Olympiad problems

2009 Abels Math Contest (Norwegian MO) Final, 1a

Show that there exist infinitely many integers that cannot be written as the difference between two perfect squares.

1976 Polish MO Finals, 3

Tags: geometry , 3d geometry , tetrahedron

Prove that for each tetrahedron, the three products of pairs of opposite edges are sides of a triangle.

2016 Bangladesh Mathematical Olympiad, 1

Tags: number theory , modular arithmetic

(a) Show that $n(n + 1)(n + 2)$ is divisible by $6$. (b) Show that $1^{2015} + 2^{2015} + 3^{2015} + 4^{2015} + 5^{2015} + 6^{2015}$ is divisible by $7$.

2003 Tournament Of Towns, 1

Tags: 3d geometry , pyramid , sphere , geometry

A triangular pyramid $ABCD$ is given. Prove that $\frac Rr > \frac ah$, where $R$ is the radius of the circumscribed sphere, $r$ is the radius of the inscribed sphere, $a$ is the length of the longest edge, $h$ is the length of the shortest altitude (from a vertex to the opposite face).

1998 Romania Team Selection Test, 2

Tags: rectangle , pigeonhole principle , geometry

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. [i]Bogdan Enescu[/i]

2000 Harvard-MIT Mathematics Tournament, 9

Tags: algebra

Edward’s formula for the stock market predicts correctly that the price of HMMT is directly proportional to a secret quantity $x$ and inversely proportional to $y$, the number of hours he slept the night before. If the price of HMMT is $\$12$ when $x = 8$ and $y = 4$, how many dollars does it cost when $x = 4$ and $y = 8$?

2020 Purple Comet Problems, 24

Tags: ellipse

Points $E$ and $F$ lie on diagonal $\overline{AC}$ of square $ABCD$ with side length $24$, such that $AE = CF = 3\sqrt2$. An ellipse with foci at $E$ and $F$ is tangent to the sides of the square. Find the sum of the distances from any point on the ellipse to the two foci.

2023 Stanford Mathematics Tournament, 9

Tags:

Suppose $a$ and $b$ are positive integers with a curious property: $(a^3 - 3ab +\tfrac{1}{2})^n + (b^3 +\tfrac{1}{2})^n$ is an integer for at least $3$, but at most finitely many different choices of positive integers $n$. What is the least possible value of $a+b$?

2019 Baltic Way, 2

Tags: algebra

Let $(F_n)$ be the sequence defined recursively by $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ for $n\geq 2$. Find all pairs of positive integers $(x,y)$ such that $$5F_x-3F_y=1.$$

2024 Chile Classification NMO Seniors, 1

Tags: number theory

Bus tickets from a transportation company are numbered with six digits, ranging from 000000 to 999999. A ticket is considered "lucky" if the sum of the first three digits equals the sum of the last three digits. For example, ticket 721055 is lucky, whereas 003101 is not. Determine how many consecutive tickets a person must buy to guarantee obtaining at least one lucky ticket, regardless of the starting ticket number.

2018 Peru MO (ONEM), 1

Tags:

1) Find a $4$-digit number $\overline{PERU}$ such that $\overline{PERU}=(P+E+R+U)^U$. Also prove that there is only one number satisfying this property.

2007 VJIMC, Problem 1

Tags: advanced fields

Can the set of positive rationals be split into two nonempty disjoint subsets $\mathbb Q_1$ and $\mathbb Q_2$, such that both are closed under addition, i.e. $p+q\in\mathbb Q_k$ for every $p,q\in\mathbb Q_k$, $k=1,2$? Can it be done when addition is exchanged for multiplication, i.e. $p\cdot q\in\mathbb Q_k$ for every $p,q\in\mathbb Q_k$, $k=1,2$?

2011 German National Olympiad, 6

Tags: algebra , polynomial , number theory , sequence , prime

Let $p>2$ be a prime. Define a sequence $(Q_{n}(x))$ of polynomials such that $Q_{0}(x)=1, Q_{1}(x)=x$ and $Q_{n+1}(x) =xQ_{n}(x) + nQ_{n-1}(x)$ for $n\geq 1.$ Prove that $Q_{p}(x)-x^p $ is divisible by $p$ for all integers $x.$

2010 CHMMC Winter, 7

Tags: algebra

Compute all real numbers $a$ such that the polynomial $x^4 + ax^3 + 1$ has exactly one real root.

2009 Purple Comet Problems, 3

Tags:

The [i]Purple Comet! Math Meet[/i] runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 28 + 29 + 30 + 1 + 2 + 3 = 120.$ What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?

2013 Abels Math Contest (Norwegian MO) Final, 1b

Tags: sequence , algebra , inequalities

The sequence $a_1, a_2, a_3,...$ is defined so that $a_1 = 1$ and $a_{n+1} =\frac{a_1 + a_2 + ...+ a_n}{n}+1$ for $n \ge 1$. Show that for every positive real number $b$ we can find $a_k$ so that $a_k < bk$.

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 $k$.

2012 IFYM, Sozopol, 1

Tags: area of a triangle , geometry

Find the area of a triangle with angles $\frac{1}{7} \pi$, $\frac{2}{7} \pi$, and $\frac{4}{7} \pi $, and radius of its circumscribed circle $R=1$.

2013 ITAMO, 3

Tags: absolute value , combinatorics

Each integer is colored with one of two colors, red or blue. It is known that, for every finite set $A$ of consecutive integers, the absolute value of the difference between the number of red and blue integers in the set $A$ is at most $1000$. Prove that there exists a set of $2000$ consecutive integers in which there are exactly $1000$ red numbers and $1000$ numbers blue.

2023 Brazil Undergrad MO, 6

Tags: algebra , real analysis , sequence

Determine all pairs $(c, d) \in \mathbb{R}^2$ of real constants such that there is a sequence $(a_n)_{n\geq1}$ of positive real numbers such that, for all $n \geq 1$, $$a_n \geq c \cdot a_{n+1} + d \cdot \sum_{1 \leq j < n} a_j .$$

1990 IMO Shortlist, 15

Tags: induction , combinatorics , partition , set systems

Determine for which positive integers $ k$ the set \[ X \equal{} \{1990, 1990 \plus{} 1, 1990 \plus{} 2, \ldots, 1990 \plus{} k\}\] can be partitioned into two disjoint subsets $ A$ and $ B$ such that the sum of the elements of $ A$ is equal to the sum of the elements of $ B.$

2008 Germany Team Selection Test, 3

Tags: symmetry , perpendicular bisector , geometry

Let $ ABCD$ be an isosceles trapezium. Determine the geometric location of all points $ P$ such that \[ |PA| \cdot |PC| \equal{} |PB| \cdot |PD|.\]

2004 ITAMO, 1

Tags: number theory

Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day. Given that the low temperatures in December $3$ and January $31$ were $5^\circ \text C$ and $2^\circ \text C$ respectively, find the low temperature in December $25$.

2013 Baltic Way, 12

Tags: circumcircle , trapezoid , geometry

A trapezoid $ABCD$ with bases $AB$ and $CD$ is such that the circumcircle of the triangle $BCD$ intersects the line $AD$ in a point $E$, distinct from $A$ and $D$. Prove that the circumcircle oF the triangle $ABE$ is tangent to the line $BC$.

MathLinks Contest 2nd, 4.3

Tags: combinatorics

In a country there are $100$ cities, some of which are connected by roads. For each four cities there are at least two roads between them. Also, there is no path that passes through each city exactly one time. Prove that one can choose two cities among those $100$, such that each of the $98$ remaining cities would be connected by a road with at least one of the two chosen cities.