Found problems: 85335
LMT Team Rounds 2021+, 3
Billiam is distributing his ample supply of balls among an ample supply of boxes. He distributes the balls as follows: he places a ball in the first empty box, and then for the greatest positive integer n such that all $n$ boxes from box $1$ to box $n$ have at least one ball, he takes all of the balls in those $n$ boxes and puts them into box $n +1$. He then repeats this process indefinitely. Find the number of repetitions of this process it takes for one box to have at least $2022$ balls.
2020 Francophone Mathematical Olympiad, 4
Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot 7^y = z^3$
2017 BMT Spring, 11
Naomi has a class of $100$ students who will compete with each other in five teams. Once the teams are made, each student will shake hands with every other student, except the students in his or her own team. Naomi chooses to partition the students into teams so as to maximize the number of handshakes. How many handshakes will there be?
1992 Tournament Of Towns, (326) 3
Let $n, m, k$ be natural numbers, with $m > n$. Which of the numbers is greater:
$$\sqrt{n+\sqrt{m+\sqrt{n+...}}}\,\,\, or \,\,\,\, \sqrt{m+\sqrt{n+\sqrt{m+...}}}\,\, ?$$
Note: Each of the expressions contains $k$ square root signs; $n, m$ alternate within each expression.
(N. Kurlandchik)
1982 IMO Longlists, 27
Let $O$ be a point of three-dimensional space and let $l_1, l_2, l_3$ be mutually perpendicular straight lines passing through $O$. Let $S$ denote the sphere with center $O$ and radius $R$, and for every point $M$ of $S$, let $S_M$ denote the sphere with center $M$ and radius $R$. We denote by $P_1, P_2, P_3$ the intersection of $S_M$ with the straight lines $l_1, l_2, l_3$, respectively, where we put $P_i \neq O$ if $l_i$ meets $S_M$ at two distinct points and $P_i = O$ otherwise ($i = 1, 2, 3$). What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$?
1951 Moscow Mathematical Olympiad, 192
a) Given a chain of $60$ links each weighing $1$ g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights $1$ g, $2$ g, . . . , $59$ g, $60$ g? A broken link also weighs $1$ g.
b) Given a chain of $150$ links each weighing $1$ g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights $1$ g, $2$ g, . . . , $149$ g, $150$ g? A broken link also weighs $1$ g.
1958 AMC 12/AHSME, 14
At a dance party a group of boys and girls exchange dances as follows: one boy dances with $ 5$ girls, a second boy dances with $ 6$ girls, and so on, the last boy dancing with all the girls. If $ b$ represents the number of boys and $ g$ the number of girls, then:
$ \textbf{(A)}\ b \equal{} g\qquad
\textbf{(B)}\ b \equal{} \frac{g}{5}\qquad
\textbf{(C)}\ b \equal{} g \minus{} 4\qquad
\textbf{(D)}\ b \equal{} g \minus{} 5\qquad \\
\textbf{(E)}\ \text{It is impossible to determine a relation between }{b}\text{ and }{g}\text{ without knowing }{b \plus{} g.}$
2022 Czech-Polish-Slovak Junior Match, 6
Find all integers $n \ge 4$ with the following property:
Each field of the $n \times n$ table can be painted white or black in such a way that each square of this table had the same color as exactly the two adjacent squares. (Squares are adjacent if they have exactly one side in common.)
How many different colorings of the $6 \times 6$ table fields meet the above conditions?
2021 Czech-Polish-Slovak Junior Match, 6
Let $s (n)$ denote the sum of digits of a positive integer $n$. Using six different digits, we formed three 2-digits $p, q, r$ such that $$p \cdot q \cdot s(r) = p\cdot s(q) \cdot r = s (p) \cdot q \cdot r.$$ Find all such numbers $p, q, r$.
1999 Romania Team Selection Test, 12
Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.
2016 Harvard-MIT Mathematics Tournament, 24
Let $\Delta A_1B_1C$ be a triangle with $\angle A_1B_1C = 90^{\circ}$ and $\frac{CA_1}{CB_1} = \sqrt{5}+2$. For any $i \ge 2$, define $A_i$ to be the point on the line $A_1C$ such that $A_iB_{i-1} \perp A_1C$ and define $B_i$ to be the point on the line $B_1C$ such that $A_iB_i \perp B_1C$. Let $\Gamma_1$ be the incircle of $\Delta A_1B_1C$ and for $i \ge 2$, $\Gamma_i$ be the circle tangent to $\Gamma_{i-1}, A_1C, B_1C$ which is smaller than $\Gamma_{i-1}$.
