Found problems: 85335
1984 Bulgaria National Olympiad, Problem 1
Solve the equation $5^x7^y+4=3^z$ in nonnegative integers.
2014 PUMaC Individual Finals A, 2
Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that
\[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]
2024 Chile TST Ibero., 5
Let $\triangle ABC$ be an acute-angled triangle. Let $P$ be the midpoint of $BC$, and $K$ the foot of the altitude from $A$ to side $BC$. Let $D$ be a point on segment $AP$ such that $\angle BDC = 90^\circ$. Let $E$ be the second point of intersection of line $BC$ with the circumcircle of $\triangle ADK$. Let $F$ be the second point of intersection of line $AE$ with the circumcircle of $\triangle ABC$. Prove that $\angle AFD = 90^\circ$.
2010 Romania National Olympiad, 1
Let $a,b,c$ be integers larger than $1$. Prove that
\[a(a-1)+b(b-1)+c(c-1)\le (a+b+c-4)(a+b+c-5)+4.\]
2017 F = ma, 23
23) A spring has a length of 1.0 meter when there is no tension on it. The spring is then stretched between two points 10 meters apart. A wave pulse travels between the two end points in the spring in a time of 1.0 seconds. The spring is now stretched between two points that are 20 meters apart. The new time it takes for a wave pulse to travel between the ends of the spring is closest to which of the following?
A) 0.5 seconds
B) 0.7 seconds
C) 1 second
D) 1.4 seconds
E) 2 seconds
2007 China National Olympiad, 3
Find a number $n \geq 9$ such that for any $n$ numbers, not necessarily distinct, $a_1,a_2, \ldots , a_n$, there exists 9 numbers $a_{i_1}, a_{i_2}, \ldots , a_{i_9}, (1 \leq i_1 < i_2 < \ldots < i_9 \leq n)$ and $b_i \in {4,7}, i =1, 2, \ldots , 9$ such that $b_1a_{i_1} + b_2a_{i_2} + \ldots + b_9a_{i_9}$ is a multiple of 9.
2006 Cuba MO, 1
Determine all monic polynomials $P(x)$ of degree $3$ with coefficients integers, which are divisible by $x-1$, when divided by $ x-5$ leave the same remainder as when divided by$ x+5$ and have a root between $2$ and $3$.
2017 Czech And Slovak Olympiad III A, 4
For each sequence of $n$ zeros and $n$ units, we assign a number that is a number sections of the same digits in it. (For example, sequence $00111001$ has $4$ such sections $00, 111,00, 1$.) For a given $n$ we sum up all the numbers assigned to each such sequence. Prove that the sum total is equal to $(n+1){2n \choose n} $
2017 Middle European Mathematical Olympiad, 2
Let $n \geq 3$ be an integer. A labelling of the $n$ vertices, the $n$ sides and the interior of a regular $n$-gon by $2n + 1$ distinct integers is called [i]memorable[/i] if the following conditions hold:
(a) Each side has a label that is the arithmetic mean of the labels of its endpoints.
(b) The interior of the $n$-gon has a label that is the arithmetic mean of the labels of all the vertices.
Determine all integers $n \geq 3$ for which there exists a memorable labelling of a regular $n$-gon consisting of $2n + 1$ consecutive integers.
1990 IberoAmerican, 4
Let $\Gamma_{1}$ be a circle. $AB$ is a diameter, $\ell$ is the tangent at $B$, and $M$ is a point on $\Gamma_{1}$ other than $A$. $\Gamma_{2}$ is a circle tangent to $\ell$, and also to $\Gamma_{1}$ at $M$.
a) Determine the point of tangency $P$ of $\ell$ and $\Gamma_{2}$ and find the locus of the center of $\Gamma_{2}$ as $M$ varies.
b) Show that there exists a circle that is always orthogonal to $\Gamma_{2}$, regardless of the position of $M$.
2023 Putnam, B3
A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called $\textit{zigzag}$ if $k=1$, or if $y_2-y_1, y_3-y_2, \ldots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1, X_2, \ldots, X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a\left(X_1, X_2, \ldots, X_n\right)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1, i_2, \ldots, i_k$ such that $X_{i_1}, X_{i_2}, \ldots X_{i_k}$ is zigzag. Find the expected value of $a\left(X_1, X_2, \ldots, X_n\right)$ for $n \geq 2$.
1998 Czech And Slovak Olympiad IIIA, 5
A circle $k$ and a point $A$ outside it are given in the plane. Prove that all trapezoids, whose non-parallel sides meet at $A$, have the same intersection of diagonals.
