Found problems: 85335
2015 Brazil Team Selection Test, 1
Let's call a function $f : R \to R$ [i]cool[/i] if there are real numbers $a$ and $b$ such that $f(x + a)$ is an even function and $f(x + b)$ is an odd function.
(a) Prove that every cool function is periodic.
(b) Give an example of a periodic function that is not cool.
2014 Online Math Open Problems, 6
Let $L_n$ be the least common multiple of the integers $1,2,\dots,n$. For example, $L_{10} = 2{,}520$ and $L_{30} = 2{,}329{,}089{,}562{,}800$. Find the remainder when $L_{31}$ is divided by $100{,}000$.
[i]Proposed by Evan Chen[/i]
2009 Dutch Mathematical Olympiad, 2
Consider the sequence of integers $0, 1, 2, 4, 6, 9, 12,...$ obtained by starting with zero, adding $1$, then adding $1$ again, then adding $2$, and adding $2$ again, then adding $3$, and adding $3$ again, and so on. If we call the subsequent terms of this sequence $a_0, a_1, a_2, ...$, then we have $a_0 = 0$, and $a_{2n-1} = a_{2n-2} + n$ , $a_{2n} = a_{2n-1} + n$ for all integers $n \ge 1$.
Find all integers $k \ge 0$ for which $a_k$ is the square of an integer.
2013 Harvard-MIT Mathematics Tournament, 25
The sequence $(z_n)$ of complex numbers satisfies the following properties:
[list]
[*]$z_1$ and $z_2$ are not real.
[*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$.
[*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$.
[*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list]
Find the product of all possible values of $z_1$.
2017 Romania National Olympiad, 1
Let be a surjective function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that if the sequence $ \left( f\left( x_n \right) \right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. Prove that it is continuous.
1983 Tournament Of Towns, (032) O1
A pedestrian walked for $3.5$ hours. In every period of one hour’s duration he walked $5$ kilometres. Is it true that his average speed was $5$ kilometres per hour?
(NN Konstantinov, Moscow)
2023 Purple Comet Problems, 13
In convex quadrilateral $ABCD$, $\angle BAD = \angle BCD = 90^o$, and $BC = CD$. Let $E$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Given that $\angle AED = 123^o$, find the degree measure of $\angle ABD$.
1999 Belarusian National Olympiad, 5
Determine the maximal value of $ k $, such that for positive reals $ a,b $ and $ c $ from inequality $ kabc >a^3+b^3+c^3 $ it follows that $ a,b $ and $ c $ are sides of a triangle.
2023 CUBRMC, 6
Find the sum of all positive divisors of $40081$.
2014 France Team Selection Test, 2
Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.
2018 Azerbaijan Junior NMO, 4
A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$
2020 AMC 8 -, 20
A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
$$
\begingroup
\setlength{\tabcolsep}{10pt}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c|}
\hline Tree 1 & \rule{0.4cm}{0.15mm} meters \\
Tree 2 & 11 meters \\
Tree 3 & \rule{0.5cm}{0.15mm} meters \\
Tree 4 & \rule{0.5cm}{0.15mm} meters \\
Tree 5 & \rule{0.5cm}{0.15mm} meters \\ \hline
Average height & \rule{0.5cm}{0.15mm}\text{ .}2 meters \\
\hline
\end{tabular}
\endgroup$$
$\newline \textbf{(A) }22.2 \qquad \textbf{(B) }24.2 \qquad \textbf{(C) }33.2 \qquad \textbf{(D) }35.2 \qquad \textbf{(E) }37.2$
1993 Tournament Of Towns, (367) 6
The width of a long winding river is not greater than $1$ km. This means by definition that from any point of each bank of the river one can reach the other bank swimming $1$ km or less. Is it true that a boat can move along the river so that its distances from both banks are never greater than
(a) $0.7$ km?
(b) $0.8$ km?
(Grigory Kondakov, Moscow)
You may assume that the banks consist of segments and arcs of circles.
