Found problems: 85335
1990 French Mathematical Olympiad, Problem 1
Let the sequence $u_n$ be defined by $u_0=0$ and $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for each $n\in\mathbb N_0$.
(a) Calculate $u_{1990}$.
(b) Find the number of indices $n\le1990$ for which $u_n=0$.
(c) Let $p$ be a natural number and $N=(2^p-1)^2$. Find $u_N$.
2007 Bulgarian Autumn Math Competition, Problem 10.3
For a natural number $m>1$ we'll denote with $f(m)$ the sum of all natural numbers less than $m$, which are also coprime to $m$. Find all natural numbers $n$, such that there exist natural numbers $k$ and $\ell$ which satisfy $f(n^{k})=n^{\ell}$.
2023 AMC 10, 18
A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2017 NIMO Summer Contest, 2
Joy has $33$ thin rods, one each of every integer length from $1$ cm through $30$ cm, and also three more rods with lengths $3$ cm, $7$ cm, and $15$ cm. She places those three rods on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
[i]Proposed by Michael Tang[/i]
2013 Dutch Mathematical Olympiad, 3
The sides $BC$ and $AD$ of a quadrilateral $ABCD$ are parallel and the diagonals intersect in $O$. For this quadrilateral $|CD| =|AO|$ and $|BC| = |OD|$ hold. Furthermore $CA$ is the angular bisector of angle $BCD$. Determine the size of angle $ABC$.
[asy]
unitsize(1 cm);
pair A, B, C, D, O;
D = (0,0);
B = 3*dir(180 + 72);
C = 3*dir(180 + 72 + 36);
A = extension(D, D + (1,0), C, C + dir(180 - 36));
O = extension(A, C, B, D);
draw(A--B--C--D--cycle);
draw(B--D);
draw(A--C);
dot("$A$", A, N);
dot("$B$", B, SW);
dot("$C$", C, SE);
dot("$D$", D, N);
dot("$O$", O, E);
[/asy]
Attention: the figure is not drawn to scale.
1999 National Olympiad First Round, 32
Let $ \left(a_{n} \right)_{n \equal{} 1}^{\infty }$ be a sequence on real numbers such that $ a{}_{n \plus{} 1} \equal{} a_{n} a_{n \plus{} 2}$ for every $ n\ge 1$. The number of elements in the set $ \left\{a_{n} : n\ge 1\right\}$ cannot be
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None}$
2019 AMC 8, 8
Gilda has a bag of marbles. She gives $20 \%$ of them to her friend Pedro. The, Gilda gives $10 \%$ of what is left to her other friend, Ebony. Finally, Gilda gives $25 \%$ of what is left in the bag to her brother. What percentage of her original bag does she have left?
$\textbf{(A) } 20 \qquad\textbf{(B) } 33\tfrac{1}{3} \qquad\textbf{(C) } 38 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 54$
KoMaL A Problems 2019/2020, A. 761
Let $n\ge3$ be a positive integer. We say that a set $S$ of positive integers is good if $|S|=n$, no element of S is a multiple of n, and the sum of all elements of $S$ is not a multiple of $n$ either. Find, in terms of $n$, the least positive integer $d$ for which there exists a good set $S$ such that there are exactly d nonempty subsets of $S$ the sum of whose elements is a multiple of $n$.
Proposed by Aleksandar Makelov, Burgas, Bulgaria and Nikolai Beluhov, Stara Zagora, Bulgaria
2019 CCA Math Bonanza, L2.1
Noew is writing a $15$-problem mock AIME consisting of four subjects of problems: algebra, geometry, combinatorics, and number theory. The AIME is considered [i]somewhat evenly distributed[/i] if there is at least one problem of each subject and there are at least six combinatorics problems. Two AIMEs are considered [i]similar[/i] if they have the same subject distribution (same number of each subject). How many non-similar somewhat evenly distributed mock AIMEs can Noew write?
[i]2019 CCA Math Bonanza Lightning Round #2.1[/i]
2007 May Olympiad, 4
Alex and Bruno play the following game: each one, in your turn, the player writes, exactly one digit, in the right of the last number written. The game finishes if we have a number with $6$ digits( distincts ) and Alex starts the game. Bruno wins if the number with $6$ digits is a prime number, otherwise Alex wins.
