Found problems: 85335
2002 IMO, 4
Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.
2005 Alexandru Myller, 3
Let $f:[0,\infty)\to\mathbb R$ be a continuous function s.t. $\lim_{x\to\infty}\frac {f(x)}x=0$. Let $(x_n)_n$ be a sequence of positive real numbers s.t. $\left(\frac{x_n}n\right)_n$ is bounded. Prove that $\lim_{n\to\infty}\frac{f(x_n)}n=0$.
[i]Dorin Andrica, Eugen Paltanea[/i]
2010 Contests, 1
A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$
2022 Chile National Olympiad, 4
In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?
2014 Chile TST IMO, 4
Let \( f(n) \) be a polynomial with integer coefficients. Prove that if \( f(-1) \), \( f(0) \), and \( f(1) \) are not divisible by 3, then \( f(n) \neq 0 \) for all integers \( n \).
2022-IMOC, A3
Find all functions $f:\mathbb R\to \mathbb R$ such that $$xy(f(x+y)-f(x)-f(y))=2f(xy)$$ for all $x,y\in \mathbb R.$
[i]Proposed by USJL[/i]
1996 Swedish Mathematical Competition, 4
The angles at $A,B,C,D,E$ of a pentagon $ABCDE$ inscribed in a circle form an increasing sequence. Show that the angle at $C$ is greater than $\pi/2$, and that this lower bound cannot be improved.
2020 Purple Comet Problems, 9
Let $a, b$, and $c$ be real numbers such that $3^a = 125$, $5^b = 49$,and $7^c = 8$1.
Find the product $abc$.
2023 Sharygin Geometry Olympiad, 4
Points $D$ and $E$ lie on the lateral sides $AB$ and $BC$ respectively of an isosceles triangle $ABC$ in such a way that $\angle BED = 3\angle BDE$. Let $D'$ be the reflection of $D$ about $AC$. Prove that the line $D'E$ passes through the incenter of $ABC$.
2023 CMIMC TCS, 2
After years at sail, you and your crew have found the island that houses the great treasure of Scottybeard, the greatest pirate to ever sail the high seas. The island takes the shape of a unit square, and the treasure (which we treat as a single point) could be buried under any point on the island.
To assist you in finding his treasure, Scotty has left a peculiar instrument. To use this instrument, you may draw any directed line (possibly one that never hits the island!), and the instrument will tell you whether the treasure lies to the "left" or the "right" of the line.*
[asy]
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((1,0.5)--(0.8,0.42),arrow=Arrow());
draw((0.8,0.42)--(0.6,0.34),arrow=Arrow());
draw((0.6,0.34)--(0.4,0.26),arrow=Arrow());
draw((0.4,0.26)--(0.2,0.18),arrow=Arrow());
draw((0.2,0.18)--(0,0.1));
label("``Right''", (0.5,0.55));
label("``Left''", (0.8,0.2));
[/asy]
However, Scotty also left a trap! If the instrument ever reports ``left'' three times in a row or ``right'' three times in a row, the island will suddenly sink into the sea, submerging the treasure forever and drowning you and your crew! You want to avoid this at all costs.
To minimize the amount of energy spent digging, you would like to narrow down the set of possible locations of the treasure to be as small as possible. However, Scotty left one last trick; you can only use the instrument 12 times before it breaks!
Devise an algorithm to use the instrument no more than 12 times that can never result in the island sinking and narrows the worst-case space of possible locations of the treasure to have as small an area as possible.
* [size=75]Where "left" or "right" is taken with respect to an observer walking along the line in its designated direction. There is also a probability zero chance the treasure is precisely on the line; this won't affect anything, but for the sake of clarity let's say the instrument reports "left" in this case.[/size]
[b]Scoring:[/b] An algorithm that achieves a worst-case area of $K$ will be awarded:
[list]
[*] 1 point for any $K<1$
[*] 10 points for $K=\tfrac 14$
[*] 20 points for $\tfrac 1{128}<K<\tfrac 14$
[*] 30 points for $K=\tfrac 1{128}$
[*] 50 points for $K_{\text{min}}<K<\tfrac 1 {128}$
[*] 75 points for $K=K_{\text{min}}$
[*] 100 points for $K=K_{\text{min}}$, with a proof that this is optimal
[/list]
(where $K_{\text{min}}$ is the smallest possible worst-case area, which we are not disclosing to avoid giving anything away)
[i]Proposed by Connor Gordon[/i]
2007 IMO, 4
In triangle $ ABC$ the bisector of angle $ BCA$ intersects the circumcircle again at $ R$, the perpendicular bisector of $ BC$ at $ P$, and the perpendicular bisector of $ AC$ at $ Q$. The midpoint of $ BC$ is $ K$ and the midpoint of $ AC$ is $ L$. Prove that the triangles $ RPK$ and $ RQL$ have the same area.
[i]Author: Marek Pechal, Czech Republic[/i]
2021 Switzerland - Final Round, 5
For which integers $n \ge 2$ can we arrange numbers $1,2, \ldots, n$ in a row, such that for all integers $1 \le k \le n$ the sum of the first $k$ numbers in the row is divisible by $k$?
