Found problems: 85335
2024 Belarusian National Olympiad, 11.3
In a triangle $ABC$ point $I$ is the incenter, $I_A$ - excenter, $W$ - midpoint of the arc $BAC$ of circumcircle $\omega$ of $ABC$. Point $H$ is the projection of $I_A$ on $IW$. The tangent line to the circumcircle $BIC$ in point $I$ intersects $\omega$ in $E, F$.
Prove that the perpendicular bisector to $AI$ is tangent to the circumcircle $EFH$
[i]M. Zorka[/i]
1992 National High School Mathematics League, 8
Then number of solutions to $\cos7x=\cos5x$ in $[0,\pi]$ is________.
1968 Dutch Mathematical Olympiad, 2
It holds: $N,a > 0$. Prove that $\frac12 \left(\frac{N}{a}+a \right) \ge \sqrt{N}$, and if $N \ge 1$ and $a = [\sqrt{N}]$.
Prove that if $a \ne \sqrt{N}: \frac12 \left(\frac{N}{a}+a \right)$ is a better approximation for $\sqrt{N}$ than $a$.
2013 NIMO Problems, 6
Let $ABC$ and $DEF$ be two triangles, such that $AB=DE=20$, $BC=EF=13$, and $\angle A = \angle D$. If $AC-DF=10$, determine the area of $\triangle ABC$.
[i]Proposed by Lewis Chen[/i]
2007 AMC 10, 6
At Euclid High School, the number of students taking the AMC10 was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, 78 in 2006, and is 85 in 2007. Between what two consecutive years was there the largest percentage increase?
$ \textbf{(A)}\ 2002\ \text{and}\ 2003 \qquad \textbf{(B)}\ 2003\ \text{and}\ 2004 \qquad \textbf{(C)}\ 2004\ \text{and}\ 2005 \qquad \textbf{(D)}\ 2005\ \text{and}\ 2006 \qquad \textbf{(E)}\ 2006\ \text{and}\ 2007$
2022 Kyiv City MO Round 1, Problem 3
You are given $n$ not necessarily distinct real numbers $a_1, a_2, \ldots, a_n$. Let's consider all $2^n-1$ ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to $1$?
For example, if $a = [-1, 2, 2]$, then we got $3$ once, $4$ once, $2$ twice, $-1$ once, $1$ twice, so the total number of ones here is $2$.
[i](Proposed by Anton Trygub)[/i]
2007 All-Russian Olympiad Regional Round, 9.6
Given a triangle. A variable poin $ D$ is chosen on side $ BC$. Points $ K$ and $ L$ are the incenters of triangles $ ABD$ and $ ACD$, respectively. Prove that the second intersection point of the circumcircles of triangles $ BKD$ and $ CLD$ moves along on a fixed circle (while $ D$ moves along segment $ BC$).
2024 MMATHS, 1
Let $ab^2=126, bc^2=14, cd^2=128, da^2=12.$ Find $\tfrac{bd}{ac}.$
2013 Romanian Masters In Mathematics, 3
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.
2010 LMT, 15
Let $x$ and $y$ be real numbers such that $x^2+y^2-22x-16y+113=0.$ Determine the smallest possible value of $x.$
2015 Rioplatense Mathematical Olympiad, Level 3, 5
For a positive integer number $n$ we denote $d(n)$ as the greatest common divisor of the binomial coefficients $\dbinom{n+1}{n} , \dbinom{n+2}{n} ,..., \dbinom{2n}{n}$.
Find all possible values of $d(n)$
1971 Vietnam National Olympiad, 1
Consider positive integers $m <n,p < q$ such that $(m, n) = 1, (p, q) = 1$ and satisfy the condition that if $\frac{m}{n}= tan\alpha$ and $\frac{p}{q} = tan\beta$, then $\alpha + \beta = 45^o$.
i) Given $m, n$, find $p, q$.
ii) Given $n, q$, find $m, p$.
ii) Given $m, q$, find $n, p$.
2005 Georgia Team Selection Test, 9
Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ \minus{} 1$. Prove that if \[ a_{0} \plus{} 2a_{1} \plus{} 2^{2}a_{2} \plus{} \cdots \plus{} 2^{n}a_{n} \equal{} 0,\] then $ a_{0} \plus{} a_{1} \plus{} \cdots \plus{} a_{n} > 0$.
