Found problems: 85335
2024 Kyiv City MO Round 2, Problem 4
Points $X$ and $Y$ are chosen inside an acute-angled triangle $ABC$ with altitude $AD$ so that $\angle BXA + \angle ACB = 180^\circ , \angle CYA + \angle ABC = 180^\circ$, and $CD + AY = BD + AX$. Point $M$ is chosen on the ray $BX$ so that $X$ lies on segment $BM$ and $XM = AC$, and point $N$ is chosen on the ray $CY$ so that $Y$ lies on segment $CN$ and $YN = AB$. Prove that $AM = AN$.
[i]Proposed by Mykhailo Shtandenko[/i]
2013 Harvard-MIT Mathematics Tournament, 2
Let $\{a_n\}_{n\geq 1}$ be an arithmetic sequence and $\{g_n\}_{n\geq 1}$ be a geometric sequence such that the first four terms of $\{a_n+g_n\}$ are $0$, $0$, $1$, and $0$, in that order. What is the $10$th term of $\{a_n+g_n\}$?
Kvant 2023, M2760
The checkered plane is divided into dominoes. What is the maximum $k{}$ for which it is always possible to choose a $100\times 100$ checkered square containing at least $k{}$ whole dominoes?
[i]Proposed by S. Berlov[/i]
1979 IMO Longlists, 5
Describe which positive integers do not belong to the set
\[E = \left\{ \lfloor n+ \sqrt n +\frac 12 \rfloor | n \in \mathbb N\right\}.\]
1989 Tournament Of Towns, (228) 2
The hexagon $ABCDEF$ is inscribed in a circle, $AB = BC = a, CD = DE = b$, and $EF = FA = c$. Prove that the area of triangle $BDF$ equals half the area of the hexagon.
(I.P. Nagel, Yevpatoria).
2014 ASDAN Math Tournament, 2
Let $a$ and $b$ be positive integers such that $a>b$ and the difference between $a^2+b$ and $a+b^2$ is prime. Compute all possible pairs $(a,b)$.
2008 Singapore Team Selection Test, 2
Let $ x_1, x_2,\ldots , x_n$ be positive real numbers such that $ x_1x_2\cdots x_n \equal{} 1$. Prove that
\[\sum_{i \equal{} 1}^n \frac {1}{n \minus{} 1 \plus{} x_i}\le 1.\]
2015 Purple Comet Problems, 19
Let a,b,c, and d be real numbers such that \[a^2 + 3b^2 + \frac{c^2+3d^2}{2} = a + b + c+d-1.\] Find $1000a + 100b + 10c + d$.
1986 AIME Problems, 14
The shortest distances between an interior diagonal of a rectangular parallelepiped, P, and the edges it does not meet are $2\sqrt{5}$, $\frac{30}{\sqrt{13}}$, and $\frac{15}{\sqrt{10}}$. Determine the volume of $P$.
2021 USEMO, 4
Let $ABC$ be a triangle with circumcircle $\omega$, and let $X$ be the reflection of $A$ in $B$. Line $CX$ meets $\omega$ again at $D$. Lines $BD$ and $AC$ meet at $E$, and lines $AD$ and $BC$ meet at $F$. Let $M$ and $N$ denote the midpoints of $AB$ and $ AC$.
Can line $EF$ share a point with the circumcircle of triangle $AMN?$
[i]Proposed by Sayandeep Shee[/i]
2017 Caucasus Mathematical Olympiad, 7
$10$ distinct numbers are given. Professor Odd had calculated all possible products of $1$, $3$, $5$, $7$, $9$ numbers among given numbers, and wrote down the sum of all these products. Similarly, Professor Even had calculated all possible products of $2$, $4$, $6$, $8$, $10$ numbers among given numbers, and wrote down the sum of all these products. It appears that Odd's sum is greater than Even's sum by $1$. Prove that one of $10$ given numbers is equal to $1$.
1976 IMO Longlists, 8
In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.
2004 Harvard-MIT Mathematics Tournament, 4
Andrea flips a fair coin repeatedly, continuing until she either flips two heads in a row (the sequence $\texttt{HH}$) or flips tails followed by heads (the sequence $\texttt{TH}$). What is the probability that she will stop after flipping $\texttt{HH}$?
