Olympiad problems

1999 Switzerland Team Selection Test, 9

Suppose that $P(x)$ is a polynomial with degree $10$ and integer coefficients. Prove that, there is an infinite arithmetic progression (open to bothside) not contain value of $P(k)$ with $k\in\mathbb{Z}$

1996 Spain Mathematical Olympiad, 2

Tags: isosceles , centroid , geometry

Let $G$ be the centroid of a triangle $ABC$. Prove that if $AB+GC = AC+GB$, then the triangle is isosceles

2017 Baltic Way, 15

Tags: polygon , geometry , angle

Let $n \ge 3$ be an integer. What is the largest possible number of interior angles greater than $180^\circ$ in an $n$-gon in the plane, given that the $n$-gon does not intersect itself and all its sides have the same length?

2007 Purple Comet Problems, 19

Tags: probability , number theory , relatively prime

Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left. There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right. The probability that this will [b]not[/b] happen is given by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

1985 IMO Longlists, 54

Tags: number theory , greatest common divisor , induction , algorithm , modular arithmetic , algebra , binomial theorem

Set $S_n = \sum_{p=1}^n (p^5+p^7)$. Determine the greatest common divisor of $S_n$ and $S_{3n}.$

2021 Purple Comet Problems, 25

Tags:

The area of the triangle whose altitudes have lengths $36.4$, $39$, and $42$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

1990 Tournament Of Towns, (276) 4

Tags: combinatorics , combinatorial geometry , geometry

We have “bricks” made in the following way: we take a unit cube and glue to three of its faces which have a common vertex three more cubes in such a way that the faces glued together coincide. Is it possible to construct from these bricks an $11 \times 12 \times 13$ box? (A Andjans, Riga )

1983 IMO Longlists, 36

Tags: combinatorics

The set $X$ has $1983$ members. There exists a family of subsets $\{S_1, S_2, \ldots , S_k \}$ such that: [b](i)[/b] the union of any three of these subsets is the entire set $X$, while [b](ii)[/b] the union of any two of them contains at most $1979$ members. What is the largest possible value of $k ?$

2005 International Zhautykov Olympiad, 1

Tags: combinatorics

The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table?

2015 BMT Spring, 3

Tags: algebra , system of equations , number theory

Find all integer solutions to \begin{align*} x^2+2y^2+3z^2&=36,\\ 3x^2+2y^2+z^2&=84,\\ xy+xz+yz&=-7. \end{align*}

2024 HMNT, 26

Tags: guts

A right rectangular prism of silly powder has dimensions $20 \times 24 \times 25.$ Jerry the wizard applies $10$ bouts of highdroxylation to the box, each of which increases one dimension of the silly powder by $1$ and decreases a different dimension of the silly powder by $1,$ with every possible choice of dimensions equally likely to be chosen and independent of all previous choices. Compute the expected volume of the silly powder after Jerry’s routine.

1981 Miklós Schweitzer, 10

Tags: probability , real analysis , function , integration , complex number , probability and stats

Let $ P$ be a probability distribution defined on the Borel sets of the real line. Suppose that $ P$ is symmetric with respect to the origin, absolutely continuous with respect to the Lebesgue measure, and its density function $ p$ is zero outside the interval $ [\minus{}1,1]$ and inside this interval it is between the positive numbers $ c$ and $ d$ ($ c < d$). Prove that there is no distribution whose convolution square equals $ P$. [i]T. F. Mori, G. J. Szekely[/i]

1979 Austrian-Polish Competition, 8

Tags: midpoint , geometry , 3d geometry

Let $A,B,C,D$ be four points in space, and $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. Show that $$AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2$$

2013 VJIMC, Problem 4

Tags: summation , algebra , binomial

Let $n$ and $k$ be positive integers. Evaluate the following sum $$\sum_{j=0}^k\binom kj^2\binom{n+2k-j}{2k}$$where $\binom nk=\frac{n!}{k!(n-k)!}$.

