Olympiad problems

2012 Turkmenistan National Math Olympiad, 7

Tags: algebra

If $a,b,c$ are positive real numbers and satisfy: $\frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}$ then prove that :$ \sum_{i=1}^{n} a^{2}_i \cdot \sum_{i=1}^{n} b^{2}_i =(\sum_{i=1}^{n} a_{i}b_{i})^2$

2000 Bosnia and Herzegovina Team Selection Test, 4

Tags: inequality , inequalities , algebra

Prove that for all positive real $a$, $b$ and $c$ holds: $$ \frac{bc}{a^2+2bc}+\frac{ac}{b^2+2ac}+\frac{ab}{c^2+2ab} \leq 1 \leq \frac{a^2}{a^2+2bc}+\frac{b^2}{b^2+2ac}+\frac{c^2}{c^2+2ab}$$

2021 Azerbaijan IMO TST, 3

Tags: algebra , polynomial , algorithm , lagrange s interpolation

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

1985 IMO Longlists, 88

Tags: polynomial , calculus , binomial theorem , algebra

Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that \[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]

2021 Korea National Olympiad, P5

Tags: sequence , inequalities , algebra , polynomial

A real number sequence $a_1, \cdots ,a_{2021}$ satisfies the below conditions. $$a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019)$$ Let the minimum of $a_1, \cdots ,a_{2021}$ be $m$, and the maximum of $a_1, \cdots ,a_{2021}$ be $M$. Let a 2021 degree polynomial $$P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021})$$ $|P(x)|$ is maximum in $[m, M]$ when $x=\alpha$. Show that $1<\alpha <2$.

1998 All-Russian Olympiad, 1

Tags: analytic geometry , vieta , algebra

Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.

2011 Swedish Mathematical Competition, 3

Tags: system of equations , system , algebra

Find all positive real numbers $x, y, z$, such that $$x - \frac{1}{y^2} = y - \frac{1}{z^2}= z - \frac{1}{x^2}$$

2003 Moldova National Olympiad, 10.8

Tags: algebra , logarithm

Find all integers n for which number $ \log_{2n\minus{}1}(n^2\plus{}2)$ is rational.

2009 Belarus Team Selection Test, 3

Tags: algebra , functional , functional equation

a) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+1) - f(n)$ for all $n \in N$? b) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+2) - f(n)$ for all $n \in N$? I. Voronovich

1954 Czech and Slovak Olympiad III A, 2

Tags: complex number , algebra , geometry

Let $a,b$ complex numbers. Show that if the roots of the equation $z^2+az+b=0$ and 0 form a triangle with the right angle at the origin, then $a^2=2b\neq0.$ Also determine whether the opposite implication holds.

2001 Taiwan National Olympiad, 5

Tags: function , algebra , polynomial , induction , number theory

Let $f(n)=\sum_{k=0}^{n-1}x^ky^{n-1-k}$ with, $x$, $y$ real numbers. If $f(n)$, $f(n+1)$, $f(n+2)$, $f(n+3)$, are integers for some $n$, prove $f(n)$ is integer for all $n$.

2016 Nigerian Senior MO Round 2, Problem 6

Tags: algebra , inequality , inequalities

Given that $a, b, c, d \in \mathbb{R}$, prove that $(ab+cd)^2 \leq (a^2+c^2)(b^2+d^2)$.

2008 Harvard-MIT Mathematics Tournament, 13

Tags: algebra , polynomial , factorial

Let $ P(x)$ be a polynomial with degree 2008 and leading coefficient 1 such that \[ P(0) \equal{} 2007, P(1) \equal{} 2006, P(2) \equal{} 2005, \dots, P(2007) \equal{} 0. \]Determine the value of $ P(2008)$. You may use factorials in your answer.

2023 UMD Math Competition Part I, #18

Tags: algebra

How many ordered triples of integers $(a, b, c)$ satisfy the following system? $$ \begin{cases} ab + c &= 17 \\ a + bc &= 19 \end{cases} $$ $$ \mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6 $$

1975 Swedish Mathematical Competition, 2

Tags: floor function , algebra

Is there a positive integer $n$ such that the fractional part of \[ \left(3+\sqrt{5}\right)^n >0.99 ? \]

2021 Austrian MO National Competition, 4

Tags: algebra , functional equation , functional

Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$. (Walther Janous)

2023 CUBRMC, 7

Tags: algebra , inequalities

Among all ordered pairs of real numbers $(a, b)$ satisfying $a^4 + 2a^2b + 2ab + b^2 = 960$, find the smallest possible value for $a$.

2013 IFYM, Sozopol, 4

Tags: algebra , polynomial , greatest common divisor , modular arithmetic , number theory

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

2004 India IMO Training Camp, 3

Tags: polynomial , inequalities , algebra

Suppose the polynomial $P(x) \equiv x^3 + ax^2 + bx +c$ has only real zeroes and let $Q(x) \equiv 5x^2 - 16x + 2004$. Assume that $P(Q(x)) = 0$ has no real roots. Prove that $P(2004) > 2004$

2005 Poland - Second Round, 3

Tags: convex-concave inequalities , algebra , inequalities

Prove that if the real numbers $a,b,c$ lie in the interval $[0,1]$, then \[\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1}\le 2\]

2023 All-Russian Olympiad Regional Round, 11.9

Tags: absolute value , algebra , inequalities

If $a, b, c$ are non-zero reals, prove that $|\frac{b} {a}-\frac{b} {c}|+|\frac{c} {a}-\frac{c}{b}|+|bc+1|>1$.

2013 BMT Spring, 2

Tags: algebra , geometry , inequalities

A point $P$ is given on the curve $x^4+y^4=1$. Find the maximum distance from the point $P$ to the origin.

1979 Bulgaria National Olympiad, Problem 4

Tags: polynomial , quadratic , parameterization , function , algebra

For each real number $k$, denote by $f(k)$ the larger of the two roots of the quadratic equation $$(k^2+1)x^2+10kx-6(9k^2+1)=0.$$Show that the function $f(k)$ attains a minimum and maximum and evaluate these two values.

1978 Putnam, A3

Tags: calculus , integration , algebra , polynomial , calculus computations , logarithm

Find the value of $ k\ (0<k<5)$ such that $ \int_0^{\infty} \frac{x^k}{2\plus{}4x\plus{}3x^2\plus{}5x^3\plus{}3x^4\plus{}4x^5\plus{}2x^6}\ dx$ is minimal.

IV Soros Olympiad 1997 - 98 (Russia), 9.4

Tags: inequalities , algebra

Find the smallest value of the expression $$16 \cdot \frac{x^3}{y}+\frac{y^3}{x}-\sqrt{xy}$$