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

2015 Mexico National Olympiad, 5
Tags: incenter , circumcircle , geometric transformation , reflection , algebra , geometry
Let $I$ be the incenter of an acute-angled triangle $ABC$. Line $AI$ cuts the circumcircle of $BIC$ again at $E$. Let $D$ be the foot of the altitude from $A$ to $BC$, and let $J$ be the reflection of $I$ across $BC$. Show $D$, $J$ and $E$ are collinear.

1992 Czech And Slovak Olympiad IIIA, 5
Tags: function , rational , max , algebra , irrational
The function $f : (0,1) \to R$ is defined by $f(x) = x$ if $x$ is irrational, $f(x) = \frac{p+1}{q}$ if $x =\frac{p}{q}$ , where $(p,q) = 1$. Find the maximum value of $f$ on the interval $(7/8,8/9)$.

2010 India Regional Mathematical Olympiad, 2
Tags: quadratic , algebra , polynomial
Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.

1961 Polish MO Finals, 2
Tags: inequalities , algebra
Prove that if $ a + b = 1 $, then $$ a^5 + b^5 \geq \frac{1}{16}$$

2017 Thailand Mathematical Olympiad, 8
Tags: sides of a triangle , minimum , right triangle , inequalities , algebra
Let $a, b, c$ be side lengths of a right triangle. Determine the minimum possible value of $\frac{a^3 + b^3 + c^3}{abc}$.

2016 Iran MO (3rd Round), 2
Tags: number theory , function , algebra , polynomial
We call a function $g$ [i]special [/i] if $g(x)=a^{f(x)}$ (for all $x$) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$. A function is called an [i]exponential polynomial[/i] if it is obtained from the product or sum of special functions. For instance, $2^{x}3^{x^{2}+x-1}+5^{2x}$ is an exponential polynomial. Prove that there does not exist a non-zero exponential polynomial $f(x)$ and a non-constant polynomial $P(x)$ with integer coefficients such that $$P(n)|f(n)$$ for all positive integers $n$.

2011 Abels Math Contest (Norwegian MO), 3a
Tags: inequalities , algebra , sequence
The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\le a_m +a_n$ for all positive integers $m$ and $n$. Show that $\frac{1}{a_{200}} > 4 \cdot 10^7$ . .

1988 Romania Team Selection Test, 10
Tags: polynomial , algebra
Let $ p > 2$ be a prime number. Find the least positive number $ a$ which can be represented as \[ a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X), \] where $ f(X)$ and $ g(X)$ are integer polynomials. [i]Mircea Becheanu[/i].

1989 Nordic, 3
Tags: elements of set , sequence , algebra
Let $S$ be the set of all points $t$ in the closed interval $[-1, 1]$ such that for the sequence $x_0, x_1, x_2, ...$ defined by the equations $x_0 = t, x_{n+1} = 2x_n^2-1$, there exists a positive integer $N$ such that $x_n = 1$ for all $n \ge N$. Show that the set $S$ has infinitely many elements.

1996 Moldova Team Selection Test, 5
Tags: algebra , polynomial
Find all polynomials $P(X)$ of fourth degree with real coefficients that verify the properties: [b]a)[/b] $P(-x)=P(x), \forall x\in\mathbb{R};$ [b]b)[/b] $P(x)\geq0, \forall x\in\mathbb{R};$ [b]c)[/b] $P(0)=1;$ [b]d)[/b] $P(X)$ has exactly two local minimums $x_1$ and $x_2$ such that $|x_1-x_2|=2.$

1940 Putnam, B5
Tags: polynomial , rational , algebra , root
Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer

2005 Grigore Moisil Urziceni, 1
Tags: equation , system of equations , algebra
Find the nonnegative real numbers $ a,b,c,d $ that satisfy the following system: $$ \left\{ \begin{matrix} a^3+2abc+bcd-6&=&a \\a^2b+b^2c+abd+bd^2&=&b\\a^2b+a^2c+bc^2+cd^2&=&c\\d^3+ab^2+abc+bcd-6&=&d \end{matrix} \right. $$

2014 HMNT, 8
Tags: algebra
Let $a, b, c, x$ be reals with $(a + b)(b + c)(c + a) \ne 0$ that satisfy $$\frac{a^2}{a + b}=\frac{a^2}{a + c}+ 20, \,\,\, \frac{b^2}{b + c}=\frac{b^2}{b + a}+ 14, \text{and}\,\,\, \frac{c^2}{c + a}=\frac{c^2}{c + b}+ x.$$ Compute $x$.

