Found problems: 85335
2011 IMO Shortlist, 1
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
[i]Proposed by Härmel Nestra, Estonia[/i]
2008 IMO Shortlist, 4
Let $ n$ be a positive integer. Show that the numbers
\[ \binom{2^n \minus{} 1}{0},\; \binom{2^n \minus{} 1}{1},\; \binom{2^n \minus{} 1}{2},\; \ldots,\; \binom{2^n \minus{} 1}{2^{n \minus{} 1} \minus{} 1}\]
are congruent modulo $ 2^n$ to $ 1$, $ 3$, $ 5$, $ \ldots$, $ 2^n \minus{} 1$ in some order.
[i]Proposed by Duskan Dukic, Serbia[/i]
1973 IMO Shortlist, 7
Given a tetrahedron $ABCD$, let $x = AB \cdot CD$, $y = AC \cdot BD$, and $z = AD \cdot BC$. Prove that there exists a triangle with edges $x, y, z.$
2011 NIMO Problems, 3
Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they meet, they are located 10 units from point $B$. Find all possible values for the distance between $A$ and $B$.
[i]Proposed by Isabella Grabski[/i]
2017 IFYM, Sozopol, 6
Let $A_n$ be the number of arranged n-tuples of natural numbers $(a_1,a_2…a_n)$, such that
$\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_n} =1$.
Find the parity of $A_{68}$.
PEN J Problems, 9
Show that the set of all numbers $\frac{\phi(n+1)}{\phi(n)}$ is dense in the set of all positive reals.
2021 Thailand TST, 3
Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set of rows and $C$ a set of columns, is called a [i]saddle pair[/i] if the following two conditions are satisfied:
[list]
[*] $(i)$ For each row $r^{\prime}$, there is $r \in R$ such that $a(r, c) \geqslant a\left(r^{\prime}, c\right)$ for all $c \in C$;
[*] $(ii)$ For each column $c^{\prime}$, there is $c \in C$ such that $a(r, c) \leqslant a\left(r, c^{\prime}\right)$ for all $r \in R$.
[/list]
A saddle pair $(R, C)$ is called a [i]minimal pair[/i] if for each saddle pair $\left(R^{\prime}, C^{\prime}\right)$ with $R^{\prime} \subseteq R$ and $C^{\prime} \subseteq C$, we have $R^{\prime}=R$ and $C^{\prime}=C$. Prove that any two minimal pairs contain the same number of rows.
2001 China Team Selection Test, 1
Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer
2023 Centroamerican and Caribbean Math Olympiad, 6
In a pond there are $n \geq 3$ stones arranged in a circle. A princess wants to label the stones with the numbers $1, 2, \dots, n$ in some order and then place some toads on the stones. Once all the toads are located, they start jumping clockwise, according to the following rule: when a toad reaches the stone labeled with the number $k$, it waits for $k$ minutes and then jumps to the adjacent stone.
What is the greatest number of toads for which the princess can label the stones and place the toads in such a way that at no time are two toads occupying a stone at the same time?
[b]Note:[/b] A stone is considered occupied by two toads at the same time only if there are two toads that are on the stone for at least one minute.
2015 Gulf Math Olympiad, 1
a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$.
b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$.
c) Suppose that $p$ is an odd prime, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$.
d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.
1959 AMC 12/AHSME, 37
When simplified the product $\left(1-\frac13\right)\left(1-\frac14\right)\left(1-\frac15\right)\cdots\left(1-\frac1n\right)$ becomes:
$ \textbf{(A)}\ \frac1n \qquad\textbf{(B)}\ \frac2n\qquad\textbf{(C)}\ \frac{2(n-1)}{n}\qquad\textbf{(D)}\ \frac{2}{n(n+1)}\qquad\textbf{(E)}\ \frac{3}{n(n+1)} $
Kyiv City MO Juniors 2003+ geometry, 2012.9.5
The triangle $ABC$ with $AB> AC$ is inscribed in a circle, the angle bisector of $\angle BAC$ intersects the side $BC$ of the triangle at the point $K$, and the circumscribed circle at the point $M$. The midline of $\Delta ABC$, which is parallel to the side $AB$, intersects $AM$ at the point $O$, the line $CO$ intersects the line $AB$ at the point $N$. Prove that a circle can be circumscribed around the quadrilateral $BNKM$.
