Found problems: 85335
2020 New Zealand MO, 8
For a positive integer $x$, define a sequence $a_0, a_1, a_2, . . .$ according to the following rules:
$a_0 = 1$, $a_1 = x + 1$ and $$a_{n+2} = xa_{n+1} - a_n$$ for all $n \ge 0$.
Prove that there exist infinitely many positive integers x such that this sequence does not contain a prime number.
2006 India National Olympiad, 1
In a non equilateral triangle $ABC$ the sides $a,b,c$ form an arithmetic progression. Let $I$ be the incentre and $O$ the circumcentre of the triangle $ABC$. Prove that
(1) $IO$ is perpendicular to $BI$;
(2) If $BI$ meets $AC$ in $K$, and $D$, $E$ are the midpoints of $BC$, $BA$ respectively then $I$ is the circumcentre of triangle $DKE$.
2023 SG Originals, Q4
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.
MathLinks Contest 3rd, 1
For a triangle $ABC$ and a point $M$ inside the triangle we consider the lines $AM, BM,CM$ which intersect the sides $BC, CA, AB$ in $A_1, B_1, C_1$ respectively. Take $A', B', C'$ to be the intersection points between the lines $AA_1, BB_1, CC_1$ and $B_1C_1, C_1A_1, A_1B_1$ respectively.
a) Prove that the lines $BC', CB'$ and $AA'$ intersect in a point $A_2$;
b) Define similarly points $B_2, C_2$. Find the loci of $M$ such that the triangle $A_1B_1C_1$ is similar with the triangle $A_2B_2C_2$.
1992 All Soviet Union Mathematical Olympiad, 566
Show that for any real numbers $x, y > 1$, we have $$\frac{x^2}{y - 1}+ \frac{y^2}{x - 1} \ge 8$$
2011 AIME Problems, 15
Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.
2023 BMT, 4
A grasshopper is traveling on the coordinate plane, starting at the origin $(0, 0)$. Each hop, the grasshopper chooses to move $1$ unit up, down, left, or right with equal probability. The grasshopper hops $4$ times and stops at point $P$. Compute the probability that it is possible to return to the origin from $P$ in at most $3$ hops.
1978 Putnam, A4
A [i]bypass[/i] operation on a set $S$ is a mapping $B: S\times S \rightarrow S$ with the property $B(B(w, x), B(y,z)) = B(w,z)$ for all $w, x, y, z \in S$.
(a) Prove that $B(a,b)=c$ implies $B(c,c)=c$ when $B$ is a bypass.
(b) Prove that $B(a,b)=c$ implies $B(a,x)=B(c,x)$ for all $x\in S$ when $B$ is a bypass.
(c) Construct a bypass operation $B$ on a finite set S with the following three properties
[list=i]
[*] $B(x,x)=x$ for all $x\in S$.
[*] There exist $d$ and $e$ in $S$ with $B(d,e)=d \ne e.$
[*] There exist $f$ and $g$ in $S$ with $B(f,g)\ne f.$
[/list]
2009 Iran Team Selection Test, 12
$ T$ is a subset of $ {1,2,...,n}$ which has this property : for all distinct $ i,j \in T$ , $ 2j$ is not divisible by $ i$ . Prove that :
$ |T| \leq \frac {4}{9}n + \log_2 n + 2$
2024 Austrian MO National Competition, 4
Let $ABC$ be an obtuse triangle with orthocenter $H$ and centroid $S$. Let $D$, $E$ and $F$ be the midpoints of segments $BC$, $AC$, $AB$, respectively.
Show that the circumcircle of triangle $ABC$, the circumcircle of triangle $DEF$ and the
circle with diameter $HS$ have two distinct points in common.
[i](Josef Greilhuber)[/i]
2021 Junior Balkan Team Selection Tests - Moldova, 2
Inside the parallelogram $ABCD$, point $E$ is chosen, such that $AE = DE$ and $\angle ABE = 90^o$. Point $F$ is the midpoint of the side $BC$ . Find the measure of the angle $\angle DFE$.
2023 Math Prize for Girls Problems, 11
A random triangle is produced as follows. A pair of standard dice is rolled independently three times to get three random numbers between 2 and 12, inclusive, by adding the numbers that come up on each pair rolled. Call these three random numbers $a$, $b$, and $t$. The random triangle has two sides of lengths $a$ and $b$ with the angle between them measuring $15(t - 1)$ degrees. What is the probability that the triangle is a right triangle?
2023 Belarusian National Olympiad, 10.2
A positive integers has exactly $81$ divisors, which are located in a $9 \times 9$ table such that for any two numbers in the same row or column one of them is divisible by the other one.
Find the maximum possible number of distinct prime divisors of $n$
2003 JHMMC 8, 4
A number plus $4$ is $2003$. What is the number?
