Olympiad problems

2001 Polish MO Finals, 3

Given positive integers $n_1<n_2<...<n_{2000}<10^{100}$. Prove that we can choose from the set $\{n_1,...,n_{2000}\}$ nonempty, disjont sets $A$ and $B$ which have the same number of elements, the same sum and the same sum of squares.

1997 Finnish National High School Mathematics Competition, 1

Tags: parameterization , quadratic , polynomial , product of roots , algebra , logarithm

Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$

2014 AIME Problems, 8

Tags: modular arithmetic , sumac , quadratic , number theory , greatest common divisor

The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.

2002 Germany Team Selection Test, 1

Tags: function , quadratic , algebra

Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying \[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\ 1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases} \] Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.

2009 AMC 12/AHSME, 23

Tags: quadratic , function , analytic geometry , geometric transformation , rotation , algebra , geometry

Functions $ f$ and $ g$ are quadratic, $ g(x) \equal{} \minus{} f(100 \minus{} x)$, and the graph of $ g$ contains the vertex of the graph of $ f$. The four $ x$-intercepts on the two graphs have $ x$-coordinates $ x_1$, $ x_2$, $ x_3$, and $ x_4$, in increasing order, and $ x_3 \minus{} x_2 \equal{} 150$. The value of $ x_4 \minus{} x_1$ is $ m \plus{} n\sqrt p$, where $ m$, $ n$, and $ p$ are positive integers, and $ p$ is not divisible by the square of any prime. What is $ m \plus{} n \plus{} p$? $ \textbf{(A)}\ 602\qquad \textbf{(B)}\ 652\qquad \textbf{(C)}\ 702\qquad \textbf{(D)}\ 752\qquad \textbf{(E)}\ 802$

2006 Junior Tuymaada Olympiad, 5

Tags: algebra , quadratic trinomial , trinomial , quadratic , constant , sides of a triangle

The quadratic trinomials $ f $, $ g $ and $ h $ are such that for every real $ x $ the numbers $ f (x) $, $ g (x) $ and $ h (x) $ are the lengths of the sides of some triangles, and the numbers $ f (x) -1 $, $ g (x) -1 $ and $ h (x) -1 $ are not the lengths of the sides of the triangle. Prove that at least of the polynomials $ f + g-h $, $ f + h-g $, $ g + h-f $ is constant.

2013 All-Russian Olympiad, 3

Tags: quadratic , modular arithmetic , prime number , number theory

Find all positive integers $k$ such that for the first $k$ prime numbers $2, 3, \ldots, p_k$ there exist positive integers $a$ and $n>1$, such that $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$. [i]V. Senderov[/i]

2007 Romania Team Selection Test, 3

Tags: inequalities , quadratic , combinatorics

Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true. [i]Dan Schwarz[/i]

2007 Federal Competition For Advanced Students, Part 1, 1

Tags: quadratic , number theory

In a quadratic table with $ 2007$ rows and $ 2007$ columns is an odd number written in each field. For $ 1\leq i\leq2007$ is $ Z_i$ the sum of the numbers in the $ i$-th row and for $ 1\leq j\leq2007$ is $ S_j$ the sum of the numbers in the $ j$-th column. $ A$ is the product of all $ Z_i$ and $ B$ the product of all $ S_j$. Show that $ A\plus{}B\neq0$

1975 Canada National Olympiad, 4

Tags: calculus , integration , ratio , floor function , quadratic , geometric sequence , algebra

For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.

2013 Online Math Open Problems, 28

Tags: modular arithmetic , quadratic

Let $n$ denote the product of the first $2013$ primes. Find the sum of all primes $p$ with $20 \le p \le 150$ such that (i) $\frac{p+1}{2}$ is even but is not a power of $2$, and (ii) there exist pairwise distinct positive integers $a,b,c$ for which \[ a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b) \] is divisible by $p$ but not $p^2$. [i]Proposed by Evan Chen[/i]

2001 Putnam, 3

Tags: algebra , polynomial , quadratic , quadratic formula

For each integer $m$, consider the polynomial \[ P_m(x)=x^4-(2m+4)x^2+(m-2)^2. \] For what values of $m$ is $P_m(x)$ the product of two non-consant polynomials with integer coefficients?

2020 CCA Math Bonanza, T5

Tags: quadratic

Find all pairs of real numbers $(x,y)$ satisfying both equations \[ 3x^2+3xy+2y^2 =2 \] \[ x^2+2xy+2y^2 =1. \] [i]2020 CCA Math Bonanza Team Round #5[/i]

2018 CCA Math Bonanza, T10

Tags: quadratic

The irrational number $\alpha>1$ satisfies $\alpha^2-3\alpha-1=0$. Given that there is a fraction $\frac{m}{n}$ such that $n<500$ and $\left|\alpha-\frac{m}{n}\right|<3\cdot10^{-6}$, find $m$. [i]2018 CCA Math Bonanza Team Round #10[/i]

2008 IMC, 3

Tags: algebra , polynomial , quadratic

Let $p$ be a polynomial with integer coeﬃcients and let $a_1<a_2<\cdots <a_k$ be integers. Given that $p(a_i)\ne 0\forall\; i=1,2,\cdots, k$. [list] (a) Prove $\exists\; a\in \mathbb{Z}$ such that \[ p(a_i)\mid p(a)\;\;\forall i=1,2,\dots ,k \] (b) Does there exist $a\in \mathbb{Z}$ such that \[ \prod_{i=1}^{k}p(a_i)\mid p(a) \][/list]

2005 Romania National Olympiad, 1

Tags: ratio , quadratic , analytic geometry , vector , gauss , geometry

Let $ABCD$ be a convex quadrilateral with $AD\not\parallel BC$. Define the points $E=AD \cap BC$ and $I = AC\cap BD$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB \parallel CD$ and $IC^{2}= IA \cdot AC$. [i]Virgil Nicula[/i]

1988 Flanders Math Olympiad, 4

Tags: calculus , derivative , geometry , trigonometry , function , inequalities , quadratic

Be $R$ a positive real number. If $R, 1, R+\frac12$ are triangle sides, call $\theta$ the angle between $R$ and $R+\frac12$ (in rad). Prove $2R\theta$ is between $1$ and $\pi$.

1985 Dutch Mathematical Olympiad, 1

Tags: quadratic , algebra

For some $ p$, the equation $ x^3 \plus{} px^2 \plus{} 3x \minus{} 10 \equal{} 0$ has three real solutions $ a,b,c$ such that $ c \minus{} b \equal{} b \minus{} a > 0$. Determine $ a,b,c,$ and $ p$.

PEN I Problems, 11

Tags: floor function , quadratic , function

Let $p$ be a prime number of the form $4k+1$. Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]

2014 India IMO Training Camp, 1

Tags: quadratic , quadratic formula , algebra

Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.

2012 Turkey Junior National Olympiad, 1

Tags: quadratic , calculus , integration , diophantine equation , number theory

Let $x, y$ be integers and $p$ be a prime for which \[ x^2-3xy+p^2y^2=12p \] Find all triples $(x,y,p)$.

2007 District Olympiad, 4

Tags: superior algebra , algebra , polynomial , quadratic

Let $\mathcal K$ be a field with $2^{n}$ elements, $n \in \mathbb N^\ast$, and $f$ be the polynomial $X^{4}+X+1$. Prove that: (a) if $n$ is even, then $f$ is reducible in $\mathcal K[X]$; (b) if $n$ is odd, then $f$ is irreducible in $\mathcal K[X]$. [hide="Remark."]I saw the official solution and it wasn't that difficult, but I just couldn't solve this bloody problem.[/hide]