Olympiad problems

2009 Kyiv Mathematical Festival, 6

Let $\{a_1,...,a_n\}\subset \{-1,1\}$ and $a>0$ . Denote by $X$ and $Y$ the number of collections $\{\varepsilon_1,...,\varepsilon_n\}\subset \{-1,1\}$, such that $$max_{1\le k\le n}(\varepsilon_1a_1+...+\varepsilon_ka_k) >\alpha$$ and $$\varepsilon_1a_1+...+\varepsilon_na_n>a$$ respectively. Prove that $X\le 2Y$.

1980 Spain Mathematical Olympiad, 1

Tags: geometry , geometric inequality , area of a triangle , maximum

Among the triangles that have a side of length $5$ m and the angle opposite of $30^o$, determine the one with maximum area, calculating the value of the other two angles and area of triangle.

1989 Austrian-Polish Competition, 5

Tags: geometry , 3d geometry , cube , sphere , product , maximum

Let $A$ be a vertex of a cube $\omega$ circumscribed about a sphere $k$ of radius $1$. We consider lines $g$ through $A$ containing at least one point of $k$. Let $P$ be the intersection point of $g$ and $k$ closer to $A$, and $Q$ be the second intersection point of $g$ and $\omega$. Determine the maximum value of $AP\cdot AQ$ and characterize the lines $g$ yielding the maximum.

2010 Hanoi Open Mathematics Competitions, 10

Tags: maximum , inequalities , algebra

Find the maximum value of $M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}$ , $x,y, z > 0$

2013 Hanoi Open Mathematics Competitions, 11

Tags: polynomial , minimum , maximum

The positive numbers $a, b,c, d, p, q$ are such that $(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1$ holds for all real numbers $x$. Find the smallest value of $p$ or the largest value of $q$.

2018 India PRMO, 9

Tags: trinomial , maximum , integer , root

Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$?

2017 Czech-Polish-Slovak Match, 2

Tags: board , combinatorics , maximum , square

Each of the ${4n^2}$ unit squares of a ${2n \times 2n}$ board ${(n \ge 1) }$ has been colored blue or red. A set of four different unit squares of the board is called [i]pretty [/i]if these squares can be labeled ${A,B,C,D}$ in such a way that ${A}$ and ${B}$ lie in the same row, ${C}$ and ${D}$ lie in the same row, ${A}$ and ${C}$ lie in the same column, ${B}$ and ${D}$ lie in the same column, ${A}$ and ${D}$ are blue, and ${B}$ and ${C}$ are red. Determine the largest possible number of different [i]pretty [/i]sets on such a board. (Poland)

1951 Moscow Mathematical Olympiad, 194

Tags: geometry , maximum , convex polygon , area

One side of a convex polygon is equal to $a$, the sum of exterior angles at the vertices not adjacent to this side are equal to $120^o$. Among such polygons, find the polygon of the largest area.

2017 Hanoi Open Mathematics Competitions, 5

Tags: maximum , digit , number theory

Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is (A): $1099$ (B): $2099$ (C): $1199$ (D): $2199$ (E): None of the above.

1965 All Russian Mathematical Olympiad, 062

Tags: geometry , maximum , incircle

What is the maximal possible length of the segment, being cut out by the sides of the triangle on the tangent to the inscribed circle, being drawn parallel to the base, if the triangle's perimeter equals $2p$?

2007 Sharygin Geometry Olympiad, 20

Tags: geometry , 3d geometry , pyramid , volume , maximum

The base of a pyramid is a regular triangle having side of size $1$. Two of three angles at the vertex of the pyramid are right. Find the maximum value of the volume of the pyramid.

1975 All Soviet Union Mathematical Olympiad, 208

Tags: rectangle , combinatorial geometry , combinatorics , maximum , table

a) Given a big square consisting of $7\times 7$ squares. You should mark the centres of $k$ points in such a way, that no quadruple of the marked points will be the vertices of a rectangle with the sides parallel to the sides of the given squares. What is the greatest $k$ such that the problem has solution? b) The same problem for $13\times 13$ square.

2016 German National Olympiad, 6

Tags: algebra , cyclic function , function , inequality , maximum

Let \[ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 \] be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$. Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.

1999 Nordic, 2

Tags: maximum , polygon , angle , geometry

Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon.

2018 VTRMC, 6

Tags: extremal , maximum , minimum value , sequence

For $n \in \mathbb{N}$, define $a_n = \frac{1 + 1/3 + 1/5 + \dots + 1/(2n-1)}{n+1}$ and $b_n = \frac{1/2 + 1/4 + 1/6 + \dots + 1/(2n)}{n}$. Find the maximum and minimum of $a_n - b_n$ for $1 \leq n \leq 999$.

2013 India PRMO, 13

Tags: maximum , divide , number theory

To each element of the set $S = \{1,2,... ,1000\}$ a colour is assigned. Suppose that for any two elements $a, b$ of $S$, if $15$ divides $a + b$ then they are both assigned the same colour. What is the maximum possible number of distinct colours used?

2011 Hanoi Open Mathematics Competitions, 5

Tags: algebra , maximum , inequalities

Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$. Find the greatest value of $M = abc$

1985 Tournament Of Towns, (091) T2

Tags: difference , prime , composite , combinatorics , maximum , number theory

From the set of numbers $1 , 2, 3, . . . , 1985$ choose the largest subset such that the difference between any two numbers in the subset is not a prime number (the prime numbers are $2, 3 , 5 , 7,... , 1$ is not a prime number) .

1999 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: geometry , distance , maximum , semicircle , cyclic quadrilateral

The quadrilateral $ABCD$ is inscribed in a circle of radius $1$, so that $AB$ is a diameter of the circumference and $CD = 1$. A variable point $X$ moves along the semicircle determined by $AB$ that does not contain $C$ or $D$. Determine the position of $X$ for which the sum of the distances from $X$ to lines $BC, CD$ and $DA$ is maximum.

1948 Moscow Mathematical Olympiad, 153

Tags: maximum , radical , cube , 3d geometry , combinatorial geometry , geometry , circles

* What is the radius of the largest possible circle inscribed into a cube with side $a$?

2010 Dutch IMO TST, 2

Tags: functional equation , maximum , algebra

Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.

2014 Hanoi Open Mathematics Competitions, 15

Tags: algebra , sum , inequalities , maximum

Let $a_1,a_2,...,a_9 \ge - 1$ and $a^3_1+a^3_2+...+a^3_9= 0$. Determine the maximal value of $M = a_1 + a_2 + ... + a_9$.

2011 Kyiv Mathematical Festival, 2

Tags: algebra , maximum , inequality

Find maximum of the expression $(a -b^2)(b - a^2)$, where $0 \le a,b \le 1$.

2005 Czech And Slovak Olympiad III A, 1

Tags: arithmetic sequence , sequence , algebra , maximum , recurrence relation

Consider all arithmetical sequences of real numbers $(x_i)^{\infty}=1$ and $(y_i)^{\infty} =1$ with the common first term, such that for some $k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30$, and $x_{k+1}y_{k+1} = 16$. Find all such pairs of sequences with the maximum possible $k$.

1991 All Soviet Union Mathematical Olympiad, 557

Tags: sum , absolute , maximum , algebra

The real numbers $x_1, x_2, ... , x_{1991}$ satisfy $$|x_1 - x_2| + |x_2 - x_3| + ... + |x_{1990} - x_{1991}| = 1991$$ What is the maximum possible value of $$|s_1 - s_2| + |s_2 - s_3| + ... + |s_{1990} - s_{1991}|$$ where $$s_n = \frac{x_1 + x_2 + ... + x_n}{n}?$$