2014 Singapore MO Open, 3
Let $0<a_1<a_2<\cdots <a_n$ be real numbers. Prove that
\[\left (\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}\right )^2 \leq \frac{1}{a_1}+\frac{1}{a_2-a_1}+\cdots +\frac{1}{a_n-a_{n-1}}.\]
2023 India Regional Mathematical Olympiad, 6
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $P$. The point $Q$ is chosen on the segment $BC$ so that $PQ$ is perpendicular to $AC$. Prove that the line joining the centres of the circumcircles of triangles $APD$ and $BQD$ is parallel to $AD$.
2000 Croatia National Olympiad, Problem 3
Let $m>1$ be an integer. Determine the number of positive integer solutions of the equation $\left\lfloor\frac xm\right\rfloor=\left\lfloor\frac x{m-1}\right\rfloor$.
2010 Sharygin Geometry Olympiad, 7
The line passing through the vertex $B$ of a triangle $ABC$ and perpendicular to its median $BM$ intersects the altitudes dropped from $A$ and $C$ (or their extensions) in points $K$ and $N.$ Points $O_1$ and $O_2$ are the circumcenters of the triangles $ABK$ and $CBN$ respectively. Prove that $O_1M=O_2M.$
2006 Pan African, 3
For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ and let $\{x\} = x - \lfloor x\rfloor$. If $a, b, c$ are distinct real numbers, prove that
\[\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}\]
is an integer if and only if $\{a\} + \{b\} + \{c\}$ is an integer.
2008 Hanoi Open Mathematics Competitions, 9
Consider a triangle $ABC$. For every point M $\in BC$ ,we define $N \in CA$ and $P \in AB$ such that $APMN$ is a parallelogram. Let $O$ be the intersection of $BN$ and $CP$. Find $M \in BC$ such that $\angle PMO=\angle OMN$
2007 AMC 8, 9
To complete the grid below, each of the digits 1 through 4 must occur once in each row and once in each column. What number will occupy the lower right-hand square?
\[ \begin{tabular}{|c|c|c|c|}\hline 1 & & 2 & \\ \hline 2 & 3 & & \\ \hline & &&4\\ \hline & &&\\ \hline\end{tabular} \]
$\textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ \text{cannot be determined}$
Mathley 2014-15, 2
Let $ABC$ be a triangle with a circumcircle $(K)$. A circle touching the sides $AB,AC$ is internally tangent to $(K)$ at $K_a$; two other points $K_b,K_c$ are defined in the same manner. Prove that the area of triangle $K_aK_bK_c$ does not exceed that of triangle $ABC$.
Nguyen Minh Ha, Hanoi University of Education, Xuan Thuy, Cau Giay, Hanoi.
1964 AMC 12/AHSME, 16
Let $f(x)=x^2+3x+2$ and let $S$ be the set of integers $\{0, 1, 2, \dots , 25 \}$. The number of members $s$ of $S$ such that $f(s)$ has remainder zero when divided by 6 is:
${{ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 18 }\qquad\textbf{(E)}\ 17 } $
1977 Dutch Mathematical Olympiad, 2
Four masts stand on a flat horizontal piece of land at the vertices of a square $ABCD$. The height of the mast on $A$ is $7$ meters, of the mast on $B$ $13$ meters, and of the mast on $C$ $15$ meters. Within the square there is a point $P$ on the ground equidistant from each of the tops of these three masts.
(a) What length must the sides of the square be at least for this to be possible?
(b) The distance from $P$ to the top of the mast on $D$ is equal to the distance from$ P$ to each of the tops of the three other masts. Calculate the height of the mast at $D$.
2024 Azerbaijan JBMO TST, 2
Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.
2017 India IMO Training Camp, 3
There are $n$ lamps $L_1, L_2, \dots, L_n$ arranged in a circle in that order. At any given time, each lamp is either [i]on[/i] or [i]off[/i]. Every second, each lamp undergoes a change according to the following rule:
(a) For each lamp $L_i$, if $L_{i-1}, L_i, L_{i+1}$ have the same state in the previous second, then $L_i$ is [i]off[/i] right now. (Indices taken mod $n$.)
(b) Otherwise, $L_i$ is [i]on[/i] right now.
Initially, all the lamps are [i]off[/i], except for $L_1$ which is [i]on[/i]. Prove that for infinitely many integers $n$ all the lamps will be [i]off[/i] eventually, after a finite amount of time.