Olympiad problems

MIPT student olimpiad autumn 2022, 4

Tags: geometry

In $R^n$ space is given a finite set of points $X$. It is known that for any subset $Y \subseteq X$ of at most $n+1$ points, there is a unit ball $B_Y$ containing $Y$ and not containing the origin. Prove that there is a unit a ball $B_X$ containing $X$ and not containing the origin.

1968 IMO Shortlist, 26

Tags: function , algebra , periodic function , functional equation

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$. Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.

2005 MOP Homework, 5

Tags: number theory

Find all integer solutions to $y^2(x^2+y^2-2xy-x-y)=(x+y)^2(x-y)$.

2004 Purple Comet Problems, 11

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Find the sum of all integers $x$ satisfying $1 + 8x \le 358 - 2x \le 6x + 94$.

1987 IMO Longlists, 32

Tags: number theory , inequalities

Solve the equation $28^x = 19^y +87^z$, where $x, y, z$ are integers.

1998 South africa National Olympiad, 5

Tags: greatest common divisor , number theory

Prove that \[ \gcd{\left({n \choose 1},{n \choose 2},\dots,{n \choose {n - 1}}\right)} \] is a prime if $n$ is a power of a prime, and 1 otherwise.

2018 Korea National Olympiad, 4

Tags: algebra , inequality , inequalities , sequence

Find all real values of $K$ which satisfies the following. Let there be a sequence of real numbers $\{a_n\}$ which satisfies the following for all positive integers $n$. (i). $0 < a_n < n^K$. (ii). $a_1 + a_2 + \cdots + a_n < \sqrt{n}$. Then, there exists a positive integer $N$ such that for all integers $n>N$, $$a^{2018}_1 + a^{2018}_2 + \cdots +a^{2018}_n < \frac{n}{2018}$$

2006 QEDMO 2nd, 12

Tags: induction , additive number theory

Let $a_{1}=1$, $a_{2}=2$, $a_{3}$, $a_{4}$, $\cdots$ be the sequence of positive integers of the form $2^{\alpha}3^{\beta}$, where $\alpha$ and $\beta$ are nonnegative integers. Prove that every positive integer is expressible in the form \[a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}},\] where no summand is a multiple of any other.

1966 All Russian Mathematical Olympiad, 078

Tags: convex polygon , circles , geometry

Prove that you can always pose a circle of radius $S/P$ inside a convex polygon with the perimeter $P$ and area $S$.

2014 Albania Round 2, 3

Tags: geometry

In a right $\Delta ABC$ ($\angle C = 90^{\circ} $), $CD$ is the height. Let $r_1$ and $r_2$ be the radii of inscribed circles of $\Delta ACD$ and $\Delta DCB$. Find the radius of inscribed circle of $\Delta ABC$

2024 Austrian MO Regional Competition, 1

Tags: inequalities , algebra

Let $a$, $b$ and $c$ be real numbers larger than $1$. Prove the inequality $$\frac{ab}{c-1}+\frac{bc}{a - 1}+\frac{ca}{b -1} \ge 12.$$ When does equality hold? [i](Karl Czakler)[/i]

2021 Junior Macedonian Mathematical Olympiad, Problem 4

Tags: algebra , inequality

Let $a$, $b$, $c$ be positive real numbers such that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} = \frac{27}{4}.$ Show that: $$\frac{a^3+b^2}{a^2+b^2} + \frac{b^3+c^2}{b^2+c^2} + \frac{c^3+a^2}{c^2+a^2} \geq \frac{5}{2}.$$ [i]Authored by Nikola Velov[/i]

2023 Bulgaria National Olympiad, 6

Tags: combinatorics

In a class of $26$ students, everyone is being graded on five subjects with one of three possible marks. Prove that if $25$ of these students have received their marks, then we can grade the last one in such a way that their marks differ from these of any other student on at least two subjects.

2010 Math Prize For Girls Problems, 4

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Consider the sequence of six real numbers 60, 10, 100, 150, 30, and $x$. The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$? (The median of a sequence of six real numbers is the average of the two middle numbers after all the numbers have been arranged in increasing order.)

2025 India STEMS Category A, 6

Tags: polynomial , algebra

Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots. (Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$). [i]Proposed by Malay Mahajan[/i]

2002 AIME Problems, 1

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Given that \begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& \text{y is the number formed by reversing the digits of x; and}\\ &(3)& z=|x-y|. \end{eqnarray*}How many distinct values of $z$ are possible?

2018 BMT Spring, 6

Tags: geometry

A triangle $T$ has all integer side lengths and at most one of its side lengths is greater than ten. What is the largest possible area of $T$ ?

2014 Stars Of Mathematics, 2

Tags: number theory , combinatorics

Determine all integers $n\geq 1$ for which the numbers $1,2,\ldots,n$ may be (re)ordered as $a_1,a_2,\ldots,a_n$ in such a way that the average $\dfrac {a_1+a_2+\cdots + a_k} {k}$ is an integer for all values $1\leq k\leq n$. (Dan Schwarz)

2020 HMNT (HMMO), 8

Tags: combinatorics , geometry

A bar of chocolate is made of $10$ distinguishable triangles as shown below: [center][img]https://cdn.artofproblemsolving.com/attachments/3/d/f55b0af0ce320fbfcfdbfab6a5c9c9306bfd16.png[/img][/center] How many ways are there to divide the bar, along the edges of the triangles, into two or more contiguous pieces?

2012 National Olympiad First Round, 18

Tags: number theory , prime number

If the representation of a positive number as a product of powers of distinct prime numbers contains no even powers other than $0$s, we will call the number singular. At most how many consequtive singular numbers are there? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \text{None}$

1969 AMC 12/AHSME, 30

Tags:

Let $P$ be a point of hypotenuse $AB$ (or its extension) of isosceles right triangle $ABC$. Let $s=AP^2+PB^2$. Then: $\textbf{(A) }s<2CP^2\text{ for a finite number of positions of }P$ $\textbf{(B) }s<2CP^2\text{ for an infinite number of positions of }P$ $\textbf{(C) }s=2CP^2\text{ only if }P\text{ is the midpoint of }AB\text{ or an endpoint of }AB$ $\textbf{(D) }s=2CP^2\text{ always}$ $\textbf{(E) }s>2CP^2\text{ if }P\text{ is a trisection point of }AB$

2020 Latvia Baltic Way TST, 6

Tags: combinatorics

For a natural number $n \ge 3$ we denote by $M(n)$ the minimum number of unit squares that must be coloured in a $6 \times n$ rectangle so that any possible $2 \times 3$ rectangle (it can be rotated, but it must be contained inside and cannot be cut) contains at least one coloured unit square. Is it true that for every natural $n \ge 3$ the number $M(n)$ can be expressed as $M(n)=p_n+k_n^3$, where $p_n$ is a prime and $k_n$ is a natural number?

Kyiv City MO 1984-93 - geometry, 1989.8.5

Tags: geometry , right triangle , equilateral , inscribed , construction

The student drew a right triangle $ABC$ on the board with a right angle at the vertex $B$ and inscribed in it an equilateral triangle $KMP$ such that the points $K, M, P$ lie on the sides $AB, BC, AC$, respectively, and $KM \parallel AC$. Then the picture was erased, leaving only points $A, P$ and $C$. Restore erased points and lines.

1985 IMO Longlists, 60

Tags: trigonometry , algebra

The sequence $(s_n)$, where $s_n= \sum_{k=1}^n \sin k$ for $n = 1, 2,\dots$ is bounded. Find an upper and lower bound.

2012 Denmark MO - Mohr Contest, 1

Tags: geometry , circles , area

Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$. [img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]