Found problems: 85335
2011 India IMO Training Camp, 2
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
2013 Saudi Arabia IMO TST, 4
Determine whether it is possible to place the integers $1, 2,...,2012$ in a circle in such a way that the $2012$ products of adjacent pairs of numbers leave pairwise distinct remainders when divided by $2013$.
MOAA Gunga Bowls, 2021.1
Evaluate $2\times 0+2\times 1+ 2+0\times 2 +1$.
[i]Proposed by Nathan Xiong[/i]
2009 Kazakhstan National Olympiad, 1
Prove that for any natural $n \geq 2$, the number $ \underbrace{2^{2^{\cdots^2}}}_{n \textrm{ times}}- \underbrace{2^{2^{\cdots^2}}}_{n-1 \textrm{ times}}$ is divisible by $n$.
I know, that it is a very old problem :blush: but it is a problem from olympiad.
2006 Putnam, B5
For each continuous function $f: [0,1]\to\mathbb{R},$ let $I(f)=\int_{0}^{1}x^{2}f(x)\,dx$ and $J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$
1975 Poland - Second Round, 2
In the convex quadrilateral $ ABCD $, the corresponding points $ M $ and $ N $ are chosen on the adjacent sides $ \overline{AB} $ and $ \overline{BC} $ and the intersection point of the segments $ AN $ and $ GM $ is marked by 0. Prove that if circles can be inscribed in the quadrilaterals $ AOCD $ and $ BMON $, then a circle can also be inscribed in the quadrilateral $ ABCD $.
2016 India IMO Training Camp, 3
An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$. A [i]move[/i] consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by $1$ simultaneously. Determine all positive integers $n$ such that after performing several such moves one can obtain $9$ consecutive numbers $n,(n+1),\cdots ,(n+8)$ in some order.
[asy] size(3cm);
pair A=(0,0),D=(1,0),B,C,E,F,G,H,I;
G=rotate(60,A)*D;
B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A;
draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]
2011 Danube Mathematical Competition, 4
Given a positive integer number $n$, determine the maximum number of edges a triangle-free Hamiltonian simple graph on $n$ vertices may have.
2024 South Africa National Olympiad, 3
Each of the lattice points $(x,y)$ (where $x$ and $y$ are integers) in the plane can be coloured black or white. A single strike by an $L$-shaped punch changes the colour of the four lattice points $(a,b)$, $(a+1,b)$, $(a,b+1)$ and $(a,b+2)$. All lattice points are initially coloured white. Prove that after any number of strikes, the number of black lattice points will be either zero or greater than or equal to four.
2024 ISI Entrance UGB, P7
Consider a container of the shape obtained by revolving a segment of parabola $x = 1 + y^2$ around the $y$-axis as shown below. The container is initially empty. Water is poured at a constant rate of $1\, \text{cm}^3$ into the container. Let $h(t)$ be the height of water inside container at time $t$. Find the time $t$ when the rate of change of $h(t)$ is maximum.
1956 AMC 12/AHSME, 29
The points of intersection of $ xy \equal{} 12$ and $ x^2 \plus{} y^2 \equal{} 25$ are joined in succession. The resulting figure is:
$ \textbf{(A)}\ \text{a straight line} \qquad\textbf{(B)}\ \text{an equilateral triangle} \qquad\textbf{(C)}\ \text{a parallelogram}$
$ \textbf{(D)}\ \text{a rectangle} \qquad\textbf{(E)}\ \text{a square}$
2018 Costa Rica - Final Round, F2
Consider $f (n, m)$ the number of finite sequences of $ 1$'s and $0$'s such that each sequence that starts at $0$, has exactly n $0$'s and $m$ $ 1$'s, and there are not three consecutive $0$'s or three $ 1$'s. Show that if $m, n> 1$, then
$$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$
2017 IMO Shortlist, G5
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.
