IMO 1961

Official IMO 1961 problems  ·  6/6 solved, 5 verified.

Problem 1   ✓ verified · 4m48s · Solution →

(Hungary) Solve the system of equations:

$\begin{matrix} \quad x + y + z !!! &= a ; , \ x^2 +y^2+z^2 !!! &=b^2 \ \qquad \qquad xy !!! &= z^2 \end{matrix}$

where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x, y, z$ (the solutions of the system) are distinct positive numbers.

Problem 2   ✓ verified · 11m24s · Solution →

Let a,b, and c be the lengths of a triangle whose area is S. Prove that

$a^2 + b^2 + c^2 \ge 4S\sqrt{3}$

In what case does equality hold?

Problem 3   solved · 39m28s · Solution →

Solve the equation

$\cos^n{x} - \sin^n{x} = 1$

where n is a given positive integer.

Problem 4   ✓ verified · 19m03s · Solution →

In the interior of triangle $P_1 P_2 P_3$ a point P is given. Let $Q_1,Q_2,Q_3$ be the intersections of $PP_1, PP_2,PP_3$ with the opposing edges of triangle $ABC$. Prove that among the ratios $\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}$ there exists one not larger than 2 and one not smaller than 2.

Problem 5   ✓ verified · 39m11s · Solution →

Construct a triangle ABC if the following elements are given: $AC = b, AB = c$, and $\angle AMB = \omega \left(\omega < 90^{\circ}\right)$ where M is the midpoint of BC. Prove that the construction has a solution if and only if

$b \tan{\frac{\omega}{2}} \le c < b$

In what case does equality hold?

Problem 6   ✓ verified · 3m04s · Solution →

Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?