Found problems: 85335
1996 Moldova Team Selection Test, 1
The number $n{}$ cointains $k{}$ units in binary system. Prove that $2^{n-k}{}$ divides $n!$.
2002 Germany Team Selection Test, 1
Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying
\[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\
1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases}
\]
Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.
1994 Spain Mathematical Olympiad, 5
Let $21$ pieces, some white and some black, be placed on the squares of a $3\times 7$ rectangle. Prove that there always exist four pieces of the same color standing at the vertices of a rectangle.
2009 Singapore Team Selection Test, 2
Let $H$ be the orthocentre of $\triangle ABC$ and let $P$ be a point on the circumcircle of $\triangle ABC$, distinct from $A,B,C$. Let $E$ and $F$ be the feet of altitudes from $H$ onto $AC$ and $AB$ respectively. Let $PAQB$ and $PARC$ be parallelograms. Suppose $QA$ meets $RH$ at $X$ and $RA$ meets $QH$ at $Y$. Prove that $XE$ is parallel to $YF$.
2011 Romania Team Selection Test, 2
Given a prime number $p$ congruent to $1$ modulo $5$ such that $2p+1$ is also prime, show that there exists a matrix of $0$s and $1$s containing exactly $4p$ (respectively, $4p+2$) $1$s no sub-matrix of which contains exactly $2p$ (respectively, $2p+1$) $1$s.
PEN M Problems, 20
Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?
2000 AIME Problems, 5
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n?$
2016 District Olympiad, 3
Let be a triangle $ ABC $ with $ \angle BAC = 90^{\circ } . $ On the perpendicular of $ BC $ through $ B, $ consider $ D $ such that $ AD=BC. $ Find $ \angle BAD. $
1997 Tournament Of Towns, (535) 7
You are given a balance and one copy of each of ten weights of $1, 2, 4, 8, 16, 32, 64, 128, 256$ and $512$ grams. An object weighing $M$ grams, where $M$ is a positive integer, is put on one of the pans and may be balanced in different ways by placing various combinations of the given weights on either pan of the balance.
(a) Prove that no object may be balanced in more than $89$ ways.
(b) Find a value of $M$ such that an object weighing $M$ grams can be balanced in $89$ ways.
(A Shapovalov, A Kulakov)
1987 All Soviet Union Mathematical Olympiad, 461
All the faces of a convex polyhedron are the triangles. Prove that it is possible to paint all its edges in red and blue colour in such a way, that it is possible to move from the arbitrary vertex to every vertex along the blue edges only and along the red edges only.
2001 239 Open Mathematical Olympiad, 1
Find all triples of natural numbers $ a $, $ b $, $ c $ such that
$$
\gcd (a ^ 2, b ^ 2) + \gcd (a, bc) +\gcd (b, ac) +\gcd (c, ab) = 239 ^ 2 = ab + c .
$$
Kvant 2020, M2629
The figure shows an arbitrary (green) triangle in the center. White squares were built on its sides to the outside. Some of their vertices were connected by segments, white squares were built on them again to the outside, and so on. In the spaces between the squares, triangles and quadrilaterals were formed, which were painted in different colors. Prove that
[list=a]
[*]all colored quadrilaterals are trapezoids;
[*]the areas of all polygons of the same color are equal;
[*]the ratios of the bases of one-color trapezoids are equal;
[*]if $S_0=1$ is the area of the original triangle, and $S_i$ is the area of the colored polygons at the $i^{\text{th}}$ step, then $S_1=1$, $S_2=5$ and for $n\geqslant 3$ the equality $S_n=5S_{n-1}-S_{n-2}$ is satisfied.
[/list]
[i]Proposed by F. Nilov[/i]
[center][img width="40"]https://i.ibb.co/n8gt0pV/Screenshot-2023-03-09-174624.png[/img][/center]
1974 Putnam, B3
Prove that if $a$ is a real number such that
$$\cos \pi a= \frac{1}{3},$$
then $a$ is irrational.
