Found problems: 85335
2023 HMNT, 5
Let $ABCDE$ be a convex pentagon such that
\begin{align*}
&AB+BC+CD+DE+EA=65 \text{ and} \\
&AC+CE+EB+BD+DA=72.
\end{align*}
Compute the perimeter of the convex pentagon whose vertices are the midpoints of the sides of $ABCDE.$
2001 Tournament Of Towns, 5
On the plane is a set of at least four points. If any one point from this set is removed, the resulting set has an axis of symmetry. Is it necessarily true that the whole set has an axis of symmetry?
2005 Moldova Team Selection Test, 2
Let $m\in N$ and $E(x,y,m)=(\frac{72}x)^m+(\frac{72}y)^m-x^m-y^m$, where $x$ and $y$ are positive divisors of 72.
a) Prove that there exist infinitely many natural numbers $m$ so, that 2005 divides $E(3,12,m)$ and $E(9,6,m)$.
b) Find the smallest positive integer number $m_0$ so, that 2005 divides $E(3,12,m_0)$ and $E(9,6,m_0)$.
2025 ISI Entrance UGB, 6
Let $\mathbb{N}$ denote the set of natural numbers, and let $\left( a_i, b_i \right)$, $1 \leq i \leq 9$, be nine distinct tuples in $\mathbb{N} \times \mathbb{N}$. Show that there are three distinct elements in the set $\{ 2^{a_i} 3^{b_i} \colon 1 \leq i \leq 9 \}$ whose product is a perfect cube.
2000 Polish MO Finals, 1
$PA_1A_2...A_n$ is a pyramid. The base $A_1A_2...A_n$ is a regular n-gon. The apex $P$ is placed so that the lines $PA_i$ all make an angle $60^{\cdot}$ with the plane of the base. For which $n$ is it possible to find $B_i$ on $PA_i$ for $i = 2, 3, ... , n$ such that $A_1B_2 + B_2B_3 + B_3B_4 + ... + B_{n-1}B_n + B_nA_1 < 2A_1P$?
2009 Cuba MO, 5
Prove that there are infinitely many positive integers $n$ such that $\frac{5^n-1}{n+2}$ is an integer.
2012 Today's Calculation Of Integral, 788
For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions:
(1) Find $f'(x)$.
(2) Sketch the graph of $y=f(x)$.
(3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.
2006 France Team Selection Test, 3
Let $M=\{1,2,\ldots,3 \cdot n\}$. Partition $M$ into three sets $A,B,C$ which $card$ $A$ $=$ $card$ $B$ $=$ $card$ $C$ $=$ $n .$
Prove that there exists $a$ in $A,b$ in $B, c$ in $C$ such that or $a=b+c,$ or $b=c+a,$ or $c=a+b$
[i]Edited by orl.[/i]
2023 China Team Selection Test, P13
Does there exists a positive irrational number ${x},$ such that there are at most finite positive integers ${n},$ satisfy that for any integer $1\leq k\leq n,$ $\{kx\}\geq\frac 1{n+1}?$
OMMC POTM, 2024 4
A man was born on April 1st, [b]20[/b] BCE and died on April 1st, [b]24[/b] CE. How many years did he live?
Clarification: Forget about the time he's born or died, assume he is born and died at the exact precise same time on each day
2010 Greece JBMO TST, 3
Given an acute and scalene triangle $ABC$ with $AB<AC$ and random line $(e)$ that passes throuh the center of the circumscribed circles $c(O,R)$. Line $(e)$, intersects sides $BC,AC,AB$ at points $A_1,B_1,C_1$ respectively (point $C_1$ lies on the extension of $AB$ towards $B$). Perpendicular from $A$ on line $(e)$ and $AA_1$ intersect circumscribed circle $c(O,R)$ at points $M$ and $A_2$ respectively. Prove that
a) points $O,A_1,A_2, M$ are consyclic
b) if $(c_2)$ is the circumcircle of triangle $(OBC_1)$ and $(c_3)$ is the circumcircle of triangle $(OCB_1)$, then circles $(c_1),(c_2)$ and $(c_3)$ have a common chord
2004 Switzerland Team Selection Test, 5
A brick has the shape of a cube of size $2$ with one corner unit cube removed. Given a cube of side $2^{n}$ divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.
