Found problems: 85335
1988 Spain Mathematical Olympiad, 5
A well-known puzzle asks for a partition of a cross into four parts which are to be reassembled into a square. One solution is exhibited on the picture.
[img]https://cdn.artofproblemsolving.com/attachments/9/1/3b8990baf5e37270c640e280c479b788d989ba.png[/img]
Show that there are infinitely many solutions. (Some solutions split the cross into four equal parts!)
1979 IMO Longlists, 67
A circle $C$ with center $O$ on base $BC$ of an isosceles triangle $ABC$ is tangent to the equal sides $AB,AC$. If point $P$ on $AB$ and point $Q$ on $AC$ are selected such that $PB \times CQ = (\frac{BC}{2})^2$, prove that line segment $PQ$ is tangent to circle $C$, and prove the converse.
1997 Bulgaria National Olympiad, 1
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$.
Prove that
$ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$.
2009 Paraguay Mathematical Olympiad, 5
In a triangle $ABC$, let $I$ be its incenter. The distance from $I$ to the segment $BC$ is $4 cm$ and the distance from that point to vertex $B$ is $12 cm$. Let $D$ be a point in the plane region between segments $AB$ and $BC$ such that $D$ is the center of a circumference that is tangent to lines $AB$ and $BC$ and passes through $I$. Find all possible values of the length $BD$.
2003 Junior Tuymaada Olympiad, 7
Through the point $ K $ lying outside the circle $ \omega $, the tangents are drawn $ KB $ and $ KD $ to this circle ($ B $ and $ D $ are tangency points) and a line intersecting a circle at points $ A $ and $ C $. The bisector of angle $ ABC $ intersects the segment $ AC $ at the point $ E $ and circle $ \omega $ at $ F $. Prove that $ \angle FDE = 90^\circ $.
2000 Greece Junior Math Olympiad, 4
Four pupils decided to buy some mathematical books so that
(a) everybody buys exactly three different books , and
(b) every two of the pupils buy exactly one book in common.
What are the greatest and smallest number of different books they can buy?
2010 Moldova Team Selection Test, 1
Let $ p\in\mathbb{R}_\plus{}$ and $ k\in\mathbb{R}_\plus{}$. The polynomial $ F(x)\equal{}x^4\plus{}a_3x^3\plus{}a_2x^2\plus{}a_1x\plus{}k^4$ with real coefficients has $ 4$ negative roots. Prove that $ F(p)\geq(p\plus{}k)^4$
1940 Putnam, B7
Which is greater
$$\sqrt{n}^{\sqrt{n+1}} \;\; \; \text{or}\;\;\; \sqrt{n+1}^{\sqrt{n}}$$
where $n>8?$
2024 Taiwan TST Round 1, C
Let $n \geq 5$ be a positive integer. There are $n$ stars with values $1$ to $n$, respectively. Anya and Becky play a game. Before the game starts, Anya places the $n$ stars in a row in whatever order she wishes. Then, starting from Becky, each player takes the left-most or right-most star in the row. After all the stars have been taken, the player with the highest total value of stars wins; if their total values are the same, then the game ends in a draw. Find all $n$ such that Becky has a winning strategy.
[i]
Proposed by Ho-Chien Chen[/i]
2007 Sharygin Geometry Olympiad, 5
Reconstruct a triangle, given the incenter, the midpoint of some side and the foot of the altitude drawn on this side.
1985 IMO Longlists, 89
Given that $n$ elements $a_1, a_2,\dots, a_n$ are organized into $n$ pairs $P_1, P_2, \dots, P_n$ in such a way that two pairs $P_i, P_j$ share exactly one element when $(a_i, a_j)$ is one of the pairs, prove that every element is in exactly two of the pairs.
2006 China Northern MO, 2
$p$ is a prime number that is greater than $2$. Let $\{ a_{n}\}$ be a sequence such that $ na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}$.
Show that if $a_{1}=5$, the $16 \mid a_{81}$.
2021 JHMT HS, 10
Parallelogram $JHMT$ satisfies $JH=11$ and $HM=6,$ and point $P$ lies on $\overline{MT}$ such that $JP$ is an altitude of $JHMT.$ The circumcircles of $\triangle{HMP}$ and $\triangle{JMT}$ intersect at the point $Q\neq M.$ Let $A$ be the point lying on $\overline{JH}$ and the circumcircle of $\triangle{JMT}.$ If $MQ=10,$ then the perimeter of $\triangle{JAM}$ can be expressed in the form $\sqrt{a}+\tfrac{b}{c},$ where $a, \ b,$ and $c$ are positive integers, $a$ is not divisible by the square of any prime, and $b$ and $c$ are relatively prime. Find $a+b+c.$
2012 AMC 10, 16
Three runners start running simultaneously from the same point on a $500$-meter circular track. They each run clockwise around the course maintaining constant speeds of $4.4$, $4.8$, and $5.0$ meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?
