Found problems: 85335
2012 Junior Balkan Team Selection Tests - Romania, 2
Let us choose arbitrarily $n$ vertices of a regular $2n$-gon and color them red. The remaining vertices are colored blue. We arrange all red-red distances into a nondecreasing sequence and do the same with the blue-blue distances. Prove that the two sequences thus obtained are identical.
2011 Bosnia And Herzegovina - Regional Olympiad, 4
For positive integer $n$, prove that at least one of the numbers $$A=2n-1 , B=5n-1, C=13n-1$$ is not perfect square
2012 Peru IMO TST, 2
Let $a, b, c$ be the lengths of the sides of a triangle, and $h_a, h_b, h_c$ the lengths of the heights corresponding to the sides $a, b, c,$ respectively. If $t \geq \frac{1} {2}$ is a real number, show that there is a triangle with sidelengths $$ t\cdot a + h_a, \ t\cdot b + h_b , \ t\cdot c + h_c.$$
1955 Putnam, A7
Consider the function $f$ defined by the differential equation \[ f'' (x) = (x^3 + ax) f(x) \] and the initial conditions $f(0) = 1, f'(0) = 0.$ Prove that the roots of $f$ are bounded above but unbounded below.
2015 Iran Team Selection Test, 1
$a,b,c,d$ are positive numbers such that $\sum_{cyc} \frac{1}{ab} =1$. Prove that :
$abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}$
2012 CHMMC Spring, 1
Let $a, b, c$ be positive integers. Suppose that $(a + b)(a + c) = 77$ and $(a + b)(b + c) = 56$. Find $(a + c)(b + c)$.
1984 Bundeswettbewerb Mathematik, 3
Let $a$ and $b$ be positive integers. Show that if $a \cdot b$ is even, then there are positive integers $c$ and $d$ with $a^2 + b^2 + c^2 = d^2$; if, on the other hand, $a\cdot b$ is odd, there are no such positive integers $c$ and $d$.
1957 AMC 12/AHSME, 33
If $ 9^{x \plus{} 2} \equal{} 240 \plus{} 9^x$, then the value of $ x$ is:
$ \textbf{(A)}\ 0.1 \qquad
\textbf{(B)}\ 0.2\qquad
\textbf{(C)}\ 0.3\qquad
\textbf{(D)}\ 0.4\qquad
\textbf{(E)}\ 0.5$
2017 SDMO (High School), 2
There are $5$ accents in French, each applicable to only specific letters as follows:
[list]
[*] The cédille: ç
[*] The accent aigu: é
[*] The accent circonflexe: â, ê, î, ô, û
[*] The accent grave: à, è, ù
[*] The accent tréma: ë, ö, ü
[/list]
Cédric needs to write down a phrase in French. He knows that there are $3$ words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If $n$ is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of $n$.
2017 Online Math Open Problems, 21
Let $\mathbb{Z}_{\geq 0}$ be the set of nonnegative integers. Let $f: \mathbb{Z}_{\geq0} \to \mathbb{Z}_{\geq0}$ be a function such that, for all $a,b \in \mathbb{Z}_{\geq0}$: \[f(a)^2+f(b)^2+f(a+b)^2=1+2f(a)f(b)f(a+b).\]
Furthermore, suppose there exists $n \in \mathbb{Z}_{\geq0}$ such that $f(n)=577$. Let $S$ be the sum of all possible values of $f(2017)$. Find the remainder when $S$ is divided by $2017$.
[i]Proposed by Zack Chroman[/i]
1996 All-Russian Olympiad Regional Round, 10.2
Is it true that from an arbitrary triangle you can cut three equal figures, the area of each of which is more than a quarter of the area triangle?
1969 AMC 12/AHSME, 30
Let $P$ be a point of hypotenuse $AB$ (or its extension) of isosceles right triangle $ABC$. Let $s=AP^2+PB^2$. Then:
$\textbf{(A) }s<2CP^2\text{ for a finite number of positions of }P$
$\textbf{(B) }s<2CP^2\text{ for an infinite number of positions of }P$
$\textbf{(C) }s=2CP^2\text{ only if }P\text{ is the midpoint of }AB\text{ or an endpoint of }AB$
$\textbf{(D) }s=2CP^2\text{ always}$
$\textbf{(E) }s>2CP^2\text{ if }P\text{ is a trisection point of }AB$
2018 May Olympiad, 1
Juan makes a list of $2018$ numbers. The first is $ 1$. Then each number is obtained by adding to the previous number, one of the numbers $ 1$, $2$, $3$, $4$, $5$, $6$, $7$, $ 8$ or $9$. Knowing that none of the numbers in the list ends in $0$, what is the largest value you can have the last number on the list?