2021 JBMO Shortlist, N3
For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$.
Find the largest possible value of $T_A$.
2006 VJIMC, Problem 3
For a function $f:[0,1]\to\mathbb R$ the secant of $f$ at points $a,b\in[0,1]$, $a<b$, is the line in $\mathbb R^2$ passing through $(a,f(a))$ and $(b,f(b))$. A function is said to intersect its secant at $a,b$ if there exists a point $c\in(a,b)$ such that $(c,f(c))$ lies on the secant of $f$ at $a,b$.
1. Find the set $\mathcal F$ of all continuous functions $f$ such that for any $a,b\in[0,1]$, $a<b$, the function $f$ intersects its secant at $a,b$.
2. Does there exist a continuous function $f\notin\mathcal F$ such that for any rational $a,b\in[0,1],a<b$, the function $f$ intersects its secant at $a,b$?
1973 AMC 12/AHSME, 3
The stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime numbers. For such representations of the even number 126, the largest possible difference between the two primes is
$ \textbf{(A)}\ 112 \qquad
\textbf{(B)}\ 100 \qquad
\textbf{(C)}\ 92 \qquad
\textbf{(D)}\ 88 \qquad
\textbf{(E)}\ 80$
1967 AMC 12/AHSME, 24
The number of solution-pairs in the positive integers of the equation $3x+5y=501$ is:
$\textbf{(A)}\ 33\qquad
\textbf{(B)}\ 34\qquad
\textbf{(C)}\ 35\qquad
\textbf{(D)}\ 100\qquad
\textbf{(E)}\ \text{none of these}$
2012 Online Math Open Problems, 28
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum $c$ so that for any $r<c,$ the fly can always avoid being caught?
[i]Author: Anderson Wang[/i]
2018 IFYM, Sozopol, 2
$n > 1$ is an odd number and $a_1, a_2, . . . , a_n$ are positive integers such that $gcd(a_1, a_2, . . . , a_n) = 1$. If
$d = gcd (a_1^n + a_1.a_2. . . a_n, a_2^n + a_1.a_2. . . a_n, . . . , a_n^n + a_1.a_2. . . a_n) $
find all possible values of $d$.
2009 Denmark MO - Mohr Contest, 2
Solve the system of equations $$\begin{cases} \dfrac{1}{x+y}+ x = 3 \\ \\ \dfrac{x}{x+y}=2 \end{cases}$$
2003 District Olympiad, 3
Consider an array $n \times n$ ($n\ge 2$) with $n^2$ integers. In how many ways one can complete the array if the product of the numbers on any row and column is $5$ or $-5$?
1998 Romania Team Selection Test, 1
Let $ABC$ be an equilateral triangle and $n\ge 2$ be an integer. Denote by $\mathcal{A}$ the set of $n-1$ straight lines which are parallel to $BC$ and divide the surface $[ABC]$ into $n$ polygons having the same area and denote by $\mathcal{P}$ the set of $n-1$ straight lines parallel to $BC$ which divide the surface $[ABC]$ into $n$ polygons having the same perimeter.
Prove that the intersection $\mathcal{A} \cap \mathcal{P}$ is empty.
[i]Laurentiu Panaitopol[/i]
2020 Durer Math Competition Finals, 15
The function $f$ is defined on positive integers : if $n$ has prime factorization $p^{k_1}_{1} p^{k_2}_{2} ...p^{k_t}_{t}$ then $f(n) = (p_1-1)^{k_1+1}(p_2-1)^{k_2+1}...(p_t-1)^{k_t+1}$. If we keep using this function repeatedly, starting from any positive integer $n$, we will always get to $1$ after some number of steps. What is the smallest integer $n$ for which we need exactly $6$ steps to get to $1$?
PEN A Problems, 8
The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.
2021 Brazil Team Selection Test, 2
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 $\angle ACB = 90^{\circ}$.
2013 India IMO Training Camp, 1
For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$. For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$. Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.