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Stage 6 · Algebra

10 problems · National Olympiad, first round · mathsolympiadprep.com

The answer key prints on its own page at the end.

Let $P$ denote the set of all subsets of $\{1, \ldots, 23\}$ . A subset $S \subseteq P$ is called good if whenever $A, B$ are sets in $S$ , the set $(A \backslash B) \cup (B \backslash A)$ is also in $S$ . (Here, $A \backslash B$ denotes the set of all elements in $A$ that are not in $B$ , and $B \backslash A$ denotes the set of all elements in $B$ that are not in $A$ .) What fraction of the good subsets of $P$ have between 2015 and 3015 elements, inclusive?
If your answer is a decimal number or a fraction (of the form $m / n$ , where $m$ and $n$ are positive integers), then your score on this problem will be equal to $\max \{0,25-\lfloor 1000|A-N|\rfloor\}$ , where $N$ is your answer and $A$ is the actual answer. Otherwise, your score will be zero.

HMMT February

Algebra Solution and answer checking →
Let $k$ be a real number such that the product of real roots of the equation
$X^{4}+2 X^{3}+(2+2 k) X^{2}+(1+2 k) X+2 k=0$
is $-2013$ . Find the sum of the squares of these real roots.

Selection tests for the Balkan Mathematical Olympiad 2013

Algebra Solution and answer checking →
Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$ , if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.

Asia Pacific Math Olympiad

Algebra Solution and answer checking →
Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.

China Team Selection Test

Algebra Solution and answer checking →
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+\to\mathbb{R}^+$ that satisfy $\Big(1+yf(x)\Big)\Big(1-yf(x+y)\Big)=1$ for all $x,y\in\mathbb{R}^+$ .

Czech-Polish-Slovak Match

Algebra Solution and answer checking →
The angles $A$ , $B$ and $C$ of a triangle are measured in degrees, and the lengths of the opposite sides are $a$ , $b$ and $c$ respectively. Prove that
$60 \le \frac{aA + bB + cC}{a+b+c} < 90.$

BMO Round 1 2014

Algebra Solution and answer checking →
Let $n \ge 2$ be an integer. Let $x_1,\ldots,x_n$ be positive real numbers whose product is $n-1$ . Prove that

KöMaL problem A 2025

Algebra Solution and answer checking →
Andriy and Olesya write a natural number each on a chalkboard. It turns out, that number, written by Olesya, has sum of digits $2018$ and has precisely $1$ digit less than Andriy's number. It is also known, that difference of numbers, written by him, equals to one-digit number. What can be the number, written by Andriy?

Ukrainian National Mathematical Olympiad

Algebra Solution and answer checking →
If $x$ , $y$ , $z$ are positive numbers with $x + y + z = 1$ , show that:
$\begin{align*} a)\ 1 - \frac{x^2-yz}{x^2+x} &= \frac{(1-y)(1-z)}{x^2+x}; \\ b)\ \frac{x^2-yz}{x^2+x} + \frac{y^2-zx}{y^2+y} + \frac{z^2-xy}{z^2+z} &\le 0. \end{align*}$

Romanian Mathematical Olympiad - District Round

Algebra Solution and answer checking →
Let $a_1, a_2, \dots, a_n$ be a sequence of real numbers lying between $1$ and $-1$ , i.e. $-1 < a_i < 1$ , for $1 \le i \le n$ , and such that
(i) $a_1 + a_2 + \dots + a_n = 0$ ;
(ii) $a_1^2 + a_2^2 + \dots + a_n^2 = 40$ .
Determine the smallest possible value of $n$ .

China Hong Kong Mathematical Olympiad

Algebra Solution and answer checking →

Answer key — Stage 6 · Algebra

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

$18839183877670041942218307147122500601235/47691684840486192422095701784512492731212$ open
$4027$ open
$P(x)=\pm x^d+c \text{, where } c \text {is an integer and }d\text{ is a positive integer.}$ open
$(a, n) = (a, 1)$ open
$f(x) = \frac{1}{x + a}$ open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution 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.