Olympiad problems

2016 Israel Team Selection Test, 1

Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \geq 2$.

1979 IMO Longlists, 60

Tags: optimization , calculus , lagrange , maximization , algebra

Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$

2016 Israel Team Selection Test, 1

Tags: calculus , lagrange , inequalities

Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \geq 2$.

1979 IMO Shortlist, 11

Tags: optimization , calculus , inequality , maximization , lagrange

Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.

2011 Gheorghe Vranceanu, 2

Tags: real analysis , function , lagrange , darboux , weierstrass , darboux s property

Let $ f:[0,1]\longrightarrow (0,\infty ) $ be a continuous function and $ \left( b_n \right)_{n\ge 1} $ be a sequence of numbers from the interval $ (0,1) $ that converge to $ 0. $ [b]a)[/b] Demonstrate that for any fixed $ n, $ the equation $ F(x)=b_nF(1)+\left( 1-b_n\right) F(0) $ has an unique solution, namely $ x_n, $ where $ F $ is a primitive of $ f. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{x_n}{b_n} . $

1987 Traian Lălescu, 1.2

Tags: real analysis , lagrange , function

Let $ I $ be a real interval, and $ f:I\longrightarrow\mathbb{R} $ be a continuous function. Prove that $ f $ is monotone if and only if $ \min(\left( f(a),f(b)\right) \le\frac{1}{b-a}\int_a^b f(x)dx \le\max\left( f(a),f(b) \right) , $ for any distinct $ a,b\in I. $

2006 Grigore Moisil Urziceni, 3

Tags: real analysis , function , rolle , lagrange , primitivability , primitive

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that admits a primitive $ F. $ [b]a)[/b] Show that there exists a real number $ c $ such that $ f(c)-F(c)>1 $ if $ \lim_{x\to\infty } \frac{1+F(x)}{e^x} =-\infty . $ [b]b)[/b] Prove that there exists a real number $ c' $ such that $ f(c') -(F(c'))^2<1. $ [i]Cristinel Mortici[/i]

1985 Traian Lălescu, 2.1

Tags: real analysis , function , integral , lagrange

Let $ f:[-1,1]\longrightarrow\mathbb{R} $ a derivable function and a non-negative integer $ n. $ Show that there is a $ c\in [-1,1] $ so that: $$ \int_{-1}^1 x^{2n+1} f(x)dx =\frac{2}{2n+3}f'(c). $$

2012 Centers of Excellency of Suceava, 4

Tags: real analysis , function , calculus , derivative , lagrange , convexity

Let be two real numbers $ a<b $ and a differentiable function $ f:[a,b]\longrightarrow\mathbb{R} $ that has a bounded derivative. Show that if $ \frac{f(b)-f(a)}{b-a} $ is equal to the global supremum or infimum of $ f', $ then $ f $ is polynomial with degree $ 1. $ [i]Cătălin Țigăeru[/i]

1979 IMO Longlists, 29

Tags: calculus , inequality , maximization , optimization , lagrange

Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.

1979 IMO Shortlist, 20

Tags: calculus , algebra , maximization , optimization , lagrange

Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$