Found problems: 85335
2002 AMC 10, 13
Participation in the local soccer league this year is $10\%$ higher than last year. The number of males increased by $5\%$ and the number of females increased by $20\%$. What fraction of the soccer league is now female?
$\textbf{(A) }\dfrac13\qquad\textbf{(B) }\dfrac4{11}\qquad\textbf{(C) }\dfrac25\qquad\textbf{(D) }\dfrac49\qquad\textbf{(E) }\dfrac12$
2022 OMpD, 2
Let $ABCD$ be a rectangle. The point $E$ lies on side $ \overline{AB}$ and the point $F$ is lies side $ \overline{AD}$, such that $\angle FEC=\angle CEB$ and $\angle DFC=\angle CFE$. Determine the measure of the angle $\angle FCE$ and the ratio $AD/AB$.
2020/2021 Tournament of Towns, P1
Let us say that a circle intersects a quadrilateral [i]properly[/i] if it intersects each of the quadrilateral’s sides at two distinct interior points. Is it true that for each convex quadrilateral there exists a circle which intersects it properly?
[i]Alexandr Perepechko[/i]
2004 Croatia National Olympiad, Problem 3
The altitudes of a tetrahedron meet at a single point. Prove that this point, the centroid of one face of the tetrahedron, the foot of the altitude on that face, and the three points dividing the other three altitudes in ratio $2:1$ (closer to the feet) all lie on a sphere.
1986 Greece Junior Math Olympiad, 1
Find all pairs of integers $(x,y)$ such that $$(x+1)(y+1)(x+y)(x^2+y^2)=16x^2y^2$$
2002 AMC 10, 25
When $ 15$ is appended to a list of integers, the mean is increased by $ 2$. When $ 1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $ 1$. How many integers were in the original list?
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
2006 MOP Homework, 3
Let $X=\{A_{1},...,A_{n}\}$ be a set of distinct 3-element subsets of the set $\{1,2,...,36\}$ such that
(a) $A_{i},A_{j}$ have nonempty intersections for all $i,j$
(b) The intersection of all elements of $X$ is the empty set.
Show that $n\leq 100$.
Determine the number of such sets $X$ when $n=100$
1977 Vietnam National Olympiad, 5
The real numbers $a_0, a_1, ... , a_{n+1}$ satisfy $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_k + a_{k+1}| \le 1$ for $k = 1, 2, ... , n$. Show that $|a_k| \le \frac{ k(n + 1 - k)}{2}$ for all $k$.
2007 Nordic, 1
Find a solution to the equation $x^2-2x-2007y^2=0$ in positive integers.
2002 Portugal MO, 6
On March $6$, $2002$, the celebrations of the $500$th anniversary of the birth of by mathematician Pedro Nunes. That morning, only ten people entered the Viva bookstore for science. Each of these people bought exactly $3$ different books. Furthermore, any two people bought at least one copy of the same book. The Adventures of Mathematics by Pedro Nunes was one of the books that achieved the highest number of sales in this morning. What is the smallest value this number could have taken?
2007 ITest, 43
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers: \begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} She notes that two of them have exactly $8$ positive divisors each. Find the common prime divisor of those two integers.
2016 Purple Comet Problems, 22
In $\triangle{ABC}$, $cos\angle{A} =\frac{2}{3}$, $cos\angle{B} =\frac{1}{9}$, and $BC = 24$. Find the length $AC$.
2012 Pan African, 1
The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?
2021 AIME Problems, 10
Consider the sequence $(a_k)_{k\ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k\ge 1$, if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then
\[a_{k+1} = \frac{m + 18}{n+19}.\]
Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\frac{t}{t+1}$ for some positive integer $t$.
2020 Dutch IMO TST, 4
Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.
2016 BMT Spring, 8
A regular unit $7$-simplex is a polytope in $7$-dimensional space with $8$ vertices that are all exactly a distance of $ 1$ apart. (It is the $7$-dimensional analogue to the triangle and the tetrahedron.) In this $7$-dimensional space, there exists a point that is equidistant from all $8$ vertices, at a distance $d$. Determine $d$.
1982 IMO Longlists, 21
Al[u][b]l[/b][/u] edges and all diagonals of regular hexagon $A_1A_2A_3A_4A_5A_6$ are colored blue or red such that each triangle $A_jA_kA_m, 1 \leq j < k < m\leq 6$ has at least one red edge. Let $R_k$ be the number of red segments $A_kA_j, (j \neq k)$. Prove the inequality
\[\sum_{k=1}^6 (2R_k-7)^2 \leq 54.\]
2004 India IMO Training Camp, 2
Define a function $g: \mathbb{N} \mapsto \mathbb{N}$ by the following rule:
(a) $g$ is nondecrasing
(b) for each $n$, $g(n)$ i sthe number of times $n$ appears in the range of $g$,
Prove that $g(1) = 1$ and $g(n+1) = 1 + g( n +1 - g(g(n)))$ for all $n \in \mathbb{N}$
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 \( n \).
1937 Moscow Mathematical Olympiad, 035
Given three points that are not on the same straight line. Three circles pass through each pair of the points so that the tangents to the circles at their intersection points are perpendicular to each other. Construct the circles.
2012 AMC 12/AHSME, 22
Distinct planes $p_1,p_2,....,p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $ P =\bigcup_{j=1}^{k}p_{j} $. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$?
$ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 24 $
1992 National High School Mathematics League, 12
The maximum value of function $f(x)=\sqrt{x^4-3x^2-6x+13}-\sqrt{x^4-x^2+1}$ is________.
2012 Baltic Way, 5
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ for which
\[f(x + y) = f(x - y) + f(f(1 - xy))\]
holds for all real numbers $x$ and $y$.
2013 AMC 10, 3
Square $ ABCD $ has side length $ 10 $. Point $ E $ is on $ \overline{BC} $, and the area of $ \bigtriangleup ABE $ is $ 40 $. What is $ BE $?
$\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad $
[asy]
pair A,B,C,D,E;
A=(0,0);
B=(0,50);
C=(50,50);
D=(50,0);
E = (30,50);
draw(A--B);
draw(B--E);
draw(E--C);
draw(C--D);
draw(D--A);
draw(A--E);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
label("A",A,SW);
label("B",B,NW);
label("C",C,NE);
label("D",D,SE);
label("E",E,N);
[/asy]
1963 Swedish Mathematical Competition., 3
What is the remainder on dividing $1234^{567} + 89^{1011}$ by $12$?