Olympiad problems

1988 China Team Selection Test, 1

Tags: quadratic , inequalities

Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds: \[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\] Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).

1960 Putnam, A4

Tags: geometry , locus

Given two points, $P$ and $Q$, on the same side of a line $L$, the problem is to find a third point $R$ so that $PR+ RQ+RS$ is minimal, where $S$ is the unique point on $L$ such that $RS$ is perpendicular to $L.$ Consider all cases.

2004 USA Team Selection Test, 1

Tags: inequalities , quadratic , algebra , polynomial , trigonometry , geometry , parallelogram

Suppose $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ are real numbers such that \[ (a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 -1)(b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 - 1) > (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n - 1)^2. \] Prove that $a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 > 1$ and $b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 > 1$.

1993 AMC 12/AHSME, 20

Tags: quadratic , complex number , algebra , quadratic formula

Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true? $ \textbf{(A)}\ \text{For all positive real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(B)}\ \text{For all negative real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(C)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and rational.} \\ \qquad\textbf{(D)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and irrational.} \\ \qquad\textbf{(E)}\ \text{For all complex numbers}\ k,\ \text{neither root is real.} $

2019 Purple Comet Problems, 28

Tags: algebra

There are positive integers $m$ and $n$ such that $m^2 -n = 32$ and $\sqrt[5]{m +\sqrt{n}}+ \sqrt[5]{m -\sqrt{n}}$ is a real root of the polynomial $x^5 - 10x^3 + 20x - 40$. Find $m + n$.

2016 ASDAN Math Tournament, 3

Tags: team test

Moor has $2016$ white rabbit candies. He and his $n$ friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that $n$ is a positive integer (Moor has at least $1$ friend), how many possible values of $n$ exist?

2020 LMT Fall, A5 B19

Tags:

Ada is taking a math test from 12:00 to 1:30, but her brother, Samuel, will be disruptive for two ten-minute periods during the test. If the probability that her brother is not disruptive while she is solving the challenge problem from 12:45 to 1:00 can be expressed as $\frac{m}{n}$, find $m+n$. [i]Proposed by Ada Tsui[/i]

2020 Middle European Mathematical Olympiad, 3#

Tags: geometry , circumcircle , incenter

Let $ABC$ be an acute scalene triangle with circumcircle $\omega$ and incenter $I$. Suppose the orthocenter $H$ of $BIC$ lies inside $\omega$. Let $M$ be the midpoint of the longer arc $BC$ of $\omega$. Let $N$ be the midpoint of the shorter arc $AM$ of $\omega$. Prove that there exists a circle tangent to $\omega$ at $N$ and tangent to the circumcircles of $BHI$ and $CHI$.

2007 AIME Problems, 1

Tags:

How many positive perfect squares less than $10^{6}$ are multiples of $24$?

2008 Croatia Team Selection Test, 3

Tags: circumcircle , geometry

Point $ M$ is taken on side $ BC$ of a triangle $ ABC$ such that the centroid $ T_c$ of triangle $ ABM$ lies on the circumcircle of $ \triangle ACM$ and the centroid $ T_b$ of $ \triangle ACM$ lies on the circumcircle of $ \triangle ABM$. Prove that the medians of the triangles $ ABM$ and $ ACM$ from $ M$ are of the same length.

2022 Kosovo National Mathematical Olympiad, 1

Tags: combinatorics

Ana has a scale that shows which side weight more or if both side are equal. She has $4$ weights which look the same but they weight $1001g, 1002g, 1004g$ and $1005g$, respectively. Is it possible for Ana to find out the weight of each of them with only $4$ measurements?

2007 Junior Balkan Team Selection Tests - Romania, 2

Tags: power of a point , radical axis , geometry

Let $w_{1}$ and $w_{2}$ be two circles which intersect at points $A$ and $B$. Consider $w_{3}$ another circle which cuts $w_{1}$ in $D,E$, and it is tangent to $w_{2}$ in the point $C$, and also tangent to $AB$ in $F$. Consider $G \in DE \cap AB$, and $H$ the symetric point of $F$ w.r.t $G$. Find $\angle{HCF}$.

2010 CHMMC Fall, 9

Tags: algebra

Let $a_0, a_1, . . . ,a_n$ be such that $a_n \ne 0$ and $$(1 + x + x^3)^{342} (1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{341} =\sum^{n}_{i=0}a_ix^i.$$ Compute the number of odd terms in the sequence $a_0, a_1, . . . ,a_n$.

