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Stage 8 · Mixed

10 problems · IMO Shortlist mid-range; USAMO P2/P5 · mathsolympiadprep.com

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

1. Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

   Combinatorics Solution and answer checking →

2. Does there exist a field such that its multiplicative group is isomorphic to its additive group?

   Algebra Solution and answer checking →

3. Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A \cup B = S$, $A \cap B = \emptyset$, and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be the number of linear partitions of $S$. For each positive integer $n$, find the maximum of $L_S$ over all sets $S$ of $n$ points.

   Combinatorics Solution and answer checking →

4. Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions:

   i. $f(x) = a_0 $x^{2n}$ + a_2 $x^{2n}$ - 2} + $\cdots + a_{2n}$ - 2}
   x^2 + $a_{2n},$ a_0 > 0$;
   ii. $\$\sum_{j=0}^n a_{2j} a_{2n}$ - 2j} $\leq \left( \right.$
   $$\begin{array}{c} 2n\\ n\end{array}$$ $\left. \right)$ a_0 $a_{2n}\$;$
   iii. All the roots of $f(x)$ are imaginary numbers with no real part.

   Algebra Solution and answer checking →

5. Points $A$, $V_1$, $V_2$, $B$, $U_2$, $U_1$ lie fixed on a circle $\Gamma$, in that order, and such that $BU_2 > AU_1 > BV_2 > AV_1$.

   Let $X$ be a variable point on the arc $V_1 V_2$ of $\Gamma$ not containing $A$ or $B$. Line $XA$ meets line $U_1 V_1$ at $C$, while line $XB$ meets line $U_2 V_2$ at $D$. Let $O$ and $\rho$ denote the circumcenter and circumradius of $\triangle XCD$, respectively.

   Prove there exists a fixed point $K$ and a real number $c$, independent of $X$, for which $OK^2 - \rho^2 = c$ always holds regardless of the choice of $X$.

   Geometry Solution and answer checking →

6. For each integer $n\ge 1,$ compute the smallest possible value of $$\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor$$ over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$

   *

   Combinatorics Solution and answer checking →

7. Let $n \geq 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_{1}$, $\ldots, x_{n}$ satisfying $$x_{1}\left(1-x_{2}\right)=x_{2}\left(1-x_{3}\right)=\ldots=x_{n-1}\left(1-x_{n}\right)=x_{n}\left(1-x_{1}\right)=a$$

   Algebra Solution and answer checking →

8. Given positive integer $n \ge 5$ and a convex polygon $P$, namely $A_1A_2...A_n$. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $A_iA_jA_kA_l (1\le i<j<k<l\le n)$ not on diagonals of $P$, such that the $\tbinom{n}{4}$ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

   Geometry Solution and answer checking →

9. Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties:

   (i) $f(1) = 2$;

   (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

   Algebra Solution and answer checking →

10. Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$:

    (i) move the last digit of $a$ to the first position to obtain the numb er $b$;
    (ii) square $b$ to obtain the number $c$;
    (iii) move the first digit of $c$ to the end to obtain the number $d$.

    (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.)

    Find all numbers $a$ for which $d\left( a\right) =a^2$.

    *

    Number theory Solution and answer checking →

Answer key — Stage 8 · Mixed

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

1. $$$\boxed{ \begin{cases} 1 & \text{if }n\text{ is a power of }2 \\ 2 & \text{otherwise} \end{cases} }.$$$ open
2. There exist no such field. open
3. $\binom{n}{2} + 1$ open
4. $f(x) = a_0 (x^2 + \alpha^2)^n \text{ where } a_0 > 0 \text{ and } \alpha \in \mathbb{R} \setminus \{0\}$ open
5. $K \text{ is the intersection of } AB' \text{ and } BA', \text{ and } c \text{ is a constant}$ open
6. $\lfloor \log_2(n) \rfloor + 1$ open
7. $(-\infty, \frac{1}{4}] \cup \{\frac{1}{4 \cos^{2} \frac{k\pi}{n}}; k \in \mathbb{N}, 1 \leq k < \frac{n}{2}\}$ open
8. $\text{Proven}$ open
9. $/text{yes}$ open
10. $a = \underbrace{2\dots2}_{n \ge 0}1, \qquad a = 2, \qquad a = 3.$ open

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