Found problems: 85335
2006 Iran Team Selection Test, 2
Suppose $n$ coins are available that their mass is unknown. We have a pair of balances and every time we can choose an even number of coins and put half of them on one side of the balance and put another half on the other side, therefore a [i]comparison[/i] will be done. Our aim is determining that the mass of all coins is equal or not. Show that at least $n-1$ [i]comparisons[/i] are required.
2001 Junior Balkan MO, 1
Solve the equation $a^3+b^3+c^3=2001$ in positive integers.
[i]Mircea Becheanu, Romania[/i]
2020 Brazil Cono Sur TST, 2
Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.
2005 Germany Team Selection Test, 3
We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.
2024 Bulgarian Spring Mathematical Competition, 10.4
A graph $G$ is called $\textit{divisibility graph}$ if the vertices can be assigned distinct positive integers such that between two vertices assigned $u, v$ there is an edge iff $\frac{u} {v}$ or $\frac{v} {u}$ is a positive integer. Show that for any positive integer $n$ and $0 \leq e \leq \frac{n(n-1)}{2}$, there is a $\textit{divisibility graph}$ with $n$ vertices and $e$ edges.
[hide=Remark on source of 10.3] It appears to be Kvant 2022 Issue 10 M2719, so it will not be posted; the same problem was also used as 9.4.
2021 AIME Problems, 6
For any finite set $S$, let $|S|$ denote the number of elements in $S$. FInd the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy
$$|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$$
1985 Dutch Mathematical Olympiad, 4
A convex hexagon $ ABCDEF$ is such that each of the diagonals $ AD,BE,CF$ divides the hexagon into two parts of equal area. Prove that these three diagonals are concurrent.
2003 Estonia Team Selection Test, 1
Two treasure-hunters found a treasure containing coins of value $a_1< a_2 < ... < a_{2003}$ (the quantity of coins of each value is unlimited). The first treasure-hunter forms all the possible sets of different coins containing odd number of elements, and takes the most valuable coin of each such set. The second treasure-hunter forms all the possible sets of different coins containing even number of elements, and takes the most valuable coin of each such set. Which one of them is going to have more money and how much more?
(H. Nestra)
2023 Dutch IMO TST, 1
Find all prime numbers $p$ such that the number
$$3^p+4^p+5^p+9^p-98$$
has at most $6$ positive divisors.
2017 USAJMO, 1
Prove that there are infinitely many distinct pairs $(a, b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.
2000 BAMO, 2
Let $ABC$ be a triangle with $D$ the midpoint of side $AB, E$ the midpoint of side $BC$, and $F$ the midpoint of side $AC$. Let $k_1$ be the circle passing through points $A, D$, and $F$, let $k_2$ be the circle passing through points $B, E$, and $D$, and let $k_3$ be the circle passing through $C, F$, and $E$. Prove that circles $k_1, k_2$, and $k_3$ intersect in a point.
2021 Saudi Arabia Training Tests, 18
Let $ABC$ be a triangle with $AB < AC$ and incircle $(I)$ tangent to $BC$ at $D$. Take $K$ on $AD$ such that $CD = CK$. Suppose that $AD$ cuts $(I)$ at $G$ and $BG$ cuts $CK$ at $L$. Prove that K is the midpoint of $CL$.
2019 IMO Shortlist, G5
Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$.
Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$.
Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$.
(Hungary)
2005 Slovenia Team Selection Test, 2
Find all functions $f : R^+ \to R^+$ such that $x^2(f(x)+ f(y)) = (x+y)f (f(x)y)$ for any $x,y > 0$.
2023 Stanford Mathematics Tournament, 1
For all positive integers $n > 1$, let $f(n)$ denote the largest odd proper divisor of $n$ (a proper divisor of $n$ is a positive divisor of $n$ except for $n$ itself). Given that $N=20^{23}\cdot23^{20}$, compute
\[\frac{f(N)}{f(f(f(N)))}.\]
1997 Estonia National Olympiad, 3
The points $A, B, M$ and $N$ are on a circle with center $O$ such that the radii $OA$ and $OB$ are perpendicular to each other, and $MN$ is parallel to $AB$ and intersects the radius $OA$ at $P$. Find the radius of the circle if $|MP|= 12$ and $|P N| = 2 \sqrt{14}$
2007 Today's Calculation Of Integral, 210
Evaluate $\int_{1}^{\pi}\left(x^{3}\ln x-\frac{6}{x}\right)\sin x\ dx$.
2003 JHMMC 8, 28
How many of the positive divisors of $120$ are divisible by $4$?
2019 Romania National Olympiad, 3
Find all natural numbers $ n\ge 4 $ that satisfy the property that the affixes of any nonzero pairwise distinct complex numbers $ a,b,c $ that verify the equation
$$ (a-b)^n+(b-c)^n+(c-a)^n=0, $$
represent the vertices of an equilateral triangle in the complex plane.
2009 Belarus Team Selection Test, 3
Find all real numbers $a$ for which there exists a function $f: R \to R$ asuch that $x + f(y) =a(y + f(x))$ for all real numbers $x,y\in R$.
I.Voronovich
2013 Purple Comet Problems, 12
How many four-digit positive integers have no adjacent equal even digits? For example, count numbers such as $1164$ and $2035$ but not $6447$ or $5866$.
2019-IMOC, A3
Find all $3$-tuples of positive reals $(a,b,c)$ such that
$$\begin{cases}a\sqrt[2019]b-c=a\\b\sqrt[2019]c-a=b\\c\sqrt[2019]a-b=c\end{cases}$$
1985 IMO Longlists, 79
Let $a, b$, and $c$ be real numbers such that
\[\frac{1}{bc-a^2} + \frac{1}{ca-b^2}+\frac{1}{ab-c^2} = 0.\]
Prove that
\[\frac{a}{(bc-a^2)^2} + \frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2} = 0.\]
2003 Junior Balkan MO, 1
Let $n$ be a positive integer. A number $A$ consists of $2n$ digits, each of which is 4; and a number $B$ consists of $n$ digits, each of which is 8. Prove that $A+2B+4$ is a perfect square.
2007 AMC 12/AHSME, 21
The first $ 2007$ positive integers are each written in base $ 3$. How many of these base-$ 3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.)
$ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 101 \qquad \textbf{(C)}\ 102 \qquad \textbf{(D)}\ 103 \qquad \textbf{(E)}\ 104$