2014 CentroAmerican, 3

Tags: inequalities

Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal, \[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of \[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]

2019 Miklós Schweitzer, 8

Tags: real analysis

Let $f: \mathbb{R} \to \mathbb{R}$ be a measurable function such that $f(x+t) - f(x)$ is locally integrable for every $t$ as a function of $x$. Prove that $f$ is locally integrable.

2002 National Olympiad First Round, 22

Tags: modular arithmetic

If $2^n$ divides $5^{256} - 1$, what is the largest possible value of $n$? $ \textbf{a)}\ 8 \qquad\textbf{b)}\ 10 \qquad\textbf{c)}\ 11 \qquad\textbf{d)}\ 12 \qquad\textbf{e)}\ \text{None of above} $

2011 District Olympiad, 3

Tags: linear algebra

Let $A,B\in \mathcal{M}_2(\mathbb{C})$ two non-zero matrices such that $AB+BA=O_2$ and $\det(A+B)=0$. Prove $A$ and $B$ have null traces.

2009 Turkey Team Selection Test, 2

Tags: inequalities , geometry

In a triangle $ ABC$ incircle touches the sides $ AB$, $ AC$ and $ BC$ at $ C_1$, $ B_1$ and $ A_1$ respectively. Prove that $ \sqrt {\frac {AB_1}{AB}} \plus{} \sqrt {\frac {BC_1}{BC}} \plus{} \sqrt {\frac {CA_1}{CA}}\leq\frac {3}{\sqrt {2}}$ is true.

1991 Swedish Mathematical Competition, 2

Tags: inequalities , algebra

$x, y$ are positive reals such that $x - \sqrt{x} \le y - 1/4 \le x + \sqrt{x}$. Show that $y - \sqrt{y} \le x - 1/4 \le y + \sqrt{y}$.

2020 Mexico National Olympiad, 6

Tags: vector , algebra

Let $n\ge 2$ be a positive integer. Let $x_1, x_2, \dots, x_n$ be non-zero real numbers satisfying the equation \[\left(x_1+\frac{1}{x_2}\right)\left(x_2+\frac{1}{x_3}\right)\dots\left(x_n+\frac{1}{x_1}\right)=\left(x_1^2+\frac{1}{x_2^2}\right)\left(x_2^2+\frac{1}{x_3^2}\right)\dots\left(x_n^2+\frac{1}{x_1^2}\right).\] Find all possible values of $x_1, x_2, \dots, x_n$. [i]Proposed by Victor Domínguez[/i]

2004 IMO Shortlist, 3

Tags: graph theory , combinatorics , algorithm , game

The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\ge 4}$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from the complete graph on $n$ vertices (where each pair of vertices are joined by an edge). [i]Proposed by Norman Do, Australia[/i]

1990 Poland - Second Round, 6

Tags: geometry , combinatorics , combinatorial geometry , convex polygon , area

For any convex polygon $ W $ with area 1, let us denote by $ f(W) $ the area of the convex polygon whose vertices are the centers of all sides of the polygon $ W $. For each natural number $ n \geq 3 $, determine the lower limit and the upper limit of the set of numbers $ f(W) $ when $ W $ runs through the set of all $ n $ convex angles with area 1.

1998 Austrian-Polish Competition, 4

Tags: inequalities , sum , algebra

For positive integers $m, n$, denote $$S_m(n)=\sum_{1\le k \le n} \left[ \sqrt[k^2]{k^m}\right]$$ Prove that $S_m(n) \le n + m (\sqrt[4]{2^m}-1)$

2016 Turkey EGMO TST, 5

Tags: combinatorics , periodic sequence , algebra , sequence

A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that \[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \] Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.

1996 Putnam, 5

Tags: floor function

Given a finite binary string $S$ of symbols $X,O$ we define $\Delta(S)=n(X)-n(O)$ where $n(X),n(O)$ respectively denote number of $X$'s and $O$'s in a string. For example $\Delta(XOOXOOX)=3-4=-1$. We call a string $S$ $\emph{balanced}$ if every substring $T$ of $S$ has $-2\le \Delta(T)\le 2$. Find number of balanced strings of length $n$.