Found problems: 85335
2016 HMNT, 9
Let the sequence $a_i$ be defined as $a_{i+1} = 2^{a_i}$. Find the number of integers $1 \le n \le 1000$ such that if $a_0 = n$, then $100$ divides $a_{1000} - a_1$.
2012 Hanoi Open Mathematics Competitions, 1
[b]Q1.[/b] Assum that $a-b=-(a-b).$ Then:
$(A) \; a=b; \qquad (B) \; a<b; \qquad (C) \; a>b \qquad (D) \; \text{ It is impossible to compare those of a and b.}$
2022 Dutch Mathematical Olympiad, 5
Kira has $3$ blocks with the letter $A$, $3$ blocks with the letter $B$, and $3$ blocks with the letter $C$. She puts these $9$ blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence $ABCAABCBC$ the blocks with the letter A have distances $1, 3$, and $4$ between one another, the blocks with the letter $B$ have distances $2, 4$, and $6$ between one another, and the blocks with the letter $C$ have distances $2, 4$, and $6$ between one another. Altogether, we got distances of $1, 2, 3, 4$, and $6$; these are $5$ distinct distances. What is the maximum number of distinct distances that can occur?
2014 Thailand TSTST, 3
Let $S$ be the set of all 3-tuples $(a, b, c)$ of positive integers such that $a + b + c = 2013$. Find $$\sum_{(a,b,c)\in S} abc.$$
2021 Cono Sur Olympiad, 4
In a heap there are $2021$ stones. Two players $A$ and $B$ play removing stones of the pile, alternately starting with $A$. A valid move for $A$ consists of remove $1, 2$ or $7$ stones. A valid move for B is to remove $1, 3, 4$ or $6$ stones. The player who leaves the pile empty after making a valid move wins. Determine if some of the players have a winning strategy. If such a strategy exists, explain it.
2019 European Mathematical Cup, 3
In an acute triangle $ABC$ with $|AB| \not= |AC|$, let $I$ be the incenter and $O$ the circumcenter. The incircle is tangent to $\overline{BC}, \overline{CA}$ and $\overline{AB}$ in $D,E$ and $F$ respectively. Prove that if the line parallel to $EF$ passing through $I$, the line parallel to $AO$ passing through $D$ and the altitude from $A$ are concurrent, then the point of concurrence is the orthocenter of the triangle $ABC$.
[i]Proposed by Petar Nizié-Nikolac[/i]
1993 AMC 12/AHSME, 9
Country $\mathcal{A}$ has $c\%$ of the world's population and owns $d\%$ of the world's wealth. Country $\mathcal{B}$ has $e\%$ of the world's population and $f\%$ of its wealth. Assume that the citizens of $\mathcal{A}$ share the wealth of $\mathcal{A}$ equally, and assume that those of $\mathcal{B}$ share the wealth of $\mathcal{B}$ equally. Find the ratio of the wealth of a citizen of $\mathcal{A}$ to the wealth of a citizen of $\mathcal{B}$.
$ \textbf{(A)}\ \frac{cd}{ef} \qquad\textbf{(B)}\ \frac{ce}{df} \qquad\textbf{(C)}\ \frac{cf}{de} \qquad\textbf{(D)}\ \frac{de}{cf} \qquad\textbf{(E)}\ \frac{df}{ce} $
1967 IMO Shortlist, 2
Find all real solutions of the system of equations:
\[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$
2014 National Olympiad First Round, 4
What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white and $2$ blue balls?
$
\textbf{(A)}\ \dfrac{2}{5}
\qquad\textbf{(B)}\ \dfrac{3}{7}
\qquad\textbf{(C)}\ \dfrac{16}{35}
\qquad\textbf{(D)}\ \dfrac{10}{21}
\qquad\textbf{(E)}\ \dfrac{5}{14}
$
2018 Harvard-MIT Mathematics Tournament, 3
How many noncongruent triangles are there with one side of length $20,$ one side of length $17,$ and one $60^{\circ}$ angle?
2024 Baltic Way, 20
Positive integers $a$, $b$ and $c$ satisfy the system of equations
\begin{align*}
(ab-1)^2&=c(a^2+b^2)+ab+1,\\
a^2+b^2&=c^2+ab.
