Found problems: 85335
1954 AMC 12/AHSME, 49
The difference of the squares of two odd numbers is always divisible by $ 8$. If $ a>b$, and $ 2a\plus{}1$ and $ 2b\plus{}1$ are the odd numbers, to prove the given statement we put the difference of the squares in the form:
$ \textbf{(A)}\ (2a\plus{}1)^2\minus{}(2b\plus{}1)^2 \\
\textbf{(B)}\ 4a^2\minus{}4b^2\plus{}4a\minus{}4b \\
\textbf{(C)}\ 4[a(a\plus{}1)\minus{}b(b\plus{}1)] \\
\textbf{(D)}\ 4(a\minus{}b)(a\plus{}b\plus{}1) \\
\textbf{(E)}\ 4(a^2\plus{}a\minus{}b^2\minus{}b)$
2023 Lusophon Mathematical Olympiad, 2
Let $D$ be a point on the inside of triangle $ABC$ such that $AD=CD$, $\angle DAB=70^{\circ}$, $\angle DBA=30^{\circ}$ and $\angle DBC=20^{\circ}$. Find the measure of angle $\angle DCB$.
2012 ELMO Shortlist, 4
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
2003 Gheorghe Vranceanu, 4
Prove that among any $ 16 $ numbers smaller than $ 101 $ there are four of them that have the property that the sum of two of them is equal to the sum of the other two.
2023 Estonia Team Selection Test, 3
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
1996 Putnam, 5
Given a finite binary string $S$ of symbols $X,O$ we define $\Delta(S)=n(X)-n(O)$ where $n(X),n(O)$ respectively denote number of $X$'s and $O$'s in a string. For example $\Delta(XOOXOOX)=3-4=-1$. We call a string $S$ $\emph{balanced}$ if every substring $T$ of $S$ has $-2\le \Delta(T)\le 2$. Find number of balanced strings of length $n$.
Kyiv City MO 1984-93 - geometry, 1992.7.2
Inside a right angle is given a point $A$. Construct an equilateral triangle, one of the vertices of which is point $A$, and two others lie on the sides of the angle (one on each side).
2020 Canadian Junior Mathematical Olympiad, 1
Let $a_1, a_2, a_3, . . .$ be a sequence of positive real numbers that satisfies $a_1 = 1$ and $a^2_{n+1} + a_{n+1} = a_n$ for every natural number $n$. Prove that $a_n \ge \frac{1}{n}$ for every natural number $n$.
2015 Oral Moscow Geometry Olympiad, 6
In an acute-angled isosceles triangle $ABC$, altitudes $CC_1$ and $BB_1$ intersect the line passing through the vertex $A$ and parallel to the line $BC$, at points $P$ and $Q$. Let $A_0$ be the midpoint of side $BC$, and $AA_1$ the altitude. Lines $A_0C_1$ and $A_0B_1$ intersect line $PQ$ at points $K$ and $L$. Prove that the circles circumscribed around triangles $PQA_1, KLA_0, A_1B_1C_1$ and a circle with a diameter $AA_1$ intersect at one point.
1966 IMO Longlists, 16
We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XY Z$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q.$
2009 Purple Comet Problems, 13
How many subsets of the set $\{1, 2, 3, \ldots, 12\}$ contain exactly one or two prime numbers?
2003 Iran MO (3rd Round), 23
Find all homogeneous linear recursive sequences such that there is a $ T$ such that $ a_n\equal{}a_{n\plus{}T}$ for each $ n$.
2011 Albania Team Selection Test, 3
In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$.
[b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$.
[b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.
2010 F = Ma, 19
Consider the following graphs of position [i]vs.[/i] time.
[asy]
size(500);
picture pic;
// Rectangle
draw(pic,(0,0)--(20,0)--(20,15)--(0,15)--cycle);
label(pic,"0",(0,0),S);
label(pic,"2",(4,0),S);
label(pic,"4",(8,0),S);
label(pic,"6",(12,0),S);
label(pic,"8",(16,0),S);
label(pic,"10",(20,0),S);
label(pic,"-15",(0,2),W);
label(pic,"-10",(0,4),W);
label(pic,"-5",(0,6),W);
label(pic,"0",(0,8),W);
label(pic,"5",(0,10),W);
label(pic,"10",(0,12),W);
label(pic,"15",(0,14),W);
label(pic,rotate(90)*"x (m)",(-2,7),W);
label(pic,"t (s)",(11,-2),S);
// Tick Marks
draw(pic,(4,0)--(4,0.3));
draw(pic,(8,0)--(8,0.3));
draw(pic,(12,0)--(12,0.3));
draw(pic,(16,0)--(16,0.3));
draw(pic,(20,0)--(20,0.3));
draw(pic,(4,15)--(4,14.7));
draw(pic,(8,15)--(8,14.7));
draw(pic,(12,15)--(12,14.7));
draw(pic,(16,15)--(16,14.7));
draw(pic,(20,15)--(20,14.7));
draw(pic,(0,2)--(0.3,2));
draw(pic,(0,4)--(0.3,4));
draw(pic,(0,6)--(0.3,6));
draw(pic,(0,8)--(0.3,8));
draw(pic,(0,10)--(0.3,10));
draw(pic,(0,12)--(0.3,12));
draw(pic,(0,14)--(0.3,14));
draw(pic,(20,2)--(19.7,2));
draw(pic,(20,4)--(19.7,4));
draw(pic,(20,6)--(19.7,6));
draw(pic,(20,8)--(19.7,8));
draw(pic,(20,10)--(19.7,10));
draw(pic,(20,12)--(19.7,12));
draw(pic,(20,14)--(19.7,14));
// Path
add(pic);
path A=(0,14)--(20,14);
draw(A);
label("I.",(8,-4),3*S);
path B=(0,6)--(20,6);
picture pic2=shift(30*right)*pic;
draw(shift(30*right)*B);
label("II.",(38,-4),3*S);
add(pic2);
path C=(0,12)--(20,14);
picture pic3=shift(60*right)*pic;
draw(shift(60*right)*C);
label("III.",(68,-4),3*S);
add(pic3);
[/asy]
Which of the graphs could be the motion of a particle in the given potential?