2012 Vietnam National Olympiad, 2
Tags: quadratic , polynomial , arithmetic sequence , algebra
Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.

2017 Junior Balkan Team Selection Tests - Romania, 1
Tags: inequalities , algebra
If $a, b, c \in [-1, 1]$ satisfy $a + b + c + abc = 0$, prove that $a^2 + b^2 + c^2 \ge 3(a + b + c)$ . When does the equality hold?

2022 IMO Shortlist, A7
Tags: polynomial , algebra
For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?

2004 239 Open Mathematical Olympiad, 1
Tags: polynomial , algebra , function
Given non-constant linear functions $p_1(x), p_2(x), \dots p_n(x)$. Prove that at least $n-2$ of polynomials $p_1p_2\dots p_{n-1}+p_n, p_1p_2\dots p_{n-2} p_n + p_{n-1},\dots p_2p_3\dots p_n+p_1$ have a real root.

2025 Kosovo National Mathematical Olympiad`, P2
Tags: inequality , algebra , inequalities
Let $x$ and $y$ be real numbers where at least one of them is bigger than $2$ and $xy+4 > 2(x+y)$ holds. Show that $xy>x+y$.

2007 ITest, 23
Tags: algebra
Find the product of the non-real roots of the equation \[(x^2-3)^2+5(x^2-3)+6=0.\] $\begin{array}{@{\hspace{0em}}l@{\hspace{13.7em}}l@{\hspace{13.7em}}l} \textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }-1\\\\ \textbf{(D) }2&\textbf{(E) }-2&\textbf{(F) }3\\\\ \textbf{(G) }-3&\textbf{(H) }4&\textbf{(I) }-4\\\\ \textbf{(J) }5&\textbf{(K) }-5&\textbf{(L) }6\\\\ \textbf{(M) }-6&\textbf{(N) }3+2i&\textbf{(O) }3-2i\\\\ \textbf{(P) }\dfrac{-3+i\sqrt3}2&\textbf{(Q) }8&\textbf{(R) }-8\\\\ \textbf{(S) }12&\textbf{(T) }-12&\textbf{(U) }42\\\\ \textbf{(V) }\text{Ying} & \textbf{(W) }2007 &\end{array}$

2023 Taiwan TST Round 2, A
Tags: function equation , algebra
Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that $$f\left(xy+f(y)\right)f(x)=x^2f(y)+f(xy)$$ for all $x,y \in \mathbb{R}$ [i]Proposed by chengbilly[/i]

2002 Switzerland Team Selection Test, 9
Tags: inequalities , algebra
For each real number $a$ and integer $n \ge 1$ prove the inequality $a^n +\frac{1}{a^n} -2 \ge n^2 \left(a +\frac{1}{a} -2\right)$ and find the cases of equality.

1950 Moscow Mathematical Olympiad, 174
Tags: weighing , combinatorics , algebra
a) Given $555$ weights: of $1$ g, $2$ g, $3$ g, . . . , $555$ g, divide them into three piles of equal mass. b) Arrange $81$ weights of $1^2, 2^2, . . . , 81^2$ (all in grams) into three piles of equal mass.

2014 European Mathematical Cup, 4
Tags: function , algebra
Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that for all $x,y\in{{\mathbb{R}}}$ holds $f(x^2)+f(2y^2)=(f(x+y)+f(y))(f(x-y)+f(y))$ [i]Proposed by Matija Bucić[/i]

2015 India National Olympiad, 3
Tags: function , trigonometry , functional equation , algebra
Find all real functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2+yf(x))=xf(x+y)$.

2016 Junior Regional Olympiad - FBH, 2
Tags: root , algebra
If $$w=\sqrt{1+\sqrt{-3+2\sqrt{3}}}-\sqrt{1-\sqrt{-3+2\sqrt{3}}}$$ prove that $w=\sqrt{3}-1$