Question 1
Let $A,B,C$ be the angles of triangle $ABC$. Prove that
$$\sin\frac A2\sin\frac B2\sin\frac C2\lt \frac14.$$
$$\sin\frac A2\sin\frac B2\sin\frac C2\lt \frac14.$$
Question bank VMO 1970
Exam papers
5 questions
Question 2
Find all natural numbers which divide $N=1890$ but are not divisible by $45$. How many are there?
Question 3
To every point $(x,y)$ in the coordinate plane is assigned a real number $f(x,y)$. Suppose that:
(a) $f(x,0)=ax$, where $a$ is a nonzero constant;
(b) if two distinct points $(x_1,y_1),(x_2,y_2)$ have the same value of $f$, then $f$ is constant on the whole line through them.
Prove that:
(a) for every real $\alpha$, the set where $f(x,y)=\alpha$ is a line $D_\alpha$, and all these lines are parallel;
(b) $f(x,y)=ax+by$ for all $x,y$, where $b$ is a constant.
(a) $f(x,0)=ax$, where $a$ is a nonzero constant;
(b) if two distinct points $(x_1,y_1),(x_2,y_2)$ have the same value of $f$, then $f$ is constant on the whole line through them.
Prove that:
(a) for every real $\alpha$, the set where $f(x,y)=\alpha$ is a line $D_\alpha$, and all these lines are parallel;
(b) $f(x,y)=ax+by$ for all $x,y$, where $b$ is a constant.
Question 4
A fixed circle has center $O$, radius $R$, two fixed perpendicular diameters $AB,CD$, and tangent $At$ at $A$. A point $M$ moves on the circle; the lines $BM,DM$ meet $At$ at $P,Q$, respectively.
(a) Prove that the distance between $P,Q$ is the fourth proportional to the three lengths $AP,AQ$ and the diameter. Express the relation by the simplest possible formula.
(b) Construct with ruler and compass a point $M$ for which $BQ$ and $DP$ are parallel.
(c) Let $OP$ and $BQ$ meet at $N$. Guess the locus of $N$ and prove it.
(d) Let $BP,BQ$ meet the tangent at $D$ at $P',Q'$, respectively. State and prove a relation between $P',Q'$. Let $DP,DQ$ meet $BC$ at $P'',Q''$. State and prove a relation between $P'',Q''$.
Editorial clarification: use configurations where the named lines and finite intersections are defined. The unqualified source wording in (a), including its order of the three lengths, is preserved for discussion in the solution.
(a) Prove that the distance between $P,Q$ is the fourth proportional to the three lengths $AP,AQ$ and the diameter. Express the relation by the simplest possible formula.
(b) Construct with ruler and compass a point $M$ for which $BQ$ and $DP$ are parallel.
(c) Let $OP$ and $BQ$ meet at $N$. Guess the locus of $N$ and prove it.
(d) Let $BP,BQ$ meet the tangent at $D$ at $P',Q'$, respectively. State and prove a relation between $P',Q'$. Let $DP,DQ$ meet $BC$ at $P'',Q''$. State and prove a relation between $P'',Q''$.
Editorial clarification: use configurations where the named lines and finite intersections are defined. The unqualified source wording in (a), including its order of the three lengths, is preserved for discussion in the solution.
Question 5
Let $ABCDEFGH$ be a cube. A plane $P$ passes through $A$ and makes equal angles with the three edges $AB,AD,AE$.
(a) Find the common cosine of these three equal angles and construct the orthogonal projection of the cube onto $P$.
(b) Use this projection to discover as many noteworthy relations as possible between $P$ and the lines joining pairs of the eight vertices, and the planes containing triples of those vertices.
(a) Find the common cosine of these three equal angles and construct the orthogonal projection of the cube onto $P$.
(b) Use this projection to discover as many noteworthy relations as possible between $P$ and the lines joining pairs of the eight vertices, and the planes containing triples of those vertices.