXY Z$.

2018 VTRMC, 3

Tags: function , algebra

Prove that there is no function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $f(f(n))=n+1.$ Here $\mathbb{N}$ is the positive integers $\{1,2,3,\dots\}.$

2019 Harvard-MIT Mathematics Tournament, 10

Tags: HMMT , algebra

Prove that for all positive integers $n$, all complex roots $r$ of the polynomial \[P(x) = (2n)x^{2n} + (2n-1)x^{2n-1} + \dots + (n+1)x^{n+1} + nx^n + (n+1)x^{n-1} + \dots + (2n-1)x + 2n\] lie on the unit circle (i.e. $|r| = 1$).

II Soros Olympiad 1995 - 96 (Russia), 11.1

Tags: algebra , logarithms

Solve the equation $$\log_{10} (x^3+x)=\log_2 x.$$

2007 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: combinatorics , Sets

Let $n > 2$ be a natural number. A subset $A$ of $R$ is said $n$-[i]small [/i]if there exist $n$ real numbers $t_1 , t_2 , ..., t_n$ such that the sets $t_1 + A , t_2 + A ,... , t_n + A$ are different . Show that $R$ can not be represented as a union of $ n - 1$ $n$-[i]small [/i] sets . Notation : if $r \in R$ and $B \subset R$ , then $r + B = \{ r + b | b \in B\}$ .

2005 Taiwan TST Round 1, 1

Tags: quadratics , algebra , polynomial , function , inequalities , complex numbers , triangle inequality

Let $f(x)=Ax^2+Bx+C$, $g(x)=ax^2+bx+c$ be two quadratic polynomial functions with real coefficients that satisfy the relation \[|f(x)| \ge |g(x)|\] for all real $x$. Prove that $|b^2-4ac| \le |B^2-4AC|.$ My solution was nearly complete...

1986 National High School Mathematics League, 4

Tags: geometry , 3D geometry , tetrahedron

None face of a tetrahedron is isosceles triangle. How many kinds of lengths of edges do the tetrahedron have at least? $\text{(A)}3\qquad\text{(B)}4\qquad\text{(C)}5\qquad\text{(D)}6$

1991 India Regional Mathematical Olympiad, 3

Tags:

A four-digit number has the following properties: (a) It is a perfect square; (b) Its first two digits are equal (c) Its last two digits are equal. Find all such four-digit numbers.

2018 ASDAN Math Tournament, 3

Tags: Geometry Tiebreaker , 2018

In $\vartriangle ABC$, $AC > AB$. $B$ is reflected across $\overline{AC}$ to a point $D$, and $C$ is reflected across $\overline{AD}$ to a point $E$. Suppose that $AC = 6\sqrt3 + 6$, $BC = 6$, and $\overline{BC} \parallel \overline{AE}$. Compute $AB$.

2014 Benelux, 3

Tags: inequalities , number theory unsolved , number theory

For all integers $n\ge 2$ with the following property: [list] [*] for each pair of positive divisors $k,~\ell <n$, at least one of the numbers $2k-\ell$ and $2\ell-k$ is a (not necessarily positive) divisor of $n$ as well.[/list]

2020 Baltic Way, 9

Tags: combinatorics , combinatorics proposed

Each vertex $v$ and each edge $e$ of a graph $G$ are assigned numbers $f(v)\in\{1,2\}$ and $f(e)\in\{1,2,3\}$, respectively. Let $S(v)$ be the sum of numbers assigned to the edges incident to $v$ plus the number $f(v)$. We say that an assignment $f$ is [i]cool [/i]if $S(u) \ne S(v)$ for every pair $(u,v)$ of adjacent (i.e. connected by an edge) vertices in $G$. Prove that for every graph there exists a cool assignment.

2020 BMT Fall, 23

Tags: trigonometry , algebra

Let $0 < \theta < 2\pi$ be a real number for which $\cos (\theta) + \cos (2\theta) + \cos (3\theta) + ...+ \cos (2020\theta) = 0$ and $\theta =\frac{\pi}{n}$ for some positive integer $n$. Compute the sum of the possible values of $n \le 2020$.

1984 IMO Longlists, 2

Tags: modular arithmetic , geometry unsolved , geometry

Given a regular convex $2m$- sided polygon $P$, show that there is a $2m$-sided polygon $\pi$ with the same vertices as $P$ (but in different order) such that $\pi$ has exactly one pair of parallel sides.

2006 Flanders Math Olympiad, 2

Tags: geometry , geometric transformation , reflection , trigonometry , algebra , linear equation

Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point on $\left[AB\right]$. $Q$ is the point on $BC$ such that $PQ$ is perpendicular to $AB$. $R$ is the point on $AC$ such that $QR$ is perpendicular to $BC$. And $S$ is the point on $AB$ such that $RS$ is perpendicular to $AC$. $Q'$ is the point on $BC$ such that $PQ'$ is perpendicular to $BC$. $R'$ is the point on $AC$ such that $Q'R'$ is perpendicular to $AC$. And $S'$ is the point on $AB$ such that $R'S'$ is perpendicular to $AB$. Determine $\frac{|PB|}{|AB|}$ if $S=S'$.

2004 All-Russian Olympiad, 3

Tags: induction , inequalities , combinatorics unsolved , combinatorics

In a country there are several cities; some of these cities are connected by airlines, so that an airline connects exactly two cities in each case and both flight directions are possible. Each airline belongs to one of $k$ flight companies; two airlines of the same flight company have always a common final point. Show that one can partition all cities in $k+2$ groups in such a way that two cities from exactly the same group are never connected by an airline with each other.

2005 Harvard-MIT Mathematics Tournament, 1

Tags: symmetry

How many real numbers $x$ are solutions to the following equation? \[ |x-1| = |x-2| + |x-3| \]

1966 AMC 12/AHSME, 34

Tags: geometry , geometric transformation , rotation , AMC

Let $r$ be the speed in miles per hour at which a wheel, $11$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\tfrac{1}{4}$ of a second, the speed $r$ is increased by $5$ miles per hour. The $r$ is: $\text{(A)}\ 9\qquad \text{(B)}\ 10\qquad \text{(C)}\ 10\tfrac{1}{2}\qquad \text{(D)}\ 11\qquad \text{(E)}\ 12$

1999 Miklós Schweitzer, 1

Tags: geometry

Call a subset of the plane a circular set iff there exists a point such that for every ray starting from it, the ray intersects the subset once. show that the plane is a countable union of circular sets. [hide=idea] let H be a transcendence basis of R over Q. let $\{h_1,h_2,...\}$ be a subset of H. let $K_n$ be the field of real numbers that are algebraic over $H\setminus\{h_n\}$. $K_n\times K_n$ can be covered by a circular set $J_n$. $R\times R\subseteq \cup (K_n\times K_n) \subseteq \cup J_n \subseteq R\times R$ the first inclusion proof: x,y algebraically depend on H, so they depend on H', where H' is a finite subset of H. $\exists n$ st $h_n\notin H'$ $(x,y)\in K_n\times K_n$[/hide]