Olympiad problems

2005 Spain Mathematical Olympiad, 1

Let $a$ and $b$ be integers. Demonstrate that the equation $$(x-a)(x-b)(x-3) +1 = 0$$ has an integer solution.

2017 Princeton University Math Competition, A7

Tags: geometry , cyclic quadrilateral , circumcircle , number theory , relatively prime

Let $ACDB$ be a cyclic quadrilateral with circumcenter $\omega$. Let $AC=5$, $CD=6$, and $DB=7$. Suppose that there exists a unique point $P$ on $\omega$ such that $\overline{PC}$ intersects $\overline{AB}$ at a point $P_1$ and $\overline{PD}$ intersects $\overline{AB}$ at a point $P_2$, such that $AP_1=3$ and $P_2B=4$. Let $Q$ be the unique point on $\omega$ such that $\overline{QC}$ intersects $\overline{AB}$ at a point $Q_1$, $\overline{QD}$ intersects $\overline{AB}$ at a point $Q_2$, $Q_1$ is closer to $B$ than $P_1$ is to $B$, and $P_2Q_2=2$. The length of $P_1Q_1$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

2016 NZMOC Camp Selection Problems, 4

Tags: number theory , divisible

A quadruple $(p, a, b, c)$ of positive integers is a[i] karaka quadruple[/i] if $\bullet$ $p$ is an odd prime number $\bullet$ $a, b$ and $c$ are distinct, and $\bullet$ $ab + 1$, $bc + 1$ and $ca + 1$ are divisible by $p$. (a) Prove that for every karaka quadruple $(p, a, b, c)$ we have $p + 2 \le\frac{a + b + c}{3}$. (b) Determine all numbers $p$ for which a karaka quadruple $(p, a, b, c)$ exists with $p + 2 =\frac{a + b + c}{3}$

2010 Postal Coaching, 1

Tags: number theory

Does there exist an increasing sequence of positive integers $a_1 , a_2 ,\cdots$ with the following two properties? (i) Every positive integer $n$ can be uniquely expressed in the form $n = a_j - a_i$ , (ii) $\frac{a_k}{k^3}$ is bounded.

2009 Princeton University Math Competition, 1

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

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 $a + b + c$.

2022 AMC 10, 7

Tags: number theory

The least common multiple of a positive integer $n$ and 18 is 180, and the greatest common divisor of $n$ and 45 is 15. What is the sum of the digits of $n$? $\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }12$

2000 Portugal MO, 3

Tags: number theory , divide , divisor

Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.

2021 European Mathematical Cup, 3

Tags: factorial , perfect square , number theory

Let $\ell$ be a positive integer. We say that a positive integer $k$ is [i]nice [/i] if $k!+\ell$ is a square of an integer. Prove that for every positive integer $n \geqslant \ell$, the set $\{1, 2, \ldots,n^2\}$ contains at most $n^2-n +\ell$ nice integers. \\ \\ (Théo Lenoir)

2008 German National Olympiad, 6

Tags: number theory

Find all real numbers $ x$ such that $ 4x^5 \minus{} 7$ and $ 4x^{13} \minus{} 7$ are both perfect squares.

2015 Taiwan TST Round 3, 2

Tags: number theory

Consider the permutation of $1,2,...,n$, which we denote as $\{a_1,a_2,...,a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions: (1)$a_1=1$ (2)$|a_i-a_{i-1}|\le2, i=1,2,...,n-1$ what is the residue when we divide $f(2015)$ by $4$ ?

1998 Slovenia National Olympiad, Problem 2

Tags: number theory

A four-digit number has the property that the units digit equals the tens digit increased by $1$, the hundreds digit equals twice the tens digit, and the thousands digit is at least twice the units. Determine this four-digit number, knowing that it is twice a prime number.

2014 Argentine National Olympiad, Level 3, 2.

Tags: number theory

Given several numbers, one of them, $a$, is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$. This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called [i]good[/i] if there are $m$ or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.

2000 Bulgaria National Olympiad, 3

Tags: induction , modular arithmetic , number theory

Let $ p$ be a prime number and let $ a_1,a_2,\ldots,a_{p \minus{} 2}$ be positive integers such that $ p$ doesn't $ a_k$ or $ {a_k}^k \minus{} 1$ for any $ k$. Prove that the product of some of the $ a_i$'s is congruent to $ 2$ modulo $ p$.

