Found problems: 85335
2006 Peru MO (ONEM), 3
A pair $(m, n)$ of positive integers is called “[i]linked[/i]” if $m$ divides $3n + 1$ and $n$ divides $3m + 1$. If $a, b, c$ are distinct positive integers such that $(a, b)$ and $( b, c)$ are linked pairs, prove that the number $1$ belongs to the set $\{a, b, c\}$
2016 BMT Spring, 10
Define $T_n =\sum^{n}){i=1} i(n + 1 - i)$. Find $\lim_{n\to \infty} \frac{T_n}{n^3}$.
1971 AMC 12/AHSME, 21
If $\log_2(\log_3(\log_4 x))=\log_3(\log_4(\log_2 y))=\log_4(\log_2(\log_3 z))=0$, then the sum $x+y+z$ is equal to
$\textbf{(A) }50\qquad\textbf{(B) }58\qquad\textbf{(C) }89\qquad\textbf{(D) }111\qquad \textbf{(E) }1296$
2006 Baltic Way, 9
To every vertex of a regular pentagon a real number is assigned. We may perform the following operation repeatedly: we choose two adjacent vertices of the pentagon and replace each of the two numbers assigned to these vertices by their arithmetic mean. Is it always possible to obtain the position in which all five numbers are zeroes, given that in the initial position the sum of all five numbers is equal to zero?
2011 Indonesia TST, 1
For all positive integer $n$, define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$.
Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$-digit integers $n$ (in decimal notation).
2017 Bulgaria National Olympiad, 3
Let $M$ be a set of $2017$ positive integers. For any subset $A$ of $M$ we define $f(A) := \{x\in M\mid \text{ the number of the members of }A\,,\, x \text{ is multiple of, is odd }\}$.
Find the minimal natural number $k$, satisfying the condition: for any $M$, we can color all the subsets of $M$ with $k$ colors, such that whenever $A\neq f(A)$, $A$ and $f(A)$ are colored with different colors.
2017 AMC 10, 24
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$?
$\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$
2003 Argentina National Olympiad, 4
The trapezoid $ABCD$ of bases $AB$ and $CD$, has $\angle A = 90^o, AB = 6, CD = 3$ and $AD = 4$. Let $E, G, H$ be the circumcenters of triangles $ABC, ACD, ABD$, respectively. Find the area of the triangle $EGH$.
1996 AMC 8, 22
The horizontal and vertical distances between adjacent points equal $1$ unit. The area of triangle $ABC$ is
[asy]
for (int a = 0; a < 5; ++a)
{
for (int b = 0; b < 4; ++b)
{
dot((a,b));
}
}
draw((0,0)--(3,2)--(4,3)--cycle);
label("$A$",(0,0),SW);
label("$B$",(3,2),SE);
label("$C$",(4,3),NE);
[/asy]
$\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/2 \qquad \text{(C)}\ 3/4 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 5/4$
1997 Pre-Preparation Course Examination, 5
Let $H$ be the orthocenter of the triangle $ABC$ and $P$ an arbitrary point on circumcircle of triangle. $BH$ meets $AC$ at $E$. $PAQB$ and $PARC$ are two parallelograms and $AQ$ meets $HR$ at $X$. Show that $EX \parallel AP$.
1998 Irish Math Olympiad, 1
Find all positive integers $ n$ having exactly $ 16$ divisors $ 1\equal{}d_1<d_2<...<d_{16}\equal{}n$ such that $ d_6\equal{}18$ and $ d_9\minus{}d_8\equal{}17.$
2016 Croatia Team Selection Test, Problem 1
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real $x,y$:
$$ f(x^2) + xf(y) = f(x) f(x + f(y)) \, . $$
2019 Philippine TST, 6
Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.
1990 Kurschak Competition, 3
We would like to give a present to one of $100$ children. We do this by throwing a biased coin $k$ times, after predetermining who wins in each possible outcome of this lottery.
Prove that we can choose the probability $p$ of throwing heads, and the value of $k$ such that, by distributing the $2^k$ different outcomes between the children in the right way, we can guarantee that each child has the same probability of winning.