Which player has the winning strategy?
Russian TST 2017, P3
Let $K=(V, E)$ be a finite, simple, complete graph. Let $\phi: E \to \mathbb{R}^2$ be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle are collinear. Show that the range of $\phi$ is contained in a line.
Kyiv City MO Juniors 2003+ geometry, 2014.7.4
In the quadrilateral $ABCD$ the condition $AD = AB + CD$ is fulfilled. The bisectors of the angles $BAD$ and $ADC$ intersect at the point $P $, as shown in Fig. Prove that $BP = CP$.
[img]https://cdn.artofproblemsolving.com/attachments/3/1/67268635aaef9c6dc3363b00453b327cbc01f3.png[/img]
(Maria Rozhkova)
1969 Miklós Schweitzer, 3
Let $ f(x)$ be a nonzero, bounded, real function on an Abelian group $ G$, $ g_1,...,g_k$ are given elements of $ G$ and $ \lambda_1,...,\lambda_k$ are real numbers. Prove that if \[ \sum_{i=1}^k \lambda_i f(g_ix) \geq 0\] holds for all $ x \in G$, then \[ \sum_{i=1}^k \lambda_i \geq 0.\]
[i]A. Mate[/i]
2015 Polish MO Finals, 1
In triangle $ABC$ the angle $\angle A$ is the smallest. Points $D, E$ lie on sides $AB, AC$ so that $\angle CBE=\angle DCB=\angle BAC$. Prove that the midpoints of $AB, AC, BE, CD$ lie on one circle.
2019 IberoAmerican, 6
Let $a_1, a_2, \dots, a_{2019}$ be positive integers and $P$ a polynomial with integer coefficients such that, for every positive integer $n$,
$$P(n) \text{ divides } a_1^n+a_2^n+\dots+a_{2019}^n.$$
Prove that $P$ is a constant polynomial.
2014 Greece Team Selection Test, 4
Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.
2004 Romania National Olympiad, 4
In the interior of a cube of side $6$ there are $1001$ unit cubes with the faces parallel to the faces of the given cube. Prove that there are $2$ unit cubes with the property that the center of one of them lies in the interior or on one of the faces of the other cube.
[i]Dinu Serbanescu[/i]
2006 Princeton University Math Competition, 1
What is the greatest possible number of edges in a planar graph with $12$ vertices? A planar graph is one that can be drawn in a plane with none of the edges crossing (they intersect only at vertices).
2009 Kosovo National Mathematical Olympiad, 1
Find the graph of the function $y=x+|1-x^3|$.
2006 National Olympiad First Round, 6
What is the sum of $3+3^2+3^{2^2} + 3^{2^3} + \dots + 3^{2^{2006}}$ in $\mod 11$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 10
$
2010 Benelux, 4
Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and
\[a^3 + b^3 = p^n\mbox{.}\]
[i](2nd Benelux Mathematical Olympiad 2010, Problem 4)[/i]
1993 All-Russian Olympiad, 3
Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?
2016 AMC 8, 19
The sum of $25$ consecutive even integers is $10,000$. What is the largest of these $25$ consecutive integers?
$\textbf{(A)}\mbox{ }360\qquad\textbf{(B)}\mbox{ }388\qquad\textbf{(C)}\mbox{ }412\qquad\textbf{(D)}\mbox{ }416\qquad\textbf{(E)}\mbox{ }424$
1973 All Soviet Union Mathematical Olympiad, 176
Given $n$ points, $n > 4$. Prove that tou can connect them with arrows, in such a way, that you can reach every point from every other point, having passed through one or two arrows. (You can connect every pair with one arrow only, and move along the arrow in one direction only.)
2022 MIG, 21
Let $T(p)$ denote the number of right triangles with integer side lengths and one of its side lengths being $p$. Which of the following values of $p$ produces the greatest possible value of $T(p)$ among all five answer choices?
$\textbf{(A) }24\qquad\textbf{(B) }27\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }54$