Olympiad problems

1968 IMO, 3

Tags: algebra , system of equations , polynomial , IMO , IMO 1968 , Discriminant

Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.

1968 IMO Shortlist, 26

Tags: function , algebra , periodic function , functional equation , IMO , IMO 1968

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$. Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.

1968 IMO, 4

Tags: geometry , 3D geometry , tetrahedron , triangle inequality , IMO , IMO 1968

Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

1968 IMO Shortlist, 3

Tags: geometry , 3D geometry , tetrahedron , triangle inequality , IMO , IMO 1968

Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

1968 IMO Shortlist, 2

Tags: trigonometry , algebra , Triangle , triangle inequality , IMO , IMO 1968

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

2010 Morocco TST, 4

Tags: trigonometry , algebra , Triangle , triangle inequality , IMO , IMO 1968

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

1968 IMO, 6

Tags: floor function , number theory , equation , Summation , IMO , IMO 1968

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] [hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]

1968 IMO Shortlist, 22

Tags: number theory , decimal representation , Digits , equation , IMO , IMO 1968

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

1968 IMO, 1

Tags: trigonometry , algebra , Triangle , triangle inequality , IMO , IMO 1968

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

1968 IMO Shortlist, 4

Tags: algebra , system of equations , polynomial , IMO , IMO 1968 , Discriminant

Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.

1968 IMO, 2

Tags: number theory , decimal representation , Digits , equation , IMO , IMO 1968

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

1968 IMO Shortlist, 15

Tags: floor function , number theory , equation , Summation , IMO , IMO 1968

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] [hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]

1968 IMO, 5

Tags: function , algebra , periodic function , functional equation , IMO , IMO 1968

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$. Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.