2021 Harvard-MIT Mathematics Tournament., 8

Tags: geometry

Two circles with radii $71$ and $100$ are externally tangent. Compute the largest possible area of a right triangle whose vertices are each on at least one of the circles.

1997 Akdeniz University MO, 5

Tags: geometry , perimeter , incenter

An $ABC$ triangle divide by a $d$ line such that, new two pieces' areas and perimeters are equal. Prove that $ABC$'s incenter lies $d$

2017 ASDAN Math Tournament, 1

Tags:

In $\triangle ABC$, we have $\angle ABC=20^\circ$. In addition, $D$ is drawn on $\overline{AB}$ such that $AC=CD=BD$. Compute $\angle ACD$ in degrees.

2023 Bulgaria National Olympiad, 3

Tags: algebra , polynomial , abstract algebra , inequalities

Let $f(x)$ be a polynomial with positive integer coefficients. For every $n\in\mathbb{N}$, let $a_{1}^{(n)}, a_{2}^{(n)}, \dots , a_{n}^{(n)}$ be fixed positive integers that give pairwise different residues modulo $n$ and let \[g(n) = \sum\limits_{i=1}^{n} f(a_{i}^{(n)}) = f(a_{1}^{(n)}) + f(a_{2}^{(n)}) + \dots + f(a_{n}^{(n)})\] Prove that there exists a constant $M$ such that for all integers $m>M$ we have $\gcd(m, g(m))>2023^{2023}$.

2010 Math Prize For Girls Problems, 10

Tags: analytic geometry , geometry , geometric transformation , congruent triangles

The triangle $ABC$ lies on the coordinate plane. The midpoint of $\overline{AB}$ has coordinates $(-16, -63)$, the midpoint of $\overline{AC}$ has coordinates $(13, 50)$, and the midpoint of $\overline{BC}$ has coordinates $(6, -85)$. What are the coordinates of point $A$?

MathLinks Contest 4th, 4.3

Tags: combinatorics

Given is a graph $G$. An [i]empty [/i] subgraph of $G$ is a subgraph of $G$ with no edges between its vertices. An edge of $G$ is called [i]important [/i] if and only if the removal of this edge will increase the size of the maximal empty subgraph. Suppose that two important edges in $G$ have a common endpoint. Prove there exists a cycle of odd length in $G$.

2016 Taiwan TST Round 1, 3

Tags: binomial coefficient , function , algebra , number theory , surjective function

Let $\mathbb{Z}^+$ denote the set of all positive integers. Find all surjective functions $f:\mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ that satisfy all of the following conditions: for all $a,b,c \in \mathbb{Z}^+$, (i)$f(a,b) \leq a+b$; (ii)$f(a,f(b,c))=f(f(a,b),c)$ (iii)Both $\binom{f(a,b)}{a}$ and $\binom{f(a,b)}{b}$ are odd numbers.(where $\binom{n}{k}$ denotes the binomial coefficients)

2015 ASDAN Math Tournament, 1

Tags: algebra test

Given that $xy+x+y=5$ and $x+1=2$, compute $y+1$.

2024 Thailand Mathematical Olympiad, 4

Tags: combinatorics

In a table with $88$ rows and $253$ columns, each cell is colored either purple or yellow. Suppose that for each yellow cell $c$, $$x(c)y(c)\geq184.$$ Where $x(c)$ is the number of purple cells that lie in the same row as $c$, and $y(c)$ is the number of purple cells that lie in the same column as $c$.\\ Find the least possible number of cells that are colored purple.

2018 AMC 12/AHSME, 25

Tags:

For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$? $\textbf{(A)} \text{ 12} \qquad \textbf{(B)} \text{ 14} \qquad \textbf{(C)} \text{ 16} \qquad \textbf{(D)} \text{ 18} \qquad \textbf{(E)} \text{ 20}$

2013 China Team Selection Test, 2

Tags: circumcircle , geometric transformation , homothety , trigonometry , euler , geometry

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.