1984 IMO Longlists, 31
Let $f_1(x) = x^3+a_1x^2+b_1x+c_1 = 0$ be an equation with three positive roots $\alpha>\beta>\gamma > 0$. From the equation $f_1(x) = 0$, one constructs the equation $f_2(x) = x^3 +a_2x^2 +b_2x+c_2 = x(x+b_1)^2 -(a_1x+c_1)^2 = 0$. Continuing this process, we get equations $f_3,\cdots, f_n$. Prove that
\[\lim_{n\to\infty}\sqrt[2^{n-1}]{-a_n} = \alpha\]
1989 USAMO, 5
Let $u$ and $v$ be real numbers such that
\[ (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. \]
Determine, with proof, which of the two numbers, $u$ or $v$, is larger.
1987 Bundeswettbewerb Mathematik, 1
Let $p>3$ be a prime and $n$ a positive integer such that $p^n$ has $20$ digits. Prove that at least one digit appears more than twice in this number.
LMT Speed Rounds, 2010.18
Let $l$ be a line and $A$ be a point such that $A$ is not on $l.$ Let $P$ be a point on $l$ such that segment $AP$ and line $l$ for a $60^{\circ}$ angle and $AP=1.$ Extend segment $AP$ past $P$ to a point $B$ on the other side of $l.$ Then, let the perpendicular from $B$ to $l$ have foot $M,$ and extend $BM$ past $M$ to $C.$ Finally, extend $CP$ past $P$ to $D.$ Given that $\frac{BP}{AP}=\frac{CM}{BM}=\frac{DP}{CP}=2,$ determine the are of triangle $BPD.$
2012 Tournament of Towns, 3
In the parallelogram $ABCD$, the diagonal $AC$ touches the incircles of triangles $ABC$ and $ADC$ at $W$ and $Y$ respectively, and the diagonal $BD$ touches the incircles of triangles $BAD$ and $BCD$ at $X$ and $Z$ respectively. Prove that either $W,X, Y$ and $Z$ coincide, or $WXYZ$ is a rectangle.
1999 Harvard-MIT Mathematics Tournament, 8
Squares $ABKL$, $BCMN$, $CAOP$ are drawn externally on the sides of a triangle $ABC$. The line segments $KL$, $MN$, $OP$, when extended, form a triangle $A'B'C'$. Find the area of $A'B'C'$ if $ABC$ is an equilateral triangle of side length $2$.
2022 HMNT, 2
Alice is thinking of a positive real number $x$, and Bob is thinking of a positive real number $y$. Given that $x^{\sqrt{y}}=27$ and $(\sqrt{x})^y=9$, compute $xy$.
1990 All Soviet Union Mathematical Olympiad, 522
Two grasshoppers sit at opposite ends of the interval $[0, 1]$. A finite number of points (greater than zero) in the interval are marked. A move is for a grasshopper to select a marked point and jump over it to the equidistant point the other side. This point must lie in the interval for the move to be allowed, but it does not have to be marked. What is the smallest $n$ such that if each grasshopper makes $n$ moves or less, then they end up with no marked points between them?
2012-2013 SDML (Middle School), 4
A bucket filled with $25$ identical blocks weighs $35$ pounds. After three of the blocks are removed, the bucket of blocks weighs $31$ pounds. What is the weight in pounds of the empty bucket?
$\text{(A) }\frac{2}{3}\text{ lbs}\qquad\text{(B) }1\frac{1}{3}\text{ lbs}\qquad\text{(C) }1\frac{2}{3}\text{ lbs}\qquad\text{(D) }2\frac{1}{3}\text{ lbs}\qquad\text{(E) }2\frac{2}{3}\text{ lbs}$
KoMaL A Problems 2024/2025, A. 906
Let $\mathcal{V}_c$ denote the infinite parallel ruler with the parallel edges being at distance $c$ from each other. The following construction steps are allowed using ruler $\mathcal V_c$:
[list]
[*] the line through two given points;
[*] line $\ell'$ parallel to a given line $\ell $at distance $c$ (there are two such lines, both of which can be constructed using this step);
[*] for given points $A$ and $B$ with $|AB|\ge c$ two parallel lines at distance $c$ such that one of them passes through $A$, and the other one passes through $B$ (if $|AB|>c$, there exists two such pairs of parallel lines, and both can be constructed using this step).
[/list]
On the perimeter of a circular piece of paper three points are given that form a scalene triangle. Let $n$ be a given positive integer. Prove that based on the three points and $n$ there exists $C>0$ such that for any $0<c\le C$ it is possible to construct $n$ points using only $\mathcal V_c$ on one of the excircles of the triangle.
[i]We are not allowed to draw anything outside our circular paper. We can construct on the boundary of the paper; it is allowed to take the intersection point of a line with the boundary of the paper.[/i]
[i]Proposed by Áron Bán-Szabó[/i]
1956 AMC 12/AHSME, 26
Which one of the following combinations of given parts does not determine the indicated triangle?
$ \textbf{(A)}\ \text{base angle and vertex angle; isosceles triangle}$
$ \textbf{(B)}\ \text{vertex angle and the base; isosceles triangle}$
$ \textbf{(C)}\ \text{the radius of the circumscribed circle; equilateral triangle}$
$ \textbf{(D)}\ \text{one arm and the radius of the inscribed circle; right triangle}$
$ \textbf{(E)}\ \text{two angles and a side opposite one of them; scalene triangle}$