Found problems: 15925
1950 AMC 12/AHSME, 17
The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:
\[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline
x&0&1&2&3&4\\\hline
y&100&90&70&40&0\\\hline
\end{tabular}\]
$\textbf{(A)}\ y=100-10x \qquad
\textbf{(B)}\ y=100-5x^2 \qquad
\textbf{(C)}\ y=100-5x-5x^2 \qquad\\
\textbf{(D)}\ y=20-x-x^2 \qquad
\textbf{(E)}\ \text{None of these}$
2016 PAMO, 4
Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that
$\frac{1}{(x+1)^2+y^2+1}$ $+$ $\frac{1}{(y+1)^2+z^2+1}$ $+$ $\frac{1}{(z+1)^2+x^2+1}$ $\leq$ ${\frac{1}{2}}$.
2019 Belarusian National Olympiad, 9.7
Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients such that $P(Q(x)^2)=P(x)\cdot Q(x)^2$.
[i](I. Voronovich)[/i]
2003 Argentina National Olympiad, 2
On the blackboard are written the $2003$ integers from $1$ to $2003$. Lucas must delete $90$ numbers. Next, Mauro must choose $37$ from the numbers that remain written. If the $37$ numbers Mauro chooses form an arithmetic progression, Mauro wins. If not, Lucas wins. Decide if Lucas can choose the $90$ numbers he erases so that victory is assured.
2023 JBMO TST - Turkey, 3
Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$f(x+f(x))=f(-x)$ and for all $x \leq y$ it satisfies $f(x) \leq f(y)$
2002 Italy TST, 3
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that
$(\text{i})$ $x$ and $y$ are relatively prime;
$(\text{ii})$ $x$ divides $y^2+m;$
$(\text{iii})$ $y$ divides $x^2+m.$
2020 ELMO Problems, P5
Let $m$ and $n$ be positive integers. Find the smallest positive integer $s$ for which there exists an $m \times n$ rectangular array of positive integers such that
[list]
[*]each row contains $n$ distinct consecutive integers in some order,
[*]each column contains $m$ distinct consecutive integers in some order, and
[*]each entry is less than or equal to $s$.
[/list]
[i]Proposed by Ankan Bhattacharya.[/i]
1940 Moscow Mathematical Olympiad, 058
Solve the system $\begin{cases} (x^3 + y^3)(x^2 + y^2) = 2b^5 \\
x + y = b \end{cases}$ in $C$
2023 Estonia Team Selection Test, 2
Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.
2023 Vietnam National Olympiad, 3
Find the maximum value of the positive real number $k$ such that the inequality
$$\frac{1}{kab+c^2} +\frac{1} {kbc+a^2} +\frac{1} {kca+b^2} \geq \frac{k+3}{a^2+b^2+c^2} $$holds for all positive real numbers $a,b,c$ such that $a^2+b^2+c^2=2(ab+bc+ca).$
2016 BMT Spring, 7
Find the coefficient of $x^2$ in the following polynomial
$$(1 -x)^2(1 + 2x)^2(1 - 3x)^2... (1 -11x)^2.$$
2023 LMT Spring, 10
The sequence $a_0,a_1,a_2,...$ is defined such that $a_0 = 2+ \sqrt3$, $a_1 =\sqrt{5-2\sqrt5}$, and
$$a_n a_{n-1}a_{n-2} - a_n + a_{n-1} + a_{n-2} = 0.$$
Find the least positive integer $n$ such that $a_n = 1$.
1990 IMO, 3
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
1979 IMO Shortlist, 3
Find all polynomials $f(x)$ with real coefficients for which
\[f(x)f(2x^2) = f(2x^3 + x).\]
2016 BMT Spring, 2
Find an integer pair of solutions $(x, y)$ to the following system of equations.
$$\log_2 (y^x) = 16$$
$$\log_2 (x^y) = 8$$
2024 Baltic Way, 4
Find the largest real number $\alpha$ such that, for all non-negative real numbers $x$, $y$ and $z$, the following inequality holds:
\[
(x+y+z)^3 + \alpha (x^2z + y^2x + z^2y) \geq \alpha (x^2y + y^2z + z^2x).
\]
2013 Romania Team Selection Test, 3
Determine all injective functions defined on the set of positive integers into itself satisfying the following condition: If $S$ is a finite set of positive integers such that $\sum\limits_{s\in S}\frac{1}{s}$ is an integer, then $\sum\limits_{s\in S}\frac{1}{f\left( s\right) }$ is also an integer.
2021 Greece JBMO TST, 1
If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$
2020 Princeton University Math Competition, B1
The function $f(x) = x^2 + (2a + 3)x + (a^2 + 1)$ only has real zeroes. Suppose the smallest possible value of $a$ can be written in the form $p/q$, where $p, q$ are relatively prime integers. Find $|p| + |q|$.
1999 China Team Selection Test, 2
For a fixed natural number $m \geq 2$, prove that
[b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\]
[b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.
2024 Saint Petersburg Mathematical Olympiad, 2
A strongman Bambula can carry several weights at the same time, if their total weight does not exceed $200$ kg, and these weights are no more than three. On the way to work, he injured his finger and found that he could now carry no more than two weights (and still no more than $200$ kg). At what minimum $k$ is the statement true: [i]any set of $100$ weights that Bambula could previously carry in $50$ runs, with a sore finger, he will be able to carry in no more than $k$ runs?[/i]
2011 Polish MO Finals, 3
Prove that it is impossible for polynomials $f_1(x),f_2(x),f_3(x),f_4(x)\in \mathbb{Q}[x]$ to satisfy \[f_1^2(x)+f_2^2(x)+f_3^2(x)+f_4^2(x) = x^2+7.\]
2010 Danube Mathematical Olympiad, 5
Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$, with $x_1+x_2+\ldots +x_n=n$, for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value.
1986 Tournament Of Towns, (122) 4
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.
2009 Cono Sur Olympiad, 1
The four circles in the figure determine 10 bounded regions. $10$ numbers summing to $100$ are written in these regions, one in each region. The sum of the numbers contained in each circle is equal to $S$ (the same quantity for each of the four circles). Determine the greatest and smallest possible values of $S$.