Found problems: 85335
MBMT Team Rounds, 2020.30
Let the number of ways for a rook to return to its original square on a $4\times 4$ chessboard in 8 moves if it starts on a corner be $k$. Find the number of positive integers that are divisors of $k$. A "move" counts as shifting the rook by a positive number of squares on the board along a row or column. Note that the rook may return back to its original square during an intermediate step within its 8-move path.
[i]Proposed by Bradley Guo[/i]
1991 Baltic Way, 13
An equilateral triangle is divided into $25$ equal equilateral triangles labelled by $1$ through $25$. Prove that one can find two triangles having a common side whose labels differ by more than $3$.
2011 Bogdan Stan, 1
Let be the matrix $ A=\begin{pmatrix} 1& 2& -1\\ 2&2 &0\\1& 4& -3 \end{pmatrix} . $
[b]a)[/b] Show that the equation $ AX=\begin{pmatrix} 2\\ 1\\5 \end{pmatrix} $ has infinite solutions in $ \mathcal{M}_1^3\left( \mathbb{C} \right) . $
[b]b)[/b] Find the rank of the adugate of $ A. $
III Soros Olympiad 1996 - 97 (Russia), 11.1
Find the smallest positive root of the equation $$\{tg x\}=\sin x. $$
($\{a\}$ is the fractional part of $a$, $\{a\}$ is equal to the difference between $ a$ and the largest integer not exceeding $a$.)
1999 Kazakhstan National Olympiad, 4
Seven dwarfs live in one house and each has its own hat. One morning one day, two dwarfs inadvertently exchanged hats. At any time, any three gnomes can sit down at the round table and exchange hats clockwise. Is it possible that by evening all the gnomes will be with their hats.
1998 Argentina National Olympiad, 1
Jorge writes a list with an even number of integers, not all equal to $0$ (there may be repeated numbers). Show that Martin can cross out a number from the list, of his choice, so that it is impossible for Jorge to separate the remaining numbers into two groups in such a way that the sum of all the numbers in one group is equal to the sum of all the others. numbers from the other group.
2002 India IMO Training Camp, 17
Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.
Ukraine Correspondence MO - geometry, 2015.11
Let $ABC$ be an non- isosceles triangle, $H_a$, $H_b$, and $H_c$ be the feet of the altitudes drawn from the vertices $A, B$, and $C$, respectively, and $M_a$, $M_b$, and $M_c$ be the midpoints of the sides $BC$, $CA$, and $AB$, respectively. The circumscribed circles of triangles $AH_bH_c$ and $AM_bM_c$ intersect for second time at point $A'$. The circumscribed circles of triangles $BH_cH_a$ and $BM_cM_a$ intersect for second time at point $B'$. The circumscribed circles of triangles $CH_aH_b$ and $CM_aM_b$ intersect for second time at point $C'$. Prove that points $A', B'$ and $C'$ lie on the same line.
2009 Greece Junior Math Olympiad, 3
Consider the numbers$$A= \frac{1}{4}\cdot \frac{3}{6}\cdot \frac{5}{8}\cdot ...\frac{595}{598}\cdot \frac{597}{600}$$and$$B= \frac{2}{5}\cdot \frac{4}{7}\cdot \frac{6}{9}\cdot ...\frac{596}{599}\cdot \frac{598}{601}$$. Prove that:
(a) $A < B$,
(b) $A < \frac{1}{5990}$
1995 Iran MO (2nd round), 3
In a quadrilateral $ABCD$ let $A', B', C'$ and $D'$ be the circumcenters of the triangles $BCD, CDA, DAB$ and $ABC$, respectively. Denote by $S(X, YZ)$ the plane which passes through the point $X$ and is perpendicular to the line $YZ.$ Prove that if $A', B', C'$ and $D'$ don't lie in a plane, then four planes $S(A, C'D'), S(B, A'D'), S(C, A'B')$ and $S(D, B'C')$ pass through a common point.
2007 Thailand Mathematical Olympiad, 2
In a dance party there are $n$ girls and $n$ boys, and some $m$ songs are played. Each song is danced to by at least one pair of a boy and a girl, who both receive a [i]malai [/i] each. Prove that for all positive integers $k \le n$, it is possible to select $k$ boys and $n - k$ girls so that the $n$ selected people received at least $m$ malai in total.
