Olympiad problems

2008 China Team Selection Test, 3

Tags: trapezoid , geometry , trigonometry , number theory , modular arithmetic , analytic geometry , induction

Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)

2011 Purple Comet Problems, 11

Tags: probability , pigeonhole principle , modular arithmetic

Six distinct positive integers are randomly chosen between $1$ and $2011;$ inclusive. The probability that some pair of the six chosen integers has a difference that is a multiple of $5 $ is $n$ percent. Find $n.$

2012 Online Math Open Problems, 41

Tags: modular arithmetic , function

Find the remainder when \[ \sum_{i=2}^{63} \frac{i^{2011}-i}{i^2-1}. \] is divided by 2016. [i]Author: Alex Zhu[/i]

2013 AMC 10, 19

Tags: modular arithmetic

In base $10$, the number $2013$ ends in the digit $3$. In base $9$, on the other hand, the same number is written as $(2676)_9$ and ends in the digit $6$. For how many positive integers $b$ does the base-$b$ representation of $2013$ end in the digit $3$? $\textbf{(A) }6\qquad \textbf{(B) }9\qquad \textbf{(C) }13\qquad \textbf{(D) }16\qquad \textbf{(E) }18\qquad$

1995 Vietnam Team Selection Test, 2

Tags: rational root theorem , search , polynomial , algebra , modular arithmetic

Find all integers $ k$ such that for infinitely many integers $ n \ge 3$ the polynomial \[ P(x) =x^{n+ 1}+ kx^n - 870x^2 + 1945x + 1995\] can be reduced into two polynomials with integer coefficients.

2008 Germany Team Selection Test, 2

Tags: triplets , counting , combinatorics , modular arithmetic

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

2005 MOP Homework, 4

Tags: binomial theorem , number theory , algebra , modular arithmetic

Let $p$ be an odd prime. Prove that \[\sum^{p-1}_{k=1} k^{2p-1} \equiv \frac{p(p+1)}{2}\pmod{p^2}.\]

2000 IMO Shortlist, 7

Tags: coefficient , modular arithmetic , polynomial , algebra

For a polynomial $ P$ of degree 2000 with distinct real coefficients let $ M(P)$ be the set of all polynomials that can be produced from $ P$ by permutation of its coefficients. A polynomial $ P$ will be called [b]$ n$-independent[/b] if $ P(n) \equal{} 0$ and we can get from any $ Q \in M(P)$ a polynomial $ Q_1$ such that $ Q_1(n) \equal{} 0$ by interchanging at most one pair of coefficients of $ Q.$ Find all integers $ n$ for which $ n$-independent polynomials exist.

2010 Vietnam Team Selection Test, 1

Tags: number theory , modular arithmetic

Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that: \[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\] Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$

1990 India National Olympiad, 4

Tags: combinatorics , modular arithmetic

Consider the collection of all three-element subsets drawn from the set $ \{1,2,3,4,\dots,299,300\}$. Determine the number of those subsets for which the sum of the elements is a multiple of 3.

2007 JBMO Shortlist, 5

Tags: quadratic , modular arithmetic , number theory

Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.

PEN E Problems, 20

Tags: modular arithmetic , number theory

Verify that, for each $r \ge 1$, there are infinitely many primes $p$ with $p \equiv 1 \; \pmod{2^r}$.

2005 Iran Team Selection Test, 3

Tags: function , modular arithmetic , combinatorics , induction , analytic geometry

Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.

2012 China National Olympiad, 2

Tags: number theory , modular arithmetic , inequalities

Consider a square-free even integer $n$ and a prime $p$, such that 1) $(n,p)=1$; 2) $p\le 2\sqrt{n}$; 3) There exists an integer $k$ such that $p|n+k^2$. Prove that there exists pairwise distinct positive integers $a,b,c$ such that $n=ab+bc+ca$. [i]Proposed by Hongbing Yu[/i]

2012 Online Math Open Problems, 24

Tags: modular arithmetic , function

Find the number of ordered pairs of positive integers $(a,b)$ with $a+b$ prime, $1\leq a, b \leq 100$, and $\frac{ab+1}{a+b}$ is an integer. [i]Author: Alex Zhu[/i]

2001 China National Olympiad, 3

Tags: limit , modular arithmetic , number theory

Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\neq0$ such that (i) $m<2a$; (ii) $2n|(2am-m^2+n^2)$; (iii) $n^2-m^2+2mn\leq2a(n-m)$. For $(m, n)\in A$, let \[f(m,n)=\frac{2am-m^2-mn}{n}.\] Determine the maximum and minimum values of $f$.

2014 Contests, 1

Tags: digit , modular arithmetic , number theory

Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$. [list=a] [*] Determine its units digit. [*] Determine its tens digit. [/list]

2006 USAMO, 3

Tags: polynomial , modular arithmetic , algebra

For integral $m$, let $p(m)$ be the greatest prime divisor of $m.$ By convention, we set $p(\pm 1) = 1$ and $p(0) = \infty.$ Find all polynomials $f$ with integer coefficients such that the sequence \[ \{p \left( f \left( n^2 \right) \right) - 2n \}_{n \geq 0} \] is bounded above. (In particular, this requires $f \left (n^2 \right ) \neq 0$ for $n \geq 0.$)

1997 AIME Problems, 1

Tags: difference of squares , 3d geometry , matrix , geometry , algebra , linear algebra , modular arithmetic

How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?

PEN A Problems, 42

Tags: divisibility theory , modular arithmetic

Suppose that $2^n +1$ is an odd prime for some positive integer $n$. Show that $n$ must be a power of $2$.

2012 Peru IMO TST, 6

Tags: modular arithmetic , number theory

Let $p$ be an odd prime number. For every integer $a,$ define the number $S_a = \sum^{p-1}_{j=1} \frac{a^j}{j}.$ Let $m,n \in \mathbb{Z},$ such that $S_3 + S_4 - 3S_2 = \frac{m}{n}.$ Prove that $p$ divides $m.$ [i]Proposed by Romeo Meštrović, Montenegro[/i]

2020 March Advanced Contest, 1

Tags: number theory , modular arithmetic , function

In terms of $a$, $b$, and a prime $p$, find an expression which gives the number of $x \in \{0, 1, \ldots, p-1\}$ such that the remainder of $ax$ upon division by $p$ is less than the remainder of $bx$ upon division by $p$.

2004 Germany Team Selection Test, 3

Tags: counting , function , combinatorics , modular arithmetic , decimal representation

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

2014 Contests, 1

Tags: modular arithmetic , number theory

Let $p$ be an odd prime.Positive integers $a,b,c,d$ are less than $p$,and satisfy $p|a^2+b^2$ and $p|c^2+d^2$.Prove that exactly one of $ac+bd$ and $ad+bc$ is divisible by $p$

2013 District Olympiad, 4

Tags: algebra , induction , trigonometry , modular arithmetic

Let $n\in {{\mathbb{N}}^{*}}$. Prove that $2\sqrt{{{2}^{n}}}\cos \left( n\arccos \frac{\sqrt{2}}{4} \right)$ is an odd integer.