Olympiad problems

1987 IMO Longlists, 20

Tags: inequalities , pigeonhole principle , linear algebra , inequality system , IMO Shortlist , IMO 1987 , IMO

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$. [i](IMO Problem 3)[/i] [i]Proposed by Germany, FR[/i]

1965 IMO Shortlist, 1

Tags: inequalities , trigonometry , IMO , IMO 1965

Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality \[ 2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}. \]

1963 IMO, 1

Tags: algebra , equation , parametric equation , IMO , IMO 1963

Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.

1990 IMO, 2

Tags: graph theory , combinatorics , Extremal combinatorics , Coloring , IMO , IMO 1990

Let $ n \geq 3$ and consider a set $ E$ of $ 2n \minus{} 1$ distinct points on a circle. Suppose that exactly $ k$ of these points are to be colored black. Such a coloring is [b]good[/b] if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $ n$ points from $ E$. Find the smallest value of $ k$ so that every such coloring of $ k$ points of $ E$ is good.

1998 IMO, 4

Tags: number theory , Divisibility , polynomial , IMO , IMO 1998 , david monk , Hi

Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.

2017 IMO Shortlist, N1

Tags: number theory , IMO , IMO 2017 , IMO Shortlist , Sequence

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$. [i]Proposed by Stephan Wagner, South Africa[/i]

1986 IMO Shortlist, 12

Tags: invariant , algorithm , combinatorics , pentagon , game strategy , IMO , IMO 1986

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

1972 IMO Longlists, 26

Tags: algebra , inequality system , IMO , IMO 1972

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

1969 IMO Shortlist, 45

Tags: combinatorics , combinatorial geometry , counting , convex quadrilateral , IMO , IMO 1969

Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.

1967 IMO, 5

Tags: algebra , Sequence , algebraic identities , IMO , IMO 1967

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

1966 IMO, 3

Tags: geometry , 3D geometry , tetrahedron , sphere , IMO , IMO 1966

Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

2010 IMO, 2

Tags: geometry , IMO Shortlist , Inversion , IMO 2010 , P2 , IMO

Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$. [i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]

1972 IMO Longlists, 15

Tags: factorial , number theory , binomial coefficients , recurrence relation , IMO , IMO 1972

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

1988 IMO, 1

Tags: algebra , polynomial , Vieta , Inequality , roots , IMO , IMO 1988

Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

1961 IMO Shortlist, 4

Tags: ratio , inequalities , geometry , Triangle , Intersection , IMO , IMO 1961

Consider triangle $P_1P_2P_3$ and a point $p$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers \[ \dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3} \] at least one is $\leq 2$ and at least one is $\geq 2$

1978 IMO, 2

Tags: geometry , 3D geometry , sphere , Locus problems , Locus , IMO , IMO 1978

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.

2000 IMO Shortlist, 8

Tags: geometry , homothety , circumcircle , Triangle , incircle , IMO , imo 2000

Let $ AH_1, BH_2, CH_3$ be the altitudes of an acute angled triangle $ ABC$. Its incircle touches the sides $ BC, AC$ and $ AB$ at $ T_1, T_2$ and $ T_3$ respectively. Consider the symmetric images of the lines $ H_1H_2, H_2H_3$ and $ H_3H_1$ with respect to the lines $ T_1T_2, T_2T_3$ and $ T_3T_1$. Prove that these images form a triangle whose vertices lie on the incircle of $ ABC$.

2018 IMO, 5

Tags: IMO , number theory , imo 2018 , Lifting the Exponent , IMO Shortlist , Hi

Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number $$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$ is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$. [i]Proposed by Bayarmagnai Gombodorj, Mongolia[/i]

2005 IMO, 3

Tags: inequalities , algebra , IMO 2005 , IMO Shortlist , IMO , hojoo lee , Hi

Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that \[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \] [i]Hojoo Lee, Korea[/i]

1977 Germany Team Selection Test, 1

Tags: rearrangement inequality , Inequality , permutation , algebra , IMO , IMO 1975

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

2014 IMO, 3

Tags: geometry , circumcircle , IMO , IMO 2014 , quadrilateral , tangent circles , tangent

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[ \angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.

1976 IMO Longlists, 26

Tags: geometry , 3D geometry , combinatorics , packing , IMO , imo 1976

A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.

1971 IMO Longlists, 43

Tags: linear algebra , matrix , algebra , combinatorial inequality , combinatorics , IMO , IMO 1971

Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.

1998 IMO, 5

Tags: geometry , incenter , trigonometry , Triangle , IMO , IMO 1998

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

1988 IMO Longlists, 7

Tags: linear algebra , combinatorics , Set systems , Coloring , IMO , IMO 1988

Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $ \frac {n}{2}$ zeros?