This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 15925

2001 IMO Shortlist, 4
Tags: functional equation , algebra , function
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying \[ f(xy)(f(x) - f(y)) = (x-y)f(x)f(y) \] for all $x,y$.

2014 BMT Spring, 7
Tags: algebra
If $f(x, y) = 3x^2 + 3xy + 1$ and $f(a, b) + 1 = f(b, a) = 42$, then determine $|a + b|$.

1998 Slovenia National Olympiad, Problem 2
Tags: function , algebra
find all functions $f(x)$ satisfying: $(\forall x\in R) f(x)+xf(1-x)=x^2+1$

2016 India Regional Mathematical Olympiad, 5
Tags: algebra , inequalities , holder inequality , symmetric inequality , am-gm
Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that $$(x^3+2y)(y^3+2z)(z^3+2x) \ge 27.$$

2013 Baltic Way, 4
Tags: algebra , inequality
Prove that the following inequality holds for all positive real numbers $x,y,z$: \[\dfrac{x^3}{y^2+z^2}+\dfrac{y^3}{z^2+x^2}+\dfrac{z^3}{x^2+y^2}\ge \dfrac{x+y+z}{2}.\]

1953 Polish MO Finals, 5
Tags: geometry , algebra
From point $ O $ a car starts on a straight road and travels with constant speed $ v $. A cyclist who is located at a distance $ a $ from point $ O $ and at a distance $ b $ from the road wants to deliver a letter to this car. What is the minimum speed a cyclist should ride to reach his goal?

1983 Canada National Olympiad, 2
Tags: functional equation , function , algebra , logarithm
For each $r\in\mathbb{R}$ let $T_r$ be the transformation of the plane that takes the point $(x, y)$ into the point $(2^r x; r2^r x+2^r y)$. Let $F$ be the family of all such transformations (i.e. $F = \{T_r : r\in\mathbb{R}\}$). Find all curves $y = f(x)$ whose graphs remain unchanged by every transformation in $F$.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.6
Tags: functional equation , algebra
Find all functions $f:R\to R$ such that for any real $x, y$ , $$f(x+2^y)=f(2^x)+f(y)$$

2012 China National Olympiad, 2
Tags: linear algebra , polynomial , symmetry , geometric transformation , geometry , algebra , matrix
Let $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there?

2025 Spain Mathematical Olympiad, 6
Tags: functional equation , algebra
Let $\mathbb{R}_{\neq 0}$ be the set of nonzero real numbers. Find all functions $f:\mathbb{R}_{\neq 0}\rightarrow\mathbb{R}_{\neq 0}$ such that, for all $x,y\in\mathbb{R}_{\neq 0}$, \[(x-y)f(y^2)+f\left(xy\,f\left(\frac{x^2}{y}\right)\right)=f(y^2f(y)).\]

2010 Bundeswettbewerb Mathematik, 2
Tags: floor function , algebra , recurrence relation , sequence , radical
The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor $ for $n \ge 1$. Find all numbers that appear more than twice at this sequence.

1969 IMO Longlists, 38
Tags: trigonometry , algebra , series summation , trigonometric identities
$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$

1997 Tournament Of Towns, (545) 6
Tags: polynomial , algebra
Prove that if $F(x)$ and $G(x)$ are polynomials with coefficients $0$ and $1$ such that $$F(x)G(x) = 1 +x + x^2 +...+ x^{n-1}$$ holds for some $n > 1$, then one of them can be represented in the form $$ (1 +x + x^2 +...+ x^{k-1}) T(x)$$ for some $k > 1$ where $T(x)$ is a polynomial with coefficients $0$ and $1$. (V Senderov, M Vialiy)

1987 Swedish Mathematical Competition, 1
Tags: algebra , sum
Sixteen real numbers are arranged in a magic square of side $4$ so that the sum of numbers in each row, column or main diagonal equals $k$. Prove that the sum of the numbers in the four corners of the square is also $k$.

2020 Tournament Of Towns, 1
Tags: algebra , parabola , analytic geometry , segment
Consider two parabolas $y = x^2$ and $y = x^2 - 1$. Let $U$ be the set of points between the parabolas (including the points on the parabolas themselves). Does $U$ contain a line segment of length greater than $10^6$ ? Alexey Tolpygo

VMEO IV 2015, 10.1
Tags: algebra , rational
Given a real number $\alpha$ satisfying $\alpha^3 = \alpha + 1$. Determine all $4$-tuples of rational numbers $(a, b, c, d)$ satisfying: $a\alpha^2 + b\alpha+ c = \sqrt{d}.$

2019 ELMO Shortlist, A4
Tags: functional equation , algebra
Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$ [i]Proposed by Carl Schildkraut[/i]

Kvant 2022, M2684
Tags: algebra
Let $a_1,a_2,\ldots,a_n$ ($n\geq 2$) be nonnegative real numbers whose sum is $\frac{n}{2}$. For every $i=1,\ldots,n$ define $$b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}$$ where $a_{j+n}=a_j$ for every $j$. Prove that $b_i\geq 1$ holds for at least one index $i$.

2002 Tuymaada Olympiad, 2
Tags: induction , algebra , function
Find all the functions $f(x),$ continuous on the whole real axis, such that for every real $x$ \[f(3x-2)\leq f(x)\leq f(2x-1).\] [i]Proposed by A. Golovanov[/i]

2015 Singapore Junior Math Olympiad, 3
Tags: combinatorics , algebra , sequence
There are $30$ children, $a_1,a_2,...,a_{30}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.

2011 IMC, 4
Tags: algebra , polynomial , induction
Let $f$ be a polynomial with real coefficients of degree $n$. Suppose that $\displaystyle \frac{f(x)-f(y)}{x-y}$ is an integer for all $0 \leq x<y \leq n$. Prove that $a-b | f(a)-f(b)$ for all distinct integers $a,b$.

2008 Indonesia TST, 2
Tags: inequalities , algebra , sequence , sum
Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$ for all positive integers $n$.

2005 USAMTS Problems, 3
Tags: geometry , algebra , analytic geometry , quadratic , trapezoid , complex number , quadratic formula
Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.

2013 Junior Balkan Team Selection Tests - Moldova, 4
Tags: angle , algebra
A train from stop $A$ to stop $B$ is traveled in $X$ minutes ($0 <X <60$). It is known that when starting from $A$, as well as when arriving at $B$, the angle formed by the hour and the minute had measure equal to $X$ degrees. Find $X $.

1990 IMO Longlists, 88
Tags: cauchy schwarz , holder , inequality , 4-variable inequality , algebra
Let $ w, x, y, z$ are non-negative reals such that $ wx \plus{} xy \plus{} yz \plus{} zw \equal{} 1$. Show that $ \frac {w^3}{x \plus{} y \plus{} z} \plus{} \frac {x^3}{w \plus{} y \plus{} z} \plus{} \frac {y^3}{w \plus{} x \plus{} z} \plus{} \frac {z^3}{w \plus{} x \plus{} y}\geq \frac {1}{3}$.