Found problems: 85335
2019 Yasinsky Geometry Olympiad, p1
It is known that in the triangle $ABC$ the distance from the intersection point of the angle bisector to each of the vertices of the triangle does not exceed the diameter of the circle inscribed in this triangle. Find the angles of the triangle $ABC$.
(Grigory Filippovsky)
LMT Team Rounds 2010-20, B5
Given the following system of equations
$a_1 + a_2 + a_3 = 1$
$a_2 + a_3 + a_4 = 2$
$a_3 + a_4 + a_5 = 3$
$...$
$a_{12} + a_{13} + a_{14} = 12$
$a_{13} + a_{14} + a_1 = 13$
$a_{14 }+ a_1 + a_2 = 14$
find the value of $a_{14}$.
1952 Putnam, A3
Develop necessary and sufficient conditions which ensure that $r_1, r_2, r_3$ and $r_1^2, r_2^2, r_3^2$ are simultaneously roots of the equation $x^3 + ax^2 + bx + c = 0.$
2018-2019 Fall SDPC, 2
Find all pairs of positive integers $(m,n)$ such that $2^m-n^2$ is the square of an integer.
2004 Bundeswettbewerb Mathematik, 2
Let $k$ be a positive integer. In a circle with radius $1$, finitely many chords are drawn. You know that every diameter of the circle intersects at most $k$ of these chords.
Prove that the sum of the lengths of all these chords is less than $k \cdot \pi$.
2017 Saint Petersburg Mathematical Olympiad, 6
In acute-angled triangle $ABC$, the height $AH$ and median $BM$ were drawn. Point $D$ lies on the circumcircle of triangle $BHM$ such that $AD \parallel BM$ and $B, D$ are on opposite sides of line $AC$. Prove that $BC=BD$.
Indonesia Regional MO OSP SMA - geometry, 2003.3
The points $P$ and $Q$ are the midpoints of the edges $AE$ and $CG$ on the cube $ABCD.EFGH$ respectively. If the length of the cube edges is $1$ unit, determine the area of the quadrilateral $DPFQ$ .
1978 IMO Longlists, 50
A variable tetrahedron $ABCD$ has the following properties:
Its edge lengths can change as well as its vertices, but the opposite edges remain equal $(BC = DA, CA = DB, AB = DC)$; and the vertices $A,B,C$ lie respectively on three fixed spheres with the same center $P$ and radii $3, 4, 12$. What is the maximal length of $PD$?
1995 India Regional Mathematical Olympiad, 7
Show that for any real number $x$:
\[ x^2 \sin{x} + x \cos{x} + x^2 + \frac{1}{2} > 0 . \]
2014 Thailand TSTST, 3
Define $a_k=2^{2^{k-2013}}+k$ for all integers $k$. Simplify $$(a_0+a_1)(a_1-a_0)(a_2-a_1)\cdots(a_{2013}-a_{2012}).$$
2023 IFYM, Sozopol, 5
Let $r \geq 2023$ be a rational number. The real numbers $a, b$, and $c$ satisfy
\[
4a^2 + 4b^2 + 9c^2 = r.
\]
Does there exist a value of $r$ for which the number of rational triples $(a,b,c)$ that achieve the maximum possible value of $4ab + 6bc - 6ac$ is:
a) zero
b) finite, but non-zero?
2000 Rioplatense Mathematical Olympiad, Level 3, 2
In a triangle $ABC$, points $D, E$ and $F$ are considered on the sides $BC, CA$ and $AB$ respectively, such that the areas of the triangles $AFE, BFD$ and $CDE$ are equal. Prove that $$\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}$$
Note: $(XYZ)$ is the area of triangle $XYZ$.
2007 Bosnia Herzegovina Team Selection Test, 2
Find all pairs of integers $(x,y)$ such that
$x(x+2)=y^2(y^2+1)$
PEN O Problems, 46
Suppose $p$ is a prime with $p \equiv 3 \; \pmod{4}$. Show that for any set of $p-1$ consecutive integers, the set cannot be divided two subsets so that the product of the members of the one set is equal to the product of the members of the other set.
