Found problems: 85335
1998 Gauss, 11
Kalyn cut rectangle R from a sheet of paper and then cut figure S from R. All the cuts were made
parallel to the sides of the original rectangle. In comparing R to S
(A) the area and perimeter both decrease
(B) the area decreases and the perimeter increases
(C) the area and perimeter both increase
(D) the area increases and the perimeter decreases
(E) the area decreases and the perimeter stays the same
2012-2013 SDML (Middle School), 10
Two of the diagonals of a regular pentagon are selected at random. What is the probability that the two selected diagonals intersect inside the pentagon?
$\text{(A) }\frac{2}{5}\qquad\text{(B) }\frac{1}{5}\qquad\text{(C) }\frac{7}{10}\qquad\text{(D) }\frac{3}{5}\qquad\text{(E) }\frac{1}{2}$
1998 Romania National Olympiad, 3
Suppose $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function for which the inequality $f'(x) \leq f'(x+\frac{1}{n})$ holds for every $x\in\mathbb{R}$ and every $n\in\mathbb{N}$.Prove that f is continiously differentiable
2018 AIME Problems, 2
The number \(n\) can be written in base \(14\) as \(\underline{a}\) \(\underline{b}\) \(\underline{c}\), can be written in base \(15\) as \(\underline{a}\) \(\underline{c}\) \(\underline{b}\), and can be written in base \(6\) as \(\underline{a}\) \(\underline{c}\) \(\underline{a}\) \(\underline{c}\), where \(a > 0\). Find the base-\(10\) representation of \(n\).
2023 BMT, 8
A circle intersects equilateral triangle $\vartriangle XY Z$ at $A,$ $B$, $C$, $D$, $E$, and $F$ such that points $X$, $A$, $B$, $Y$ , $C$, $D$, $Z$, $E$, and $F$ lie on the equilateral triangle in that order. If $AC^2 +CE^2 +EA^2 = 1900$ and $BD^2 + DF^2 + FB^2 = 2092$, compute the positive difference between the areas of triangles $\vartriangle ACE$ and $\vartriangle BDF$.
1998 Tournament Of Towns, 2
$ABCD$ is a parallelogram. A point $M$ is found on the side $AB$ or its extension such that $\angle MAD = \angle AMO$ where $O$ is the intersection point of the diagonals of the parallelogram. Prove that $MD = MG$.
(M Smurov)
2013 India IMO Training Camp, 3
For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.
2017 NMTC Junior, 4
a) $a,b,c,d$ are positive reals such that $abcd=1$. Prove that \[\sum_{cyc} \frac{1+ab}{1+a}\geq 4.\]
(b)In a scalene triangle $ABC$, $\angle BAC =120^{\circ}$. The bisectors of angles $A,B,C$ meets the opposite sides in $P,Q,R$ respectively. Prove that the circle on $QR$ as diameter passes through the point $P$.
Russian TST 2020, P1
Determine all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ satisfying $xf(xf(2y))=y+xyf(x)$ for all $x,y>0$.
2007 Kazakhstan National Olympiad, 4
Find all functions $ f :\mathbb{R}\to\mathbb{R} $, satisfying the condition
$f (xf (y) + f (x)) = 2f (x) + xy$
for any real $x$ and $y$.
2022 JHMT HS, 7
Find the least positive integer $N$ such that there exist positive real numbers $a_1,a_2,\dots,a_N$ such that
\[ \sum_{k=1}^{N}ka_k=1 \quad \text{and} \quad \sum_{k=1}^{N}\frac{a_k^2}{k}\leq \frac{1}{2022^2}. \]
1998 Harvard-MIT Mathematics Tournament, 6
How many pairs of positive integers $(a,b)$ with $a\leq b$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{6}$?
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.
2003 APMO, 1
Let $a,b,c,d,e,f$ be real numbers such that the polynomial
\[ p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f \]
factorises into eight linear factors $x-x_i$, with $x_i>0$ for $i=1,2,\ldots,8$. Determine all possible values of $f$.