2010 Turkey Team Selection Test, 2
Show that
\[ \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) \]
for all positive real numbers $a, \: b, \: c.$
2012 Princeton University Math Competition, A2
Let $a, b, c$ be real numbers such that $a+b+c=abc$. Prove that $\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge \frac{3}{4}$.
2013 HMNT, 3
The digits $1,2,3,4, 5,6$ are randomly chosen (without replacement) to form the three-digit numbers $M = \overline{ABC}$ and $N = \overline{DEF}$. For example, we could have $M = 413$ and $N = 256$. Find the expected value of $M \cdot N$.
2017 Olympic Revenge, 3
Let $n$ a positive integer. We call a pair $(\pi ,C)$ composed by a permutation $\pi$$:$ {$1,2,...n$}$\rightarrow${$1,2,...,n$} and a binary function $C:$ {$1,2,...,n$}$\rightarrow${$0,1$} "revengeful" if it satisfies the two following conditions:
$1)$For every $i$ $\in$ {$1,2,...,n$}, there exist $j$ $\in$ $S_{i}=${$i, \pi(i),\pi(\pi(i)),...$} such that $C(j)=1$.
$2)$ If $C(k)=1$, then $k$ is one of the $v_{2}(|S_{k}|)+1$ highest elements of $S_{k}$, where $v_{2}(t)$ is the highest nonnegative integer such that $2^{v_{2}(t)}$ divides $t$, for every positive integer $t$.
Let $V$ the number of revengeful pairs and $P$ the number of partitions of $n$ with all parts powers of $2$.
Determine $\frac{V}{P}$.
2025 USA IMO Team Selection Test, 6
Prove that there exists a real number $\varepsilon>0$ such that there are infinitely many sequences of integers $0<a_1<a_2<\hdots<a_{2025}$ satisfying
\[\gcd(a_1^2+1, a_2^2+1,\hdots, a_{2025}^2+1) > a_{2025}^{1+\varepsilon}.\]
[i]Pitchayut Saengrungkongka[/i]
2009 Indonesia TST, 1
a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime?
b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?
1992 IMO Longlists, 57
For positive numbers $a, b, c$ define $A = \frac{(a + b + c)}{3}$, $G = \sqrt[3]{abc}$, $H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.$ Prove that
\[ \left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.\]
2014 Taiwan TST Round 3, 2
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.
2023 AMC 12/AHSME, 11
What is the degree measure of the acute angle formed by lines with slopes $2$ and $\tfrac{1}{3}$?
$\textbf{(A)}~30\qquad\textbf{(B)}~37.5\qquad\textbf{(C)}~45\qquad\textbf{(D)}~52.5\qquad\textbf{(E)}~60$
2006 Costa Rica - Final Round, 1
Consider the set $S=\{1,2,...,n\}$. For every $k\in S$, define $S_{k}=\{X \subseteq S, \ k \notin X, X\neq \emptyset\}$. Determine the value of the sum \[S_{k}^{*}=\sum_{\{i_{1},i_{2},...,i_{r}\}\in S_{k}}\frac{1}{i_{1}\cdot i_{2}\cdot...\cdot i_{r}}\] [hide]in fact, this problem was taken from an austrian-polish[/hide]
2019 Polish Junior MO Finals, 5.
In the every cell of the board $5\times5$ there is one of the numbers: $-1$, $0$, $1$. It is true that in every $2 \times 2$ square there are three numbers summing up to $0$. Determine the maximal sum of all numbers in a board.
2006 Princeton University Math Competition, 6
Suppose that $P(x)$ is a polynomial with the property that there exists another polynomial $Q(x)$ to satisfy $P(x)Q(x)=P(x^2)$. $P(x)$ and $Q(x)$ may have complex coefficients. If $P(x)$ is a quintic with distinct complex roots $r_1,\dots,r_5$, find all possible values of $|r_1|+\dots+|r_5|$.