Olympiad problems

2012 Baltic Way, 7

On a $2012 \times 2012$ board, some cells on the top-right to bottom-left diagonal are marked. None of the marked cells is in a corner. Integers are written in each cell of this board in the following way. All the numbers in the cells along the upper and the left sides of the board are 1's. All the numbers in the marked cells are 0's. Each of the other cells contains a number that is equal to the sum of its upper neighbour and its left neighbour. Prove that the number in the bottom right corner is not divisible by 2011.

2009 Italy TST, 3

Tags: number theory , geometry

Find all pairs of integers $(x,y)$ such that \[ y^3=8x^6+2x^3y-y^2.\]

I Soros Olympiad 1994-95 (Rus + Ukr), 11.8

Tags: algebra , arithmetic progression

Let's write down a segment of a series of integers from $0$ to $1995$. Among the numbers written out, two have been crossed out. Let's consider the longest arithmetic progression contained among the remaining $1994$ numbers. Let $K$ be the length of the progression. Which two numbers must be crossed out so that the value of $K$ is the smallest?

2009 Today's Calculation Of Integral, 478

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^{\frac{\pi}{4}} \{(x\sqrt{\sin x}\plus{}2\sqrt{\cos x})\sqrt{\tan x}\plus{}(x\sqrt{\cos x}\minus{}2\sqrt{\sin x})\sqrt{\cot x}\}\ dx.$

1970 All Soviet Union Mathematical Olympiad, 142

Tags: number theory , digit

All natural numbers containing not more than $n$ digits are divided onto two groups. The first contains the numbers with the even sum of the digits, the second -- with the odd sum. Prove that if $0<k<n$ than the sum of the $k$-th powers of the numbers in the first group equals to the sum of the $k$-th powers of the numbers in the second group.

2022 IMO Shortlist, G5

Tags: geometry

Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

2007 Sharygin Geometry Olympiad, 4

Tags: midpoint , geometry , circumcircle , locus , locus problems

Given a triangle $ABC$. An arbitrary point $P$ is chosen on the circumcircle of triangle $ABH$ ($H$ is the orthocenter of triangle $ABC$). Lines $AP$ and $BP$ meet the opposite sidelines of the triangle at points $A' $ and $B'$, respectively. Determine the locus of midpoints of segments $A'B'$.

1998 Putnam, 6

Tags: geometry

Let $A,B,C$ denote distinct points with integer coefficients in $\mathbb{R}^2$. Prove that if \[(|AB|+|BC|)^2<8\cdot[ABC]+1\] then $A,B,C$ are three vertices of a square. Here $|XY|$ is the length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$.

1984 All Soviet Union Mathematical Olympiad, 389

Tags: sequence , limit , algebra , recurrence relation

Given a sequence $\{x_n\}$, $$x_1 = x_2 = 1, x_{n+2} = x^2_{n+1} - \frac{x_n}{2}$$ Prove that the sequence has limit and find it.

2015 China Team Selection Test, 1

Tags: perpendicular bisector , angle bisector , geometry

$\triangle{ABC}$ is isosceles with $AB = AC >BC$. Let $D$ be a point in its interior such that $DA = DB+DC$. Suppose that the perpendicular bisector of $AB$ meets the external angle bisector of $\angle{ADB}$ at $P$, and let $Q$ be the intersection of the perpendicular bisector of $AC$ and the external angle bisector of $\angle{ADC}$. Prove that $B,C,P,Q$ are concyclic.

2024 Dutch IMO TST, 4

Tags: combinatorics

Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and island where the fire starts at.

2003 Alexandru Myller, 4

Tags: function , calculus , integration

[b]a)[/b] Prove that the function $ 1\le t\mapsto\int_{1}^t\frac{\sin x}{x^n} dx $ has an horizontal asymptote, for any natural number $ n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty }\lim_{t\to\infty }\int_{1}^t\frac{\sin x}{x^n} . $ [i]Mihai Piticari[/i]

2015 Purple Comet Problems, 6

Tags:

There are digits a and b so that the 15-digit number 7a7ba7ab7ba7b77 is divisible by 99. Find 10a + b.

