Found problems: 85335
2005 Junior Balkan Team Selection Tests - Romania, 12
Find all positive integers $n$ and $p$ if $p$ is prime and \[ n^8 - p^5 = n^2+p^2 . \]
[i]Adrian Stoica[/i]
2013 239 Open Mathematical Olympiad, 8
Prove that if you choose $10^{100}$ points on a circle and arrange numbers from $1$ to $10^{100}$ on them in some order, then you can choose $100$ pairwise disjoint chords with ends at the selected points such that the sums of the numbers at the ends of all of them are equal to each other.
MathLinks Contest 5th, 4.2
Given is a unit cube in space. Find the maximal integer $n$ such that there are $n$ points, satisfying the following conditions:
(a) All points lie on the surface of the cube;
(b) No face contains all these points;
(c) The $n$ points are the vertices of a polygon.
2010 JBMO Shortlist, 3
Consider a triangle ${ABC}$ and let ${M}$ be the midpoint of the side ${BC.}$ Suppose ${\angle MAC=\angle ABC}$ and ${\angle BAM=105^{\circ}.}$ Find the measure of ${\angle ABC}$.
2015 Saint Petersburg Mathematical Olympiad, 3
There are weights with mass $1,3,5,....,2i+1,...$ Let $A(n)$ -is number of different sets with total mass equal $n$( For example $A(9)=2$, because we have two sets $9=9=1+3+5$). Prove that $A(n) \leq A(n+1)$ for $n>1$
1991 Bundeswettbewerb Mathematik, 1
Determine all solutions of the equation $4^x + 4^y + 4^z = u^2$ for integers $x,y,z$ and $u$.
STEMS 2021 CS Cat B, Q2
Given two forests $A$ and $B$ with \(V(A) = V(B)\), that is the graphs are over same vertex set. Suppose $A$ has [b]strictly more[/b] edges than $B$. Prove that there exists an edge of $A$ which if included in the edge set of $B$, then $B$ will still remain a forest. Graphs are undirected
2015 India IMO Training Camp, 2
Let $A$ be a finite set of pairs of real numbers such that for any pairs $(a,b)$ in $A$ we have $a>0$. Let $X_0=(x_0, y_0)$ be a pair of real numbers(not necessarily from $A$). We define $X_{j+1}=(x_{j+1}, y_{j+1})$ for all $j\ge 0$ as follows: for all $(a,b)\in A$, if $ax_j+by_j>0$ we let $X_{j+1}=X_j$; otherwise we choose a pair $(a,b)$ in $A$ for which $ax_j+by_j\le 0$ and set $X_{j+1}=(x_j+a, y_j+b)$. Show that there exists an integer $N\ge 0$ such that $X_{N+1}=X_N$.
2019 Polish MO Finals, 1
Let $ABC$ be an acute triangle. Points $X$ and $Y$ lie on the segments $AB$ and $AC$, respectively, such that $AX=AY$ and the segment $XY$ passes through the orthocenter of the triangle $ABC$. Lines tangent to the circumcircle of the triangle $AXY$ at points $X$ and $Y$ intersect at point $P$. Prove that points $A, B, C, P$ are concyclic.
2014 Postal Coaching, 4
Denote by $F_n$ the $n^{\text{th}}$ Fibonacci number $(F_1=F_2=1)$.Prove that if $a,b,c$ are positive integers such that $a| F_b,b|F_c,c|F_a$,then either $5$ divides each of $a,b,c$ or $12$ divides each of $a,b,c$.
1988 Polish MO Finals, 1
The real numbers $x_1, x_2, ... , x_n$ belong to the interval $(0,1)$ and satisfy $x_1 + x_2 + ... + x_n = m + r$, where $m$ is an integer and $r \in [0,1)$. Show that $x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2$.
1967 All Soviet Union Mathematical Olympiad, 093
Given natural number $k$ with a property "if $n$ is divisible by $k$, than the number, obtained from $n$ by reversing the order of its digits is also divisible by $k$". Prove that the $k$ is a divisor of $99$.
