Found problems: 85335
2018 ELMO Shortlist, 1
Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$?
[i]Proposed by Daniel Liu[/i]
2021 AIME Problems, 2
In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
pair A, B, C, D, E, F;
A = (0,3);
B=(0,0);
C=(11,0);
D=(11,3);
E=foot(C, A, (9/4,0));
F=foot(A, C, (35/4,3));
draw(A--B--C--D--cycle);
draw(A--E--C--F--cycle);
filldraw(A--(9/4,0)--C--(35/4,3)--cycle,gray*0.5+0.5*lightgray);
dot(A^^B^^C^^D^^E^^F);
label("$A$", A, W);
label("$B$", B, W);
label("$C$", C, (1,0));
label("$D$", D, (1,0));
label("$F$", F, N);
label("$E$", E, S);
[/asy]
2013 IFYM, Sozopol, 2
Find the perimeter of the base of a regular triangular pyramid with volume 99 and apothem 6.
2022 CMIMC, 2.4 1.2
A shipping company charges $.30l+.40w+.50h$ dollars to process a right rectangular prism-shaped box with dimensions $l,w,h$ in inches. The customers themselves are allowed to label the three dimensions of their box with $l,w,h$ for the purpose of calculating the processing fee. A customer finds that there are two different ways to label the dimensions of their box $B$ to get a fee of $\$8.10$, and two different ways to label $B$ to get a fee of $\$8.70$. None of the faces of $B$ are squares. Find the surface area of $B$, in square inches.
[i]Proposed by Justin Hsieh[/i]
2022 HMNT, 9
Call a positive integer $n$ quixotic if the value of
\[\operatorname{lcm}(1,2,...,n)\cdot\left(\frac11+\frac12+\frac13+\dots+\frac1n\right)\]is divisible by 45. Compute the tenth smallest quixotic integer.
2011 Princeton University Math Competition, A5 / B8
Let $d(n)$ denote the number of divisors of $n$ (including itself). You are given that \[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\] Find $p(6)$, where $p(x)$ is the unique polynomial with rational coefficients satisfying \[p(\pi) = \sum_{n=1}^{\infty} \frac{d(n)}{n^2}.\]
2022 Moldova EGMO TST, 6
Let $ABC$ be a triangle with $\angle ABC=130$. Point $D$ on side $AC$ is the foot of the perpendicular from $B$. Points $E$ and $F$ are on sides $(AB)$ and $(BC)$ such that $DE=DF$ and $AEFC$ is cyclic. Find $\angle EDF$.
2015 Thailand Mathematical Olympiad, 9
Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$
2011 Irish Math Olympiad, 2
In a tournament with $n$ players, $n$ < 10, each player plays once against each other player scoring 1 point for a win and 0 points for a loss. Draws do not occur. In a particular tournament only one player ended with an odd number of points and was ranked fourth. Determine whether or not this is possible. If so, how many wins did the player have?
2022 IOQM India, 4
Consider the set of all 6-digit numbers consisting of only three digits, $a,b,c$ where $a,b,c$ are distinct. Suppose the sum of all these numbers is $593999406$. What is the largest remainder when the three digit number $abc$ is divided by $100$?
2015 BMT Spring, P2
Suppose that fixed circle $C_1$ with radius $a > 0$ is tangent to the fixed line $\ell$ at $A$. Variable circle $C_2$, with center $X$, is externally tangent to $C_1$ at $B \ne A$ and $\ell$ at $C$. Prove that the set of all $X$ is a parabola minus a point
1996 Dutch Mathematical Olympiad, 4
A line $l$ intersects the segment $AB$ perpendicular to $C$. Three circles are drawn successively with $AB, AC$ and $BC$ as the diameter. The largest circle intersects $l$ in $D$. The segments $DA$ and $DB$ still intersect the two smaller circles in $E$ and $F$.
a. Prove that quadrilateral $CFDE$ is a rectangle.
b. Prove that the line through $E$ and $F$ touches the circles with diameters $AC$ and $BC$ in $E$ and $F$.
