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

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Found problems: 15925

2021 239 Open Mathematical Olympiad, 4
Tags: algebra , inequalities
Different positive $a, b, c$ are such that $a^{239} = ac- 1$ and $b^{239} = bc- 1$.Prove that $238^2 (ab)^{239} <1$.

1996 All-Russian Olympiad Regional Round, 11.5
Tags: algebra
Given the function $f(x) =|4 - 4|x||- 2$. How many solutions does the equation $f(f(x)) = x$ have?

2000 Harvard-MIT Mathematics Tournament, 3
Tags: algebra , geometry
A twelve foot tree casts a five foot shadow. How long is Henry’s shadow (at the same time of day) if he is five and a half feet tall?

2001 Baltic Way, 11
Tags: function , algebra
The real-valued function $f$ is defined for all positive integers. For any integers $a>1, b>1$ with $d=\gcd (a, b)$, we have \[f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right) \] Determine all possible values of $f(2001)$.

2012 Indonesia TST, 1
Tags: function , polynomial , induction , algebra
Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that \[f(x+t) - f(x) = P(x)\] for all $x \in \mathbb{R}$.

1969 IMO Shortlist, 48
Tags: inequality system , algebra , maximization
$(NET 3)$ Let $x_1, x_2, x_3, x_4,$ and $x_5$ be positive integers satisfying \[x_1 +x_2 +x_3 +x_4 +x_5 = 1000,\] \[x_1 -x_2 +x_3 -x_4 +x_5 > 0,\] \[x_1 +x_2 -x_3 +x_4 -x_5 > 0,\] \[-x_1 +x_2 +x_3 -x_4 +x_5 > 0,\] \[x_1 -x_2 +x_3 +x_4 -x_5 > 0,\] \[-x_1 +x_2 -x_3 +x_4 +x_5 > 0\] $(a)$ Find the maximum of $(x_1 + x_3)^{x_2+x_4}$ $(b)$ In how many different ways can we choose $x_1, . . . , x_5$ to obtain the desired maximum?

2004 Romania National Olympiad, 3
Tags: function , integration , calculus , inequalities , polynomial , linear algebra , algebra
Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \] [i]Ion Rasa[/i]

2016 NIMO Summer Contest, 9
Tags: algebra , floor function , function
Compute the number of real numbers $t$ such that \[t = 50 \sin(t - \lfloor t \rfloor).\] Here $\lfloor \cdot\rfloor$ denotes the greatest integer function. [i]Proposed by David Altizio[/i]

2023 Indonesia TST, 1
Tags: algebra
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

1957 AMC 12/AHSME, 34
Tags: inequalities , parameterization , analytic geometry , graphing lines , slope , algebra , function
The points that satisfy the system $ x \plus{} y \equal{} 1,\, x^2 \plus{} y^2 < 25,$ constitute the following set: $ \textbf{(A)}\ \text{only two points} \qquad \\ \textbf{(B)}\ \text{an arc of a circle}\qquad \\ \textbf{(C)}\ \text{a straight line segment not including the end\minus{}points}\qquad \\ \textbf{(D)}\ \text{a straight line segment including the end\minus{}points}\qquad \\ \textbf{(E)}\ \text{a single point}$

Kvant 2025, M2827
Tags: algebra , inequalities
It is known about positive numbers $a, b, c$ that it is possible to form a triangle from segments of length $a^{2024}, b^{2024}, c^{2024}$. Prove that it is possible to reduce one of the numbers $a, b, c$ by $2024$ times and obtain the numbers $a', b', c'$ so that segments with lengths $a', b', c'$ can also be formed into a triangle. [i]L. Shatunov[/i]

2021 Saudi Arabia IMO TST, 11
Tags: algebra , inequality , 4-variable inequality , inequalities
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

