Found problems: 85335
2014 ELMO Shortlist, 5
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]
2005 Bulgaria Team Selection Test, 4
Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_{i}$, $i \in \{1,2, \dots, 2005 \}$.
2008 Rioplatense Mathematical Olympiad, Level 3, 3
Consider a collection of stones whose total weight is $65$ pounds and each of whose stones is at most $w$ pounds. Find the largest number $w$ for which any such collection of stones can be divided into two groups whose total weights differ by at most one pound.
Note: The weights of the stones are not necessarily integers.
2021 Mediterranean Mathematics Olympiad, 4
Let $x_1,x_2,x_3,x_4,x_5$ ve non-negative real numbers, so that
$x_1\le4$ and
$x_1+x_2\le13$ and
$x_1+x_2+x_3\le29$ and
$x_1+x_2+x_3+x_4\le54$ and
$x_1+x_2+x_3+x_4+x_5\le90$.
Prove that $\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5}\le20$.
2018 Rioplatense Mathematical Olympiad, Level 3, 1
Determine if there are $2018$ different positive integers such that the sum of their squares is a perfect cube and the sum of their cubes is a perfect square.
2007 JBMO Shortlist, 2
$\boxed{\text{A2}}$ Prove that for all Positive reals $a,b,c$ $\frac{a^2-bc}{2a^2+bc}+\frac{b^2-ca}{2b^2+ca}+\frac{c^2-ab}{2c^2+ab}\leq 0$
1998 All-Russian Olympiad Regional Round, 8.7
Let $O$ be the center of a circle circumscribed about an acute angle triangle $ABC$, $S_A$, $S_B$, $S_C$ - circles with center O, tangent to sides $BC$, $CA$, $AB$ respectively. Prove that the sum of three angles : between the tangents to $S_A$ drawn from point $A$, to $S_B$ from point $B$ and to $S_C$ - from point $C$, is equal to $180^o$.
2001 APMO, 3
Two equal-sized regular $n$-gons intersect to form a $2n$-gon $C$. Prove that the sum of the sides of $C$ which form part of one $n$-gon equals half the perimeter of $C$.
[i]Alternative formulation:[/i]
Let two equal regular $n$-gons $S$ and $T$ be located in the plane such that their intersection $S\cap T$ is a $2n$-gon (with $n\ge 3$). The sides of the polygon $S$ are coloured in red and the sides of $T$ in blue.
Prove that the sum of the lengths of the blue sides of the polygon $S\cap T$ is equal to the sum of the lengths of its red sides.
2005 AMC 10, 6
At the beginning of the school year, Lisa’s goal was to earn an A on at least $ 80\%$ of her $ 50$ quizzes for the year. She earned an A on $ 22$ of the first $ 30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
2000 French Mathematical Olympiad, Exercise 1
We are given $b$ white balls and $n$ black balls ($b,n>0$) which are to be distributed among two urns, at least one in each. Let $s$ be the number of balls in the first urn, and $r$ the number of white ones among them. One randomly chooses an urn and randomly picks a ball from it.
(a) Compute the probability $p$ that the drawn ball is white.
(b) If $s$ is fixed, for which $r$ is $p$ maximal?
(c) Find the distribution of balls among the urns which maximizes $p$.
(d) Give a generalization for larger numbers of colors and urns.
2019 China Team Selection Test, 4
Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that
1) $f(0,x)$ is non-decreasing ;
2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ;
3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ;
4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .
2015 Purple Comet Problems, 10
Find the sum of all the real values of x satisfying $(x+\frac{1}{x}-17)^2$ $= x + \frac{1}{x} + 17.$
2022 IOQM India, 6
Let $x,y,z$ be positive real numbers such that $x^2 + y^2 = 49, y^2 + yz + z^2 = 36$ and $x^2 + \sqrt{3}xz + z^2 = 25$. If the value of $2xy + \sqrt{3}yz + zx$ can be written as $p \sqrt{q}$ where $p,q \in \mathbb{Z}$ and $q$ is squarefree, find $p+q$.
1946 Moscow Mathematical Olympiad, 115
Prove that if $\alpha$ and $\beta$ are acute angles and $\alpha$ < $\beta$ , then $\frac{tan \alpha}{\alpha} < \frac{tan \beta}{\beta} $
1958 Kurschak Competition, 2
Show that if $m$ and $n$ are integers such that $m^2 + mn + n^2$ is divisible by $9$, then they must both be divisible by $3$.
2016 HMIC, 4
Let $P$ be an odd-degree integer-coefficient polynomial. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.
[i]Victor Wang[/i]
2019 USAMTS Problems, 4
Princess Pear has $100$ jesters with heights $1, 2, \dots, 100$ inches. On day $n$ with $1 \leq n \leq 100$, Princess Pear holds a court with all her jesters with height at most $n$ inches, and she receives two candied cherries from every group of $6$ jesters with a median height of $n - 50$ inches. A jester can be part of multiple groups.
On day $101$, Princess Pear summons all $100$ jesters to court one final time. Every group of $6$ jesters with a median height of 50.5 inches presents one more candied cherry to the Princess. How many candied cherries does Princess Pear receive in total?
Please provide a numerical answer (with justification).
2017 ASDAN Math Tournament, 3
Let $f(x)=x^4+2x+1$. Find the slope of the tangent line to the curve at $(0,1)$.
2002 China Team Selection Test, 1
In acute triangle $ ABC$, show that:
$ \sin^3{A}\cos^2{(B \minus{} C)} \plus{} \sin^3{B}\cos^2{(C \minus{} A)} \plus{} \sin^3{C}\cos^2{(A \minus{} B)} \leq 3\sin{A} \sin{B} \sin{C}$
and find out when the equality holds.
2013 ELMO Shortlist, 9
Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$.
Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$.
Find a closed form for $a_n$.
[i]Proposed by Bobby Shen[/i]
2024 India National Olympiad, 6
For each positive integer $n \ge 3$, define $A_n$ and $B_n$ as
\[A_n = \sqrt{n^2 + 1} + \sqrt{n^2 + 3} + \cdots + \sqrt{n^2+2n-1}\]
\[B_n = \sqrt{n^2 + 2} + \sqrt{n^2 + 4} + \cdots + \sqrt{n^2 + 2n}.\]
Determine all positive integers $n\ge 3$ for which $\lfloor A_n \rfloor = \lfloor B_n \rfloor$.
Note. For any real number $x$, $\lfloor x\rfloor$ denotes the largest integer $N\le x$.
[i]Anant Mudgal and Navilarekallu Tejaswi[/i]
2017 Harvard-MIT Mathematics Tournament, 4
[b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for
which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.
2005 Kyiv Mathematical Festival, 5
The plane is dissected by broken lines into some regions. It is possible to paint the map formed by these regions in three colours so that any neighbouring regions will have different colours. Call by knots the points which belong to at least two segments of broken lines. One of the segments connecting two knots is erased and replaced by arbitrary broken line connecting the same knots. Prove that it is possible to paint new map in three colours so that any neighbouring regions will have different colours.
VII Soros Olympiad 2000 - 01, 8.1
If there are as many boys in the class as there are girls in the class now, the percentage of girls will decrease by $1.4$ times. Find out what percentage of the students in the class were boys.
2002 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be an isosceles triangle such that $AB = AC$ and $\angle A = 20^o$. Let $M$ be the foot of the altitude from $C$ and let $N$ be a point on the side $AC$ such that $CN =\frac12 BC$. Determine the measure of the angle $AMN$.