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Stage 7 · Mixed

10 problems · National olympiad second round; IMO P1/P4 · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. $P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$?

   1. A-3
   2. B-2
   3. C$2\sqrt 3$
   4. D$3\sqrt 3$
   5. E$\text{None of above}$

   Algebra Solution and answer checking →

2. Given triangle $ABC$, let $M$ be the midpoint of side $AB$ and $N$ be the midpoint of side $AC$. A circle is inscribed inside quadrilateral $NMBC$, tangent to all four sides, and that circle touches $MN$ at point $X.$ The circle inscribed in triangle $AMN$ touches $MN$ at point $Y$, with $Y$ between $X$ and $N$. If $XY=1$ and $BC=12$, find, with proof, the lengths of the sides $AB$ and $AC$.

   Geometry Solution and answer checking →

3. When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons?

   Geometry Solution and answer checking →

4. The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure?
   

   Geometry Solution and answer checking →

5. Solve the following system of equations for real $x,y$ and $z$:
   $\text{}$
   x &=& $\sqrt{2y+3}\\$
   y &=& $\sqrt{2z+3}\\$
   z &=& $\text{2x+3 .}$

   Algebra Solution and answer checking →

6. Given a positive integer $k$, a pigeon and a seagull play a game on an $n\times n$ board. The pigeon goes first, and they take turns doing the operations. The pigeon will choose $m$ grids and lay an egg in each grid he chooses. The seagull will choose a $k\times k$ grids and eat all the eggs inside them. If at any point every grid in the $n\times n$ board has an egg in it, then the pigeon wins. Else, the seagull wins. For every integer $n\geq k$, find all $m$ such that the pigeon wins.

   Proposed by amano_hina

   Combinatorics Solution and answer checking →

7. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$f(f(m)+n)+f(m)=f(n)+f(3 m)+2014$$ for all integers $m$ and $n$. (Netherlands) Answer. There is only one such function, namely $n \longmapsto 2 n+1007$.

   Algebra Solution and answer checking →

8. Find the smallest real number $C$, such that for any positive integers $x \neq y$ holds the following:

   $$\min(\{\sqrt{x^2 + 2y}\}, \{\sqrt{y^2 + 2x}\})<C$$

   Here $\{x\}$ denotes the fractional part of $x$. For example, $\{3.14\} = 0.14$.

   Proposed by Anton Trygub

   Algebra Solution and answer checking →

9. A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$.

   Number theory Solution and answer checking →

10. Let $m > 1$ be an integer, $n$ is an odd number satisfying $3\le n < 2m,$ number $a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$ satisfies $(1)$ for any $1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$ is a permutation of $1,2,3,\cdots,m; (2)$ for any $\text{1 < i m, 1 j n 1, |a i,j a i, j 1 | 1}$ holds. Find the minimal value of $M$, where $\text{M max 1 < i < m j 1 n a i,j .}$

    Combinatorics Solution and answer checking →

Answer key — Stage 7 · Mixed

Worked solutions for every problem are on the site, one page per problem.

1. $-2$ open
2. $AB = 12$ open
3. $\frac{7 - 3\sqrt{5}}{2}$ open
4. $100\pi - 40 \text{ kr.}$ open
5. $x = y = z = 3$ open
6. $k \leq n \leq 2k - 1$ open
7. $f(n) = 2n + 1007$ open
8. $\varphi - 1$ open
9. $324$ open
10. $7$ open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.