1994 Taiwan National Olympiad, 6
Tags: polynomial , algebra
For $-1\leq x\leq 1$ and $n\in\mathbb N$ define $T_{n}(x)=\frac{1}{2^{n}}[(x+\sqrt{1-x^{2}})^{n}+(x-\sqrt{1-x^{2}})^{n}]$. a)Prove that $T_{n}$ is a monic polynomial of degree $n$ in $x$ and that the maximum value of $|T_{n}(x)|$ is $\frac{1}{2^{n-1}}$. b)Suppose that $p(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\in\mathbb{R}[x]$ is a monic polynomial of degree $n$ such that $p(x)>-\frac{1}{2^{n-1}}$ forall $x$, $-1\leq x\leq 1$. Prove that there exists $x_{0}$, $-1\leq x_{0}\leq 1$ such that $p(x_{0})\geq\frac{1}{2^{n-1}}$.

2007 IMO Shortlist, 1
Tags: sequence , algebra
Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define \[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \} \] and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$. (a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, \[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*) \] (b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*). [i]Author: Michael Albert, New Zealand[/i]

2003 Greece JBMO TST, 2
Tags: sum , algebra
Calculate if $n\in N$ with $n>2$ the value of $$B=\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+...+\sqrt{1+\frac{1}{(n-1)^2}+\frac{1}{n^2}} $$

2011 All-Russian Olympiad, 3
Tags: modular arithmetic , algebra , polynomial , calculus , integration , number theory
Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$. [i]A. Golovanov[/i]

2002 India IMO Training Camp, 3
Tags: quadratic , algebra
Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$. How many quadratics are there of the form $ax^2+2bx+c$, with equal roots, and such that $a,b,c$ are distinct elements of $X$?

2018 Romania National Olympiad, 4
Tags: algebra
Let $n \in \mathbb{N}_{\geq 2}.$ For any real numbers $a_1,a_2,...,a_n$ denote $S_0=1$ and for $1 \leq k \leq n$ denote $$S_k=\sum_{1 \leq i_1 < i_2 < ... <i_k \leq n}a_{i_1}a_{i_2}...a_{i_k}$$ Find the number of $n-$tuples $(a_1,a_2,...a_n)$ such that $$(S_n-S_{n-2}+S_{n-4}-...)^2+(S_{n-1}-S_{n-3}+S_{n-5}-...)^2=2^nS_n.$$

1969 All Soviet Union Mathematical Olympiad, 117
Tags: digit , sequence , algebra
Given a finite sequence of zeros and ones, which has two properties: a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap); b) if you add any digit to the right of the sequence, then property (a) will no longer hold true. Prove that the first four digits of our sequence coincide with the last four

2014 ELMO Shortlist, 5
Tags: function , algebra
Let $\mathbb R^\ast$ denote the set of nonzero reals. Find all functions $f: \mathbb R^\ast \to \mathbb R^\ast$ satisfying \[ f(x^2+y)+1=f(x^2+1)+\frac{f(xy)}{f(x)} \] for all $x,y \in \mathbb R^\ast$ with $x^2+y\neq 0$. [i]Proposed by Ryan Alweiss[/i]

1939 Moscow Mathematical Olympiad, 048
Tags: factoring , algebra , factor
Factor $a^{10} + a^5 + 1$ into nonconstant polynomials with integer coefficients

2019 PUMaC Algebra B, 8
Tags: algebra
A [i]weak binary representation[/i] of a nonnegative integer $n$ is a representation $n=a_0+2\cdot a_1+2^2\cdot a_2+\dots$ such that $a_i\in\{0,1,2,3,4,5\}$. Determine the number of such representations for $513$.

2024 5th Memorial "Aleksandar Blazhevski-Cane", P2
Tags: inequalities , algebra
Let $x,y$ and $z$ be positive real numbers such that $xy+z^2=8$. Determine the smallest possible value of the expression $$\frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}.$$

2016 Azerbaijan Team Selection Test, 1
Tags: algebra
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

2010 Iran MO (3rd Round), 3
Tags: polynomial , search , inequalities , algebra
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)