Olympiad problems

2004 India IMO Training Camp, 2

Determine all integers $a$ such that $a^k + 1$ is divisible by $12321$ for some $k$

Kvant 2021, M2673

Tags: combinatorics , probability

There are $n{}$ passengers in the queue to board a $n{}$-seat plane. The first one in the queue is an absent-minded old lady who, after boarding the plane, sits down at a randomly selected place. Each subsequent passenger sits in his seat if it is free, and in a random seat otherwise. How many passengers will be out of their seats on average? [i]Proposed by A. Zaslavsky[/i]

1991 Greece Junior Math Olympiad, 4

Tags: algebra

Let $x+y=a$ and $xy=b$. Calculate exression $ x^4+y^4$ in terms of $a$ and $b$.

2022 Czech-Austrian-Polish-Slovak Match, 4

Tags: number theory

Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.

2023 Turkey MO (2nd round), 3

Tags: combinatorics

Let a $9$-digit number be balanced if it has all numerals $1$ to $9$. Let $S$ be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of $k$ such that any two subsequences of $S$ which has consecutive $k$ numerals are different from each other.

2020 Taiwan TST Round 3, 5

Tags: moving points , geometry

Let $O$ and $H$ be the circumcenter and the orthocenter, respectively, of an acute triangle $ABC$. Points $D$ and $E$ are chosen from sides $AB$ and $AC$, respectively, such that $A$, $D$, $O$, $E$ are concyclic. Let $P$ be a point on the circumcircle of triangle $ABC$. The line passing $P$ and parallel to $OD$ intersects $AB$ at point $X$, while the line passing $P$ and parallel to $OE$ intersects $AC$ at $Y$. Suppose that the perpendicular bisector of $\overline{HP}$ does not coincide with $XY$, but intersect $XY$ at $Q$, and that points $A$, $Q$ lies on the different sides of $DE$. Prove that $\angle EQD = \angle BAC$. [i]Proposed by Shuang-Yen Lee[/i]

2009 District Olympiad, 3

Tags: limit , induction , real analysis

Let $(x_n)_{n\ge 1}$ a sequence defined by $x_1=2,\ x_{n+1}=\sqrt{x_n+\frac{1}{n}},\ (\forall)n\in \mathbb{N}^*$. Prove that $\lim_{n\to \infty} x_n=1$ and evaluate $\lim_{n\to \infty} x_n^n$.

2018 PUMaC Live Round, Calculus 3

Tags: calculus

Let $\mathcal{R}(f(x))$ denote the number of distinct real roots of $f(x)$. Compute $$\sum_{a=1}^{1009}\sum_{b=1010}^{2018}\mathcal{R}(x^{2018}-ax^{2016}+b).$$

1966 All Russian Mathematical Olympiad, 073

Tags: inequalities , geometric inequality , geometry

a) Points $B$ and $C$ are inside the segment $[AD]$. $|AB|=|CD|$. Prove that for all of the points P on the plane holds inequality $$|PA|+|PD|>|PB|+|PC|$$ b) Given four points $A,B,C,D$ on the plane. For all of the points $P$ on the plane holds inequality $$|PA|+|PD| > |PB|+|PC|.$$ Prove that points $B$ and C are inside the segment $[AD]$ and$ |AB|=|CD|$.

2010 Saudi Arabia BMO TST, 2

Tags: right triangle , geometry , geometric inequality

Show that in any triangle $ABC$ with $\angle A = 90^o$ the following inequality holds $$(AB -AC)^2(BC^2 + 4AB \cdot AC)^ 2 < 2BC^6.$$

2011 Moldova Team Selection Test, 4

Tags: algebra

Let $n$ be an integer satisfying $n\geq2$. Find the greatest integer not exceeding the expression: $E=1+\sqrt{1+\frac{2^2}{3!}}+\sqrt[3]{1+\frac{3^2}{4!}}+\dots+\+\sqrt[n]{1+\frac{n^2}{(n+1)!}}$

The Golden Digits 2024, P3

Tags: combinatorics , geometry

On the surface of a sphere, a non-intersecting closed curve is drawn. It divides the surface of the sphere in two regions, which are coloured red and blue. Prove that there exist two antipodes of different colours. [i]Note: the curve is colourless.[/i] [i]Proposed by Vlad Spătaru[/i]

2005 Greece National Olympiad, 4

Tags: circumcircle , geometry

Let $OX_1 , OX_2$ be rays in the interior of a convex angle $XOY$ such that $\angle XOX_1=\angle YOY_1< \frac{1}{3}\angle XOY$. Points $K$ on $OX_1$ and $L$ on $OY_1$ are fixed so that $OK=OL$, and points $A$, $B$ are vary on rays $(OX , (OY$ respectively such that the area of the pentagon $OAKLB$ remains constant. Prove that the circumcircle of the triangle $OAB$ passes from a fixed point, other than $O$.

