Found problems: 85335
1963 Dutch Mathematical Olympiad, 3
Twenty numbers $a_1,a_2,..,a_{20}$ satisfy:
$$a_k \ge 7k \,\,\,\,\, for \,\,\,\,\, k = 1,2,..., 20$$
$$a_1+a_2+...+a_{20}=1518$$
Prove that among the numbers $k = 1,2,... ,20$ there are no more than seventeen, for which $a_k \ge 20k -2k^2$.
2007 May Olympiad, 1
Determine the largest natural number that has all its digits different and is a multiple of $5$, $8$ and $11$.
1959 Czech and Slovak Olympiad III A, 1
Construct a triangle $ABC$ with the right angle at vertex $C$ given lengths of its medians $m_a$, $m_b$. Discuss conditions of solvability.
2021 Taiwan TST Round 2, A
Prove that if non-zero complex numbers $\alpha_1,\alpha_2,\alpha_3$ are distinct and noncollinear on the plane, and satisfy $\alpha_1+\alpha_2+\alpha_3=0$, then there holds
\[\sum_{i=1}^{3}\left(\frac{|\alpha_{i+1}-\alpha_{i+2}|}{\sqrt{|\alpha_i|}}\left(\frac{1}{\sqrt{|\alpha_{i+1}|}}+\frac{1}{\sqrt{|\alpha_{i+2}|}}-\frac{2}{\sqrt{|\alpha_{i}|}}\right)\right)\leq 0......(*)\]
where $\alpha_4=\alpha_1, \alpha_5=\alpha_2$. Verify further the sufficient and necessary condition for the equality holding in $(*)$.
2018 India PRMO, 20
Determine the sum of all possible positive integers $n, $ the product of whose digits equals $n^2 -15n -27$.
ICMC 6, 2
Show that if the distance between opposite edges of a tetrahedron is at least $1$, then its volume is at least $\frac{1}{3}$.
[i]Proposed by Simeon Kiflie[/i]
2016 Gulf Math Olympiad, 2
Let $x$ be a real number that satisfies $x^1 + x^{-1} = 3$
Prove that $x^n + x^{-n}$ is an positive integer , then prove that the positive integer $x^{3^{1437}}+x^{3^{-1437}}$ is divisible by at least $1439 \times 2^{1437}$ positive integers
2025 Bulgarian Winter Tournament, 12.4
Prove that a graph containing a copy of each possible tree on $n$ vertices as a subgraph has at least $n(\ln n - 2)$ edges.
2008 Bulgarian Autumn Math Competition, Problem 8.1
Solve the equation $|x-m|+|x+m|=x$ depending on the value of the parameter $m$.
2010 National Chemistry Olympiad, 12
Commercial vinegar is a $5.00\%$ by mass aqueous solution of acetic acid, $\ce{CH3CO2H}$ $(M=60.0)$. What is the molarity of acetic acid in vinegar? [density of vinegar = 1.00g/mL)
$ \textbf{(A)}\hspace{.05in}0.833 M\qquad\textbf{(B)}\hspace{.05in}1.00 M\qquad\textbf{(C)}\hspace{.05in}1.20 M\qquad\textbf{(D)}\hspace{.05in}3.00 M\qquad$
2009 Kyrgyzstan National Olympiad, 9
For any positive $ a_1 ,a_2 ,...,a_n$ prove that $ \frac {{a_1 }} {{a_2 \plus{} a_3 }} \plus{} \frac {{a_2 }} {{a_3 \plus{} a_4 }} \plus{} ... \plus{} \frac {{a_n }} {{a_1 \plus{} a_2 }} > \frac {n} {4}$ holds.
Kvant 2021, M2681
Find all pairs of distinct rational numbers $(a,b)$ such that $a^a=b^b$.
[i]Proposed by I. Dorofeev[/i]
2007 Portugal MO, 5
Rua do Antonio has $100$ houses numbered from $1$ to $100$. Any house numbered with the difference between the numbers of two houses of the same color is a different color. Prove that on Rua do Antonio there are houses of at least five different colors.
1950 AMC 12/AHSME, 17
The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:
\[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline
x&0&1&2&3&4\\\hline
y&100&90&70&40&0\\\hline
\end{tabular}\]
$\textbf{(A)}\ y=100-10x \qquad
\textbf{(B)}\ y=100-5x^2 \qquad
\textbf{(C)}\ y=100-5x-5x^2 \qquad\\
\textbf{(D)}\ y=20-x-x^2 \qquad
\textbf{(E)}\ \text{None of these}$
2019 Canada National Olympiad, 3
You have a $2m$ by $2n$ grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place $mn$ counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
2012 JBMO ShortLists, 2
Let $a$ , $b$ , $c$ be positive real numbers such that $abc=1$ . Show that :
\[\frac{1}{a^3+bc}+\frac{1}{b^3+ca}+\frac{1}{c^3+ab} \leq \frac{ \left (ab+bc+ca \right )^2 }{6}\]
2000 Belarus Team Selection Test, 5.1
Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$).
If $BC = a, AM = m_a, AL = l_a$, prove the inequalities:
(a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$
(b) $2l_a \le a\cot \frac{a}{2} $.
2009 QEDMO 6th, 1
Solve $y^5 - x^2 = 4$ in integers numbers $x,y$.
2006 German National Olympiad, 3
For which positive integer n can you color the numbers 1,2...2n with n colors, such that every color is used twice and the numbers 1,2,3...n occur as difference of two numbers of the same color exatly once.
2023 Ukraine National Mathematical Olympiad, 10.8
Consider a complete graph on $4046$ nodes, whose edges are colored in some colors. Let's call this graph $k$-good if we can split all its nodes into $2023$ pairs so that there are exactly $k$ distinct colors among the colors of $2023$ edges that connect the nodes from the same pairs. Is it possible that the graph is $999$-good and $1001$-good but not $1000$-good?
[i]Proposed by Anton Trygub[/i]
2008 Harvard-MIT Mathematics Tournament, 4
In a triangle $ ABC$, take point $ D$ on $ BC$ such that $ DB \equal{} 14, DA \equal{} 13, DC \equal{} 4$, and the circumcircle of $ ADB$ is congruent to the circumcircle of $ ADC$. What is the area of triangle $ ABC$?
2009 ISI B.Stat Entrance Exam, 9
Consider $6$ points located at $P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)$. Let $R$ be the region consisting of [i]all[/i] points in the plane whose distance from $P_0$ is smaller than that from any other $P_i$, $i=1,2,3,4,5$. Find the perimeter of the region $R$.
2016 Saudi Arabia BMO TST, 3
Show that there are infinitely many positive integers $n$ such that $n$ has at least two prime divisors and $20^n + 16^n$ is divisible by $n^2$.
1952 Miklós Schweitzer, 10
Let $ n$ be a positive integer. Prove that, for $ 0<x<\frac{\pi}{n\plus{}1}$,
$ \sin{x}\minus{}\frac{\sin{2x}}{2}\plus{}\cdots\plus{}(\minus{}1)^{n\plus{}1}\frac{\sin{nx}}{n}\minus{}\frac{x}{2}$
is positive if $ n$ is odd and negative if $ n$ is even.
1988 Czech And Slovak Olympiad IIIA, 5
Find all numbers $a \in (-2, 2)$ for which the polynomial $x^{154}-ax^{77}+1$ is a multiple of the polynomial $x^{14}-ax^{7}+1$.