Olympiad problems

2015 BMT Spring, 7

At Durant University, an A grade corresponds to raw scores between $90$ and $100$, and a B grade corresponds to raw scores between $80$ and $90$. Travis has $3$ equally weighted exams in his math class. Given that Travis earned an A on his first exam and a B on his second (but doesn't know his raw score for either), what is the minimum score he needs to have a $90\%$ chance of getting an A in the class? Note that scores on exams do not necessarily have to be integers.

I Soros Olympiad 1994-95 (Rus + Ukr), 9.8

Tags: inequalities , algebra

Let $f(x) =x^2-2x$. Find all $x$ for which $f(f(x))<3$.

2007 Harvard-MIT Mathematics Tournament, 20

Tags: algebra , vieta , polynomial

For $a$ a positive real number, let $x_1$, $x_2$, $x_3$ be the roots of the equation $x^3-ax^2+ax-a=0$. Determine the smallest possible value of $x_1^3+x_2^3+x_3^3-3x_1x_2x_3$.

1997 Slovenia National Olympiad, Problem 1

Tags: algebra

Prove that if real numbers $a,b,c,d$ satisfy $a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$, then they also satisfy $a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4$.

2007 Iran Team Selection Test, 1

Tags: polynomial , inequalities , algebra

Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]

1983 Bulgaria National Olympiad, Problem 5

Tags: solve equation , complex number , polynomial , algebra

Can the polynomials $x^{5}-x-1$ and $x^{2}+ax+b$ , where $a,b\in Q$, have common complex roots?

2019 India Regional Mathematical Olympiad, 6

Tags: coordinate geometry , parabola , conic , algebra , analytic geometry , geometry

Let $k$ be a positive real number. In the $X-Y$ coordinate plane, let $S$ be the set of all points of the form $(x,x^2+k)$ where $x\in\mathbb{R}$. Let $C$ be the set of all circles whose center lies in $S$, and which are tangent to $X$-axis. Find the minimum value of $k$ such that any two circles in $C$ have at least one point of intersection.

2019 Spain Mathematical Olympiad, 3

Tags: algebra , polynomial

The real numbers $a$, $b$ and $c$ verify that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct real roots; these roots are equal to $\tan y$, $\tan 2y$ and $\tan 3y$, for some real number $y$. Find all possible values of $y$, $0\leq y < \pi$.

2012 Centers of Excellency of Suceava, 3

Tags: algebra , newton s binom

Let $ a,b,n $ be three natural numbers. Prove that there exists a natural number $ c $ satisfying: $$ \left( \sqrt{a} +\sqrt{b} \right)^n =\sqrt{ c+(a-b)^n} +\sqrt{c} $$ [i]Dan Popescu[/i]

2009 IMO Shortlist, 5

Tags: number theory , permutation , algebra , polynomial

Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

1995 Kurschak Competition, 2

Tags: polynomial , algebra

Consider a polynomial in $n$ variables with real coefficients. We know that if every variable is $\pm1$, the value of the polynomial is positive, or negative if the number of $-1$'s is even, or odd, respectively. Prove that the degree of this polynomial is at least $n$.

1987 IMO Longlists, 59

Tags: trigonometry , complex number , algebra

It is given that $a_{11}, a_{22}$ are real numbers, that $x_1, x_2, a_{12}, b_1, b_2$ are complex numbers, and that $a_{11}a_{22}=a_{12}\overline{a_{12}}$ (Where $\overline{a_{12}}$ is he conjugate of $a_{12}$). We consider the following system in $x_1, x_2$: \[\overline{x_1}(a_{11}x_1 + a_{12}x_2) = b_1,\]\[\overline{x_2}(a_{12}x_1 + a_{22}x_2) = b_2.\] [b](a) [/b]Give one condition to make the system consistent. [b](b) [/b]Give one condition to make $\arg x_1 - \arg x_2 = 98^{\circ}.$

1998 Vietnam National Olympiad, 3

Tags: algebra , modular arithmetic

The sequence $\{a_{n}\}_{n\geq 0}$ is defined by $a_{0}=20,a_{1}=100,a_{n+2}=4a_{n+1}+5a_{n}+20(n=0,1,2,...)$. Find the smallest positive integer $h$ satisfying $1998|a_{n+h}-a_{n}\forall n=0,1,2,...$

1992 India National Olympiad, 1

Tags: trig identities , algebra , trigonometry

In a triangle $ABC$, $\angle A = 2 \cdot \angle B$. Prove that $a^2 = b (b+c)$.

2023 Brazil National Olympiad, 4

Tags: real number , algebra , inequalities , system of equations

Let $x, y, z$ be three real distinct numbers such that $$\begin{cases} x^2-x=yz \\ y^2-y=zx \\ z^2-z=xy \end{cases}$$ Show that $-\frac{1}{3} < x,y,z < 1$.

2007 AMC 10, 23

Tags: difference of squares , special factorizations , algebra

How many ordered pairs $ (m,n)$ of positive integers, with $ m > n$, have the property that their squares differ by $ 96$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$

2000 Bundeswettbewerb Mathematik, 4

Tags: algebra , 3d geometry , geometry

Consider the sums of the form $\sum_{k=1}^{n} \epsilon_k k^3,$ where $\epsilon_k \in \{-1, 1\}.$ Is any of these sums equal to $0$ if [b](a)[/b] $n=2000;$ [b](b)[/b] $n=2001 \ ?$

2005 Thailand Mathematical Olympiad, 16

Tags: sum , algebra

Compute the sum of roots of $(2 - x)^{2005} + x^{2005} = 0$.

1978 Swedish Mathematical Competition, 2

Tags: algebra , number theory , sum , digit

Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find \[ s_1 + s_2 + \cdots + s_n \]

2017 Iran Team Selection Test, 1

Tags: algebra , inequalities

Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality: $$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$ [i]Proposed by Mohammad Jafari[/i]

2021 Moldova Team Selection Test, 5

Tags: algebra , function , geometry

Let $ABC$ be an equilateral triangle. Find all positive integers $n$, for which the function $f$, defined on all points $M$ from the circle $S$ circumscribed to triangle $ABC$, defined by the formula $f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n$, is a constant function.

2012 Princeton University Math Competition, A6

Tags: algebra

Let an be a sequence such that $a_0 = 0$ and: $a_{3n+1} = a_{3n} + 1 = a_n + 1$ $a_{3n+2} = a_{3n} + 2 = a_n + 2$ for all natural numbers $n$. How many $n$ less than $2012$ have the property that $a_n = 7$?

2013 BMT Spring, 2

Tags: algebra

Find the sum of all positive integers $N$ such that $s =\sqrt[3]{2 + \sqrt{N}} + \sqrt[3]{2 - \sqrt{N}}$ is also a positive integer

2019 Thailand TST, 3

Tags: polynomial , algebra

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

2023 CMI B.Sc. Entrance Exam, 3

Tags: polynomial , algebra

Consider the polynomial $p(x) = x^4 + ax^3 + bx^2 + cx + d$. It is given that $p(x)$ has its only root at $x = r$ i.e $p(r) = 0$. $\textbf{(a)}$ Show that if $a, b, c, d$ are rational then $r$ is rational. $\textbf{(b)}$ Show that if $a, b, c, d$ are integers then $r$ is an integer. [hide=Hint](Hint: Consider the roots of $p'(x)$ )[/hide]