Found problems: 85335
JOM 2024, 5
Do there exist infinitely many triplets of positive integers $(a, b, c)$ such that
the following two conditions hold:
1. $\gcd(a, b, c) = 1$;
2. $a+b+c, a^2+b^2+c^2$ and $abc$ are all perfect squares?
[i](Proposed by Ivan Chan Guan Yu)[/i]
2011 Purple Comet Problems, 18
Let $a$ be a positive real number such that $\tfrac{a^2}{a^4-a^2+1}=\tfrac{4}{37}$. Then $\tfrac{a^3}{a^6-a^3+1}=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1993 India Regional Mathematical Olympiad, 4
Let $ABCD$ be a rectangle with $AB = a$ and $BC = b$. Suppose $r_1$ is the radius of the circle passing through $A$ and $B$ touching $CD$; and similarly $r_2$ is the radius of the circle passing through $B$ and $C$ and touching $AD$. Show that \[ r_1 + r_2 \geq \frac{5}{8} ( a + b) . \]
1981 Romania Team Selection Tests, 5.
Consider the set $S$ of lattice points with positive coordinates in the plane. For each point $P(a,b)$ from $S$, we draw a segment between it and each of the points in the set \[S(P)=\{(a+b,c)\mid c\in\mathbb{Z}, \, c>a+b\}.\] Show that there is no colouring of the points in $S$ with a finite number of colours such that every two points joined by a segment are coloured with different colours.
[i]Ioan Tomescu[/i]
1997 Romania Team Selection Test, 4
Let $n\ge 2$ be an integer and let $P(X)=X^n+a_{n-1}X^{n-1}+\ldots +a_1X+1$ be a polynomial with positive integer coefficients. Suppose that $a_k=a_{n-k}$ for all $k\in 1,2,\ldots,n-1$. Prove that there exist infinitely many pairs of positive integers $x,y$ such that $x|P(y)$ and $y|P(x)$.
[i]Remus Nicoara[/i]
2011 Northern Summer Camp Of Mathematics, 4
Find all positive integers $n$ such that $7^n+147$ is a perfect square.
2016 Purple Comet Problems, 13
One afternoon Elizabeth noticed that twice as many cars on the expressway carried only a driver as compared to the number of cars that carried a driver and one passenger. She also noted that twice as many cars carried a driver and one passenger as those that carried a driver and two passengers. Only 10% of the cars carried a driver and three passengers, and no car carried more than four people. Any car containing at least three people was allowed to use the fast lane. Elizabeth calculated that $\frac{m}{n}$ of the people in cars on the expressway were allowed to ride in the fast lane, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2023 ELMO Shortlist, G4
Let \(D\) be a point on segment \(PQ\). Let \(\omega\) be a fixed circle passing through \(D\), and let \(A\) be a variable point on \(\omega\). Let \(X\) be the intersection of the tangent to the circumcircle of \(\triangle ADP\) at \(P\) and the tangent to the circumcircle of \(\triangle ADQ\) at \(Q\). Show that as \(A\) varies, \(X\) lies on a fixed line.
[i]Proposed by Elliott Liu and Anthony Wang[/i]
2022 Stanford Mathematics Tournament, 1
If $f(x)=x^4+4x^3+7x^2+6x+2022$, compute $f'(3)$.
2016 Costa Rica - Final Round, A3
Let $x$ and $y$ be two positive real numbers, such that $x + y = 1$. Prove that $$\left(1 +\frac{1}{x}\right)\left(1 +\frac{1}{y}\right) \ge 9$$
2014 Balkan MO, 1
Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds.
[i]UK - David Monk[/i]
1998 Harvard-MIT Mathematics Tournament, 4
Let $f(x)=1+\dfrac{x}{2}+\dfrac{x^2}{4}+\dfrac{x^3}{8}+\cdots,$ for $-1\leq x \leq 1$. Find $\sqrt{e^{\int\limits_0^1 f(x)dx}}$.
2006 Moldova Team Selection Test, 3
Let $a,b,c$ be sides of the triangle. Prove that
\[ a^2\left(\frac{b}{c}-1\right)+b^2\left(\frac{c}{a}-1\right)+c^2\left(\frac{a}{b}-1\right)\geq 0 . \]
2010 Miklós Schweitzer, 6
Is there a continuous function $ f: \mathbb {R} ^ {2} \rightarrow \mathbb {R} $ for every $ d \in \mathbb {R} $ we have $ g_{m,d}(x) = f (x, m x + d) $ is strictly monotonic on $ \mathbb {R} $ if $ m \ge 0, $ and not monotone on any non-empty open interval if $ m <0? $
2004 Olympic Revenge, 1
$ABC$ is a triangle and $D$ is an internal point such that $\angle DAB=\angle DBC =\angle DCA$. $O_a$ is the circumcenter of $DBC$. $O_b$ is the circumcenter of $DAC$. $O_c$ is the circumcenter of $DAB$. Show that if the area of $ABC$ and $O_aO_bO_c$ are equal then $ABC$ is equilateral.
2006 Vietnam Team Selection Test, 3
In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.
Find the minimum value of a [i]good[/i] number $m$.
1948 Putnam, A2
Two spheres in contact have a common tangent cone. These three surfaces divide the space into various parts, only one of which is bounded by all three surfaces, it is "ring-shaped." Being given the radii of the spheres, $r$ and $R$, find the volume of the "ring-shaped" part. (The desired expression is a rational function of $r$ and $R.$)
PEN K Problems, 30
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+f(f(n))+f(n)=3n.\]
1953 Poland - Second Round, 6
Given a circle and two tangents to this circle. Draw a third tangent to the circle in such a way that its segment contained by the given tangents has the given length $ d $.
2016 CCA Math Bonanza, L2.2
In triangle $ABC$, $AB=7$, $AC=9$, and $BC=8$. The angle bisector of $\angle{BAC}$ intersects side $BC$ at $D$, and the angle bisector of $\angle{ABC}$ intersects $AD$ at $E$. Compute $AE^2$.
[i]2016 CCA Math Bonanza Lightning #2.2[/i]
2019 Singapore MO Open, 4
Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$.
[i]Proposed by fattypiggy123[/i]
1999 China Team Selection Test, 2
For a fixed natural number $m \geq 2$, prove that
[b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\]
[b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.
2020 Taiwan TST Round 2, 1
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
(Nigeria)
PEN J Problems, 8
Prove that for any $ \delta\in[0,1]$ and any $ \varepsilon>0$, there is an $ n\in\mathbb{N}$ such that $ \left |\frac{\phi (n)}{n}-\delta\right| <\varepsilon$.
VMEO IV 2015, 11.2
Let $ABC$ be a triangle with two isogonal points $ P$ and $Q$ . Let $D, E$ be the projection of $P$ on $AB$, $AC$. $G$ is the projection of $Q$ on $BC$. $U$ is the projection of $G$ on $DE$, $ L$ is the projection of $P$ on $AQ$, $K$ is the symmetric of $L$ wrt $UG$.
Prove that $UK$ passes through a fixed point when $P$ and $Q$ vary.