Found problems: 85335
2000 AIME Problems, 14
In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP=PQ=QB=BC.$ Angle $ACB$ is $r$ times as large as angle $APQ,$ where $r$ is a positive real number. Find the greatest integer that does not exceed $1000r.$
2003 JHMMC 8, 26
Given that $5^3+5^3 + 5^3 + 5^3 + 5^3 = 5^J$ and $3^2 + 3^2 + 3^2 = 3^N$ , what is the value of $J^
N$ ?
2000 Baltic Way, 15
Let $n$ be a positive integer not divisible by $2$ or $3$. Prove that for all integers $k$, the number $(k+1)^n-k^n-1$ is divisible by $k^2+k+1$.
1987 Greece National Olympiad, 1
We color all points of a plane using $3$ colors. Prove that there are at least two points of the plane having same colours and with distance among them $1$.
2022 ITAMO, 6
Let $ABC$ be a non-equilateral triangle and let $R$ be the radius of its circumcircle. The incircle of $ABC$ has $I$ as its centre and is tangent to side $CA$ in point $D$ and to side $CB$ in point $E$.
Let $A_1$ be the point on line $EI$ such that $A_1I=R$, with $I$ being between $A_1$ and $E$. Let $B_1$ be the point on line $DI$ such that $B_1I=R$, with $I$ being between $B_1$ and $D$. Let $P$ be the intersection of lines $AA_1$ and $BB_1$.
(a) Prove that $P$ belongs to the circumcircle of $ABC$.
(b) Let us now also suppose that $AB=1$ and $P$ coincides with $C$. Determine the possible values of the perimeter of $ABC$.
2013 Math Prize For Girls Problems, 6
Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.
2020 LIMIT Category 1, 4
The total number of solutions of $xyz=2520$
(A)$2520$
(B)$2160$
(C)$540$
(D)None of these
2008 Harvard-MIT Mathematics Tournament, 19
Let $ ABCD$ be a regular tetrahedron, and let $ O$ be the centroid of triangle $ BCD$. Consider the point $ P$ on $ AO$ such that $ P$ minimizes $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$. Find $ \sin \angle PBO$.
2025 Ukraine National Mathematical Olympiad, 11.6
Oleksii chose $11$ pairwise distinct positive integer numbers not exceeding $2025$. Prove that among them, it is possible to choose two numbers \(a < b\) such that the number \(b\) gives an even remainder when divided by the number \(a\).
[i]Proposed by Anton Trygub[/i]
2024 Korea Junior Math Olympiad (First Round), 17.
Find the number of $n$ that follow the following:
$ \bigstar $ The number of integers $ (x,y,z) $ following this equation is not a multiple of 4.
$ 2n=x^2+2y^2+2x^2+2xy+2yz $
2014 Sharygin Geometry Olympiad, 3
Points $M$ and $N$ are the midpoints of sides $AC$ and $BC$ of a triangle $ABC$. It is known that $\angle MAN = 15^o$ and $\angle BAN = 45^o$. Find the value of angle $\angle ABM$.
(A. Blinkov)
2017 Puerto Rico Team Selection Test, 6
Find all functions $f: R \to R$ such that $f (xy) \le yf (x) + f (y)$, for all $x, y\in R$.
2021 Durer Math Competition Finals, 3
On the evening of Halloween a group of $n$ kids collected $k$ bars of chocolate of the same type. At the end of the evening they wanted to divide the bars so that everybody gets the same amount of chocolate, and none of the bars is broken into more than two pieces. For which $n$ and $k$ is this possible?
1997 Romania National Olympiad, 2
Let $A$ be a square matrix of odd order (at least $3$) whose entries are odd integers. Prove that if $A$ is invertible, then it is not possible for all the minors of the entries of a row to have equal absolute values.
1987 IMO Longlists, 1
Let $x_1, x_2,\cdots, x_n$ be $n$ integers. Let $n = p + q$, where $p$ and $q$ are positive integers. For $i = 1, 2, \cdots, n$, put
\[S_i = x_i + x_{i+1} +\cdots + x_{i+p-1} \text{ and } T_i = x_{i+p} + x_{i+p+1} +\cdots + x_{i+n-1}\]
(it is assumed that $x_{i+n }= x_i$ for all $i$). Next, let $m(a, b)$ be the number of indices $i$ for which $S_i$ leaves the remainder $a$ and $T_i$ leaves the remainder $b$ on division by $3$, where $a, b \in \{0, 1, 2\}$. Show that $m(1, 2)$ and $m(2, 1)$ leave the same remainder when divided by $3.$
2023 Francophone Mathematical Olympiad, 4
Find all integers $n \geqslant 0$ such that $20n+2$ divides $2023n+210$.
2004 Putnam, A6
Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0\le x\le1,0\le y\le1.$ Show that
$\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx$
$\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.$
2023 Durer Math Competition (First Round), 4
Let $k$ be a circle with diameter $AB$ and centre $O$. Let C be an arbitrary point on the circle different from $A$ and $B$. Let $D$ be the point for which $O$, $B$, $D$ and $C$ (in this order) are the four vertices of a parallelogram. Let $E$ be the intersection of the line $BD$ and the circle $k$, and let $F$ be the orthocenter of the triangle $OAC$. Prove that the points $O, D, E, C, F$ lie on a circle.
1996 IMO Shortlist, 5
Let $ P(x)$ be the real polynomial function, $ P(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d.$ Prove that if $ |P(x)| \leq 1$ for all $ x$ such that $ |x| \leq 1,$ then
\[ |a| \plus{} |b| \plus{} |c| \plus{} |d| \leq 7.\]
2015 Estonia Team Selection Test, 12
Call an $n$-tuple $(a_1, . . . , a_n)$ [i]occasionally periodic [/i] if there exist a nonnegative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+p+j}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, . . . , a_n)$ with elements from set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in \{1, 2, . . . , k\}$.
2020-21 KVS IOQM India, 10
Let $A$ and $B$ be two finite sets such that there are exactly $144$ sets which are subsets of $A$ or subsets of $B$. Find the number of elements in $A \cup B$.
2020 Iran MO (3rd Round), 4
What is the maximum number of subsets of size $5$, taken from the set $A=\{1,2,3,...,20\}$ such that any $2$ of them share exactly $1$ element.
1941 Moscow Mathematical Olympiad, 086
Given three points $H_1, H_2, H_3$ on a plane. The points are the reflections of the intersection point of the heights of the triangle $\vartriangle ABC$ through its sides. Construct $\vartriangle ABC$.
2016 Israel Team Selection Test, 1
Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \geq 2$.
1974 Chisinau City MO, 77
Is it possible to simultaneously take away on eight three-ton vehicles $50$ stones, the weight of which is respectively equal to $416, 418, 420, .., 512, 514$ kg?