Olympiad problems

PEN A Problems, 23

Tags: number theory , quadratic , gauss , modular arithmetic , relatively prime , divisibility theory

(Wolstenholme's Theorem) Prove that if \[1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\] is expressed as a fraction, where $p \ge 5$ is a prime, then $p^{2}$ divides the numerator.

2004 AIME Problems, 12

Tags: geometry , trapezoid , algebra , quadratic , analytic geometry

Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6$, $BC=5=DA$, and $CD=4$. Draw circles of radius 3 centered at $A$ and $B$, and circles of radius 2 centered at $C$ and $D$. A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}p$, where $k$, $m$, $n$, and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p$.

2008 JBMO Shortlist, 7

Tags: modular arithmetic , quadratic , number theory

Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $.

1989 IMO Longlists, 24

Tags: geometry , quadratic , number theory

Let $ a, b, c, d$ be positive integers such that $ ab \equal{} cd$ and $ a\plus{}b \equal{} c \minus{} d.$ Prove that there exists a right-angled triangle the measure of whose sides (in some unit) are integers and whose area measure is $ ab$ square units.

1994 Turkey MO (2nd round), 5

Tags: calculus , integration , quadratic , number theory

Find the set of all ordered pairs $(s,t)$ of positive integers such that \[t^{2}+1=s(s+1).\]

1993 All-Russian Olympiad, 3

Tags: algebra , quadratic

Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?

2000 Putnam, 3

Tags: reflection , geometry , geometric transformation , rectangle , quadratic , analytic geometry

The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area $5$, and the polygon $P_2P_4P_6P_8$ is a rectangle of area $4$, find the maximum possible area of the octagon.

2011 AIME Problems, 15

Tags: vieta , quadratic , yuh. , polynomial , algebra

For some integer $m$, the polynomial $x^3-2011x+m$ has the three integer roots $a$, $b$, and $c$. Find $|a|+|b|+|c|$.

1971 AMC 12/AHSME, 18

Tags: ratio , quadratic

The current in a river is flowing steadily at $3$ miles per hour. A motor boat which travels at a constant rate in still water goes downstream $4$ miles and then returns to its starting point. The trip takes one hour, excluding the time spent in turning the boat around. The ratio of the downstream to the upstream rate is $\textbf{(A) }4:3\qquad\textbf{(B) }3:2\qquad\textbf{(C) }5:3\qquad\textbf{(D) }2:1\qquad \textbf{(E) }5:2$

2017 All-Russian Olympiad, 2

Tags: polynomial , number theory , quadratic , algebra

$a,b,c$ - different natural numbers. Can we build quadratic polynomial $P(x)=kx^2+lx+m$, with $k,l,m$ are integer, $k>0$ that for some integer points it get values $a^3,b^3,c^3$ ?

2024 Belarusian National Olympiad, 11.2

Tags: polynomial , algebra , quadratic

$29$ quadratic polynomials $f_1(x), \ldots, f_{29}(x)$ and $15$ real numbers $x_1<x_2<\ldots<x_{15}$ are given. Prove that for some two given polynomials $f_i(x)$ and $f_j(x)$ the following inequality holds: $$\sum_{k=1}^{14} (f_i(x_{k+1})-f_i(x_k))(f_j(x_{k+1})-f_j(x_k))>0$$ [i]A. Voidelevich[/i]

2007 Romania Team Selection Test, 3

Tags: combinatorics , inequalities , quadratic

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]

1991 Arnold's Trivium, 87

Tags: derivative , function , calculus , quadratic , vector , linear algebra , matrix

Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.

2005 District Olympiad, 4

Tags: incenter , angle bisector , geometry , circumcircle , quadratic , perpendicular bisector , ratio

In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.

2007 JBMO Shortlist, 1

Tags: function , quadratic , algebra , inequalities

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

2001 Korea - Final Round, 1

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

Given an odd prime $p$, find all functions $f:Z \rightarrow Z$ satisfying the following two conditions: (i) $f(m)=f(n)$ for all $m,n \in Z$ such that $m\equiv n\pmod p$; (ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$.

2012 Indonesia TST, 3

Tags: geometry , quadratic , rectangle , vector

The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$. Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.

2009 AMC 12/AHSME, 17

Tags: quadratic , geometric series

Let $ a\plus{}ar_1\plus{}ar_1^2\plus{}ar_1^3\plus{}\cdots$ and $ a\plus{}ar_2\plus{}ar_2^2\plus{}ar_2^3\plus{}\cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $ r_1$, and the sum of the second series is $ r_2$. What is $ r_1\plus{}r_2$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac{1\plus{}\sqrt{5}}{2}\qquad \textbf{(E)}\ 2$

2001 Austria Beginners' Competition, 2

Tags: algebra , quadratic , quadratic polynomial

Consider the quadratic equation $x^2-2mx-1=0$, where $m$ is an arbitrary real number. For what values of $m$ does the equation have two real solutions, such that the sum of their cubes is equal to eight times their sum.

2013 Greece National Olympiad, 2

Tags: quadratic , diophantine equation , number theory

Solve in integers the following equation: \[y=2x^2+5xy+3y^2\]

2001 CentroAmerican, 2

Tags: arithmetic sequence , function , quadratic , absolute value

Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2\plus{}bx\plus{}c\equal{}0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2\plus{}bx\plus{}a\equal{}0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a\plus{}c\equal{}0$.

1986 China Team Selection Test, 3

Tags: quadratic , inequalities

Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that: i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$ ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.

2010 Hanoi Open Mathematics Competitions, 6

Tags: number theory , quadratic , integer

Let $a,b$ be the roots of the equation $x^2-px+q = 0$ and let $c, d$ be the roots of the equation $x^2 - rx + s = 0$, where $p, q, r,s$ are some positive real numbers. Suppose that $M =\frac{2(abc + bcd + cda + dab)}{p^2 + q^2 + r^2 + s^2}$ is an integer. Determine $a, b, c, d$.