2025 CMIMC Team, 5
Suppose we have a uniformly random function from $\{1, 2, 3, \ldots, 25\}$ to itself. Find the expected value of $$\sum_{x=1}^{25} (f(f(x))-x)^2.$$
2014-2015 SDML (High School), 10
A circle is inscribed in an equilateral triangle. Three nested sequences of circles are then constructed as follows: each circle touches the previous circle and has two edges of the triangle as tangents. This is represented by the figure below.
[asy]
import olympiad;
pair A, B, C;
A = dir(90);
B = dir(210);
C = dir(330);
draw(A--B--C--cycle);
draw(incircle(A,B,C));
draw(incircle(A,2/3*A+1/3*B,2/3*A+1/3*C));
draw(incircle(A,8/9*A+1/9*B,8/9*A+1/9*C));
draw(incircle(A,26/27*A+1/27*B,26/27*A+1/27*C));
draw(incircle(A,80/81*A+1/81*B,80/81*A+1/81*C));
draw(incircle(A,242/243*A+1/243*B,242/243*A+1/243*C));
draw(incircle(B,2/3*B+1/3*A,2/3*B+1/3*C));
draw(incircle(B,8/9*B+1/9*A,8/9*B+1/9*C));
draw(incircle(B,26/27*B+1/27*A,26/27*B+1/27*C));
draw(incircle(B,80/81*B+1/81*A,80/81*B+1/81*C));
draw(incircle(B,242/243*B+1/243*A,242/243*B+1/243*C));
draw(incircle(C,2/3*C+1/3*B,2/3*C+1/3*A));
draw(incircle(C,8/9*C+1/9*B,8/9*C+1/9*A));
draw(incircle(C,26/27*C+1/27*B,26/27*C+1/27*A));
draw(incircle(C,80/81*C+1/81*B,80/81*C+1/81*A));
draw(incircle(C,242/243*C+1/243*B,242/243*C+1/243*A));
[/asy]
What is the ratio of the area of the largest circle to the combined area of all the other circles?
$\text{(A) }\frac{8}{1}\qquad\text{(B) }\frac{8}{3}\qquad\text{(C) }\frac{9}{1}\qquad\text{(D) }\frac{9}{3}\qquad\text{(E) }\frac{10}{3}$
ICMC 7, 2
Fredy starts at the origin of the Euclidean plane. Each minute, Fredy may jump a positive integer distance to another lattice point, provided the jump is not parallel to either axis. Can Fredy reach any given lattice point in 2023 jumps or less?
[i]Proposed by Tony Wang[/i]
2015 India Regional MathematicaI Olympiad, 5
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I.$ Let the internal angle bisectors of $\angle A,\angle B,\angle C$ meet $\Gamma$ in $A',B',C'$ respectively. Let $B'C'$ intersect $AA'$ at $P,$ and $AC$ in $Q.$ Let $BB'$ intersect $AC$ in $R.$ Suppose the quadrilateral $PIRQ$ is a kite; that is, $IP=IR$ and $QP=QR.$ Prove that $ABC$ is an equilateral triangle.
2018 Taiwan TST Round 1, 2
Given a scalene triangle $ \triangle ABC $. $ B', C' $ are points lie on the rays $ \overrightarrow{AB}, \overrightarrow{AC} $ such that $ \overline{AB'} = \overline{AC}, \overline{AC'} = \overline{AB} $. Now, for an arbitrary point $ P $ in the plane. Let $ Q $ be the reflection point of $ P $ w.r.t $ \overline{BC} $. The intersections of $ \odot{\left(BB'P\right)} $ and $ \odot{\left(CC'P\right)} $ is $ P' $ and the intersections of $ \odot{\left(BB'Q\right)} $ and $ \odot{\left(CC'Q\right)} $ is $ Q' $. Suppose that $ O, O' $ are circumcenters of $ \triangle{ABC}, \triangle{AB'C'} $ Show that
1. $ O', P', Q' $ are colinear
2. $ \overline{O'P'} \cdot \overline{O'Q'} = \overline{OA}^{2} $
1985 IMO Longlists, 32
A collection of $2n$ letters contains $2$ each of $n$ different letters. The collection is partitioned into $n$ pairs, each pair containing $2$ letters, which may be the same or different. Denote the number of distinct partitions by $u_n$. (Partitions differing in the order of the pairs in the partition or in the order of the two letters in the pairs are not considered distinct.) Prove that $u_{n+1}=(n+1)u_n - \frac{n(n-1)}{2} u_{n-2}.$
[i]Similar Problem :[/i]
A pack of $2n$ cards contains $n$ pairs of $2$ identical cards. It is shuffled and $2$ cards are dealt to each of $n$ different players. Let $p_n$ be the probability that every one of the $n$ players is dealt two identical cards. Prove that $\frac{1}{p_{n+1}}=\frac{n+1}{p_n} + \frac{n(n-1)}{2p_{n-2}}.$
2001 Junior Balkan Team Selection Tests - Moldova, 6
Let the nonnegative numbers $a_1, a_2,... a_9$, where $a_1 = a_9 = 0$ and let at least one of the numbers is nonzero.
