Question bank Math Olympiad Vault 1964

Find all value ranges of $x$ for which $$
\cos x+\operatorname{sen} x\gt 1
$$ same problem for $$
\cos x+|\operatorname{sen} x|\gt 1
$$

An equilateral triangle inscribed in a circle of center $O$ and radius equal to $4~\mathrm{cm}$, a right angle is rotated around $O$. Find the area of the common part to the given triangle and the one obtained in that turn.

A microgroove rotates at speed of $33 \frac{1}{3}$ revolutions per minute and its hearing lasts 24 min $30 \mathrm{~s}$. The engraved part has an outer diameter $29 \mathrm{~cm}$ and 11.5 $\mathrm{cm}$ inner diameter. With these data, calculate the length of the engraved groove.

How many three-digit numbers (i.e. greater than $99$ and less than $1000$) are there that have their central number higher than the other two? How many of them also have the three different figures?

It is well known that if $\frac{p}{q} = \frac{r}{s}$, both reasons are equal to $\frac{p - r}{q - s}$. We now write the equality $$
\frac{3x - b}{3x - 5b} = \frac{3a - 4b}{3a - 8b}
$$ by the previous property, both fractions must be equal to $$
\frac{3x - 5b - 3a + 8b}{3x - b - 3a + 4b} = \frac{3x - 3a + 3b}{3x - 3a + 3b} = 1
$$ while the proposals are usually different from the unit. clearly explain what this result is due to.

If $m$, $k$, $n$ are natural numbers and $n \gt 1$, prove that we cannot have $m(m + 1) = k^n$.

In the triangle $ABC$, the length of the altitude from $A$ is not less than $BC$, and the length of the altitude from $B$ is not less than $AC$. Find the angles.

What is the smallest number of tetrahedrons into which a cube can be partitioned?

A circle is inscribed in $ABCD$. $AB$ is parallel to $CD$, and $BC = AD$. The diagonals $AC$, $BD$ meet at $E$. The circles inscribed in $ABE$, $BCE$, $CDE$, $DAE$ have radius $r_1$, $r_2$, $r_3$, $r_4$ respectively. Prove that $1 / r_1 + 1 / r_3 = 1 / r_2 + 1 / r_4$.

Given a lattice of regular hexagons. A bug crawls from vertex $A$ to vertex $B$ along edges of the hexagons, taking the shortest possible path (or one of them). Prove that it travels a distance at least $AB/2$ in one direction. If it travels exactly $AB/2$ in one direction, how many edges does it traverse?

Given $n$ odd and a set of integers $\{a_1\}$, $\{a_2\}$, ..., $\{a_n\}$, derive a new set $(\{a_1\} + \{a_2\}) / 2$, $(\{a_2\} + \{a_3\}) / 2$, ..., $(\{a_{n - 1}\} + \{a_n\}) / 2$, $(\{a_n\} + \{a_1\}) / 2$. However many times we repeat this process for a particular starting set we always get integers. Prove that all the numbers in the starting set are equal.

For example, if we started with $5, 9, 1$, we would get $7, 5, 3$, and then $6, 4, 5$, and then $5, 4, 5, 5.5$. The last set does not consist entirely of integers.

Reduce each of the first billion natural numbers (billion $= 10^9$) to a single digit by taking its digit sum repeatedly. Do we get more 1s than 2s?

$ABCD$ is a convex quadrilateral. $A'$ is the foot of the perpendicular from $A$ to the diagonal $BD$, $B'$ is the foot of the perpendicular from $B$ to the diagonal $AC$, and so on. Prove that $A'B'C'D'$ is similar to $ABCD$.

A circle center $O$ is inscribed in $ABCD$ (touching every side). Prove that $\angle AOB + \angle COD$ equals $180$ degrees.

The natural numbers $a$, $b$, $n$ are such that for every natural number $k$ not equal to $b$, $b - k$ divides $a - k^n$. Prove that $a = b^n$.

a) Find the smallest square with last digit not $0$ which becomes another square (not zero) by the deletion of its last two digits.


b) Find all squares, not containing the digits $0$ or $5$, such that if the second digit is deleted the resulting number divides the original one.

a. The convex hexagon $ABCDEF$ has all angles equal. Prove that $AB - DE = EF - BC = CD - FA$.


b. Given six lengths $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$, $a_{6}$ satisfying $a_{1} - a_{4} = a_{5} - a_{2} = a_{3} - a_{6}$, show that you can construct a hexagon with sides $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$, $a_{6}$ and equal angles.

Find all possible integer solutions for $\sqrt{x + \sqrt{x + \ldots + \sqrt{x}}} = y$, where there are 1998 square roots.

How many (algebraically) different expressions can we obtain by placing parentheses in $a_1 / a_2 / \ldots / a_n$?