2011 Olympic Revenge, 3
Let $E$ to be an infinite set of congruent ellipses in the plane, and $r$ a fixed line. It is known that each line parallel to $r$ intersects at least one ellipse belonging to $E$. Prove that there exist infinitely many triples of ellipses belonging to $E$, such that there exists a line that intersect the triple of ellipses.
2013 Pan African, 3
Let $ABCDEF$ be a convex hexagon with $\angle A= \angle D$ and $\angle B=\angle E$ . Let $K$ and $L$
be the midpoints of the sides $AB$ and $DE$ respectively. Prove that the sum of the areas of triangles $FAK$, $KCB$ and $CFL$ is equal to half of the area of the hexagon if and only if
\[\frac{BC}{CD}=\frac{EF}{FA}.\]
2011 Canadian Students Math Olympiad, 4
Circles $\Gamma_1$ and $\Gamma_2$ have centers $O_1$ and $O_2$ and intersect at $P$ and $Q$. A line through $P$ intersects $\Gamma_1$ and $\Gamma_2$ at $A$ and $B$, respectively, such that $AB$ is not perpendicular to $PQ$. Let $X$ be the point on $PQ$ such that $XA=XB$ and let $Y$ be the point within $AO_1 O_2 B$ such that $AYO_1$ and $BYO_2$ are similar. Prove that $2\angle{O_1 AY}=\angle{AXB}$.
[i]Author: Matthew Brennan[/i]
2010 Kyrgyzstan National Olympiad, 2
Fifteen pairwise coprime positive integers chosen so that each of them less than 2010. Show that at least one of them is prime.
2025 All-Russian Olympiad, 9.1
Several line segments parallel to the sides of a rectangular sheet of paper were drawn on it. These segments divided the sheet into several rectangles, inside of which there are no drawn lines. Petya wants to draw one diagonal in each of the rectangles, dividing it into two triangles, and color each triangle either black or white. Is it always possible to do this in such a way that no two triangles of the same color share a segment of their boundary?
2010 Today's Calculation Of Integral, 633
Let $f(x)$ be a differentiable function. Find the value of $x$ for which
\[\{f(x)\}^2+(e+1)f(x)+1+e^2-2\int_0^x f(t)dt-2f(x)\int_0^x f(t)dt+2\left\{\int_0^x f(t)dt\right\}^2\]
is minimized.
[i]1978 Tokyo Medical College entrance exam[/i]
Russian TST 2018, P3
Kirill has $n{}$ identical footballs and two infinite rows of baskets, each numbered with consecutive natural numbers. In one row the baskets are red, in the other they are blue. Kirill puts all the balls into baskets so that the number of balls in the either row of baskets does not increase. Denote by $A{}$ the number of ways to arrange the balls so that the first blue basket contains more balls than any red one, and by $B{}$ the number of arrangements so that the number of some blue basket corresponds with the number of balls in it. Prove that $A = B$.
2024 New Zealand MO, 2
Consider the sequence $a_{1}, a_{2}, a_{3},\ldots$ defined by $a_{1}=2024^{2024}$ and for each positive integer $n$, $$a_{n+1}=\left|a_{n}-\sqrt{2}\right|.$$ Prove that there exists an integer $k$ such that $a_{k+2}=a_k$.
[i]Here [/i]$\left|x\right|$[i] denotes the absolute value of [/i]$x$.
Kyiv City MO 1984-93 - geometry, 1993.10.4
Prove theat for an arbitrary triangle holds the inequality $$a \cos A+ b \cos B + c \cos C \le p ,$$ where $a, b, c$ are the sides of the triangle, $A, B, C$ are the angles, $p$ is the semiperimeter.
2006 Tournament of Towns, 5
A square is dissected into $n$ congruent non-convex polygons whose sides are parallel to the sides of the square, and no two of these polygons are parallel translates of each other. What is the maximum value of $n$? (4)
1991 Federal Competition For Advanced Students, P2, 6
Find the number of ten-digit natural numbers (which do not start with zero) containing no block $ 1991$.