Olympiad problems

2019 Dutch IMO TST, 2

Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.

1936 Moscow Mathematical Olympiad, 026

Tags: consecutive , algebra , Product

Find $4$ consecutive positive integers whose product is $1680$.

2013 IMAC Arhimede, 6

Tags: inequalities , algebra , Sums and Products , Product , Sum

Let $p$ be an odd positive integer. Find all values of the natural numbers $n\ge 2$ for which holds $$\sum_{i=1}^{n} \prod_{j\ne i} (x_i-x_j)^p\ge 0$$ where $x_1,x_2,..,x_n$ are any real numbers.

The numbers $ a_i $, $ b_i $, $ c_i $, $ d_i $ satisfy the conditions $ 0\leq c_i \leq a_i \leq b_i \leq d_i $ and $ a_i+b_i = c_i+d_i $ for $ i=1,2 ,\ldots,n$. Prove that $$ \prod_{i=1}^n a_i + \prod_{i=1}^n b_i \leq \prod_{i=1}^n c_i + \prod_{i=1}^n d_i$$

2001 Estonia Team Selection Test, 4

Tags: Sum , Product , algebra , combinatorics

Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.

2015 Caucasus Mathematical Olympiad, 1

Tags: number theory , prime , Natural Numbers , Product , prime numbers

Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.

2019 Tournament Of Towns, 4

Tags: combinatorics , Coloring , Sum , Product

Each segment whose endpoints are the vertices of a given regular $100$-gon is colored red, if the number of vertices between its endpoints is even, and blue otherwise. (For example, all sides of the $100$-gon are red.) A number is placed in every vertex so that the sum of their squares is equal to $1$. On each segment the product of the numbers at its endpoints is written. The sum of the numbers on the blue segments is subtracted from the sum of the numbers on the red segments. What is the greatest possible result? (Ilya Bogdanov)

1997 Dutch Mathematical Olympiad, 1

Tags: number theory , sum of digits , Product

For each positive integer $n$ we define $f (n)$ as the product of the sum of the digits of $n$ with $n$ itself. Examples: $f (19) = (1 + 9) \times 19 = 190$, $f (97) = (9 + 7) \times 97 = 1552$. Show that there is no number $n$ with $f (n) = 19091997$.

1976 IMO, 1

Tags: number theory , Product , maximization , IMO , imo 1976

Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$

2002 Singapore Senior Math Olympiad, 2

Tags: geometry , distance , Product , circumcircle , incircle

The vertices of a triangle inscribed in a circle are the points of tangency of a triangle circumscribed about the circle. Prove that the product of the perpendicular distances from any point on the circle to the sides of the inscribed triangle is the same as the product of the perpendicular distances from the same point to the sides of the circumscribed triangle.

1992 Austrian-Polish Competition, 5

Tags: geometry , Fixed point , fixed , circle , Product

Given a circle $k$ with center $M$ and radius $r$, let $AB$ be a fixed diameter of $k$ and let $K$ be a fixed point on the segment $AM$. Denote by $t$ the tangent of $k$ at $A$. For any chord $CD$ through $K$ other than $AB$, denote by $P$ and Q the intersection points of $BC$ and $BD$ with $t$, respectively. Prove that $AP\cdot AQ$ does not depend on $CD$.

1997 Israel National Olympiad, 3

Tags: inequalities , Product , primes , number theory

Let $n?$ denote the product of all primes smaller than $n$. Prove that $n? > n$ holds for any natural number $n > 3$.

1958 February Putnam, B2

Tags: Putnam , number theory , square , Product

Prove that the product of four consecutive positive integers cannot be a perfect square or cube.

1997 Abels Math Contest (Norwegian MO), 2b

Tags: isosceles , circle , Product , independent , geometry

Let $A,B,C$ be different points on a circle such that $AB = AC$. Point $E$ lies on the segment $BC$, and $D \ne A$ is the intersection point of the circle and line $AE$. Show that the product $AE \cdot AD$ is independent of the choice of $E$.

2016 Lusophon Mathematical Olympiad, 1

Tags: number theory , Prime factor , Product , coprime , positive integers

Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?

1992 All Soviet Union Mathematical Olympiad, 575

Tags: 3D geometry , geometry , Plane , fixed , Product , sphere

A plane intersects a sphere in a circle $C$. The points $A$ and $B$ lie on the sphere on opposite sides of the plane. The line joining $A$ to the center of the sphere is normal to the plane. Another plane $p$ intersects the segment $AB$ and meets $C$ at $P$ and $Q$. Show that $BP\cdot BQ$ is independent of the choice of $p$.

2018 Hanoi Open Mathematics Competitions, 13

Tags: sum of digits , Product , Digits , number theory , Sum

For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively. 1) Find all values of n such that $n = P(n)$: 2) Determine all values of n such that $n = S(n) + P(n)$.

1992 IMO Longlists, 47

Tags: floor function , algebra , Product , Calculate , IMO Shortlist , IMO Longlist

Evaluate \[\left \lfloor \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \ \right \rfloor\]

2020 Taiwan TST Round 3, 1

Tags: algebra , Product , IMO Shortlist , IMO Shortlist 2019 , determinant

Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that \[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.\]

1936 Moscow Mathematical Olympiad, 030

Tags: combinatorics , number theory , Product

How many ways are there to represent $10^6$ as the product of three factors? Factorizations which only differ in the order of the factors are considered to be distinct.

2018 Saudi Arabia BMO TST, 2

Tags: combinatorics , Sum , Product

Suppose that $2018$ numbers $1$ and $-1$ are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all $2018$ products is negative. Find all possible values of sum of $2018$ given numbers.

1998 Switzerland Team Selection Test, 8

Tags: geometry , Product , Equilateral

Let $\vartriangle ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP,BP,CP$ meet the sides $BC,CA,AB$ in the points $X,Y,Z$ respectively. Prove that $XY \cdot YZ\cdot ZX \ge XB\cdot YC\cdot ZA$.

2025 AIME, 4

Tags: AIME , AIME II , 2025 AIME II , 2025 AMC , AMC , Product , logarithms

The product \[\prod^{63}_{k=4} \frac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = \frac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot \frac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot \frac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots \frac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\] is equal to $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

1999 Bosnia and Herzegovina Team Selection Test, 5

Tags: set , Product , Sum , number theory , combinatorics

For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that $a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$ $b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$ where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$

1986 Tournament Of Towns, (122) 4

Tags: algebra , Sum , reciprocal , Product

Consider subsets of the set $1 , 2,..., N$. For each such subset we can compute the product of the reciprocals of each member. Find the sum of all such products.