Found problems: 85335
2014 HMNT, 10
Let $ABCDEF$ be a convex hexagon with the following properties.
(a) $\overline{AC}$ and $\overline{AE}$ trisect $\angle BAF$.
(b) $\overline{BE} \parallel \overline{CD}$ and $\overline{CF} \parallel \overline{DE}$.
(c) $AB = 2AC = 4AE = 8AF$.
Suppose that quadrilaterals $ACDE$ and $ADEF$ have area $2014$ and $1400$, respectively. Find the area of quadrilateral $ABCD$.
2015 Regional Olympiad of Mexico Southeast, 1
Find all integers $n>1$ such that every prime that divides $n^6-1$ also divides $n^5-n^3-n^2+1$.
1994 AIME Problems, 9
A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$
2012 Princeton University Math Competition, B4
For a set $S$ of integers, define $\max (S)$ to be the maximal element of $S$. How many non-empty subsets $S \subseteq \{1, 2, 3, ... , 10\}$ satisfy $\max (S) \le |S| + 2$?
2013 Kyiv Mathematical Festival, 1
There are $24$ apples in $4$ boxes. An optimistic worm is convinced that he can eat no more than half of the apples such that there will be $3$ boxes with equal number of apples. Is it possible that he is wrong?
2017 Germany Team Selection Test, 2
In a convex quadrilateral $ABCD$, $BD$ is the angle bisector of $\angle{ABC}$. The circumcircle of $ABC$ intersects $CD,AD$ in $P,Q$ respectively and the line through $D$ parallel to $AC$ cuts $AB,AC$ in $R,S$ respectively. Prove that point $P,Q,R,S$ lie on a circle.
1990 Bundeswettbewerb Mathematik, 3
Given any five nonnegative real numbers with the sum $1$, show that they can be arranged around a circle in such a way that the five products of two consecutive numbers sum up to at most $1/5$.
2013 Hong kong National Olympiad, 2
For any positive integer $a$, define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$. Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$.
2006 All-Russian Olympiad Regional Round, 9.6
In an acute triangle $ABC$, the angle bisector$AD$ and altitude $BE$ are drawn. Prove that angle $CED$ is greater than $45^o$.
2020 LMT Fall, B1
Four $L$s are equivalent to three $M$s. Nine $M$s are equivalent to fourteen $T$ s. Seven $T$ s are equivalent to two $W$ s. If Kevin has thirty-six $L$s, how many $W$ s would that be equivalent to?
LMT Guts Rounds, 10
A two digit prime number is such that the sum of its digits is $13.$ Determine the integer.
2018-IMOC, G5
Suppose $I,O,H$ are incenter, circumcenter, orthocenter of $\vartriangle ABC$ respectively. Let $D = AI \cap BC$,$E = BI \cap CA$, $F = CI \cap AB$ and $X$ be the orthocenter of $\vartriangle DEF$. Prove that $IX \parallel OH$.
2019 Saint Petersburg Mathematical Olympiad, 5
Baron Munchhausen has a collection of stones, such that they are of $1000$ distinct whole weights, $2^{1000}$ stones of every weight. Baron states that if one takes exactly one stone of every weight, then the weight of all these $1000$ stones chosen will be less than $2^{1010}$, and there is no other way to obtain this weight by picking another set of stones of the collection.
Can this statement happen to be true?
[i](М. Антипов)[/i]
[hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]
2006 All-Russian Olympiad, 6
Let $K$ and $L$ be two points on the arcs $AB$ and $BC$ of the circumcircle of a triangle $ABC$, respectively, such that $KL\parallel AC$. Show that the incenters of triangles $ABK$ and $CBL$ are equidistant from the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$.
2008 China Team Selection Test, 1
Let $ ABC$ be a triangle, let $ AB > AC$. Its incircle touches side $ BC$ at point $ E$. Point $ D$ is the second intersection of the incircle with segment $ AE$ (different from $ E$). Point $ F$ (different from $ E$) is taken on segment $ AE$ such that $ CE \equal{} CF$. The ray $ CF$ meets $ BD$ at point $ G$. Show that $ CF \equal{} FG$.
1992 Romania Team Selection Test, 10
In a tetrahedron $VABC$, let $I$ be the incenter and $A',B',C'$ be arbitrary points on the edges $AV,BV,CV$, and let $S_a,S_b,S_c,S_v$ be the areas of triangles $VBC,VAC,VAB,ABC$, respectively. Show that points $A',B',C',I$ are coplanar if and only if $\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v$
2016 APMC, 3
Let $a_1,a_2,\cdots$ be a strictly increasing sequence on positive integers.
Is it always possible to partition the set of natural numbers $\mathbb{N}$ into infinitely many subsets with infinite cardinality $A_1,A_2,\cdots$, so that for every subset $A_i$, if we denote $b_1<b_2<\cdots$ be the elements of $A_i$, then for every $k\in \mathbb{N}$ and for every $1\le i\le a_k$, it satisfies $b_{i+1}-b_{i}\le k$?
1908 Eotvos Mathematical Competition, 3
A regular polygon of 10 sides (a regular decagon) may be inscribed in a circle in the following two distinct ways: Divide the circumference into $10$ equal arcs and
(1) join each division point to the next by straight line segments,
(2) join each division point to the next but two by straight line segments. (See figures).
Prove that the difference in the side lengths of these two decagons is equal to the radius of their circumscribed circle.
[img]https://cdn.artofproblemsolving.com/attachments/7/9/41c38d08f4f89e07852942a493df17eaaf7498.png[/img]
2024 Princeton University Math Competition, A3 / B5
Let $\sigma$ be a permutation of the set $S := \{1, 2, \ldots , 100\},$ such that $\sigma(a+b) \equiv \sigma(a)+\sigma(b) \pmod{100}$ whenever $a, b, a + b \in S.$ Denote by $f(s)$ the sum of the distinct values $\sigma(s)$ can take over all possible $\sigma$s satisfying the given condition. What is the nonnegative difference between the maximum and minimum value $f$ takes on when ranging over all $s \in S$?
1993 IberoAmerican, 3
Let $\mathbb{N}^*=\{1,2,\ldots\}$. Find al the functions $f: \mathbb{N}^*\rightarrow \mathbb{N}^*$ such that:
(1) If $x<y$ then $f(x)<f(y)$.
(2) $f\left(yf(x)\right)=x^2f(xy)$ for all $x,y \in\mathbb{N}^*$.
1995 China Team Selection Test, 3
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.)
Does the same result hold with a degree 3 or degree 5 polynomial?
2019 China Team Selection Test, 4
Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that
1) $f(0,x)$ is non-decreasing ;
2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ;
3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ;
4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .
2023 Dutch Mathematical Olympiad, 3
Felix chooses a positive integer as the starting number and writes it on the board. He then repeats the next step: he replaces the number $n$ on the board by $\frac12n$ if $n$ is even and by $n^2 + 3$ if $n$ is odd. For how many choices of starting numbers below $2023$ will Felix never write a number of more than four digits on the board?
1980 IMO Shortlist, 19
Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]
2016 AMC 10, 13
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
$\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$