$ \textbf{(A)}\ 1,000
\qquad\textbf{(B)}\ 1,250
\qquad\textbf{(C)}\ 2,500
\qquad\textbf{(D)}\ 5,000
\qquad\textbf{(E)}\ 10,000
$
2015 ASDAN Math Tournament, 9
A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws.
2010 AMC 12/AHSME, 5
Lucky Larry's teacher asked him to substitute numbers for $ a$, $ b$, $ c$, $ d$, and $ e$ in the expression $ a\minus{}(b\minus{}(c\minus{}(d\plus{}e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence. The numbers Larry substituted for $ a$, $ b$, $ c$, and $ d$ were $ 1$, $ 2$, $ 3$, and $ 4$, respectively. What number did Larry substitute for $ e$?
$ \textbf{(A)}\ \minus{}5\qquad\textbf{(B)}\ \minus{}3\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$
2017-2018 SDML (Middle School), 13
In the diagram, two circles, each with center D, have radii of $1$ and $2$. The total area of the shaded region is $\frac{5}{12}$ of the area of the larger circle. How many degrees are in the measure of $\angle ADC$?
[asy]
int angle = 100;
path A = arc(0, 1, 0, angle);
path B = arc(0, 1, angle, 360);
path C = arc(0, 2, 0, angle);
path D = arc(0, 2, angle, 360);
filldraw(C -- origin -- cycle, gray);
filldraw(D -- origin -- cycle, white);
filldraw(A -- origin -- cycle, white);
filldraw(B -- origin -- cycle, gray);
label("$D$", origin, NE);
label("$C$", (2, 0), E);
label("$A$", (2, 0) * dir(angle), N);
[/asy]
$\mathrm{(A) \ } 100 \qquad \mathrm{(B) \ } 105 \qquad \mathrm {(C) \ } 110 \qquad \mathrm{(D) \ } 115 \qquad \mathrm{(E) \ } 120$
2006 Korea - Final Round, 1
In a triangle $ABC$ with $AB\not = AC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ , respectively. Line $AD$ meets the incircle again at $P$ . The line $EF$ and the line through $P$ perpendicular to $AD$ meet at $Q$. Line $AQ$ intersects $DE$ at $X$ and $DF$ at $Y$ . Prove that $A$ is the midpoint of $XY$.
2021 Swedish Mathematical Competition, 4
Give examples of a function $f : R \to R$ that satisfies $0 < f(x) < f(x + f(x)) <\sqrt2 x$, for all positive $x$,
and show that there is no function $f : R \to R$ that satisfies $x < f(x + f(x)) <\sqrt2 f(x)$, for all positive $x$.
2017 All-Russian Olympiad, 8
In a non-isosceles triangle $ABC$,$O$ and $I$ are circumcenter and incenter,respectively.$B^\prime$ is reflection of $B$ with respect to $OI$ and lies inside the angle $ABI$.Prove that the tangents to circumcirle of $\triangle BB^\prime I$ at $B^\prime$,$I$ intersect on $AC$. (A. Kuznetsov)
2023 Junior Balkan Team Selection Tests - Romania, P1
Let $a$ and $b$ be two distinct positive integers with the same parity. Prove that the fraction $\frac{a!+b!}{2^a}$ is not an integer.
2020 CHMMC Winter (2020-21), 11
Let $n \ge 3$ be a positive integer. Suppose that $\Gamma$ is a unit circle passing through a point $A$. A regular $3$-gon, regular $4$-gon, \dots, regular $n$-gon are all inscribed inside $\Gamma$ such that $A$ is a common vertex of all these regular polygons. Let $Q$ be a point on $\Gamma$ such that $Q$ is a vertex of the regular $n$-gon, but $Q$ is not a vertex of any of the other regular polygons. Let $\mathcal{S}_n$ be the set of all such points $Q$. Find the number of integers $3 \le n \le 100$ such that
\[
\prod_{Q \in \mathcal{S}_n} |AQ| \le 2.
\]
1979 IMO Longlists, 73
In a plane a finite number of equal circles are given. These circles are mutually nonintersecting (they may be externally tangent). Prove that one can use at most four colors for coloring these circles so that two circles tangent to each other are of different colors. What is the smallest number of circles that requires four colors?
1978 IMO Longlists, 1
The set $M = \{1, 2, . . . , 2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2, \dots, M_k,$ where $n \ge k^3 + k.$ Prove that there exist even numbers $2j_1, 2j_2, \dots, 2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ $(1 \le i \le k)$ such that the numbers $2j_1 - 1, 2j_2 - 1, \dots, 2j_{k+1} - 1$ are also in one and the same subset $M_j (1 \le j \le k).$
1997 Romania National Olympiad, 3
Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$
a) Prove that there exist nonconstant functions in $\mathcal{F}.$
b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.