\end{align*}
a) Prove that $c+1$ is a perfect square.
b) Find all such triples $(a,b,c)$.
2014 Contests, 4
In an election, there are a total of $12$ candidates. An election committee has $6$ members voting. It is known that at most two candidates voted by any two committee members are the same. Find the maximum number of committee members.
2020 USMCA, 7
Let $ABCD$ be a convex quadrilateral, and let $\omega_A$ and $\omega_B$ be the incircles of $\triangle ACD$ and $\triangle BCD$, with centers $I$ and $J$. The second common external tangent to $\omega_A$ and $\omega_B$ touches $\omega_A$ at $K$ and $\omega_B$ at $L$. Prove that lines $AK$, $BL$, $IJ$ are concurrent.
1995 India Regional Mathematical Olympiad, 6
Let $A_1A_2A_3 \ldots A_{21}$ be a 21-sided regular polygon inscribed in a circle with centre $O$. How many triangles $A_iA_jA_k$, $1 \leq i < j < k \leq 21$, contain the centre point $O$ in their interior?
2022 Canada National Olympiad, 4
Call a set of $n$ lines [i]good[/i] if no $3$ lines are concurrent. These $n$ lines divide the Euclidean plane into regions (possible unbounded). A [i]coloring[/i] is an assignment of two colors to each region, one from the set $\{A_1, A_2\}$ and the other from $\{B_1, B_2, B_3\}$, such that no two adjacent regions (adjacent meaning sharing an edge) have the same $A_i$ color or the same $B_i$ color, and there is a region colored $A_i, B_j$ for any combination of $A_i, B_j$.
A number $n$ is [i]colourable[/i] if there is a coloring for any set of $n$ good lines. Find all colourable $n$.
2007 IMO, 5
Let $a$ and $b$ be positive integers. Show that if $4ab - 1$ divides $(4a^{2} - 1)^{2}$, then $a = b$.
[i]Author: Kevin Buzzard and Edward Crane, United Kingdom [/i]
1972 Bulgaria National Olympiad, Problem 5
In a circle with radius $R$, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral.
(a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed.
(b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$. When does equality hold?
[i]H. Lesov[/i]
2014 Federal Competition For Advanced Students, 3
Let $a_n$ be a sequence de fined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot 3^n$ for $n \ge 0$.
Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$.
1997 Pre-Preparation Course Examination, 2
Two circles $O, O'$ meet each other at points $A, B$. A line from $A$ intersects the circle $O$ at $C$ and the circle $O'$ at $D$ ($A$ is between $C$ and $D$). Let $M,N$ be the midpoints of the arcs $BC, BD$, respectively (not containing $A$), and let $K$ be the midpoint of the segment $CD$. Show that $\angle KMN = 90^\circ$.
2023 Ukraine National Mathematical Olympiad, 10.6
Let $P(x), Q(x), R(x)$ be polynomials with integer coefficients, such that $P(x) = Q(x)R(x)$. Let's denote by $a$ and $b$ the largest absolute values of coefficients of $P, Q$ correspondingly. Does $b \le 2023a$ always hold?
[i]Proposed by Dmytro Petrovsky[/i]
2005 Today's Calculation Of Integral, 40
Evaluate
\[\int_0^1 x^{2005}e^{-x^2}dx\]
2007 AMC 8, 22
A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4.5 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6.2 \qquad
\textbf{(E)}\ 7$
2016 IOM, 6
In a country with $n$ cities, some pairs of cities are connected by one-way flights operated by one of two companies $A$ and $B$. Two cities can be connected by more than one flight in either direction. An $AB$-word $w$ is called implementable if there is a sequence of connected flights whose companies’ names form the word $w$. Given that every $AB$-word of length $ 2^n $ is implementable, prove that every finite $AB$-word is implementable. (An $AB$-word of length $k$ is an arbitrary sequence of $k$ letters $A $ or $B$; e.g. $ AABA $ is a word of length $4$.)
2006 Bosnia and Herzegovina Team Selection Test, 2
It is given a triangle $\triangle ABC$. Determine the locus of center of rectangle inscribed in triangle $ABC$ such that one side of rectangle lies on side $AB$.
2010 Macedonia National Olympiad, 1
Solve the equation
\[ x^3+2y^3-4x-5y+z^2=2012, \]
in the set of integers.