2022 EGMO, 5
For all positive integers $n$, $k$, let $f(n, 2k)$ be the number of ways an $n \times 2k$ board can be fully covered by $nk$ dominoes of size $2 \times 1$. (For example, $f(2, 2)=2$ and $f(3, 2)=3$.) Find all positive integers $n$ such that for every positive integer $k$, the number $f(n, 2k)$ is odd.
2023 LMT Fall, 10
Aidan and Andrew independently select distinct cells in a $4 $ by $4$ grid, as well as a direction (either up, down, left, or right), both at random. Every second, each of them will travel $1$ cell in their chosen direction. Find the probability that Aidan and Andrew willmeet (be in the same cell at the same time) before either one of them hits an edge of the grid. (If Aidan and Andrew cross paths by switching cells, it doesn’t count as meeting.)
2015 Chile National Olympiad, 6
On the plane, a closed curve with simple auto intersections is drawn continuously. In the plane a finite number is determined in this way from disjoint regions. Show that each of these regions can be completely painted either white or blue, so that every two regions that share a curve segment at its edges, they always have different colors.
Clarification: a car intersection is simple if looking at a very small disk around from her, the curve looks like a junction $\times$.
2021 Kyiv Mathematical Festival, 1
Is it possible to mark four points on the plane so that the distances between any point and three other points form an arithmetic progression? (V. Brayman)
2011 Belarus Team Selection Test, 2
Positive real $a,b,c$ satisfy the condition $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=1+\frac{1}{6}\left( \frac{a}{c}+\frac{b}{a}+\frac{c}{b} \right)$$ Prove that $$\frac{a^3bc}{b+c}+\frac{b^3ca}{a+c}+\frac{c^3ab}{a+b}\ge \frac{1}{6}(ab+bc+ca)^2$$
I.Voronovich
2008 Tournament Of Towns, 3
Acute triangle $A_1A_2A_3$ is inscribed in a circle of radius $2$. Prove that one can choose points $B_1, B_2, B_3$ on the arcs $A_1A_2, A_2A_3, A_3A_1$ respectively, such that the numerical value of the area of the hexagon $A_1B_1A_2B_2A_3B_3$ is equal to the numerical value of the perimeter of the triangle $A_1A_2A_3.$
2014 South East Mathematical Olympiad, 7
Let $\omega_{1}$ be a circle with centre $O$. $P$ is a point on $\omega_{1}$. $\omega_{2}$ is a circle with centre $P$, with radius smaller than $\omega_{1}$. $\omega_{1}$ meets $\omega_{2}$ at points $T$ and $Q$. Let $TR$ be a diameter of $\omega_{2}$. Draw another two circles with $RQ$ as the radius, $R$ and $P$ as the centres. These two circles meet at point $M$, with $M$ and $Q$ lie on the same side of $PR$. A circle with centre $M$ and radius $MR$ intersects $\omega_{2}$ at $R$ and $N$. Prove that a circle with centre $T$ and radius $TN$ passes through $O$.
2021 Romania EGMO TST, P3
Determine all pairs of positive integers $(m,n)$ for which an $m\times n$ rectangle can be tiled with (possibly rotated) L-shaped trominos.
2018 Online Math Open Problems, 24
Let $p = 101$ and let $S$ be the set of $p$-tuples $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \to \{0, 1, \dots, p-1\}$ such that
[list]
[*] $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$ for all $a, b \in S$, and
[*] $f(a) = f(b)$ whenever all components of $a-b$ are divisible by $p$.
[/list]
Compute the number of positive integer divisors of $N$. (Here addition and subtraction in $\mathbb{Z}^p$ are done component-wise.)
[i]Proposed by Ankan Bhattacharya[/i]
2010 Indonesia TST, 3
Let $ x$, $ y$, and $ z$ be integers satisfying the equation \[ \dfrac{2008}{41y^2}\equal{}\dfrac{2z}{2009}\plus{}\dfrac{2007}{2x^2}.\] Determine the greatest value that $ z$ can take.
[i]Budi Surodjo, Jogjakarta[/i]
2019 ELMO Shortlist, N4
A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$.
[i]Proposed by Carl Schildkraut and Holden Mui[/i]