2024 China Team Selection Test, 7
For coprime positive integers $a,b$,denote $(a^{-1}\bmod{b})$ by the only integer $0\leq m<b$ such that $am\equiv 1\pmod{b}$
(1)Prove that for pairwise coprime integers $a,b,c$, $1<a<b<c$,we have\[(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})>\sqrt a.\]
(2)Prove that for any positive integer $M$,there exists pairwise coprime integers $a,b,c$, $M<a<b<c$ such that
\[(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})< 100\sqrt a.\]
2013 Romania National Olympiad, 1
The right prism $ABCA'B'C'$, with $AB = AC = BC = a$, has the property that there exists an unique point $M \in (BB')$ so that $AM \perp MC'$. Find the measure of the angle of the straight line $AM$ and the plane $(ACC')$ .
2010 Victor Vâlcovici, 2
Let $ f:[2,\infty )\rightarrow\mathbb{R} $ be a differentiable function satisfying $ f(2)=0 $ and
$$ \frac{df}{dx}=\frac{2}{x^2+f^4{x}} , $$
for any $ x\in [2,\infty ) . $ Show that there exists $ \lim_{x\to\infty } f(x) $ and is at most $ \ln 3. $
[i]Gabriel Daniilescu[/i]
PEN A Problems, 56
Let $a, b$, and $c$ be integers such that $a+b+c$ divides $a^2 +b^2 +c^2$. Prove that there are infinitely many positive integers $n$ such that $a+b+c$ divides $a^n +b^n +c^n$.
2014 Contests, 1
Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$.
[i]Proposed by Evan Chen[/i]
2023 BMT, 14
Right triangle $\vartriangle ABC$ with $\angle A = 30^o$ and $\angle B = 90^o$ is inscribed in a circle $\omega_1$ with radius $4$. Circle $\omega_2$ is drawn to be the largest circle outside of $\vartriangle ABC$ that is tangent to both $\overline{BC}$ and $\omega_1$, and circles $\omega_3$ and $\omega_4$ are drawn this same way for sides $\overline{AC}$ and $\overline{AB}$, respectively. Suppose that the intersection points of these smaller circles with the bigger circle are noted as points $D$, $E$, and $F$. Compute the area of triangle $\vartriangle DEF$.
1956 Putnam, B7
The polynomials $P(z)$ and $Q(z)$ with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials
$$P(z)+1 \;\; \text{and} \;\; Q(z)+1.$$
Prove that $P(z)=Q(z).$
Russian TST 2015, P1
Let $n>4$ be a natural number. Prove that \[\sum_{k=2}^n\sqrt[k]{\frac{k}{k-1}}<n.\]
1950 AMC 12/AHSME, 14
For the simultaneous equations
\[ 2x\minus{}3y\equal{}8\]
\[ 6y\minus{}4x\equal{}9\]
$\textbf{(A)}\ x=4,y=0 \qquad
\textbf{(B)}\ x=0,y=\dfrac{3}{2}\qquad
\textbf{(C)}\ x=0,y=0 \qquad\\
\textbf{(D)}\ \text{There is no solution} \qquad
\textbf{(E)}\ \text{There are an infinite number of solutions}$
1993 China Team Selection Test, 1
For all primes $p \geq 3,$ define $F(p) = \sum^{\frac{p-1}{2}}_{k=1}k^{120}$ and $f(p) = \frac{1}{2} - \left\{ \frac{F(p)}{p} \right\}$, where $\{x\} = x - [x],$ find the value of $f(p).$
1972 IMO Longlists, 5
Given a pyramid whose base is an $n$-gon inscribable in a circle, let $H$ be the projection of the top vertex of the pyramid to its base. Prove that the projections of $H$ to the lateral edges of the pyramid lie on a circle.
Kvant 2021, M2669
Prove that for any natural number $n{}$ the numbers $1,2,\ldots,n$ can be divided into several groups so that the sum of the numbers in each group is equal to a power of three.
[i]Proposed by V. Novikov[/i]