2019 LIMIT Category C, Problem 12
$\lim_{x\to0}x\left\lfloor\frac1x\right\rfloor=?$
2018 CMIMC Individual Finals, 2
John has a standard four-sided die. Each roll, he gains points equal to the value of the roll multiplied by the number of times he has now rolled that number; for example, if his first rolls were $3,3,2,3$, he would have $3+6+2+9=20$ points. Find the expected number of points John will have after rolling the die 25 times.
2025 Belarusian National Olympiad, 9.6
Numbers $a,b,c$ are lengths of sides of some triangle. Prove the inequality$$\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c} \geq \frac{a+b}{2c}+\frac{b+c}{2a}+\frac{c+a}{2b}$$
[i]M. Karpuk[/i]
2000 Chile National Olympiad, 1
Professor David proposed to his wife to calculate the steps of an escalator that worked in a shopping mall, asking him to walk up counting the steps that rise from the bottom to the end. The teacher, in turn, left with his wife, but walking the twice as fast, so that the woman advanced one step each time her husband advanced $2$. When The lady arrived at the top reported that she had counted $21$ steps, while the teacher counted $28$ of them. How many steps are there in sight on the ladder at any given time?
[hide=original wording]El profesor David propuso a su senora calcular los escalones de una escalera mecanica que funcionaba en un centro comercial, pidiendole que caminara hacia arriba contando los escalones que subiera desde la base hasta el final. El profesor a su vez, partio junto a su senora, pero caminando el doble de rapido, de modo que la senora avanzaba un escalon cada vez que su marido avanzaba 2. Cuando la senora llego arriba informo que habıa contado 21 escalones, mientras que el profesor conto 28 de ellos, ¿Cuantos escalones hay a la vista en la escalera en un instante cualquiera?[/hide]
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 $PD$?
2016 KOSOVO TST, 1
Solve equation :
$\sqrt{x+\sqrt{4x+\sqrt{16x}+..+\sqrt{4^nx+3}}}-\sqrt{x}=1$
2017 Harvard-MIT Mathematics Tournament, 1
Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0, 1, \dots, 9$. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll?
1997 Croatia National Olympiad, Problem 1
Find the last four digits of each of the numbers $3^{1000}$ and $3^{1997}$.
2016 ASDAN Math Tournament, 26
The Euclidean Algorithm on inputs $a$ and $b$ is a way to find the greatest common divisor $\gcd(a,b)$. Suppose WLOG that $a>b$. On each step of the Euclidan Algorithm, we solve the equation $a=bq+r$ for integers $q,r$ such that $0\leq r<b$, and repeat on $b$ and $r$. Thus $\gcd(a,b)=\gcd(b,r)$, and we repeat. If $r=0$, we are done. For example, $\gcd(100,15)=\gcd(15,10)=\gcd(10,5)=5$, because $100=15\cdot6+10$, $15=10\cdot1+5$, and $10=5\cdot2+0$. Thus, the Euclidean Algorithm here takes $3$ steps. What is the largest number of steps that the Euclidean Algorithm can take on some integer inputs $a,b$ where $0<a,b<10^{2016}$?
Let $C$ be the actual answer and $A$ be the answer you submit. If $\tfrac{|A-C|}{C}>\tfrac{1}{2}$, then your score will be $0$. Otherwise, your score will be given by $\max\{0,\lceil25-2(\tfrac{|A-C|}{20})^{1/2.2}\rceil\}$.
1985 IMO Longlists, 36
Determine whether there exist $100$ distinct lines in the plane having exactly $1985$ distinct points of intersection
2015 Sharygin Geometry Olympiad, 2
Prove that an arbitrary triangle with area $1$ can be covered by an isosceles triangle with area less than $\sqrt{2}$.
ICMC 5, 6
Is it possible to cover a circle of area $1$ with finitely many equilateral triangles whose areas sum to $1.01$, all pointing in the same direction?
[i]Proposed by Ethan Tan[/i]