2014 PUMaC Number Theory B, 2
What is the last digit of ${17^{17^{17^{17}}}}$?
2004 Singapore Team Selection Test, 2
Let $0 < a, b, c < 1$ with $ab + bc + ca = 1$. Prove that
\[\frac{a}{1-a^2} + \frac{b}{1-b^2} + \frac{c}{1-c^2} \geq \frac {3 \sqrt{3}}{2}.\]
Determine when equality holds.
2023 USA EGMO Team Selection Test, 5
Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \]
[i]Carl Schildkraut[/i]
1966 IMO, 6
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
2019 Jozsef Wildt International Math Competition, W. 61
If $a$, $b$, $c \in \mathbb{R}$ then$$\sum \limits_{cyc} \sqrt{(c+a)^2b^2+c^2a^2}+\sqrt{5}\left |\sum \limits_{cyc} \sqrt{ab}\right |\geq \sum \limits_{cyc}\sqrt{(ab+2bc+ca)^2+(b+c)^2a^2}$$
CNCM Online Round 1, 6
In triangle $\triangle ABC$ with $BC = 1$, the internal angle bisector of $\angle A$ intersects $BC$ at $D$. $M$ is taken to be the midpoint of $BC$. Point $E$ is chosen on the boundary of $\triangle ABC$ such that $ME$ bisects its perimeter. The circumcircle $\omega$ of $\triangle DEC$ is taken, and the second intersection of $AD$ and $\omega$ is $K$, as well as the second intersection of $ME$ and $\omega$ being $L$. If $B$ lies on line $KL$ and $ED$ is parallel to $AB$, then the perimeter of $\triangle ABC$ can be written as a real number $S$. Compute $\lfloor 1000S\rfloor$.
Proposed by Albert Wang (awang11)
2017-IMOC, A1
Prove that for all $a,b>0$ with $a+b=2$, we have
$$\left(a^n+1\right)\left(b^n+1\right)\ge4$$
for all $n\in\mathbb N_{\ge2}$.
2023 Regional Olympiad of Mexico West, 6
There are $2023$ guinea pigs placed in a circle, from which everyone except one of them, call it $M$, has a mirror that points towards one of the $2022$ other guinea pigs. $M$ has a lantern that will shoot a light beam towards one of the guinea pigs with a mirror and will reflect to the guinea pig that the mirror is pointing and will keep reflecting with every mirror it reaches. Isaías will re-direct some of the mirrors to point to some other of the $2023$ guinea pigs. In the worst case scenario, what is the least number of mirrors that need to be re-directed, such that the light beam hits $M$ no matter the starting point of the light beam?
1989 Mexico National Olympiad, 4
Find the smallest possible natural number $n = \overline{a_m ...a_2a_1a_0} $ (in decimal system) such that the number $r = \overline{a_1a_0a_m ..._20} $ equals $2n$.
2010 India IMO Training Camp, 2
Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.
2007 Stanford Mathematics Tournament, 5
How many five-letter "words" can you spell using the letters $S$, $I$, and $T$, if a "word" is defines as any sequence of letters that does not contain three consecutive consonants?
2008 National Olympiad First Round, 29
$[AB]$ and $[CD]$ are not parallel in the convex quadrilateral $ABCD$. Let $E$ and $F$ be the midpoints of $[AD]$ and $[BC]$, respectively. If $|CD|=12$, $|AB|=22$, and $|EF|=x$, what is the sum of integer values of $x$?
$
\textbf{(A)}\ 110
\qquad\textbf{(B)}\ 114
\qquad\textbf{(C)}\ 118
\qquad\textbf{(D)}\ 121
\qquad\textbf{(E)}\ \text{None of the above}
$
V Soros Olympiad 1998 - 99 (Russia), 10.1
Find some natural number $a$ such that $2a$ is a perfect square, $3a$ is a perfect cube, $5a$ is the fifth power of some natural number.