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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.

   IMO Shortlist

   Combinatorics Solution and answer checking →

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

   International Mathematics Competition

   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.

   Putnam

   Combinatorics Solution and answer checking →

4. 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 $X$.

   USA Team Selection Test

   Geometry Solution and answer checking →

5. 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 $\{1,\dots,n\}.$

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   IMO Shortlist

   Combinatorics Solution and answer checking →

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

   China Team Selection Test

   Algebra Solution and answer checking →

7. 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 $d\left( a\right) =a^2$.

   *

   IMO Shortlist

   Number theory 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.

   China Team Selection Test

   Geometry Solution and answer checking →

9. Determine all integers $n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$.

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   IMO Shortlist

   Number theory Solution and answer checking →

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

    International Mathematics Competition

    Algebra Solution and answer checking →

Answer key — Stage 8 · Mixed

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

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

Problems belong to the competitions that set them. Each question on this sheet names the competition it came from, and the licences the material is used under are listed below. Free to copy for classroom use.

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