[asy]
unitsize (2.5 cm);
pair A, B, C, D, E, F, O;
O = (0,0);
A = (-1,0);
B = (1,0);
C = (-0.3,0);
D = intersectionpoint(C--(C + (0,1)), Circle(O,1));
E = (C + reflect(A,D)*(C))/2;
F = (C + reflect(B,D)*(C))/2;
draw(Circle(O,1));
draw(Circle((A + C)/2, abs(A - C)/2));
draw(Circle((B + C)/2, abs(B - C)/2));
draw(A--B);
draw(interp(C,D,-0.4)--D);
draw(A--D--B);
dot("$A$", A, W);
dot("$B$", B, dir(0));
dot("$C$", C, SE);
dot("$D$", D, NW);
dot("$E$", E, SE);
dot("$F$", F, SW);
[/asy]
1989 IMO Longlists, 17
Let $ a \in \mathbb{R}, 0 < a < 1,$ and $ f$ a continuous function on $ [0, 1]$ satisfying $ f(0) \equal{} 0, f(1) \equal{} 1,$ and
\[ f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.\]
Determine $ f \left( \frac{1}{7} \right).$
1958 Miklós Schweitzer, 6
[b]6.[/b] Prove that if $a_n \geq 0$ and
$\frac{1}{n}\sum_{k=1}^{n} a_k \geq \sum_{k=n+1}^{2n}a_k$ $(n=1, 2, \dots)$ ,
then $\sum_{k=1}^{\infty} a_k $ is convergent and its sum is less than $2ea_1$. [b](S. 9)[/b]
1990 Bundeswettbewerb Mathematik, 1
Consider the trinomial $f(x) = x^2 + 2bx + c$ with integer coefficients $b$ and $c$.
Prove that if $f(n) \ge 0$ for all integers $n$, then $f(x) \ge 0$ even for all rational numbers $x$.
2011 Postal Coaching, 5
Let $(a_n )_{n\ge 1}$ be a sequence of integers that satisfies
\[a_n = a_{n-1} -\text{min}(a_{n-2} , a_{n-3} )\]
for all $n \ge 4$. Prove that for every positive integer $k$, there is an $n$ such that $a_n$ is divisible by $3^k$ .
2021 Turkey MO (2nd round), 2
If a polynomial with real coefficients of degree $d$ has at least $d$ coefficients equal to $1$ and has $d$ real roots, what is the maximum possible value of $d$?
(Note: The roots of the polynomial do not have to be different from each other.)
2025 India STEMS Category A, 6
Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots.
(Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$).
[i]Proposed by Malay Mahajan[/i]
V Soros Olympiad 1998 - 99 (Russia), 9.1
In the phrase given at the end of the condition of the problem, it is necessary to put a number (numeral) in place of the ellipsis, written in verbal form and in the required case, so that the statement formulated in it is true. Here is this phrase: “The number of letters in this phrase is...”
2004 Germany Team Selection Test, 1
Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations:
$x_{1}+2x_{2}+...+nx_{n}=0$,
$x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$,
...
$x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.
2020 Durer Math Competition Finals, 10
Soma has a tower of $63$ bricks , consisting of $6$ levels. On the $k$-th level from the top, there are $2k-1$ bricks (where $k = 1, 2, 3, 4, 5, 6$), and every brick which is not on the lowest level lies on precisely $2$ smaller bricks (which lie one level below) - see the figure. Soma takes away $7$ bricks from the tower, one by one. He can only remove a brick if there is no brick lying on it. In how many ways can he do this, if the order of removals is considered as well?
[img]https://cdn.artofproblemsolving.com/attachments/b/6/4b0ce36df21fba89708dd5897c43a077d86b5e.png[/img]
2007 Mathematics for Its Sake, 2
For a given natural number $ n\ge 2, $ find all $ \text{n-tuples} $ of nonnegative real numbers which have the property that each one of the numbers forming the $ \text{n-tuple} $ is the square of the sum of the other $ n-1 $ ones.
[i]Mugur Acu[/i]
2019 Balkan MO Shortlist, G2
Let be a triangle $\triangle ABC$ with $m(\angle ABC) = 75^{\circ}$ and $m(\angle ACB) = 45^{\circ}$. The angle bisector of $\angle CAB$ intersects $CB$ at point $D$. We consider the point $E \in (AB)$, such that $DE = DC$. Let $P$ be the intersection of lines $AD$ and $CE$. Prove that $P$ is the midpoint of segment $AD$.
2017 Macedonia JBMO TST, 5
Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.
2010 ISI B.Math Entrance Exam, 1
Prove that in each year , the $13^{th}$ day of some month occurs on a Friday .