Found problems: 85335
2018 ISI Entrance Examination, 6
Let, $a\geq b\geq c >0$ be real numbers such that for all natural number $n$, there exist triangles of side lengths $a^{n} , b^{n} ,c^{n}$.
Prove that the triangles are isosceles.
1995 Tournament Of Towns, (464) 2
Do there exist $100$ positive integers such that their sum is equal to their least common multiple?
(S Tokarev)
2002 National Olympiad First Round, 14
How many primes $p$ are there such that $39p + 1$ is a perfect square?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 3
\qquad\textbf{e)}\ \text{None of above}
$
2010 Silk Road, 1
In a convex quadrilateral it is known $ABCD$ that $\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^{\circ}$ and $AD = BC$. Prove that from the lengths $DB$, $CA$ and $DC$, you can make a right triangle.
2013 India IMO Training Camp, 2
In a triangle $ABC$, with $\widehat{A} > 90^\circ$, let $O$ and $H$ denote its circumcenter and orthocenter, respectively. Let $K$ be the reflection of $H$ with respect to $A$. Prove that $K, O$ and $C$ are collinear if and only if $\widehat{A} - \widehat{B} = 90^\circ$.
2022 Kazakhstan National Olympiad, 6
Given an infinite positive integer sequence $\{x_i\}$ such that $$x_{n+2}=x_nx_{n+1}+1$$ Prove that for any positive integer $i$ there exists a positive integer $j$ such that $x_j^j$ is divisible by $x_i^i$.
[i]Remark: Unfortunately, there was a mistake in the problem statement during the contest itself. In the last sentence, it should say "for any positive integer $i>1$ ..."[/i]
1955 Czech and Slovak Olympiad III A, 1
Consider a trapezoid $ABCD,AB\parallel CD,AB>CD.$ Let us denote intersections of lines as follows: $E=AC\cap BD, F=AD\cap BC.$ Let $GH$ be a line such that $G\in AD,H\in BC, E\in GH,GH\parallel AB.$ Moreover, denote $K,L$ midpoints of the bases $AB,CD$ respectively. Show that
(a) the points $K,L$ lie on the line $EF,$
(b) lines $AC,KH$ and $BD,KG$ are not parallel (denote $M=AC\cap KH,N=BD\cap KG$),
(c) the points $F,M,N$ are collinear.
2000 National Olympiad First Round, 5
$[BD]$ is a median of $\triangle ABC$. $m(\widehat{ABD})=90^\circ$, $|AB|=2$, and $|AC|=6$. $|BC|=?$
$ \textbf{(A)}\ 3
\qquad\textbf{(B)}\ 3\sqrt2
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 4\sqrt2
\qquad\textbf{(E)}\ 2\sqrt6
$
1976 All Soviet Union Mathematical Olympiad, 224
Can you mark the cube's vertices with the three-digit binary numbers in such a way, that the numbers at all the possible couples of neighbouring vertices differ in at least two digits?
MathLinks Contest 5th, 1.1
Find all pairs of positive integers $x, y$ such that $x^3 - y^3 = 2005(x^2 - y^2)$.
2015 Putnam, B3
Let $S$ be the set of all $2\times 2$ real matrices \[M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\] whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S.$
2022 Mexican Girls' Contest, 1
Let $ABCD$ be a quadrilateral, $E$ the midpoint of side $BC$, and $F$ the midpoint of side $AD$. Segment $AC$ intersects segment $BF$ at $M$ and segment $DE$ at $N$. If quadrilateral $MENF$ is also known to be a parallelogram, prove that $ABCD$ is also a parallelogram.
1995 All-Russian Olympiad, 6
In an acute-angled triangle ABC, points $A_2$, $B_2$, $C_2$ are the midpoints of the altitudes $AA_1$, $BB_1$, $CC_1$, respectively. Compute the sum of angles $B_2A_1C_2$, $C_2B_1A_2$ and $A_2C_1B_2$.
[i]D. Tereshin[/i]
2018 BMT Spring, 1
Bob has $3$ different fountain pens and $11$ different ink colors. How many ways can he fill his fountain pens with ink if he can only put one ink in each pen?
2019 HMNT, 8
Compute the number of ordered pairs of integers $(x,y)$ such that $x^2 + y^2 < 2019$ and
$$x^2 + min(x,y) = y^2 + max(x, y) .$$
2002 Hungary-Israel Binational, 1
Find the greatest exponent $k$ for which $2001^{k}$ divides $2000^{2001^{2002}}+2002^{2001^{2000}}$.
2012 Argentina National Olympiad, 1
Determine if there are triplets ($x,y,z)$ of real numbers such that
$$\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases}$$
If the answer is affirmative, find the minimum and maximum values of $z$ in such a triplet.
2008 Czech and Slovak Olympiad III A, 1
Find all pairs of real numbers $(x,y)$ satisfying:
\[x+y^2=y^3,\]\[y+x^2=x^3.\]
2022 Yasinsky Geometry Olympiad, 3
In an isosceles right triangle $ABC$ with a right angle $C$, point $M$ is the midpoint of leg $AC$. At the perpendicular bisector of $AC$, point $D$ was chosen such that $\angle CDM = 30^o$, and $D$ and $B$ lie on different sides of $AC$. Find the angle $\angle ABD$.
(Volodymyr Petruk)
1998 Vietnam Team Selection Test, 1
Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$) such that $| f(x) - P(x)| \leq c \cdot x^{1998}$ for all real $x$. Prove that $f$ is a real polynomial.
2023 IFYM, Sozopol, 7
The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at points $A_1$, $B_1$, and $C_1$. The line through the midpoints of segments $AB_1$ and $AC_1$ intersects the tangent at $A$ to the circumcircle of triangle $ABC$ at point $A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that points $A_2$, $B_2$, and $C_2$ lie on a line.
2003 All-Russian Olympiad, 4
Find the greatest natural number $N$ such that, for any arrangement of the numbers $1, 2, \ldots, 400$ in a chessboard $20 \times 20$, there exist two numbers in the same row or column, which differ by at least $N.$
2019 CCA Math Bonanza, L1.3
Points $P$ and $Q$ are chosen on diagonal $AC$ of square $ABCD$ such that $AB=AP=CQ=1$. What is the measure of $\angle{PBQ}$ in degrees?
[i]2019 CCA Math Bonanza Lightning Round #1.3[/i]
2021 AMC 10 Fall, 8
The largest prime factor of $16384$ is $2$, because $16384 = 2^{14}$. What is the sum of the digits of the largest prime factor of $16383$?
$\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }10\qquad\textbf{(D) }16\qquad\textbf{(E) }22$
2024 Moldova EGMO TST, 7
$ \frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+...+\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+...+\sqrt{10-\sqrt{99}}}=? $