(Nagel Igor)
2000 Harvard-MIT Mathematics Tournament, 12
Calculate the number of ways of choosing $4$ numbers from the set ${1,2,\cdots ,11}$ such that at least $2$ of the numbers are consecutive.
2007 Bosnia and Herzegovina Junior BMO TST, 3
Is it possible to place some circles inside a square side length $1$, such that no two circles intersect and the sum of their radii is $2007$?
2018 CMIMC Team, 2-1/2-2
Suppose that $a$ and $b$ are non-negative integers satisfying $a + b + ab + a^b = 42$. Find the sum of all possible values of $a + b$.
Let $T = TNYWR$. Suppose that a sequence $\{a_n\}$ is defined via $a_1 = 11, a_2 = T$, and $a_n = a_{n-1} + 2a_{n-2}$ for $n \ge 3$. Find $a_{19} + a_{20}$.
2009 Korea National Olympiad, 4
There are $n ( \ge 3) $ students in a class. Some students are friends each other, and friendship is always mutual. There are $ s ( \ge 1 ) $ couples of two students who are friends, and $ t ( \ge 1 ) $ triples of three students who are each friends. For two students $ x, y $ define $ d(x,y)$ be the number of students who are both friends with $ x $ and $ y $. Prove that there exist three students $ u, v, w $ who are each friends and satisfying
\[ d(u,v) + d(v,w) + d(w,u) \ge \frac{9t}{s} . \]
1996 Romania Team Selection Test, 13
Let $ x_1,x_2,\ldots,x_n $ be positive real numbers and $ x_{n+1} = x_1 + x_2 + \cdots + x_n $. Prove that
\[ \sum_{k=1}^n \sqrt { x_k (x_{n+1} - x_k)} \leq \sqrt { \sum_{k=1}^n x_{n+1}(x_{n+1}-x_k)}. \]
[i]Mircea Becheanu[/i]
1990 AMC 8, 3
What fraction of the square is shaded?
[asy]
draw((0,0)--(0,3)--(3,3)--(3,0)--cycle);
draw((0,2)--(2,2)--(2,0)); draw((0,1)--(1,1)--(1,0)); draw((0,0)--(3,3));
fill((0,0)--(0,1)--(1,1)--cycle,grey);
fill((1,0)--(1,1)--(2,2)--(2,0)--cycle,grey);
fill((0,2)--(2,2)--(3,3)--(0,3)--cycle,grey);[/asy]
$ \text{(A)}\ \frac{1}{3}\qquad\text{(B)}\ \frac{2}{5}\qquad\text{(C)}\ \frac{5}{12}\qquad\text{(D)}\ \frac{3}{7}\qquad\text{(E)}\ \frac{1}{2} $
2012 Romania National Olympiad, 4
[i]Reduced name[/i] of a natural number $A$ with $n$ digits ($n \ge 2$) a number of $n-1$ digits obtained by deleting one of the digits of $A$: For example, the [i]reduced names[/i] of $1024$ is $124$, $104$ and $120$.
Determine how many seven-digit numbers cannot be written as the sum of one natural numbers $A$ and a [i]reduced name[/i] of $A$.
2013 China Team Selection Test, 1
Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]
1987 Tournament Of Towns, (136) 1
A machine gives out five pennies for each nickel inserted into it and five nickels for each penny. Can Peter , who starts out with one penny, use the machine several times in such a way as to end up with an equal number of nickels and pennies?
