Olympiad problems

2017 Pan-African Shortlist, N1

Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.

1991 IMTS, 3

Tags: modular arithmetic , algebra

Prove that a positive integer can be expressed in the form $3x^2+y^2$ iff it can also be expressed in form $u^2+uv+v^2$, where $x,y,u,v$ are all positive integers.

Consider the set of $5$-tuples of positive integers at most $5$. We say the tuple ($a_1$, $a_2$, $a_3$, $a_4$, $a_5$) is [i]perfect[/i] if for any distinct indices $i$, $j$, $k$, the three numbers $a_i$, $a_j$ , $a_k$ do not form an arithmetic progression (in any order). Find the number of perfect $5$-tuples.

2023 Indonesia TST, A

Tags: algebra

Let $a_1, a_2, a_3, a_4, a_5$ be non-negative real numbers satisfied \[\sum_{k = 1}^{5} a_k = 20 \ \ \ \ \text{and} \ \ \ \ \sum_{k=1}^{5} a_k^2 = 100\] Find the minimum and maximum of $\text{max} \{a_1, a_2, a_3, a_4, a_5\}$

2010 Dutch IMO TST, 1

Tags: minimum , algebra , sequence

Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if (i) $a_n < a_{n+1}$ for all $n\ge 1$, (ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.

2013 BMT Spring, 13

Tags: algebra

Let $f(n)$ be a function from integers to integers. Suppose $f(11) = 1$, and $f(a)f(b) = f(a +b) + f(a - b)$, for all integers $a, b$. Find $f(2013)$.

1952 Poland - Second Round, 1

Tags: algebra , polynomial

Find the necessary and sufficient conditions that the real numbers $ a $, $ b $, $ c $ should satisfy so that the equation $$x^3 + ax^2 + bx + c = 0$$ has three real roots creating an arithmetic progression.

2014 Putnam, 5

Tags: algebra , polynomial , inequalities

Let $P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.$ Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j\ne k.$

2013 China Team Selection Test, 3

Tags: induction , modular arithmetic , algebra

Find all positive real numbers $r<1$ such that there exists a set $\mathcal{S}$ with the given properties: i) For any real number $t$, exactly one of $t, t+r$ and $t+1$ belongs to $\mathcal{S}$; ii) For any real number $t$, exactly one of $t, t-r$ and $t-1$ belongs to $\mathcal{S}$.

2024 Alborz Mathematical Olympiad, P2

Tags: function , functional equation , algebra

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integers $a$ and $b$, we have: $$f(a^2+ab)+f(b^2+ab)=(a+b)f(a+b).$$ Proposed by Heidar Shushtari

2014 NIMO Problems, 7

Tags: algebra , polynomial , quadratic

Let $P(n)$ be a polynomial of degree $m$ with integer coefficients, where $m \le 10$. Suppose that $P(0)=0$, $P(n)$ has $m$ distinct integer roots, and $P(n)+1$ can be factored as the product of two nonconstant polynomials with integer coefficients. Find the sum of all possible values of $P(2)$. [i]Proposed by Evan Chen[/i]

2011 Tokyo Instutute Of Technology Entrance Examination, 1

Tags: invariant , linear algebra , matrix , geometry , polynomial , integration , algebra

Let $f_n\ (n=1,\ 2,\ \cdots)$ be a linear transformation expressed by a matrix $\left( \begin{array}{cc} 1-n & 1 \\ -n(n+1) & n+2 \end{array} \right)$ on the $xy$ plane. Answer the following questions: (1) Prove that there exists 2 lines passing through the origin $O(0,\ 0)$ such that all points of the lines are mapped to the same lines, then find the equation of the lines. (2) Find the area $S_n$ of the figure enclosed by the lines obtained in (1) and the curve $y=x^2$. (3) Find $\sum_{n=1}^{\infty} \frac{1}{S_n-\frac 16}.$ [i]2011 Tokyo Institute of Technlogy entrance exam, Problem 1[/i]

2005 Harvard-MIT Mathematics Tournament, 10

Tags: quadratic , algebra , quadratic formula

Find the sum of the absolute values of the roots of $x^4 - 4x^3 - 4x^2 + 16x - 8 = 0$.

2014 Bulgaria JBMO TST, 6

Tags: algebra

If $a,b$ are real numbers such that $a^3 +12a^2 + 49a + 69 = 0$ and $ b^3 - 9b^2 + 28b - 31 = 0,$ find $a + b .$

2005 District Olympiad, 2

Tags: function , algebra

Find the functions $f:\mathbb{Z}\times \mathbb{Z}\to\mathbb{R}$ such that a) $f(x,y)\cdot f(y,z) \cdot f(z,x) = 1$ for all integers $x,y,z$; b) $f(x+1,x)=2$ for all integers $x$.

1974 Chisinau City MO, 77

Tags: algebra , arithmetic progression

Is it possible to simultaneously take away on eight three-ton vehicles $50$ stones, the weight of which is respectively equal to $416, 418, 420, .., 512, 514$ kg?

1994 Korea National Olympiad, Problem 1

Tags: function , algebra

Let $ S$ be the set of nonnegative integers. Find all functions $ f,g,h: S\rightarrow S$ such that $ f(m\plus{}n)\equal{}g(m)\plus{}h(n),$ for all $ m,n\in S$, and $ g(1)\equal{}h(1)\equal{}1$.

2007 German National Olympiad, 5

Tags: arithmetic sequence , finite , algebra , real number , set

Determine all finite sets $M$ of real numbers such that $M$ contains at least $2$ numbers and any two elements of $M$ belong to an arithmetic progression of elements of $M$ with three terms.

2014 Contests, 3

Tags: limit , induction , inequalities , algebra , difference of squares

Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

2000 Saint Petersburg Mathematical Olympiad, 9.1

Tags: combinatorics , algebra

On the two sides of the road two lines of trees are planted. On every tree, the number of oaks among itself and its neighbors is written. (For the first and last trees, this is the number of oaks among itself and its only neighbor). Prove that if the two sequences of numbers on the trees are equal, then sequnces of trees on the two sides of the road are equal [I]Proposed by A. Khrabrov, D. Rostovski[/i]

2004 Germany Team Selection Test, 1

Tags: vector , geometry , 3d geometry , tetrahedron , linear algebra , algebra

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

2017 Pan-African Shortlist, N1

1991 IMTS, 3

2014 HMNT, 7

2023 Indonesia TST, A

2010 Dutch IMO TST, 1

2013 BMT Spring, 13

1952 Poland - Second Round, 1

2014 Putnam, 5

2013 China Team Selection Test, 3

2024 Alborz Mathematical Olympiad, P2

2014 NIMO Problems, 7

2011 Tokyo Instutute Of Technology Entrance Examination, 1

2005 Harvard-MIT Mathematics Tournament, 10

2014 Bulgaria JBMO TST, 6

2005 District Olympiad, 2

1974 Chisinau City MO, 77

1994 Korea National Olympiad, Problem 1

2007 German National Olympiad, 5

2014 Contests, 3

2000 Saint Petersburg Mathematical Olympiad, 9.1

2004 Germany Team Selection Test, 1

2006 Lithuania National Olympiad, 1

2019 LIMIT Category B, Problem 4

2015 BMT Spring, 8

2023 Bosnia and Herzegovina Junior BMO TST, 1.