2019 PUMaC Combinatorics B, 1
How many ways can you arrange $3$ Alice’s, $1$ Bob, $3$ Chad’s, and $1$ David in a line if the Alice’s are all indistinguishable, the Chad’s are all indistinguishable, and Bob and David want to be adjacent to each other? (In other words, how many ways can you arrange $3$ A’s, $1$ B, $3$ C’s, and $1$ D in a row where the B and D are adjacent?)
1997 IMO, 3
Let $ x_1$, $ x_2$, $ \ldots$, $ x_n$ be real numbers satisfying the conditions:
\[ \left\{\begin{array}{cccc} |x_1 \plus{} x_2 \plus{} \cdots \plus{} x_n | & \equal{} & 1 & \ \\
|x_i| & \leq & \displaystyle \frac {n \plus{} 1}{2} & \ \textrm{ for }i \equal{} 1, 2, \ldots , n. \end{array} \right.
\]
Show that there exists a permutation $ y_1$, $ y_2$, $ \ldots$, $ y_n$ of $ x_1$, $ x_2$, $ \ldots$, $ x_n$ such that
\[ | y_1 \plus{} 2 y_2 \plus{} \cdots \plus{} n y_n | \leq \frac {n \plus{} 1}{2}.
\]
2018 India Regional Mathematical Olympiad, 5
Find all natural numbers $n$ such that $1+[\sqrt{2n}]~$ divides $2n$.
( For any real number $x$ , $[x]$ denotes the largest integer not exceeding $x$. )
2023 China Team Selection Test, P4
Given $m,n\in\mathbb N_+,$ define
$$S(m,n)=\left\{(a,b)\in\mathbb N_+^2\mid 1\leq a\leq m,1\leq b\leq n,\gcd (a,b)=1\right\}.$$
Prove that: for $\forall d,r\in\mathbb N_+,$ there exists $m,n\in\mathbb N_+,m,n\geq d$ and $\left|S(m,n)\right|\equiv r\pmod d.$
2013 USA TSTST, 8
Define a function $f: \mathbb N \to \mathbb N$ by $f(1) = 1$, $f(n+1) = f(n) + 2^{f(n)}$ for every positive integer $n$. Prove that $f(1), f(2), \dots, f(3^{2013})$ leave distinct remainders when divided by $3^{2013}$.
2014 JHMMC 7 Contest, 7
How many digits could possibly be the last digit of a perfect square?
1986 India National Olympiad, 2
Solve
\[ \left\{ \begin{array}{l}
\log_2 x\plus{}\log_4 y\plus{}\log_4 z\equal{}2 \\
\log_3 y\plus{}\log_9 z\plus{}\log_9 x\equal{}2 \\
\log_4 z\plus{}\log_{16} x\plus{}\log_{16} y\equal{}2 \\
\end{array} \right.\]
2024 Kosovo Team Selection Test, P2
Let $\omega$ be a circle and let $A$ be a point lying outside of $\omega$. The tangents from $A$ to $\omega$ touch $\omega$ at points $B$ and $C$. Let $M$ be the midpoint of $BC$ and let $D$ a point on the side $BC$ different from $M$. The circle with diameter $AD$ intersects $\omega$ at points $X$ and $Y$ and the circumcircle of $\bigtriangleup ABC$ again at $E$. Prove that $AD$, $EM$, and $XY$ are concurrent.
1998 Romania National Olympiad, 2
Let $ABCD$ be a cyclic quadrilateral. Show that $\vert \overline{AC} - \overline{BD} \vert \le \vert \overline{AB}-\overline{CD} \vert$ and determine when does equality hold.
LMT Team Rounds 2021+, B16
Bob plants two saplings. Each day, each sapling has a $1/3$ chance of instantly turning into a tree. Given that the expected number of days it takes both trees to grow is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$.
[i]Proposed by Powell Zhang[/i]
India EGMO 2021 TST, 5
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
2017 Spain Mathematical Olympiad, 6
In the triangle $ABC$, the respective mid points of the sides $BC$, $AB$ and $AC$ are $D$, $E$ and $F$. Let $M$ be the point where the internal bisector of the angle $\angle ADB$ intersects the side $AB$, and $N$ the point where the internal bisector of the angle $\angle ADC$ intersects the side $AC$. Also, let $O$ be the intersection point of $AD$ and $MN$, $P$ the intersection point of $AB$ and $FO$, and $R$ the intersection point of $AC$ and $EO$. Prove that $PR=AD$.
2018 ITAMO, 4
$4.$ Let $N$ be an integer greater than $1$.Denote by $x$ the smallest positive integer with the following property:there exists a positive integer $y$ strictly less than $x-1$ , such that $x$ divides $N+y$.Prove that x is either $p^n$ or $2p$ , where $p$ is a prime number and $n$ is a positive integer