Olympiad problems

2015 India IMO Training Camp, 2

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2016 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: circles , geometry

Circle of radius $R_1$ is inscribed in an acute angle $\alpha$. Second circle with radius $R_2$ touches one of the sides forming the angle $\alpha$ in same point as first circle and intersects the second side in points $A$ and $B$, such that centers of both circles lie inside angle $\alpha$. Prove that $$AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}$$

2003 Moldova National Olympiad, 10.8

Tags: logarithm , algebra

Find all integers n for which number $ \log_{2n\minus{}1}(n^2\plus{}2)$ is rational.

2010 Baltic Way, 4

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients such that \[(x-2010)P(x+67)=xP(x) \] for every integer $x$.

1964 All Russian Mathematical Olympiad, 048

Tags: number theory , factorial

Find all the natural $n$ such that $n!$ is not divisible by $n^2$.

2021 Vietnam National Olympiad, 1

Tags: algebra

Let $(x_n)$ define by $x_1\in \left(0;\dfrac{1}{2}\right)$ and $x_{n+1}=3x_n^2-2nx_n^3$ for all $n\ge 1$. a) Prove that $(x_n)$ convergence to $0$. b) For each $n\ge 1$, let $y_n=x_1+2x_2+\cdots+n x_n$. Prove that $(y_n)$ has a limit.

2010 HMNT, 10

Tags: combinatorics

Justine has a coin which will come up the same as the last flip $\frac23$ of the time and the other side $\frac13$ of the time. She flips it and it comes up heads. She then flips it $2010$ more times. What is the probability that the last flip is heads?

2009 Thailand Mathematical Olympiad, 2

Tags: combinatorics , subset , difference

Let $k$ and $n$ be positive integers with $k < n$. Find the number of subsets of $\{1, 2, . . . , n\}$ such that the difference between the largest and smallest elements in the subset is $k$.

2020 Jozsef Wildt International Math Competition, W40

Tags: inequalities

If $0\le x_k<k$, for any $k\in\{1,2,\ldots,n\}$, $m\in\mathbb R_{\ge2}$, then prove that $$\frac1{\sqrt[m]{(1-x_1)(2-x_2)\cdots(n-x_n)}}+\frac1{\sqrt[m]{(1+x_1)(2+x_2)\cdots(n+x_n)}}\ge\frac2{\sqrt[m]{n!}}$$ [i]Proposed by Dorin Mărghidanu[/i]

2016 Sharygin Geometry Olympiad, 8

Tags: geometry

Let $ABC$ be a non-isosceles triangle, let $AA_1$ be its angle bisector and $A_2$ be the touching point of the incircle with side $BC$. The points $B_1,B_2,C_1,C_2$ are defined similarly. Let $O$ and $I$ be the circumcenter and the incenter of triangle $ABC$. Prove that the radical center of the circumcircle of the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ lies on the line $OI$.

2010 Contests, 1

Tags: combinatorics , ceiling function

In a country, there are some two-way roads between the cities. There are $2010$ roads connected to the capital city. For all cities different from the capital city, there are less than $2010$ roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number is even. $k$ roads connected to the capital city will be deleted. It is wanted that whatever the road network is, if we can reach from one city to another at the beginning, then we can reach after the deleting process also. Find the maximum value of $k.$

2007 Alexandru Myller, 4

Tags: discrete math

At a math contest which has $ 5 $ problems, each candidate has solved $ 3 $ problems. Among these candidates, for any group of $ 5 $ candidates that we might choose, we see that there is a problem which all members of the group had solved it. Prove that there is a problem solved by all candidates.

1977 All Soviet Union Mathematical Olympiad, 245

Tags: sum , combinatorics

Given a set of $n$ positive numbers. For each its nonempty subset consider the sum of all the subset's numbers. Prove that you can divide those sums onto $n$ groups in such a way, that the least sum in every group is not less than a half of the greatest sum in the same group.

2023 Princeton University Math Competition, A8

Tags: algebra

Given a positive integer $m,$ define the polynomial $$P_m(z) = z^4-\frac{2m^2}{m^2+1} z^3+\frac{3m^2-2}{m^2+1}z^2-\frac{2m^2}{m^2+1}z+1.$$ Let $S$ be the set of roots of the polynomial $P_5(z)\cdot P_7(z)\cdot P_8(z) \cdot P_{18}(z).$ Let $w$ be the point in the complex plane which minimizes $\sum_{z \in S} |z-w|.$ The value of $\sum_{z \in S} |z-w|^2$ equals $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b.$ Compute $a+b.$

2016 Argentina National Olympiad, 1

Tags: number theory , arithmetic sequence , perfect power

Find an arithmetic progression of $2016$ natural numbers such that neither is a perfect power but its multiplication is a perfect power. Clarification: A perfect power is a number of the form $n^k$ where $n$ and $k$ are both natural numbers greater than or equal to $2$.

2017 Math Prize for Girls Problems, 4

Tags:

If $\mathrm{MATH} + \mathrm{WITH} = \mathrm{GIRLS}$, compute the smallest possible value of $\mathrm{GIRLS}$. Here $\mathrm{MATH}$ and $\mathrm{WITH}$ are 4-digit numbers and $\mathrm{GIRLS}$ is a 5-digit number (all with nonzero leading digits). Different letters represent different digits.

2024 Romania Team Selection Tests, P3

Tags: combinatorics , graph theory

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

1999 May Olympiad, 5

Tags: combinatorial geometry , combinatorics

There are $12$ points that are vertices of a regular polygon with $12$ sides. Rafael must draw segments that have their two ends at two of the points drawn. He is allowed to have each point be an endpoint of more than one segment and for the segments to intersect, but he is prohibited from drawing three segments that are the three sides of a triangle in which each vertex is one of the $12$ starting points. Find the maximum number of segments Rafael can draw and justify why he cannot draw a greater number of segments.

2003 Chile National Olympiad, 5

Tags: number theory , divide , divisible

Prove that there is a natural number $N$ of the form $11...1100...00$ which is divisible by $2003$. (The natural numbers are: $1,2,3,...$)

2012 Today's Calculation Of Integral, 799

Tags: limit , logarithm , strong induction , calculus , integration , calculus computations , induction

Let $n$ be positive integer. Define a sequence $\{a_k\}$ by \[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\] (1) Find $a_2$ and $a_3$. (2) Find the general term $a_k$. (3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$. 50 points

2004 Tournament Of Towns, 2

Tags: combinatorics

What is the maximal number of checkers that can be placed on an $8\times 8$ checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker?

1985 All Soviet Union Mathematical Olympiad, 403

Tags: algebra , trigonometry , inequalities

Find all the pairs $(x,y)$ such that $|\sin x-\sin y| + \sin x \sin y \le 0$.

2009 Princeton University Math Competition, 1

Tags: algebra , relatively prime , quadratic , quadratic formula , number theory , greatest common divisor , ratio

If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$.

2015 Postal Coaching, Problem 5

Tags: incenter , geometry

Let $ABCD$ be a convex quadrilateral. In the triangle $ABC$ let $I$ and $J$ be the incenter and the excenter opposite to vertex $A$, respectively. In the triangle $ACD$ let $K$ and $L$ be the incenter and the excenter opposite to vertex $A$, respectively. Show that the lines $IL$ and $JK$, and the bisector of the angle $BCD$ are concurrent.

2015 Purple Comet Problems, 4

Tags:

Six boxes are numbered $1$, $2$, $3$, $4$, $5$, and $6$. Suppose that there are $N$ balls distributed among these six boxes. Find the least $N$ for which it is guaranteed that for at least one $k$, box number $k$ contains at least $k^2$ balls.