Found problems: 85335
2024 Euler Olympiad, Round 1, 8
Let $P$ be a point inside a square $ABCD,$ such that $\angle BPC = 135^\circ $ and the area of triangle $ADP$ is twice as much as the area of triangle $PCD.$ Find $\frac {AP}{DP}.$
[i]Proposed by Andria Gvaramia, Georgia [/i]
1998 IMO Shortlist, 7
Let $ABC$ be a triangle such that $\angle ACB=2\angle ABC$. Let $D$ be the point on the side $BC$ such that $CD=2BD$. The segment $AD$ is extended to $E$ so that $AD=DE$. Prove that \[ \angle ECB+180^{\circ }=2\angle EBC. \]
2023 Czech and Slovak Olympiad III A., 4
Let $(a_n)_{n = 0}^{\infty} $ be a sequence of positive integers such that for every $n \geq 0$ it is true that
$$a_{n+2} = a_0 a_1 + a_1 a_2 + ... + a_n a_{n+1} - 1 $$
a) Prove that there exist a prime number which divides infinitely many $a_n$
b) Prove that there exist infinitely many such prime numbers
2009 Benelux, 2
Let $n$ be a positive integer and let $k$ be an odd positive integer. Moreover, let $a,b$ and $c$ be integers (not necessarily positive) satisfying the equations
\[a^n+kb=b^n+kc=c^n+ka \]
Prove that $a=b=c$.
Denmark (Mohr) - geometry, 1995.3
From the vertex $C$ in triangle $ABC$, draw a straight line that bisects the median from $A$. In what ratio does this line divide the segment $AB$?
[img]https://1.bp.blogspot.com/-SxWIQ12DIvs/XzcJv5xoV0I/AAAAAAAAMY4/Ezfe8bd7W-Mfp2Qi4qE_gppbh9Fzvb4XwCLcBGAsYHQ/s0/1995%2BMohr%2Bp3.png[/img]
2023 Dutch BxMO TST, 5
Find all pairs of prime numbers $(p,q)$ for which
\[2^p = 2^{q-2} + q!.\]
2003 Tournament Of Towns, 1
There is $3 \times 4 \times 5$ - box with its faces divided into $1 \times 1$ - squares. Is it possible to place numbers in these squares so that the sum of numbers in every stripe of squares (one square wide) circling the box, equals $120$?
2013 ELMO Shortlist, 6
Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$.
[i]Proposed by Evan Chen[/i]
2009 Saint Petersburg Mathematical Olympiad, 6
Some cities in country are connected by road, and from every city goes $\geq 2008$ roads. Every road is colored in one of two colors. Prove, that exists cycle without self-intersections ,where $\geq 504$ roads and all roads are same color.
2021 MIG, 25
Thelma writes a list of four digits consisting of $1$, $3$, $5$, and $7$, and each digit can appear one time, multiples time, or not at all. The list has a unique [i]mode[/i], or the number that appears the most. Thelma removes two numbers of that mode from the list; her list now has no unique mode! How many lists are possible? Suppose that all possible lists are unordered.
$\textbf{(A) }18\qquad\textbf{(B) }24\qquad\textbf{(C) }30\qquad\textbf{(D) }36\qquad\textbf{(E) }48$
I Soros Olympiad 1994-95 (Rus + Ukr), 9.7
Given an acute triangle $ABC$, in which $\angle BAC <30^o$. On sides $AC$ and $AB$ are taken respectively points $D$ and $E$ such that $\angle BDC=\angle BDE = \angle ADE = 60^o$. Prove that the centers of the circles. inscribed in triangles $ADE$, $BDE$ and $BCD$ do not lie on the same line.
1972 Bundeswettbewerb Mathematik, 3
The arithmetic mean of two different positive integers $x,y$ is a two digit integer. If one interchanges the digits, the geometric mean of these numbers is archieved.
a) Find $x,y$.
b) Show that a)'s solution is unique up to permutation if we work in base $g=10$, but that there is no solution in base $g=12$.
c) Give more numbers $g$ such that a) can be solved; give more of them such that a) can't be solved, too.
2002 AMC 10, 5
Circles of radius $ 2$ and $ 3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.
