Found problems: 85335
2024 ELMO Shortlist, G4
In quadrilateral $ABCD$ with incenter $I$, points $W,X,Y,Z$ lie on sides $AB, BC,CD,DA$ with $AZ=AW$, $BW=BX$, $CX=CY$, $DY=DZ$. Define $T=\overline{AC}\cap\overline{BD}$ and $L=\overline{WY}\cap\overline{XZ}$. Let points $O_a,O_b,O_c,O_d$ be such that $\angle O_aZA=\angle O_aWA=90^\circ$ (and cyclic variants), and $G=\overline{O_aO_c}\cap\overline{O_bO_d}$. Prove that $\overline{IL}\parallel\overline{TG}$.
[i]Neal Yan[/i]
1983 IMO Longlists, 31
Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$.
2014 AMC 12/AHSME, 19
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?
$\textbf{(A) }6\qquad
\textbf{(B) }12\qquad
\textbf{(C) }24\qquad
\textbf{(D) }48\qquad
\textbf{(E) }78\qquad$
2003 Singapore Team Selection Test, 3
Determine all functions $f : Z\to Z$, where $Z$ is the set of integers, such that $$f(m + f(f(n))) = -f(f(m + 1)) - n$$ for all integers $m$ and $n$.
2023 Belarusian National Olympiad, 9.3
The triangle $ABC$ has perimeter $36$, and the length of $BC$ is $9$. Point $M$ is the midpoint of $AC$, and $I$ is the incenter.
Find the angle $MIC$.
2011 China Second Round Olympiad, 11
A line $\ell$ with slope of $\frac{1}{3}$ insects the ellipse $C:\frac{x^2}{36}+\frac{y^2}{4}=1$ at points $A,B$ and the point $P\left( 3\sqrt{2} , \sqrt{2}\right)$ is above the line $\ell$.
[list]
[b](1)[/b] Prove that the locus of the incenter of triangle $PAB$ is a segment,
[b](2)[/b] If $\angle APB=\frac{\pi}{3}$, then find the area of triangle $PAB$.[/list]
2019 PUMaC Team Round, 1
Two unit squares are stacked on top of one another to form a $1 \times 2$ rectangle. Each of the seven edges is colored either red or blue. How many ways are there to color the edges in this way such that there is exactly one path along all-blue edges from the bottom-left corner to the top-right corner?
2014 Iran Team Selection Test, 4
$n$ is a natural number. We shall call a permutation $a_1,\dots,a_n$ of $1,\dots,n$ a quadratic(cubic) permutation if $\forall 1\leq i \leq n-1$ we have $a_ia_{i+1}+1$ is a perfect square(cube).
$(a)$ Prove that for infinitely many natural numbers $n$ there exists a quadratic permutation.
$(b)$ Prove that for no natural number $n$ exists a cubic permutation.
1989 IMO Longlists, 78
Let $ P(x)$ be a polynomial with integer coefficients such that \[ P(m_1) \equal{} P(m_2) \equal{} P(m_3) \equal{} P(m_4) \equal{} 7\] for given distinct integers $ m_1,m_2,m_3,$ and $ m_4.$ Show that there is no integer m such that $ P(m) \equal{} 14.$
1973 AMC 12/AHSME, 25
A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is
$ \textbf{(A)}\ 36\pi\minus{}34 \qquad
\textbf{(B)}\ 30\pi \minus{} 15 \qquad
\textbf{(C)}\ 36\pi \minus{} 33 \qquad$
$ \textbf{(D)}\ 35\pi \minus{} 9\sqrt3 \qquad
\textbf{(E)}\ 30\pi \minus{} 9\sqrt3$
2018 BAMO, A
Twenty-five people of different heights stand in a $5\times 5$ grid of squares, with one person in each square. We know that each row has a shortest person, suppose Ana is the tallest of these five people. Similarly, we know that each column has a tallest person, suppose Bev is the shortest of these five people.
Assuming Ana and Bev are not the same person, who is taller: Ana or Bev?
Prove that your answer is always correct.
2013 Oral Moscow Geometry Olympiad, 2
Inside the angle $AOD$, the rays $OB$ and $OC$ are drawn such that $\angle AOB = \angle COD.$ Two circles are inscribed inside the angles $\angle AOB$ and $\angle COD$ . Prove that the intersection point of the common internal tangents of these circles lies on the bisector of the angle $AOD$.
2012 National Olympiad First Round, 17
Let $D$ be a point inside $\triangle ABC$ such that $m(\widehat{BAD})=20^{\circ}$, $m(\widehat{DAC})=80^{\circ}$, $m(\widehat{ACD})=20^{\circ}$, and $m(\widehat{DCB})=20^{\circ}$.