2016 Belarus Team Selection Test, 3
Point $A,B$ are marked on the right branch of the hyperbola $y=\frac{1}{x},x>0$. The straight line $l$ passing through the origin $O$ is perpendicular to $AB$ and meets $AB$ and given branch of the hyperbola at points $D$ and $C$ respectively. The circle through $A,B,C$ meets $l$ at $F$.
Find $OD:CF$
2000 AMC 10, 6
The Fibonacci Sequence $ 1,1,2,3,5,8,13,21,\ldots$ starts with two 1s and each term afterwards is the sum of its predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci Sequence?
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 9$
2011 Argentina National Olympiad Level 2, 3
Let $ABC$ be a triangle of sides $AB = 15$, $AC = 14$ and $BC = 13$. Let $M$ be the midpoint of side $AB$ and let $I$ be the incenter of triangle $ABC$. The line $MI$ intersects the altitude corresponding to the side $AB$ of triangle $ABC$ at point $P$. Calculate the length of the segment $PC$.
Note: The incenter of a triangle is the intersection point of its angle bisectors.
2012 USA Team Selection Test, 3
Determine all positive integers $n$, $n\ge2$, such that the following statement is true:
If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.
1954 AMC 12/AHSME, 20
The equation $ x^3\plus{}6x^2\plus{}11x\plus{}6\equal{}0$ has:
$ \textbf{(A)}\ \text{no negative real roots} \qquad
\textbf{(B)}\ \text{no positive real roots} \qquad
\textbf{(C)}\ \text{no real roots} \\
\textbf{(D)}\ \text{1 positive and 2 negative roots} \qquad
\textbf{(E)}\ \text{2 positive and 1 negative root}$
2020 Novosibirsk Oral Olympiad in Geometry, 3
Maria Ivanovna drew on the blackboard a right triangle $ABC$ with a right angle $B$. Three students looked at her and said:
$\bullet$ Yura said: "The hypotenuse of this triangle is $10$ cm."
$\bullet$ Roma said: "The altitude drawn from the vertex $B$ on the side $AC$ is $6$ cm."
$\bullet$ Seva said: "The area of the triangle $ABC$ is $25$ cm$^2$."
Determine which of the students was mistaken if it is known that there is exactly one such person.
2025 6th Memorial "Aleksandar Blazhevski-Cane", P4
Let $ABCDE$ be a pentagon such that $\angle DCB < 90^{\circ} < \angle EDC$. The circle with diameter $BD$ intersects the line $BC$ again at $F$, and the circle with diameter $DE$ intersects the line $CE$ again at $G$. Prove that the second intersection ($\neq D$) of the circumcircle of $\triangle DFG$ and the circle with diameter $AD$ lies on $AC$.
Proposed by [i]Petar Filipovski[/i]
1986 Traian Lălescu, 1.2
Let $ K $ be the group of Klein. Prove that:
[b]a)[/b] There is an unique division ring (up to isomorphism), $ D, $ such that $ (D,+)\cong K. $
[b]b)[/b] There are no division rings $ A $ such that $ (A\setminus\{ 0\} ,+)\cong K. $
2004 Germany Team Selection Test, 2
Let two chords $AC$ and $BD$ of a circle $k$ meet at the point $K$, and let $O$ be the center of $k$. Let $M$ and $N$ be the circumcenters of triangles $AKB$ and $CKD$. Show that the quadrilateral $OMKN$ is a parallelogram.
2000 IMO Shortlist, 7
For a polynomial $ P$ of degree 2000 with distinct real coefficients let $ M(P)$ be the set of all polynomials that can be produced from $ P$ by permutation of its coefficients. A polynomial $ P$ will be called [b]$ n$-independent[/b] if $ P(n) \equal{} 0$ and we can get from any $ Q \in M(P)$ a polynomial $ Q_1$ such that $ Q_1(n) \equal{} 0$ by interchanging at most one pair of coefficients of $ Q.$ Find all integers $ n$ for which $ n$-independent polynomials exist.
2023 Yasinsky Geometry Olympiad, 6
In the triangle $ABC$ with sides $AC = b$ and $AB = c$, the extension of the bisector of angle $A$ intersects it's circumcircle at point with $W$. Circle $\omega$ with center at $W$ and radius $WA$ intersects lines $AC$ and $AB$ at points $D$ and $F$, respectively. Calculate the lengths of segments $CD$ and $BF$.
(Evgeny Svistunov)
[img]https://cdn.artofproblemsolving.com/attachments/7/e/3b340afc4b94649992eb2dccda50ca8f3f7d1d.png[/img]
2023 USAMTS Problems, 4
Prove that for any real numbers $1 \leq \sqrt{x} \leq y \leq x^2$, the following system of equations has a real solution $(a, b, c)$: \[a+b+c = \frac{x+x^2+x^4+y+y^2+y^4}{2}\] \[ab+ac+bc = \frac{x^3 + x^5 + x^6 + y^3 + y^5 + y^6}{2}\] \[abc=\frac{x^7+y^7}{2}\]
2010 Sharygin Geometry Olympiad, 4
Projections of two points to the sidelines of a quadrilateral lie on two concentric circles (projections of each point form a cyclic quadrilateral and the radii of circles are different). Prove that this quadrilateral is a parallelogram.