2000 Saint Petersburg Mathematical Olympiad, 10.2
Let $AA_1$ and $BB_1$ be the altitudes of acute angled triangle $ABC$. Points $K$ and $M$ are midpoints of $AB$ and $A_1B_1$ respectively. Segments $AA_1$ and $KM$ intersect at point $L$. Prove that points $A$, $K$, $L$ and $B_1$ are noncyclic.
[I]Proposed by S. Berlov[/i]
2015 Regional Olympiad of Mexico Center Zone, 1
The first $360$ natural numbers are separated into $9$ blocks in such a way that the numbers in each block are consecutive. Then, the numbers in each block are added, obtaining $9$ numbers. Is it possible to fill a $3 \times 3$ grid and form a [i]magic square[/i] with these numbers?
Note: In a magic square, the sum of the numbers written in any column, diagonal or row of the grid is the same.
2018-IMOC, C4
For a sequence $\{a_i\}_{i\ge1}$ consisting of only positive integers, prove that if for all different positive integers $i$ and $j$, we have $a_i\nmid a_j$, then
$$\{p\mid p\text{ is a prime and }p\mid a_i\text{ for some }i\}$$is a infinite set.
1992 AIME Problems, 15
Define a positive integer $ n$ to be a factorial tail if there is some positive integer $ m$ such that the decimal representation of $ m!$ ends with exactly $ n$ zeroes. How many positive integers less than $ 1992$ are not factorial tails?
2023 Stanford Mathematics Tournament, 4
If the sum of the real roots $x$ to each of the equations
\[2^{2x}-2^{x+1}+1-\frac{1}{k^2}=0\]
for $k=2,3,\dots,2023$ is $N$, what is $2^N$?
2010 AMC 10, 11
The length of the interval of solutions of the inequality $ a\le 2x\plus{}3\le b$ is $ 10$. What is $ b\minus{}a$?
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 10 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 30$
2001 Switzerland Team Selection Test, 6
A function $f : [0,1] \to R$ has the following properties:
(a) $f(x) \ge 0$ for $0 < x < 1$,
(b) $f(1) = 1$,
(c) $f(x+y) \ge f(x)+ f(y) $ whenever $x,y,x+y \in [0,1]$.
Prove that $f(x) \le 2x$ for all $x \in [0,1]$.
2010 LMT, 28
Two knights placed on distinct square of an $8\times8$ chessboard, whose squares are unit squares, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}.$ In how many ways can two identical knights be placed on distinct squares of an $8\times8$ chessboard such that they do NOT attack each other?
2019 Dürer Math Competition (First Round), P2
a) 11 kayakers row on the Danube from Szentendre to Kopaszi-gát. They do not necessarily start at the same time, but we know that they all take the same route and that each kayaker rows with a constant speed. Whenever a kayaker passes another one, they do a high five. After they all arrive, everybody claims to have done precisely $10$ high fives in total. Show that it is possible for the kayakers to have rowed in such a way that this is true.
b) At a different occasion $13$ kayakers rowed in the same manner; now after arrival everybody claims to have done precisely$ 6$ high fives. Prove that at least one kayaker has miscounted.
1949 Moscow Mathematical Olympiad, 172
Two squares are said to be [i]juxtaposed [/i] if their intersection is a point or a segment. Prove that it is impossible to [i]juxtapose [/i] to a square more than eight non-overlapping squares of the same size.
2016 ASDAN Math Tournament, 3
Denote the dot product of two sequences $\langle x_1,\dots,x_n\rangle$ and $\langle y_1,\dots,y_n\rangle$ to be
$$x_1y_1+x_2y_2+\dots+x_ny_n.$$
Let $\langle a_1,\dots,a_n\rangle$ and $\langle b_1,\dots,b_n\rangle$ be two sequences of consecutive integers (i.e. for $1\leq i,i+1\leq n$, $a_i+1=a_{i+1}$ and similarly for $b_i$). Minnie permutes the two sequences so that their dot product, $m$, is minimized. Maximilian permutes the two sequences so that their dot product, $M$, is maximized. Given that $m=4410$ and $M=4865$, compute $n$, the number of terms in each sequence.