(A) $\text{I}$
(B) $\text{III}$
(C) $\text{I and II}$
(D) $\text{I and III}$
(E) $\text{I, II, and III}$
PEN O Problems, 18
Let $p$ be an odd prime number. How many $p$-element subsets $A$ of $\{1,2,\ldots \ 2p\}$ are there, the sum of whose elements is divisible by $p$?
2024 AMC 10, 24
A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^+, A^-, B^+, B^-, C^+$, and $C^-$ is rolled. Suppose the bee occupies the point $(a, b, c)$. If the die shows $A^+$, then the bee moves to the point $(a+1, b, c)$ and if the die shows $A^-$, then the bee moves to the point $(a-1, b, c)$. Analogous moves are made with the other four outcomes. Suppose the bee starts at the point $(0, 0, 0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?
$
\textbf{(A) }\frac{1}{54} \qquad
\textbf{(B) }\frac{7}{54} \qquad
\textbf{(C) }\frac{1}{6} \qquad
\textbf{(D) }\frac{5}{18} \qquad
\textbf{(E) }\frac{2}{5} \qquad
$
2024 May Olympiad, 3
Beto has rectangular chessboard where the number of rows and columns are consecutive numbers (for example, $30$ and $31$). Ana has tiles of two colors and different sizes: the red tiles are $5 \times 7$ rectangles and the blue tiles are $3 \times 5$ rectangles. Ana realized that she can cover all the squares of Beto’s board using only red tiles, which can be rotated, but must not overlap or extend beyond the board. Then, she realized she can also do the same using only blue tiles. What is the minimum number of squares that Beto’s board can have?
2015 IMO, 6
The sequence $a_1,a_2,\dots$ of integers satisfies the conditions:
(i) $1\le a_j\le2015$ for all $j\ge1$,
(ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k<\ell$.
Prove that there exist two positive integers $b$ and $N$ for which\[\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2\]for all integers $m$ and $n$ such that $n>m\ge N$.
[i]Proposed by Ivan Guo and Ross Atkins, Australia[/i]
2019 CMIMC, 2
How many ways are there to color the vertices of a cube red, blue, or green such that no edge connects two vertices of the same color? Rotations and reflections are considered distinct colorings.
1992 IMO Longlists, 10
Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.
2008 Harvard-MIT Mathematics Tournament, 16
Point $ A$ lies at $ (0, 4)$ and point $ B$ lies at $ (3, 8)$. Find the $ x$-coordinate of the point $ X$ on the $ x$-axis maximizing $ \angle AXB$.
2002 AMC 10, 5
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
[asy]unitsize(.3cm);
defaultpen(linewidth(.8pt));
path c=Circle((0,2),1);
filldraw(Circle((0,0),3),grey,black);
filldraw(Circle((0,0),1),white,black);
filldraw(c,white,black);
filldraw(rotate(60)*c,white,black);
filldraw(rotate(120)*c,white,black);
filldraw(rotate(180)*c,white,black);
filldraw(rotate(240)*c,white,black);
filldraw(rotate(300)*c,white,black);[/asy]$ \textbf{(A)}\ \pi \qquad \textbf{(B)}\ 1.5\pi \qquad \textbf{(C)}\ 2\pi \qquad \textbf{(D)}\ 3\pi \qquad \textbf{(E)}\ 3.5\pi$
2013 All-Russian Olympiad, 4
Inside the inscribed quadrilateral $ABCD$ are marked points $P$ and $Q$, such that $\angle PDC + \angle PCB,$ $\angle PAB + \angle PBC,$ $\angle QCD + \angle QDA$ and $\angle QBA + \angle QAD$ are all equal to $90^\circ$. Prove that the line $PQ$ has equal angles with lines $AD$ and $BC$.
[i]A. Pastor[/i]
2007 Belarusian National Olympiad, 4
Each point of a circle is painted in one of the $ N$ colors ($N \geq 2$). Prove that there exists an inscribed trapezoid such that all its vertices are painted the same color.
1966 All Russian Mathematical Olympiad, 081
Given $100$ points on the plane. Prove that you can cover them with a family of circles with the sum of their diameters less than $100$ and the distance between any two of the circles more than one.