2008 South East Mathematical Olympiad, 4

Tags: number theory

Let $n$ be a positive integer. $f(n)$ denotes the number of $n$-digit numbers $\overline{a_1a_2\cdots a_n}$(wave numbers) satisfying the following conditions : (i) for each $a_i \in\{1,2,3,4\}$, $a_i \not= a_{i+1}$, $i=1,2,\cdots$; (ii) for $n\ge 3$, $(a_i-a_{i+1})(a_{i+1}-a_{i+2})$ is negative, $i=1,2,\cdots$. (1) Find the value of $f(10)$; (2) Determine the remainder of $f(2008)$ upon division by $13$.

2012 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: number theory , composite

Prove tha number $19 \cdot 8^n +17$ is composite for every positive integer $n$

2018 AMC 8, 7

Tags: number theory

The $5$-digit number $\underline{2}$ $\underline{0}$ $\underline{1}$ $\underline{8}$ $\underline{U}$ is divisible by $9$. What is the remainder when this number is divided by $8$? $\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

2022 China Team Selection Test, 5

Tags: perfect square , number theory

Show that there exist constants $c$ and $\alpha > \frac{1}{2}$, such that for any positive integer $n$, there is a subset $A$ of $\{1,2,\ldots,n\}$ with cardinality $|A| \ge c \cdot n^\alpha$, and for any $x,y \in A$ with $x \neq y$, the difference $x-y$ is not a perfect square.

2009 Germany Team Selection Test, 1

Tags: number theory

Let $p > 7$ be a prime which leaves residue 1 when divided by 6. Let $m=2^p-1,$ then prove $2^{m-1}-1$ can be divided by $127m$ without residue.

2016 Bulgaria EGMO TST, 1

Tags: number theory

Find all positive integers $x$ such that $3^x + x^2 + 135$ is a perfect square.

2002 Italy TST, 2

Tags: modular arithmetic , number theory

Prove that for each prime number $p$ and positive integer $n$, $p^n$ divides \[\binom{p^n}{p}-p^{n-1}. \]

2014 China Team Selection Test, 4

Tags: number theory , vieta jumping

Let $k$ be a fixed odd integer, $k>3$. Prove: There exist infinitely many positive integers $n$, such that there are two positive integers $d_1, d_2$ satisfying $d_1,d_2$ each dividing $\frac{n^2+1}{2}$, and $d_1+d_2=n+k$.

2012 CHMMC Fall, 3

Tags: number theory

For a positive integer $n$, let $\sigma (n)$ be the sum of the divisors of $n$ (for example $\sigma (10) = 1 + 2 + 5 + 10 = 18$). For how many $n \in \{1, 2,. .., 100\}$, do we have $\sigma (n) < n+ \sqrt{n}$?

1968 Spain Mathematical Olympiad, 7

Tags: number theory , digit , power of 2

In the sequence of powers of $2$ (written in the decimal system, beginning with $2^1 = 2$) there are three terms of one digit, another three of two digits, another three of $3$, four out of $4$, three out of $5$, etc. Clearly reason the answers to the following questions: a) Can there be only two terms with a certain number of digits? b) Can there be five consecutive terms with the same number of digits? c) Can there be four terms of n digits, followed by four with $n + 1$ digits? d) What is the maximum number of consecutive powers of $2$ that can be found without there being four among them with the same number of digits?

2023 Taiwan TST Round 2, N

Tags: number theory , algebra , polynomial

Let $f_n$ be a polynomial with real coefficients for all $n \in \mathbb{Z}$. Suppose that \[f_n(k) = f_{n+k}(k) \quad n, k \in \mathbb{Z}.\] (a) Does $f_n = f_m$ necessarily hold for all $m,n \in \mathbb{Z}$? (b) If furthermore $f_n$ is a polynomial with integer coefficients for all $n \in\mathbb{Z}$, does $f_n = f_m$ necessarily hold for all $m, n \in\mathbb{Z}$? [i]Proposed by usjl[/i]

2006 IMO Shortlist, 1

Tags: number theory , diophantine equation

Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]