Olympiad problems

1997 Brazil National Olympiad, 5

Tags: polynomial , algebra

Let $f(x)= x^2-C$ where $C$ is a rational constant. Show that exists only finitely many rationals $x$ such that $\{x,f(x),f(f(x)),\ldots\}$ is finite

Kyiv City MO Juniors 2003+ geometry, 2021.8.4

Tags: equal angles , angle , geometry

Let $BM$ be the median of the triangle $ABC$, in which $AB> BC$. Point $P$ is chosen so that $AB \parallel PC$ and $PM \perp BM$. Prove that $\angle ABM = \angle MBP$. (Mikhail Standenko)

2008 Singapore Junior Math Olympiad, 1

Tags: geometry , right triangle

In $\vartriangle ABC, \angle ACB = 90^o, D$ is the foot of the altitude from $C$ to $AB$ and $E$ is the point on the side $BC$ such that $CE = BD/2$. Prove that $AD + CE = AE$.

2023 Switzerland Team Selection Test, 4

Tags: geometry

Let $ABC$ and $AMN$ be two similar, non-overlapping triangles with the same orientation, such that $AB=AC$ and $AM=AN$. Let $O$ be the circumcentre of the triangle $MAB$. Prove that the points $O, C, N$ and $A$ lie on a circle if and only if the triangle $ABC$ is equilateral.

2014 India IMO Training Camp, 2

Tags: inequalities

Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.

2010 Brazil National Olympiad, 3

Tags: modular arithmetic , greatest common divisor , search , number theory

Find all pairs $(a, b)$ of positive integers such that \[ 3^a = 2b^2 + 1. \]

2012 China Western Mathematical Olympiad, 4

Tags: inequalities , circumcircle , geometry

$P$ is a point in the $\Delta ABC$, $\omega $ is the circumcircle of $\Delta ABC $. $BP \cap \omega = \left\{ {B,{B_1}} \right\}$,$CP \cap \omega = \left\{ {C,{C_1}} \right\}$, $PE \bot AC$，$PF \bot AB$. The radius of the inscribed circle and circumcircle of $\Delta ABC $ is $r,R$. Prove $\frac{{EF}}{{{B_1}{C_1}}} \geqslant \frac{r}{R}$.

1975 IMO, 1

Tags: rearrangement inequality , inequality , permutation , algebra

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

2000 All-Russian Olympiad, 1

Tags: quadratic , algebra

Let $a,b,c$ be distinct numbers such that the equations $x^2+ax+1=0$ and $x^2+bx+c=0$ have a common real root, and the equations $x^2+x+a=0$ and $x^2+cx+b$ also have a common real root. Compute the sum $a+b+c$.

2008 Princeton University Math Competition, B1

Tags: algebra

Solve for $x$: $x = 2 + \frac{4(2^6)}{11-3}$

2024 Moldova EGMO TST, 5

Tags: geometry , angle bisector , circumcircle , geometric inequality , inequalities

$AD$ Is the angle bisector Of $\angle BAC$ Where $D$ lies on the The circumcircle of $\triangle ABC$. Show that $2AD>AB+AC$

1969 IMO Longlists, 65

Tags: algebra , algebraic identities

$(USS 2)$ Prove that for $a > b^2,$ the identity ${\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+\cdots}}}}=\sqrt{a-\frac{3}{4}b^2}-\frac{1}{2}b}$

2007 AIME Problems, 8

Tags: polynomial , quadratic , number theory , greatest common divisor , algorithm , least common multiple , algebra

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$?

2021 Switzerland - Final Round, 4

Tags: algebra , inequality , 4-variable inequality , inequalities

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2011 QEDMO 9th, 2

Tags: polynomial , algebra , cubic

Let $a,b,c$ be the three different solutions of $x^3-x-1 = 0$. Compute $a^4+b^5+c^6-c$.

2017 MIG, 5

Tags:

Based on the fact that only $10\%$ of the total volume of an iceberg can be seen above the surface of the water, if $500\text{ m}^3$ of a certain iceberg is seen, then what is the total volume of the iceberg? $\textbf{(A) } 5\qquad\textbf{(B) } 50\qquad\textbf{(C) } 500\qquad\textbf{(D) } 5{,}000\qquad\textbf{(E) } \text{imposssible to determined}$

2019 AMC 12/AHSME, 24

Tags: complex plane , area , geometry

Let $\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3.$ Let $S$ denote all points in the complex plane of the form $a+b\omega+c\omega^2,$ where $0\leq a \leq 1,0\leq b\leq 1,$ and $0\leq c\leq 1.$ What is the area of $S$? $\textbf{(A) } \frac{1}{2}\sqrt3 \qquad\textbf{(B) } \frac{3}{4}\sqrt3 \qquad\textbf{(C) } \frac{3}{2}\sqrt3\qquad\textbf{(D) } \frac{1}{2}\pi\sqrt3 \qquad\textbf{(E) } \pi$

2010 Abels Math Contest (Norwegian MO) Final, 2b

Tags: inequalities , algebra

Show that $abc \le (ab + bc + ca)(a^2 + b^2 + c^2)^2$ for all positive real numbers $a, b$ and $c$ such that $a + b + c = 1$.

2011 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: divisor , number theory

If $n$ is a positive integer and $n+1$ is divisible with $24$, prove that sum of all positive divisors of $n$ is divisible with $24$

2015 BMT Spring, 13

Tags: combinatorics , geometry

There exist right triangles with integer side lengths such that the legs differ by $ 1$. For example, $3-4-5$ and $20-21-29$ are two such right triangles. What is the perimeter of the next smallest Pythagorean right triangle with legs differing by $ 1$?

2014 CentroAmerican, 1

Tags: combinatorics

Using squares of side 1, a stair-like figure is formed in stages following the pattern of the drawing. For example, the first stage uses 1 square, the second uses 5, etc. Determine the last stage for which the corresponding figure uses less than 2014 squares. [img]http://www.artofproblemsolving.com/Forum/download/file.php?id=49934[/img]

2005 Postal Coaching, 19

Tags: function , algebra , polynomial

Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.

2024 China National Olympiad, 2

Tags: inequalities , algebra

Find the largest real number $c$ such that $$\sum_{i=1}^{n}\sum_{j=1}^{n}(n-|i-j|)x_ix_j \geq c\sum_{j=1}^{n}x^2_i$$ for any positive integer $n $ and any real numbers $x_1,x_2,\dots,x_n.$

2010 Brazil Team Selection Test, 3

Tags: cyclic quadrilateral , projective geometry , tangent , power of a point , geometry

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

2015 Singapore MO Open, 4

Tags: number theory , fibonacci sequence

Let $f_0, f_1,...$ be the Fibonacci sequence: $f_0 = f_1 = 1, f_n = f_{n-1} + f_{n-2}$ if $n \geq 2$. Determine all possible positive integers $n$ so that there is a positive integer $a$ such that $f_n \leq a \leq f_{n+1}$ and that $a( \frac{1}{f_1}+\frac{1}{f_1f_2}+\cdots+\frac{1}{f_1f_2...f_n} )$ is an integer.