2017 Korea Winter Program Practice Test, 3

Tags: combinatorics

For a number consists of $0$ and $1$, one can perform the following operation: change all $1$ into $100$, all $0$ into $1$. For all nonnegative integer $n$, let $A_n$ be the number obtained by performing the operation $n$ times on $1$(starts with $100,10011,10011100100,\dots$), and $a_n$ be the $n$-th digit(from the left side) of $A_n$. Prove or disprove that there exists a positive integer $m$ satisfies the following: For every positive integer $l$, there exists a positive integer $k\le m$ satisfying$$a_{l+k+1}=a_1,\ a_{l+k+2}=a_2,\ \dots,\ a_{l+k+2017}=a_{2017}$$

2013 APMO, 4

Tags: polynomial , combinatorics , algebra

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

1988 Vietnam National Olympiad, 2

Tags: polynomial , algebra

Suppose $ P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0$ be a real polynomial of degree $ n > 2$ with $ a_n \equal{} 1$, $ a_{n\minus{}1} \equal{} \minus{}n$, $ a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2}$ such that all the roots of $ P$ are real. Determine the coefficients $ a_i$.

1963 All Russian Mathematical Olympiad, 028

Tags: tournament , combinatorics

Eight men had participated in the chess tournament. (Each meets each, draws are allowed, giving $1/2$ of point, winner gets $1$.) Everyone has different number of points. The second one has got as many points as the four weakest participants together. What was the result of the play between the third prizer and the chess-player that have occupied the seventh place?

2018 IFYM, Sozopol, 3

Tags: number theory

Find all positive integers $n$ for which the number $\frac{n^{3n-2}-3n+1}{3n-2}$ is whole. [hide=EDIT:] In the original problem instead of whole we search for integers, so with this change $n=1$ will be a solution. [/hide]

2025 Chile TST IMO-Cono, 2

Tags: combinatorics

At a meeting, there are $ N $ people who do not know each other. Prove that it is possible to introduce them in such a way that no three of them have the same number of acquaintances.

1966 Kurschak Competition, 2

Tags: number theory , digit

Show that the $n$ digits after the decimal point in $(5 +\sqrt{26})^n$ are all equal.

2014 China Team Selection Test, 6

Tags: combinatorics

Let $n\ge 2$ be a positive integer. Fill up a $n\times n$ table with the numbers $1,2,...,n^2$ exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most $n$. Show that there exist a $2\times 2$ square of adjacent cells such that the diagonally opposite pairs sum to the same number.

2008 Bulgarian Autumn Math Competition, Problem 9.3

Tags: natural number , number theory , algebra , polynomial , divisor

Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.

2019 AIME Problems, 10

Tags: complex number

For distinct complex numbers $z_1,z_2,\dots,z_{673}$, the polynomial \[ (x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3 \] can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$, where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$. The value of \[ \left| \sum_{1 \le j <k \le 673} z_jz_k \right| \] can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2011 May Olympiad, 2

Tags: number theory , digit

We say that a four-digit number $\overline{abcd}$ ($a \ne 0$) is [i]pora [/i] if the following terms are true : $\bullet$ $a\ge b$ $\bullet$ $ab - cd = cd -ba$. For example, $2011$ is pora because $20-11 = 11-02$ Find all the numbers around.

1991 AIME Problems, 3

Tags: inequalities , floor function , ceiling function , ratio , probability

Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives \begin{eqnarray*} &\ & \binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1+\binom{1000}{2}(0.2)^2+\cdots+\binom{1000}{1000}(0.2)^{1000}\\ &\ & = A_0 + A_1 + A_2 + \cdots + A_{1000}, \end{eqnarray*} where $A_k = \binom{1000}{k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For which $k$ is $A_k$ the largest?