2011 Bosnia and Herzegovina Junior BMO TST, 2
Prove inequality, with $a$ and $b$ nonnegative real numbers:
$\frac{a+b}{1+a+b}\leq \frac{a}{1+a} + \frac{b}{1+b} \leq \frac{2(a+b)}{2+a+b}$
2008 AIME Problems, 6
The sequence $ \{a_n\}$ is defined by
\[ a_0 \equal{} 1,a_1 \equal{} 1, \text{ and } a_n \equal{} a_{n \minus{} 1} \plus{} \frac {a_{n \minus{} 1}^2}{a_{n \minus{} 2}}\text{ for }n\ge2.
\]The sequence $ \{b_n\}$ is defined by
\[ b_0 \equal{} 1,b_1 \equal{} 3, \text{ and } b_n \equal{} b_{n \minus{} 1} \plus{} \frac {b_{n \minus{} 1}^2}{b_{n \minus{} 2}}\text{ for }n\ge2.
\]Find $ \frac {b_{32}}{a_{32}}$.
2020 MIG, 24
[asy]
size(140);
import geometry;
dot((0,0));label("$(0,0)$",(0,0),SW);
dot((4,3));
dot((5,4));label("$(5,4)$",(5,4),NE);
draw((0,0)--(7,0), EndArrow);
draw((0,0)--(0,6), EndArrow);
add(grid(5,4));
[/asy]
A leprechaun wishes to travel from the origin to a pot of gold located at the coordinate point $(5,4)$. If she can only move upwards and rightwards along the unit grid, must pass a checkpoint at $(1,2)$, and must avoid an evil thief at $(4,3)$, how many distinct paths can she take?
$\textbf{(A) }7\qquad\textbf{(B) }15\qquad\textbf{(C) }21\qquad\textbf{(D) }45\qquad\textbf{(E) }126$
2016 NIMO Problems, 2
Define the [i]hotel elevator cubic [/i]as the unique cubic polynomial $P$ for which $P(11) = 11$, $P(12) = 12$, $P(13) = 14$, $P(14) = 15$. What is $P(15)$?
[i]Proposed by Evan Chen[/i]
2023 District Olympiad, P1
Let $f:[-\pi/2,\pi/2]\to\mathbb{R}$ be a twice differentiable function which satisfies \[\left(f''(x)-f(x)\right)\cdot\tan(x)+2f'(x)\geqslant 1,\]for all $x\in(-\pi/2,\pi/2)$. Prove that \[\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.\]
1997 Baltic Way, 14
In the triangle $ABC$, $AC^2$ is the arithmetic mean of $BC^2$ and $AB^2$. Show that $\cot^2B\ge \cot A\cdot\cot C$.
2012 JBMO TST - Macedonia, 2
Let $ABCD$ be a convex quadrilateral inscribed in a circle of radius $1$. Prove that \[ 0< (AB+BC+CD+AD)-(AC+BD) < 4. \]
2023 AMC 10, 7
Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3?
$\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}$
2019 Purple Comet Problems, 8
The diagram below shows a $12$ by $20$ rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/9/e/ed6be5110d923965c64887a2ca8e858c977700.png[/img]
2009 Korea National Olympiad, 2
Let $ABC$ be a triangle and $ P, Q ( \ne A, B, C ) $ are the points lying on segments $ BC , CA $. Let $ I, J, K $ be the incenters of triangle $ ABP, APQ, CPQ $. Prove that $ PIJK $ is a convex quadrilateral.
2025 Junior Balkan Team Selection Tests - Romania, P1
Let $n\geqslant 2$ and $a_1,a_2,\ldots,a_n$ be non-zero integers such that $a_1+a_2+\cdots+a_n=a_1a_2\cdots a_n.$ Prove that \[(a_1^2-1)(a_2^2-1)\cdots(a_n^2-1)\]is a perfect square.
2025 Harvard-MIT Mathematics Tournament, 7
The number $$\frac{9^9-8^8}{1001}$$ is an integer. Compute the sum of its prime factors.
1942 Putnam, A4
Find the orthogonal trajectories of the family of conics $(x+2y)^{2} = a(x+y)$. At what angle do the curves of one family cut the curves of the other family at the origin?
2007 ITest, 58
Let $T=\text{TNFTPP}$. For natural numbers $k,n\geq 2$, we define $S(k,n)$ such that \[S(k,n)=\left\lfloor\dfrac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\dfrac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\dfrac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor.\] Compute the value of $S(10,T+55)-S(10,55)+S(10,T-55)$.