Found problems: 85335
2013 Princeton University Math Competition, 3
Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at Frist, located $2$ jumps east and $4$ jumps north of Fine Hall. How many ways can he get to McCosh without going through Frist?
2014 Postal Coaching, 1
Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$.Let $A$ be a point on $\omega_2$ not lying on the line through the centres of the two circles.Let $AB$ and $AC$ be the tangents to $\omega_1$.Lines $BP$ and $CP$ meet $\omega_2$ for the second time at points $E$ and $F$.Prove that the line $EF$,the tangent to $\omega_2$ at $A$ and the common tangent at $P$ concur.
2007 Miklós Schweitzer, 3
Denote by $\omega (n)$ the number of prime divisors of the natural number $n$ (without multiplicities). Let
$$F(x)=\max_{n\leq x} \omega (n) \,\,\,\,\,\,\,\,\,\,\,\,\, G(x)=\max_{n\leq x} \left( \omega (n) + \omega (n^2+1)\right)$$
Prove that $G(x)-F(x)\to \infty$ as $x\to\infty$.
(translated by Miklós Maróti)
2016 AMC 12/AHSME, 10
A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$?
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$
1997 Chile National Olympiad, 7
In a career in mathematics, $7$ courses are taught, among which students can choose the ones you want. Determine the number of students in the career, knowing that:
$\bullet$ No two students have chosen the same courses.
$\bullet$ Any two students have at least one course in common.
$\bullet$ If the race had one more student, it would not be possible to do both.
2016 Argentina National Olympiad, 5
Let $a$ and $b$ be rational numbers such that $a+b=a^2+b^2$ . Suppose the common value $s=a+b=a^2+b^2$ is not an integer, and let's write it as an irreducible fraction: $s=\frac{m}{n}$. Let $p$ be the smallest prime divisor of $n$. Find the minimum value of $p$.
1983 Czech and Slovak Olympiad III A, 6
Consider a circle $k$ with center $S$ and radius $r$. Denote $\mathsf M$ the set of all triangles with incircle $k$ such that the largest inner angle is twice bigger than the smallest one. For a triangle $\mathcal T\in\mathsf M$ denote its vertices $A,B,C$ in way that $SA\ge SB\ge SC$. Find the locus of points $\{B\mid\mathcal T\in\mathsf M\}$.
2022 Iran Team Selection Test, 5
Find all $C\in \mathbb{R}$ such that every sequence of integers $\{a_n\}_{n=1}^{\infty}$ which is bounded from below and for all $n\geq 2$ satisfy $$0\leq a_{n-1}+Ca_n+a_{n+1}<1$$ is periodic.
Proposed by Navid Safaei
2006 Putnam, B6
Let $k$ be an integer greater than $1.$ Suppose $a_{0}>0$ and define
\[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\]
for $n\ge 0.$ Evaluate
\[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]
2010 Macedonia National Olympiad, 4
The point $O$ is the centre of the circumscribed circle of the acute-angled triangle $ABC$. The line $AO$ cuts the side $BC$ in point $N$, and the line $BO$ cuts the side $AC$ at point $M$. Prove that if $CM=CN$, then $AC=BC$.
2019 MIG, 13
What is the remainder when $1 + 10 + 19 + 28 + \cdots + 91$ is divided by $9$?
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }8$
2014 IMO Shortlist, C9
There are $n$ circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or $\textit{vice versa}$.
Suppose that Turbo’s path entirely covers all circles. Prove that $n$ must be odd.
[i]Proposed by Tejaswi Navilarekallu, India[/i]
1939 Moscow Mathematical Olympiad, 044
Prove that $cos \frac{2\pi}{5} +cos \frac{4\pi}{5} = -\frac{1}{2}$.
2025 Kosovo National Mathematical Olympiad`, P2
Find all natural numbers $n$ such that $\frac{\sqrt{n}}{2}+\frac{10}{\sqrt{n}}$ is a natural number.
2013 Moldova Team Selection Test, 1
Let $m$ be the number of ordered solutions $(a,b,c,d,e)$ satisfying:
$1)$ $a,b,c,d,e\in \mathbb{Z}^{+}$;
$2)$ $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}=1$;
Prove that $m$ is odd.
2008 Princeton University Math Competition, A1
Find all positive real numbers $b$ for which there exists a positive real number $k$ such that $n-k \leq \left\lfloor bn \right\rfloor <n$ for all positive integers $n$.
2014 NIMO Problems, 8
The side lengths of $\triangle ABC$ are integers with no common factor greater than $1$. Given that $\angle B = 2 \angle C$ and $AB < 600$, compute the sum of all possible values of $AB$.
[i]Proposed by Eugene Chen[/i]
1996 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 6
A square $ ABCD$ is inscribed in a circle. Let $ \alpha \equal{} \angle DAB, \beta \equal{} \angle BDA,$ and $ \gamma \equal{} \angle CDB$. Then $ \angle DBC$ equals
A. $ \alpha \minus{} \beta$
B. $ \alpha \minus{} \gamma$
C. $ 90^\circ \minus{} \alpha \plus{} \beta$
D. $ 90^\circ \minus{} \alpha \plus{} \gamma$
E. $ 180^\circ \minus{} \alpha \minus{} \gamma$
2002 Croatia National Olympiad, Problem 1
Solve the equation
$$\left(x^2+3x-4\right)^3+\left(2x^2-5x+3\right)^3=\left(3x^2-2x-1\right)^3.$$
2022 Vietnam TST, 3
Let $ABCD$ be a parallelogram, $AC$ intersects $BD$ at $I$. Consider point $G$ inside $\triangle ABC$ that satisfy $\angle IAG=\angle IBG\neq 45^{\circ}-\frac{\angle AIB}{4}$. Let $E,G$ be projections of $C$ on $AG$ and $D$ on $BG$. The $E-$ median line of $\triangle BEF$ and $F-$ median line of $\triangle AEF$ intersects at $H$.
$a)$ Prove that $AF,BE$ and $IH$ concurrent. Call the concurrent point $L$.
$b)$ Let $K$ be the intersection of $CE$ and $DF$. Let $J$ circumcenter of $(LAB)$ and $M,N$ are respectively be circumcenters of $(EIJ)$ and $(FIJ)$. Prove that $EM,FN$ and the line go through circumcenters of $(GAB),(KCD)$ are concurrent.
2011 Iran MO (3rd Round), 6
$a$ is an integer and $p$ is a prime number and we have $p\ge 17$. Suppose that $S=\{1,2,....,p-1\}$ and $T=\{y|1\le y\le p-1,ord_p(y)<p-1\}$. Prove that there are at least $4(p-3)(p-1)^{p-4}$ functions $f:S\longrightarrow S$ satisfying
$\sum_{x\in T} x^{f(x)}\equiv a$ $(mod$ $p)$.
[i]proposed by Mahyar Sefidgaran[/i]
2015 HMNT, 8
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ that has center $I$. If $IA = 5, IB = 7, IC = 4, ID = 9$, find the value of $\frac{AB}{CD}$.
2015 Germany Team Selection Test, 2
Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$.
Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$.
[i](Notation: $[\cdot]$ denotes the line segment.)[/i]
2017 Puerto Rico Team Selection Test, 6
Find all functions $f: R \to R$ such that $f (xy) \le yf (x) + f (y)$, for all $x, y\in R$.
1986 IMO Shortlist, 5
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.