Olympiad problems

1959 AMC 12/AHSME, 40

In triangle $ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals: $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ \text{none of these} $

1996 Moldova Team Selection Test, 5

Tags: algebra , polynomial

Find all polynomials $P(X)$ of fourth degree with real coefficients that verify the properties: [b]a)[/b] $P(-x)=P(x), \forall x\in\mathbb{R};$ [b]b)[/b] $P(x)\geq0, \forall x\in\mathbb{R};$ [b]c)[/b] $P(0)=1;$ [b]d)[/b] $P(X)$ has exactly two local minimums $x_1$ and $x_2$ such that $|x_1-x_2|=2.$

1940 Putnam, B5

Tags: polynomial , rational , algebra , root

Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer

2016 Korea Summer Program Practice Test, 3

Tags: number theory , prime number , prime , primitive root , decimal representation

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

2011 Kyiv Mathematical Festival, 3

Tags: cut , quadrilateral , geometry

Quadrilateral can be cut into two isosceles triangles in two different ways. a) Can this quadrilateral be nonconvex? b) If given quadrilateral is convex, is it necessarily a rhomb?

2013 District Olympiad, 4

Tags: trigonometry , modular arithmetic , induction , algebra

Let $n\in {{\mathbb{N}}^{*}}$. Prove that $2\sqrt{{{2}^{n}}}\cos \left( n\arccos \frac{\sqrt{2}}{4} \right)$ is an odd integer.

1988 Romania Team Selection Test, 10

Tags: polynomial , algebra

Let $ p > 2$ be a prime number. Find the least positive number $ a$ which can be represented as \[ a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X), \] where $ f(X)$ and $ g(X)$ are integer polynomials. [i]Mircea Becheanu[/i].

2012 Balkan MO, 3

Tags: induction , floor function , combinatorics , logarithm , inequalities

Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$ Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$

2014 India IMO Training Camp, 2

Tags: combinatorics

Let $n$ be a natural number.A triangulation of a convex n-gon is a division of the polygon into $n-2$ triangles by drawing $n-3$ diagonals no two of which intersect at an interior point of the polygon.Let $f(n)$ denote the number of triangulations of a regular n-gon such that each of the triangles formed is isosceles.Determine $f(n)$ in terms of $n$.

LMT Accuracy Rounds, 2022 S5

Tags: combinatorics

A bag contains $5$ identical blue marbles and $5$ identical green marbles. In how many ways can $5$ marbles from the bag be arranged in a row if each blue marble must be adjacent to at least $1$ green marble?

1989 Nordic, 3

Tags: elements of set , sequence , algebra

Let $S$ be the set of all points $t$ in the closed interval $[-1, 1]$ such that for the sequence $x_0, x_1, x_2, ...$ defined by the equations $x_0 = t, x_{n+1} = 2x_n^2-1$, there exists a positive integer $N$ such that $x_n = 1$ for all $n \ge N$. Show that the set $S$ has infinitely many elements.

1980 IMO, 11

Tags: geometry

A triangle $(ABC)$ and a point $D$ in its plane satisfy the relations \[\frac{BC}{AD}=\frac{CA}{BD}=\frac{AB}{CD}=\sqrt{3}.\] Prove that $(ABC)$ is equilateral and $D$ is its center.

2022 Taiwan TST Round 1, N

Tags: number theory

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

1983 Tournament Of Towns, (043) A5

Tags: coloring , combinatorics , combinatorial geometry , regular polygon

$k$ vertices of a regular $n$-gon $P$ are coloured. A colouring is called almost uniform if for every positive integer $m$ the following condition is satisfied: If $M_1$ is a set of m consecutive vertices of $P$ and $M_2$ is another such set then the number of coloured vertices of $M_1$ differs from the number of coloured vertices of $M_2$ at most by $1$. Prove that for all positive integers $k$ and $n$ ($k \le n$) an almost uniform colouring exists and that it is unique within a rotation. (M Kontsevich, Moscow)

2020 Online Math Open Problems, 30

Tags:

Let $c$ be the smallest positive real number such that for all positive integers $n$ and all positive real numbers $x_1$, $\ldots$, $x_n$, the inequality \[ \sum_{k=0}^n \frac{(n^3+k^3-k^2n)^{3/2}}{\sqrt{x_1^2+\dots +x_k^2+x_{k+1}+\dots +x_n}} \leq \sqrt{3}\left(\sum_{i=1}^n \frac{i^3(4n-3i+100)}{x_i}\right)+cn^5+100n^4 \] holds. Compute $\lfloor 2020c \rfloor$. [i]Proposed by Luke Robitaille[/i]

2012 Vietnam National Olympiad, 2

Tags: quadratic , polynomial , arithmetic sequence , algebra

Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.

2015 Harvard-MIT Mathematics Tournament, 3

Tags:

Let $ABCD$ be a quadrilateral with $\angle BAD = \angle ABC = 90^{\circ}$, and suppose $AB=BC=1$, $AD=2$. The circumcircle of $ABC$ meets $\overline{AD}$ and $\overline{BD}$ at point $E$ and $F$, respectively. If lines $AF$ and $CD$ meet at $K$, compute $EK$.

2022 Portugal MO, 4

Tags: median , geometry , angle

Let $[AD]$ be a median of the triangle $[ABC]$. Knowing that $\angle ADB = 45^o$ and $\angle A CB = 30^o$, prove that $\angle BAD = 30^o$.

2015 ASDAN Math Tournament, 1

Tags:

Let $a_n$ be a sequence defined as $a_1=1$, $a_2=2$, and $a_n=a_{n-1}-a_{n-2}$. Compute $a_{2015}$.

2005 Grigore Moisil Urziceni, 1

Tags: equation , system of equations , algebra

Find the nonnegative real numbers $ a,b,c,d $ that satisfy the following system: $$ \left\{ \begin{matrix} a^3+2abc+bcd-6&=&a \\a^2b+b^2c+abd+bd^2&=&b\\a^2b+a^2c+bc^2+cd^2&=&c\\d^3+ab^2+abc+bcd-6&=&d \end{matrix} \right. $$

2014 AIME Problems, 11

Tags: analytic geometry , probability , symmetry , rotation , geometry

A token starts at the point $(0,0)$ of an $xy$-coordinate grid and them makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of $|y|=|x|$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2017 Junior Balkan Team Selection Tests - Romania, 1

Tags: inequalities , algebra

If $a, b, c \in [-1, 1]$ satisfy $a + b + c + abc = 0$, prove that $a^2 + b^2 + c^2 \ge 3(a + b + c)$ . When does the equality hold?

PEN A Problems, 72

Tags: modular arithmetic , number theory , divisibility theory

Determine all pairs $(n,p)$ of nonnegative integers such that [list] [*] $p$ is a prime, [*] $n<2p$, [*] $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. [/list]

2024 Mathematical Talent Reward Programme, 2

Tags: algebra , inequalities

Find positive reals $a,b,c$ such that: $$\sqrt{\frac{a}{b+c}} + \sqrt{\frac{b}{c+a}} + \sqrt{\frac{c}{a+b}} = 2$$

2015 District Olympiad, 2

Tags: number theory

[b]a)[/b] Show that if two non-negative integers $ p,q $ satisfy the property that both $ \sqrt{2p-q} $ and $ \sqrt{2p+q} $ are non-negative integers, then $ q $ is even. [b]b)[/b] Determine how many natural numbers $ m $ are there such that $ \sqrt{2m-4030} $ and $ \sqrt{2m+4030} $ are both natural.