2015 HMIC, 4

Tags:

Prove that there exists a positive integer $N$ such that for any positive integer $n \ge N$, there are at least $2015$ non-empty subsets $S$ of $\{ n^2 + 1, n^2 + 2, \dots, n^2 + 3n \}$ with the property that the product of the elements of $S$ is a perfect square.

2010 Contests, 3

Tags: power of a point , radical axis , geometry

Let $ABCD$ be a convex quadrilateral. such that $\angle CAB = \angle CDA$ and $\angle BCA = \angle ACD$. If $M$ be the midpoint of $AB$, prove that $\angle BCM = \angle DBA$.

2019 AIME Problems, 14

Tags: number theory

Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5$, $n$, and $n + 1$ cents, $91$ cents is the greatest postage that cannot be formed.

2016 KOSOVO TST, 3

Tags: algebra , quadratic

Equations $x^2+ax+b=0$ and $x^2+px+q=0$ have a common root.Find quadratic equation roots of which are two other roots.

2018 Dutch Mathematical Olympiad, 2

Tags: combinatorics , coloring

The numbers $1$ to $15$ are each coloured blue or red. Determine all possible colourings that satisfy the following rules: • The number $15$ is red. • If numbers $x$ and $y$ have different colours and $x + y \le 15$, then $x + y$ is blue. • If numbers $x$ and $y$ have different colours and $x \cdot y \le 15$, then $x \cdot y$ is red.

1983 IMO Longlists, 40

Tags: ratio , geometry , 3d geometry , tetrahedron , sphere , function , congruent triangles

Four faces of tetrahedron $ABCD$ are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are $a < b < c$, determine the radius of the sphere circumscribed about the tetrahedron as a function on $a, b$, and $c$. What is the ratio $c/a$ if $R = a \ ?$

2019 China Team Selection Test, 3

Tags: algebra

Find all positive integer $n$, such that there exists $n$ points $P_1,\ldots,P_n$ on the unit circle , satisfying the condition that for any point $M$ on the unit circle, $\sum_{i=1}^n MP_i^k$ is a fixed value for \\a) $k=2018$ \\b) $k=2019$.

2007 Federal Competition For Advanced Students, Part 2, 3

Tags: rhombus , geometry

Determine all rhombuses $ ABCD$ with the given length $ 2a$ of ist sides by giving the angle $ \alpha \equal{} \angle BAD$, such that there exists a circle which cuts each side of the rhombus in a chord of length $ a$.

2003 Gheorghe Vranceanu, 3

Tags: modular arithmetic , number theory

Show that $ n\equiv 0\pmod 9 $ if $ 2^n\equiv -1\pmod n, $ where $ n $ is a natural number greater than $ 3. $

2006 Regional Competition For Advanced Students, 4

Tags: algebra

Let $ <h_n>$ $ n\in\mathbb N$ a harmonic sequence of positive real numbers (that means that every $ h_n$ is the harmonic mean of its two neighbours $ h_{n\minus{}1}$ and $ h_{n\plus{}1}$ : $ h_n\equal{}\frac{2h_{n\minus{}1}h_{n\plus{}1}}{h_{n\minus{}1}\plus{}h_{n\plus{}1}}$) Show that: if the sequence includes a member $ h_j$, which is the square of a rational number, it includes infinitely many members $ h_k$, which are squares of rational numbers.

2006 Estonia National Olympiad, 5

Tags: combinatorics

Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the greatest natural number that, for each its representation as a sum of positive integers, there exists a fleet such that the summands are exactly the numbers of squares contained in individual ships.

2014 ELMO Shortlist, 6

Tags: inequalities

Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that \[ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. \][i]Proposed by Sammy Luo[/i]