[asy]unitsize(3mm);
defaultpen(linewidth(0.7)+fontsize(8));
filldraw(Circle((0,0),5),grey,black);
filldraw(Circle((-2,0),3),white,black);
filldraw(Circle((3,0),2),white,black);
dot((-2,0));
dot((3,0));
draw((-2,0)--(1,0));
draw((3,0)--(5,0));
label("$3$",(-0.5,0),N);
label("$2$",(4,0),N);[/asy]
$ \textbf{(A)}\ 3\pi \qquad
\textbf{(B)}\ 4\pi \qquad
\textbf{(C)}\ 6\pi \qquad
\textbf{(D)}\ 9\pi \qquad
\textbf{(E)}\ 12\pi$
2018 Harvard-MIT Mathematics Tournament, 7
Ben "One Hunna Dolla" Franklin is flying a kite $KITE$ such that $IE$ is the perpendicular bisector of $KT$. Let $IE$ meet $KT$ at $R$. The midpoints of $KI,IT,TE,EK$ are $A,N,M,D,$ respectively. Given that $[MAKE]=18,IT=10,[RAIN]=4,$ find $[DIME]$.
Note: $[X]$ denotes the area of the figure $X$.
2011 NIMO Problems, 5
We have eight light bulbs, placed on the eight lattice points (points with integer coordinates) in space that are $\sqrt{3}$ units away from the origin. Each light bulb can either be turned on or off. These lightbulbs are unstable, however. If two light bulbs that are at most 2 units apart are both on simultaneously, they both explode. Given that no explosions take place, how many possible configurations of on/off light bulbs exist?
[i]Proposed by Lewis Chen[/i]
2004 VTRMC, Problem 4
A $9\times9$ chess board has two squares from opposite corners and its central square removed. Is it possible to cover the remaining squares using dominoes, where each domino covers two adjacent squares? Justify your answer.
1935 Moscow Mathematical Olympiad, 016
How many real solutions does the following system have ?$\begin{cases} x+y=2 \\
xy - z^2 = 1 \end{cases}$
1950 AMC 12/AHSME, 24
The equation $ x\plus{}\sqrt{x\minus{}2}\equal{}4$ has:
$\textbf{(A)}\ \text{2 real roots} \qquad
\textbf{(B)}\ \text{1 real and 1 imaginary root} \qquad
\textbf{(C)}\ \text{2 imaginary roots} \qquad
\textbf{(D)}\ \text{No roots} \qquad
\textbf{(E)}\ \text{1 real root}$
2010 AMC 8, 10
$6$ pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered with pepperoni?
$ \textbf{(A)}\ \frac 12 \qquad\textbf{(B)}\ \frac 23 \qquad\textbf{(C)}\ \frac 34 \qquad\textbf{(D)}\ \frac 56 \qquad\textbf{(E)}\ \frac 78 $
2024 Princeton University Math Competition, A8
Let $a,b,c$ be pairwise coprime integers such a that $\tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}=\tfrac{N}{a+b+c}$ for some positive integer $N.$ What is the sum of all possible values of $N.$
2017 AMC 12/AHSME, 8
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
$\textbf{(A)} \text{ } \frac{\sqrt{3}-1}{2} \qquad \textbf{(B)} \text{ } \frac{1}{2} \qquad \textbf{(C)} \text{ } \frac{\sqrt{5}-1}{2} \qquad \textbf{(D)} \text{ } \frac{\sqrt{2}}{2} \qquad \textbf{(E)} \text{ } \frac{\sqrt{6}-1}{2}$
1980 IMO, 13
Prove that the integer $145^{n} + 3114\cdot 138^{n}$ is divisible by $1981$ if $n=1981$, and that it is not divisible by $1981$ if $n=1980$.
2014 China Team Selection Test, 6
Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote
$S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$,
$S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$,
Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$
2006 AMC 10, 10
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
$ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$
2025 Taiwan Mathematics Olympiad, 4
Find all positive integers $n$ satisfying the following: there exists a way to fill in $1, \cdots, n^2$ into a $n \times n$ grid so that each block has exactly one number, each number appears exactly once, and:
1. For all positive integers $1 \leq i < n^2$, $i$ and $i + 1$ are neighbors (two numbers neighbor each other if and only if their blocks share a common edge.)
2. Any two numbers among $1^2, \cdots, n^2$ are not in the same row or the same column.