How many integers $k$ are there such that the line $A_1B_{2016}$ intersects $\Gamma_{k}$?
V Soros Olympiad 1998 - 99 (Russia), 9.8
Calculate $f(\sqrt[3]{2}-1) $, where
$$f(x) = x^{1999} + 3x^{1998} + 4x^{1997} + 2x^{1996} + 4x^{1995} + 2x^{1994} + ...$$
$$... + 4x^3 + 2x^2 + 3x+ 1.$$
2009 Middle European Mathematical Olympiad, 1
Find all functions $ f: \mathbb{R} \to \mathbb{R}$, such that
\[ f(xf(y)) \plus{} f(f(x) \plus{} f(y)) \equal{} yf(x) \plus{} f(x \plus{} f(y))\]
holds for all $ x$, $ y \in \mathbb{R}$, where $ \mathbb{R}$ denotes the set of real numbers.
1988 Greece National Olympiad, 4
Prove that there are do not exist natural numbers $k, m$ such that numbers $k^2+2m$, $m^2+2k$ to be squares of integers.
2022 VN Math Olympiad For High School Students, Problem 8
Given the triangle $ABC$ with $T$ is its [i]Fermat–Torricelli[/i] point. Let $(N_a)$ be the circumcircle of $\triangle TBC$. Choose a point $X$ on $(N_a)$ such that $TX$ is perpendicular to $BC$. The segment $BC$ intersects $(TN_aX)$ at $D$. Similar definition of points $Y, Z, E, F$. The reflection lines of the [i]Euler[/i] line of $\triangle ABC$ wrt $BC, CA, AB$ intersect $XD, YE, ZF$ at $P, Q, R$, respectively.
Prove that: $AP$ is perpendicular to $QR$ if and only if $AB = AC$ or $2BC^2 = AB^2 + AC^2$.
1992 Cono Sur Olympiad, 1
Prove that there aren't any positive integrer numbers $x,y,z$ such that $x^2+y^2=3z^2$.
2023 LMT Fall, 16
Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base $10$)?
[i]Proposed byMuztaba Syed[/i]
2023 Irish Math Olympiad, P6
A positive integer is [i]totally square[/i] is the sum of its digits (written in base $10$) is a square number. For example, $13$ is totally square because $1 + 3 = 2^2$, but $16$ is not totally square.
Show that there are infinitely many positive integers that are not the sum of two totally square integers.
1983 AMC 12/AHSME, 29
A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A$, $B$, $C$ and $D$. Also, let the distances from $P$ to $A$, $B$ and $C$, respectively, be $u$, $v$ and $w$. What is the greatest distance that $P$ can be from $D$ if $u^2 + v^2 = w^2$?
$ \textbf{(A)}\ 1 + \sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{2}\qquad\textbf{(C)}\ 2 + \sqrt{2}\qquad\textbf{(D)}\ 3\sqrt{2}\qquad\textbf{(E)}\ 3 + \sqrt{2}$
2009 District Olympiad, 1
Let $A,B,C\in \mathcal{M}_3(\mathbb{R})$ such that $\det A=\det B=\det C$ and $\det(A+iB)=\det(C+iA)$. Prove that $\det (A+B)=\det (C+A)$.
2022 Greece JBMO TST, 4
Let $n$ be a positive integer. We are given a $3n \times 3n$ board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a $2 \times 2$ square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all $n$ for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.
Proposed by [i]Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina[/i]
2018 Math Prize for Girls Olympiad, 1
Let $P$ be a point in the plane. Suppose that $P$ is inside (or on) each of 6 circles $\omega_1$, $\omega_2$, ..., $\omega_6$ in the plane. Prove that there exist distinct $i$ and $j$ so that the center of circle $\omega_i$ is inside (or on) circle $\omega_j$.
2006 Iran Team Selection Test, 4
Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that
\[ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| \]
2023 Azerbaijan IZhO TST, 1
In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$.
Proposed by Fatemeh Sajadi
2018 Iran Team Selection Test, 3
$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree $\le n$ that satisfies the following conditions?
a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $
b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $
[i]Proposed by Mojtaba Zare[/i]