$m(\widehat{ABD})= ?$
$ \textbf{(A)}\ 5^{\circ} \qquad \textbf{(B)}\ 10^{\circ} \qquad \textbf{(C)}\ 15^{\circ} \qquad \textbf{(D)}\ 20^{\circ} \qquad \textbf{(E)}\ 25^{\circ}$
2010 HMNT, 4
An ant starts at the point $(1, 0)$. Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point $(x, y)$ with $|x| + |y| \le 2$. What is the probability that the ant ends at the point $(1, 1)$?
2016 Math Prize for Girls Problems, 8
A [i]strip[/i] is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$. Let $A'$ be a rotation of $A$ in the plane by $60^\circ$. The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$. Let $x^\circ$ be the measure (in degrees) of one interior angle of $P$. What is the greatest possible value of the number $x$?
2013 China Team Selection Test, 1
Let $n$ and $k$ be two integers which are greater than $1$. Let $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$ be non-negative real numbers such that
i) $a_1\ge a_2\ge\ldots\ge a_n$ and $a_1+a_2+\ldots+a_n=1$;
ii) For any integer $m\in\{1,2,\ldots,n\}$, we have that $c_1+c_2+\ldots+c_m\le m^k$.
Find the maximum of $c_1a_1^k+c_2a_2^k+\ldots+c_na_n^k$.
2006 VTRMC, Problem 2
Let $S(n)$ denote the number of sequences of length $n$ formed by the three letters $A,B,C$ with the restriction that the $C$'s (if any) all occur in a single block immediately following the first $B$ (if any). For example $ABCCAA$, $AAABAA$, and $ABCCCC$ are counted in, but $ACACCB$ and $CAAAAA$ are not. Derive a simple formula for $S(n)$ and use it to calculate $S(10)$.
2002 APMO, 2
Find all positive integers $a$ and $b$ such that
\[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \]
are both integers.
2021 Harvard-MIT Mathematics Tournament., 4
Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c) = P(b, c, a)$ and $P(a, a, b) = 0$ for all real $a$, $b$, and $c$. If $P(1, 2, 3) = 1$, compute $P(2, 4, 8)$.
Note: $P(x, y, z)$ is a homogeneous degree $4$ polynomial if it satisfies $P(ka, kb, kc) = k^4P(a, b, c)$ for all real $k, a, b, c$.
2022 Junior Balkan Team Selection Tests - Moldova, 2
Let n be the natural number ($n\ge 2$). All natural numbers from $1$ up to $n$ ,inclusive, are written on the board in some order: $a_1$, $a_2$ , $...$ , $a_n$. Determine all natural numbers $n$ ($n\ge 2$), for which the product
$$P = (1 + a_1) \cdot (2 + a_2) \cdot ... \cdot (n + a_n)$$
is an even number, whatever the arrangement of the numbers written on the board.
2023 Iran MO (3rd Round), 3
For numbers $a,b \in \mathbb{R}$ we consider the sets:
$$A=\{a^n | n \in \mathbb{N}\} , B=\{b^n | n \in \mathbb{N}\}$$
Find all $a,b > 1$ for which there exists two real , non-constant polynomials $P,Q$ with positive leading coefficients st for each $r \in \mathbb{R}$:
$$ P(r) \in A \iff Q(r) \in B$$
2015 Cono Sur Olympiad, 5
Determine if there exists an infinite sequence of not necessarily distinct positive integers $a_1, a_2, a_3,\ldots$ such that for any positive integers $m$ and $n$ where $1 \leq m < n$, the number $a_{m+1} + a_{m+2} + \ldots + a_{n}$ is not divisible by $a_1 + a_2 + \ldots + a_m$.
2018 Belarusian National Olympiad, 11.2
The altitudes $AA_1$, $BB_1$ and $CC_1$ are drawn in the acute triangle $ABC$. The bisector of the angle $AA_1C$ intersects the segments $CC_1$ and $CA$ at $E$ and $D$ respectively. The bisector of the angle $AA_1B$ intersects the segments $BB_1$ and $BA$ at $F$ and $G$ respectively. The circumcircles of the triangles $FA_1D$ and $EA_1G$ intersect at $A_1$ and $X$.
Prove that $\angle BXC=90^{\circ}$.
1996 ITAMO, 2
Show that the equation $a^2 + b^2 = c^2 + 3$ has infinetely many triples of integers $a, b, c$ that are solutions.
2004 India IMO Training Camp, 1
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]