(F. Nazarov, Leningrad Olympiad, 1987)
2018 Online Math Open Problems, 23
Let $ABC$ be a triangle with $BC=13, CA=11, AB=10$. Let $A_1$ be the midpoint of $BC$. A variable line $\ell$ passes through $A_1$ and meets $AC,AB$ at $B_1,C_1$. Let $B_2,C_2$ be points with $B_2B=B_2C, B_2C_1\perp AB, C_2B=C_2C, C_2B_1 \perp AC$, and define $P=BB_2\cap CC_2$. Suppose the circles of diameters $BB_2, CC_2$ meet at a point $Q\neq A_1$. Given that $Q$ lies on the same side of line $BC$ as $A$, the minimum possible value of $\dfrac{PB}{PC}+\dfrac{QB}{QC}$ can be expressed in the form $\dfrac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $\gcd (a,c)=1$ and $b$ squarefree. Determine $a+b+c$.
[i]Proposed by Vincent Huang[/i]
MOAA Gunga Bowls, 2022
[u]Set 4[/u]
[b]G10.[/b] Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is $A$. Find the integer closest to $100A$.
[b]G11.[/b] The $10$-digit number $\underline{1A2B3C5D6E}$ is a multiple of $99$. Find $A + B + C + D + E$.
[b]G12.[/b] Let $A, B, C, D$ be four points satisfying $AB = 10$ and $AC = BC = AD = BD = CD = 6$. If $V$ is the volume of tetrahedron $ABCD$, then find $V^2$.
[u]Set 5[/u]
[b]G13.[/b] Nate the giant is running a $5000$ meter long race. His first step is $4$ meters, his next step is $6$ meters, and in general, each step is $2$ meters longer than the previous one. Given that his $n$th step will get him across the finish line, find $n$.
[b]G14.[/b] In square $ABCD$ with side length $2$, there exists a point $E$ such that $DA = DE$. Let line $BE$ intersect side $AD$ at $F$ such that $BE = EF$. The area of $ABE$ can be expressed in the form $a -\sqrt{b}$ where $a$ is a positive integer and $b$ is a square-free integer. Find $a + b$.
[b]G15.[/b] Patrick the Beetle is located at $1$ on the number line. He then makes an infinite sequence of moves where each move is either moving $1$, $2$, or $3$ units to the right. The probability that he does reach $6$ at some point in his sequence of moves is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[u]Set 6[/u]
[b]G16.[/b] Find the smallest positive integer $c$ greater than $1$ for which there do not exist integers $0 \le x, y \le9$ that satisfy $2x + 3y = c$.
[b]G17.[/b] Jaeyong is on the point $(0, 0)$ on the coordinate plane. If Jaeyong is on point $(x, y)$, he can either walk to $(x + 2, y)$, $(x + 1, y + 1)$, or $(x, y + 2)$. Call a walk to $(x + 1, y + 1)$ an Brilliant walk. If Jaeyong cannot have two Brilliant walks in a row, how many ways can he walk to the point $(10, 10)$?
[b]G18.[/b] Deja vu?
Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The maximum possible area of the resulting shape is $B$. Find the integer closest to $100B$.
PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1985 AMC 8, 20
In a certain year, January had exactly four Tuesdays and four Saturdays. On what day did January $ 1$ fall that year?
\[ \textbf{(A)}\ \text{Monday} \qquad
\textbf{(B)}\ \text{Tuesday} \qquad
\textbf{(C)}\ \text{Wednesday} \qquad
\textbf{(D)}\ \text{Friday} \qquad
\textbf{(E)}\ \text{Saturday}
\]
PEN G Problems, 26
Prove that if $g \ge 2$ is an integer, then two series \[\sum_{n=0}^{\infty}\frac{1}{g^{n^{2}}}\;\; \text{and}\;\; \sum_{n=0}^{\infty}\frac{1}{g^{n!}}\] both converge to irrational numbers.