Denote the sentence $(P)$: '' For $2 \le i \le 8$ there is a number $a_i$, such that $a_{i - 1} + a_{i + 1} <ka_i $”.
a) Show that the sentence $(P)$ is true for $k = 2$.
b) Determine whether is the sentence $(P)$ true for $k = \frac{19}{10}$
2010 Kyrgyzstan National Olympiad, 2
Fifteen pairwise coprime positive integers chosen so that each of them less than 2010. Show that at least one of them is prime.
1971 AMC 12/AHSME, 24
[asy]
label("$1$",(0,0),S);
label("$1$",(-1,-1),S);
label("$1$",(-2,-2),S);
label("$1$",(-3,-3),S);
label("$1$",(-4,-4),S);
label("$1$",(1,-1),S);
label("$1$",(2,-2),S);
label("$1$",(3,-3),S);
label("$1$",(4,-4),S);
label("$2$",(0,-2),S);
label("$3$",(-1,-3),S);
label("$3$",(1,-3),S);
label("$4$",(-2,-4),S);
label("$4$",(2,-4),S);
label("$6$",(0,-4),S);
label("etc.",(0,-5),S);
//Credit to chezbgone2 for the diagram[/asy]
Pascal's triangle is an array of positive integers(See figure), in which the first row is $1$, the second row is two $1$'s, each row begins and ends with $1$, and the $k^\text{th}$ number in any row when it is not $1$, is the sum of the $k^\text{th}$ and $(k-1)^\text{th}$ numbers in the immediately preceding row. The quotient of the number of numbers in the first $n$ rows which are not $1$'s and the number of $1$'s is
$\textbf{(A) }\dfrac{n^2-n}{2n-1}\qquad\textbf{(B) }\dfrac{n^2-n}{4n-2}\qquad\textbf{(C) }\dfrac{n^2-2n}{2n-1}\qquad\textbf{(D) }\dfrac{n^2-3n+2}{4n-2}\qquad \textbf{(E) }\text{None of these}$
2007 Estonia Math Open Senior Contests, 1
Let $ a_n \equal{} 1 \plus{} 2 \plus{} ... \plus{} n$ for every $ n \ge 1$; the numbers $ a_n$ are called triangular. Prove that if $ 2a_m \equal{} a_n$ then $ a_{2m \minus{} n}$ is a perfect square.
1997 Belarusian National Olympiad, 4
$$Problem 4 $$
Straight lines $k,l,m$ intersecting each other in three different points are drawn on a classboard. Bob remembers that in some coordinate system the lines$ k,l,m$ have the equations $y = ax, y = bx$ and $y = c +2\frac{ab}{a+b}x$ (where $ab(a + b)$ is non zero).
Misfortunately, both axes are erased. Also, Bob remembers that there is missing a line $n$ ($y = -ax + c$), but he has forgotten $a,b,c$. How can he reconstruct the line $n$?
1992 IMO Shortlist, 8
Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions:
[i](i)[/i] its side lengths are $ 1, 2, 3, \ldots, 1992$ in some order;
[i](ii)[/i] the polygon is circumscribable about a circle.
[i]Alternative formulation:[/i] Does there exist a 1992-gon with side lengths $ 1, 2, 3, \ldots, 1992$ circumscribed about a circle? Answer the same question for a 1990-gon.