1966 IMO Longlists, 16
We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XY Z$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q.$
1985 Vietnam National Olympiad, 2
Find all functions $ f \colon \mathbb{Z} \mapsto \mathbb{R}$ which satisfy:
i) $ f(x)f(y) \equal{} f(x \plus{} y) \plus{} f(x \minus{} y)$ for all integers $ x$, $ y$
ii) $ f(0) \neq 0$
iii) $ f(1) \equal{} \frac {5}{2}$
2021 Kyiv Mathematical Festival, 5
Bilbo composes a number triangle of zeroes and ones in such a way: he fills the topmost row with any $n$ digits, and in other rows he always writes $0$ under consecutive equal digits and writes $1$ under consecutive distinct digits. (An example of a triangle for $n=5$ is shown below.) In how many ways can Bilbo fill the topmost row for $n=100$ so that each of $n$ rows of the triangle contains even number of ones?\[\begin{smallmatrix}1\,0\,1\,1\,0\\1\,1\,0\,1\\0\,1\,1\\1\,0\\1\end{smallmatrix}\] (O. Rudenko and V. Brayman)
2017, SRMC, 3
Prove that among any $42$ numbers from the interval $[1,10^6]$, you can choose four numbers so that for any permutation $(a, b, c, d)$ of these numbers, the inequality
$$25 (ab + cd) (ad + bc) \ge 16 (ac + bd)^ 2$$
holds.
2018 Nepal National Olympiad, 4c
[b]Problem Section #4
c) A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with
one of seven colors. Each number-color combination appears on exactly one card. John will select
a set of eight cards from the deck at random. Given that he gets at least one card of each color and
at least one card with each number, the probability that John can discard one of his cards and still
have at least one card of each color and at least one card with each number is $\frac{p}{q}$, where $p$ and $q$ are
relatively prime positive integers. Find $p+q$.
1999 China National Olympiad, 1
Let $ABC$ be an acute triangle with $\angle C>\angle B$. Let $D$ be a point on $BC$ such that $\angle ADB$ is obtuse, and let $H$ be the orthocentre of triangle $ABD$. Suppose that $F$ is a point inside triangle $ABC$ that is on the circumcircle of triangle $ABD$. Prove that $F$ is the orthocenter of triangle $ABC$ if and only if $HD||CF$ and $H$ is on the circumcircle of triangle $ABC$.
2023 Indonesia Regional, 5
Given $\triangle ABC$ and points $D$ and $E$ at the line $BC$, furthermore there are points $X$ and $Y$ inside $\triangle ABC$. Let $P$ be the intersection of line $AD$ and $XE$, and $Q$ be the intersection of line $AE$ and $YD$.
If there exist a circle that passes through $X, Y, D, E$, and
$$\angle BXE + \angle BCA = \angle CYD + \angle CBA = 180^{\circ}$$
Prove that the line $BP$, $CQ$, and the perpendicular bisector of $BC$ intersect at one point.
2002 Junior Balkan MO, 3
Find all positive integers which have exactly 16 positive divisors $1 = d_1 < d_2 < \ldots < d_{16} =n$ such that the divisor $d_k$, where $k = d_5$, equals $(d_2 + d_4) d_6$.
2023 Junior Balkan Team Selection Tests - Romania, P3
Let the equilateral triangles $ABC$ and $DEF$ be congruent with the centers $O_1$, respectively $O_2$, so that segment $AB$ intersects segments $DE$ and $DF$ at $M, N$, and the segment $AC$ intersects the segments $DF$ and $EF$ at $P$ and $Q$, respectively. We denote by $I$ the intersection point of the bisectors of the angles $EMN$ and $DPQ$ and by $J$ the intersection of the bisectors of the angles $FNM$ and $EQP$. Prove that $IJ$ is the perpendicular bisector of the segment $O_1O_2$.
2020 LMT Spring, 28
A particular country has seven distinct cities, conveniently named $C_1,C_2,\dots,C_7.$ Between each pair of cities, a direction is chosen, and a one-way road is constructed in that direction connecting the two cities. After the construction is complete, it is found that any city is reachable from any other city, that is, for distinct $1 \leq i, j \leq 7,$ there is a path of one-way roads leading from $C_i$ to $C_j.$ Compute the number of ways the roads could have been configured. Pictured on the following page are the possible configurations possible in a country with three cities, if every city is reachable from every other city.
[Insert Diagram]
[i]Proposed by Ezra Erives[/i]
2019 ELMO Shortlist, C1
Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)
[i]Proposed by Milan Haiman[/i]
1992 China Team Selection Test, 1
A triangle $ABC$ is given in the plane with $AB = \sqrt{7},$ $BC = \sqrt{13}$ and $CA = \sqrt{19},$ circles are drawn with centers at $A,B$ and $C$ and radii $\frac{1}{3},$ $\frac{2}{3}$ and $1,$ respectively. Prove that there are points $A',B',C'$ on these three circles respectively such that triangle $ABC$ is congruent to triangle $A'B'C'.$