Found problems: 85335
2022 Irish Math Olympiad, 6
6. Suppose [i]a[/i], [i]b[/i], [i]c[/i] are real numbers such that [i]a[/i] + [i]b[/i] + [i]c[/i] = 1. Prove that
\[a^3 + b^3 + c^3 + 3(1-a)(1-b)(1-c) = 1.\]
2010 National Olympiad First Round, 22
How many pairs of integers $(x,y)$ are there such that $\frac{x}{y+7}+\frac{y}{x+7}=1$?
$ \textbf{(A)}\ 18
\qquad\textbf{(B)}\ 17
\qquad\textbf{(C)}\ 15
\qquad\textbf{(D)}\ 14
\qquad\textbf{(E)}\ 11
$
1969 Miklós Schweitzer, 6
Let $ x_0$ be a fixed real number, and let $ f$ be a regular complex function in the half-plane $ \Re z>x_0$ for which there exists a nonnegative function $ F \in L_1(- \infty, +\infty)$ satisfying $ |f(\alpha+i\beta)| \leq F(\beta)$ whenever $ \alpha > x_0$ , $ -\infty <\beta < +\infty$. Prove that \[ \int_{\alpha-i \infty} ^{\alpha+i \infty} f(z)dz=0.\]
[i]L. Czach[/i]
1953 AMC 12/AHSME, 5
If $ \log_6 x\equal{}2.5$, the value of $ x$ is:
$ \textbf{(A)}\ 90 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 36\sqrt{6} \qquad\textbf{(D)}\ 0.5 \qquad\textbf{(E)}\ \text{none of these}$
2007 Tournament Of Towns, 4
There three piles of pebbles, containing 5, 49, and 51 pebbles respectively. It is allowed to combine any two piles into a new one or to split any pile consisting of even number of pebbles into two equal piles. Is it possible to have 105 piles with one pebble in each in the end?
[i](3 points)[/i]
2016 IFYM, Sozopol, 4
Prove that for each $n\geq 3$ the equation: $x^n+y^n+z^n+u^n=v^{n-1}$ has infinitely many solutions in natural numbers.
2015 PAMO, Problem 3
Let $a_1,a_2,...,a_{11}$ be integers. Prove that there are numbers $b_1,b_2,...,b_{11}$, each $b_i$ equal $-1,0$ or $1$, but not all being $0$, such that the number
$$N=a_1b_1+a_2b_2+...+a_{11}b_{11}$$
is divisible by $2015$.
2012 IMO Shortlist, N4
An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers.
[b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$.
[b]b)[/b] Decide whether $a=2$ is friendly.
1998 Mediterranean Mathematics Olympiad, 3
In a triangle $ABC$, $I$ is the incenter and $D,E, F$ are the points of tangency of the incircle with $BC,CA,AB$, respectively. The bisector of angle $BIC$ meets $BC$ at $M$, and the line $AM$ intersects $EF$ at $P$. Prove that $DP$ bisects the angle $FDE$.
2014 Math Prize For Girls Problems, 13
Deepali has a bag containing 10 red marbles and 10 blue marbles (and nothing else). She removes a random marble from the bag. She keeps doing so until all of the marbles remaining in the bag have the same color. Compute the probability that Deepali ends with exactly 3 marbles remaining in the bag.
2019 Pan-African, 4
The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.
1990 All Soviet Union Mathematical Olympiad, 533
A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?
2005 Today's Calculation Of Integral, 28
Evaluate
\[\int_0^{\frac{\pi}{4}} \frac{x\cos 5x}{\cos x}dx\]
1975 All Soviet Union Mathematical Olympiad, 213
Three flies are crawling along the perimeter of the triangle $ABC$ in such a way, that the centre of their masses is a constant point. One of the flies has already passed along all the perimeter. Prove that the centre of the flies' masses coincides with the centre of masses of the triangle $ABC$ . (The centre of masses for the triangle is the point of medians intersection.
1999 AMC 12/AHSME, 1
$ 1 \minus{} 2 \plus{} 3 \minus{} 4 \plus{} \cdots \minus{} 98 \plus{} 99 \equal{}$
$ \textbf{(A)}\minus{}\! 50 \qquad \textbf{(B)}\minus{}\! 49 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 49 \qquad \textbf{(E)}\ 50$
2019 USA IMO Team Selection Test, 1
Let $ABC$ be a triangle and let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $X$ be a point such that $\overline{AX}$ is tangent to the circumcircle of triangle $ABC$. Denote by $\omega_B$ the circle through $M$ and $B$ tangent to $\overline{MX}$, and by $\omega_C$ the circle through $N$ and $C$ tangent to $\overline{NX}$. Show that $\omega_B$ and $\omega_C$ intersect on line $BC$.
[i]Merlijn Staps[/i]
2005 China Team Selection Test, 3
Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define
\[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \]
We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if $d(a,b) \geq 2^{n-1}$ holds for any two distinct elements $a$ and $b$ of $A$. How many elements can the $\textsl{Good Subset}$ of $S_n$ at most have?
2016 Belarus Team Selection Test, 2
A point $A_1$ is marked inside an acute non-isosceles triangle $ABC$ such that $\angle A_1AB = \angle A_1BC$ and $\angle A_1AC=\angle A_1CB$. Points $B_1$ and $C_1$ are defined same way. Let $G$ be the gravity center if the triangle $ABC$.
Prove that the points $A_1,B_1,C_1,G$ are concyclic.
2025 All-Russian Olympiad, 9.3
Find all natural numbers \(n\) for which there exists an even natural number \(a\) such that the number
\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]
is a perfect square.
2000 District Olympiad (Hunedoara), 4
Let be a circle centeted at $ O, $ and $ A,B,C, $ points situated on this circle. Show that if
$$ \left|\overrightarrow{OA} +\overrightarrow{OB}\right| = \left|\overrightarrow{OB} +\overrightarrow{OC}\right| = \left|\overrightarrow{OC} +\overrightarrow{OA}\right| , $$
then $ A=B=C, $ or $ ABC $ is an equilateral triangle.
V Soros Olympiad 1998 - 99 (Russia), 11.2
Five edges of a triangular pyramid are equal to $1$. Find the sixth edge if it is known that the radius of the ball circumscribed about this pyramid is equal to $1$.
1988 AMC 8, 3
$ \frac{1}{10}+\frac{2}{20}+\frac{3}{30}= $
$ \text{(A)}\ .1\qquad\text{(B)}\ .123\qquad\text{(C)}\ .2\qquad\text{(D)}\ .3\qquad\text{(E)}\ .6 $
2007 Iran Team Selection Test, 1
Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$, such that for each $i\neq j, (a_{i},a_{j})=1$, and for each $n$, the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$?
[i]By Omid Hatami[/i]
2016 International Zhautykov Olympiad, 3
There are $60$ towns in $Graphland$ every two countries of which are connected by only a directed way. Prove that we can color four towns to red and four towns to green such that every way between green and red towns are directed from red to green
2018 Bosnia And Herzegovina - Regional Olympiad, 3
Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime