Question bank Math Olympiad Vault 2018

A grid rectangle that is not a square is cut into 8 different (non-congruent) grid polygons along the grid lines. What is its minimal possible area?

The rows and the columns of a $16 \times 16$ table are labeled from $1, 2, \dots, 16$, and the product $i \cdot j$ is written in the square in row $i$, column $j$. Several rows are chosen (at least $2$) and also several columns (at least $2$). Then the numbers at their intersections are deleted.

a) Can the sum of all deleted numbers be a prime?

b) What about the sum of all undeleted numbers?

A natural number is written on each face of a cube. To each vertex of the cube assign the product of the numbers of the three faces that have this vertex in common. Let the sum of these 8 products be 315. Determine the sum of the numbers on the faces (find all possibilities).

Write in the cells of a $4 \times 4$ table a different natural number such that the sums by rows are equal and the products by columns are also equal.

Facu and Nico play the following game with a $13 \times 13$ grid square. Facu cuts the square into rectangles having a side equal to $1$, in any way he wishes. Then Nico chooses a number $k$ among $1, 2, \ldots, 13$ and takes all the obtained rectangles $1 \times k$. How many grid cells can he take with certainty?

Given are 16 balls with weights $13$, $14$, $15$, $\ldots$, $28$ grams. Determine the balls with weights $13$, $14$, $27$, $28$ grams, by using a two-pan balance at most $26$ times.

A set of natural numbers is *regular* if each of its subsets has sum different from $1810$. Partition the numbers $452$, $453$, $\ldots$, $1809$ into a minimum number of regular sets.

Which regular $n$-gons have a triangulation consisting of isosceles triangles?

Find all natural $a$ such that $4x^2 + a$ is a prime for all $x = 0, 1, \dots, a-1$.

Given is a convex quadrilateral $ABCD$ with $AB = BD = a$ and $CD = DA = b$. Let $P$ in side $AB$ such that $DP$ is bisector of $\angle ADB$ and $Q$ in side $BC$ such that $DQ$ is bisector of $\angle CDB$. Find the circumradius of triangle $DPQ$.

There were three candidates $A$, $B$, $C$ in the elections for a provincial governor. In the first round $A$ won $44\%$ of the number of votes given for $B$ and $C$ together and $C$ had fewest votes. No candidate had the majority necessary for a first-round win, so there was a second round for $A$ and $B$. The voters in the second round were the same as in the first round, except $p\%$ of the voters for $C$ who chose not to participate in the second round; $p$ is an integer, $1 \le p \le 100$. In addition the ones who voted for $B$ in the first round did so again in the second round.

A journalist claims that, knowing all the above, one can infer who the winner is with certainty. For what values of $p$ is he right?

Note. The winner in the second round is the one who obtains more than half of the total number of votes in the second round.

A chooses 13 different 3-digit numbers. Then B selects several of these 13 and tries to obtain, by using each selected number once and the operations $+, -, \times, \div$, an expression with value strictly between 3 and 4. B wins if he succeeds in doing so; otherwise A wins. Who has a winning strategy?

Given are 51 natural numbers written in a row. Their sum is 100. An integer is *representable* if it can be expressed as the sum of several consecutive numbers in the given row. Prove that for each $k \in \{1, 2, \dots, 100\}$ one of the numbers $k$ and $100-k$ is representable.

Let $\triangle ABC$ be a triangle with perimeter $100$ and incenter $I$. The parallel to $AB$ through $I$ divides the median through $A$ in ratio $7:3$, counted from $A$. Find the length of side $AB$.

For $n \in \mathbb{N}$ let $D_2(n)$ (respectively $D_3(n)$) denote the number of divisors of $n$ that are perfect squares (respectively perfect cubes). Prove that there is an $n$ such that $D_2(n) = 999D_3(n)$.

A sequence of natural numbers is *admissible* if its terms are less or equal to $100$ and its sum is greater than $1810$. Find the least $d$ such that each admissible sequence has a subsequence sum in the interval $[1810-d, 1810+d]$.

All diagonals of a convex 10-gon are drawn. They divide the angles into 80 parts. It is known that 59 of these parts are equal. Determine the maximum of different values among the 80 angles of division. How many times does each of these values occur?

In a convex quadrilateral $ABCD$ we have that:
* $R$ and $S$ are points in the interior of the segments $CD$ and $AB$ respectively, with $AD = CR$ and $BC = AS$.
* $P$ and $Q$ are the midpoints of $DR$ and $SB$ respectively.
* $M$ is the midpoint of $AC$.
If it is known that $\angle MPC + \angle MQA = 90^\circ$, prove that $ABCD$ is a cyclic quadrilateral.

Prove that every positive integer can be expressed as a sum of powers of $3$, $4$ and $7$ in such a way that the representation does not contain two powers with the same base and the same exponent.
For example, $2 = 7^0 + 7^0$ and $22 = 3^2 + 3^2 + 4^1$ are not valid sums, but $2 = 3^0 + 7^0$ and $22 = 3^2 + 3^0 + 4^1 + 4^0 + 7^1$ are valid.

Consider the product $P_n = 1! \cdot 2! \cdot 3! \cdot \dots \cdot n!$, where $n! = 1 \cdot 2 \cdot 3 \cdot \dots \cdot n$, for every positive integer $n$.

a) Find all possible values of positive integers $m$ such that $\frac{P_{2020}}{m!}$ is a perfect square.

b) Prove that there exist infinitely many values of $n$ such that $\frac{P_n}{m!}$ is a perfect square for at least two positive integers $m$.