Found problems: 85335
1991 Putnam, B5
Let $p>2$ be a prime. How many residues $\pmod p$ are both squares and squares plus one?
2013 Flanders Math Olympiad, 3
Anton the ant takes a walk along the vertices of a cube. He starts at a vertex and stops when it reaches this point again. Between two vertices it moves over an edge, a side face diagonal or a space diagonal. During the rout it visits each of the other vertices exactly [i]once [/i] and nowhere intersects its road already traveled.
(a) Show that Anton walks along at least one edge.
(b) Show that Anton walks along at least two edges.
2024 Malaysian IMO Training Camp, 7
Let $P$ be the set of all primes. Given any positive integer $n$, define $$\displaystyle f(n) = \max_{p \in P}v_p(n)$$ Prove that for any positive integer $k\ge 2$, there exists infinitely many positive integers $m$ such that \[ f(m+1) = f(m+2) = \cdots = f(m+k) \]
[i]Proposed by Ivan Chan Guan Yu[/i]
1998 Romania Team Selection Test, 4
Consider in the plane a finite set of segments such that the sum of their lengths is less than $\sqrt{2}$. Prove that there exists an infinite unit square grid covering the plane such that the lines defining the grid do not intersect any of the segments.
[i]Vasile Pop[/i]
IV Soros Olympiad 1997 - 98 (Russia), 11.2
Find all values of the parameter $a$ for which there are exactly $1998$ integers $x$ satisfying the inequality $$x^2 -\pi x +a < 0.$$
2019 IFYM, Sozopol, 5
Let $A$ be the number of 2019-digit numbers, that is made of 2 different digits (For example $10\underbrace{1...1}_{2016}0$ is such number). Determine the highest power of 3 that divides $A$.
1995 Tournament Of Towns, (471) 5
A simple polygon in the plane is a figure bounded by a closed nonself-intersecting broken line.
(a) Do there exist two congruent simple $7$-gons in the plane such that all the seven vertices of one of the $7$-gons are the vertices of the other one and yet these two $7$-gons have no common sides?
(b) Do there exist three such $7$-gons?
(V Proizvolov)
2015 South East Mathematical Olympiad, 5
Given two points $E$ and $F$ lie on segment $AB$ and $AD$, respectively. Let the segments $BF$ and $DE$ intersects at point $C$. If it’s known that $AE+EC=AF+FC$, show that $AB+BC=AD+DC$.
2011 Bulgaria National Olympiad, 2
For each natural number $a$ we denote $\tau (a)$ and $\phi (a)$ the number of natural numbers dividing $a$ and the number of natural numbers less than $a$ that are relatively prime to $a$. Find all natural numbers $n$ for which $n$ has exactly two different prime divisors and $n$ satisfies $\tau (\phi (n))=\phi (\tau (n))$.
1986 Traian Lălescu, 2.4
Show that there is an unique group $ G $ (up to isomorphism) of order $ 1986 $ which has the property that there is at most one subgroup of it having order $ n, $ for every natural number $ n. $
2024 Philippine Math Olympiad, P6
The sequence $\{a_n\}_{n\ge 1}$ of real numbers is defined as follows:
$$a_1=1, \quad \text{and}\quad a_{n+1}=\frac{1}{2\lfloor a_n \rfloor -a_n+1} \quad \text{for all} \quad n\ge 1$$
Find $a_{2024}$.
2012 Vietnam National Olympiad, 2
Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.
2020 Iranian Geometry Olympiad, 3
In acute-angled triangle $ABC$ ($AC > AB$), point $H$ is the orthocenter and point $M$ is the midpoint of the segment $BC$. The median $AM$ intersects the circumcircle of triangle $ABC$ at $X$. The line $CH$ intersects the perpendicular bisector of $BC$ at $E$ and the circumcircle of the triangle $ABC$ again at $F$. Point $J$ lies on circle $\omega$, passing through $X, E,$ and $F$, such that $BCHJ$ is a trapezoid ($CB \parallel HJ$). Prove that $JB$ and $EM$ meet on $\omega$.
[i]Proposed by Alireza Dadgarnia[/i]
2019 MMATHS, Mixer Round
[b]p1.[/b] An ant starts at the top vertex of a triangular pyramid (tetrahedron). Each day, the ant randomly chooses an adjacent vertex to move to. What is the probability that it is back at the top vertex after three days?
[b]p2.[/b] A square “rolls” inside a circle of area $\pi$ in the obvious way. That is, when the square has one corner on the circumference of the circle, it is rotated clockwise around that corner until a new corner touches the circumference, then it is rotated around that corner, and so on. The square goes all the way around the circle and returns to its starting position after rotating exactly $720^o$. What is the area of the square?
[b]p3.[/b] How many ways are there to fill a $3\times 3$ grid with the integers $1$ through $9$ such that every row is increasing left-to-right and every column is increasing top-to-bottom?
[b]p4.[/b] Noah has an old-style M&M machine. Each time he puts a coin into the machine, he is equally likely to get $1$ M&M or $2$ M&M’s. He continues putting coins into the machine and collecting M&M’s until he has at least $6$ M&M’s. What is the probability that he actually ends up with $7$ M&M’s?
[b]p5.[/b] Erik wants to divide the integers $1$ through $6$ into nonempty sets $A$ and $B$ such that no (nonempty) sum of elements in $A$ is a multiple of $7$ and no (nonempty) sum of elements in $B$ is a multiple of $7$. How many ways can he do this? (Interchanging $A$ and $B$ counts as a different solution.)
[b]p6.[/b] A subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ of size $3$ is called special if whenever $a$ and $b$ are in the set, the remainder when $a + b$ is divided by $8$ is not in the set. ($a$ and $b$ can be the same.) How many special subsets exist?
[b]p7.[/b] Let $F_1 = F_2 = 1$, and let $F_n = F_{n-1} + F_{n-2}$ for all $n \ge 3$. For each positive integer $n$, let $g(n)$ be the minimum possible value of $$|a_1F_1 + a_2F_2 + ...+ a_nF_n|,$$ where each $a_i$ is either $1$ or $-1$. Find $g(1) + g(2) +...+ g(100)$.
[b]p8.[/b] Find the smallest positive integer $n$ with base-$10$ representation $\overline{1a_1a_2... a_k}$ such that $3n = \overline{a_1a_2 a_k1}$.
[b]p9.[/b] How many ways are there to tile a $4 \times 6$ grid with $L$-shaped triominoes? (A triomino consists of three connected $1\times 1$ squares not all in a line.)
[b]p10.[/b] Three friends want to share five (identical) muffins so that each friend ends up with the same total amount of muffin. Nobody likes small pieces of muffin, so the friends cut up and distribute the muffins in such a way that they maximize the size of the smallest muffin piece. What is the size of this smallest piece?
[u]Numerical tiebreaker problems:[/u]
[b]p11.[/b] $S$ is a set of positive integers with the following properties:
(a) There are exactly 3 positive integers missing from $S$.
(b) If $a$ and $b$ are elements of $S$, then $a + b$ is an element of $S$. (We allow $a$ and $b$ to be the same.)
How many possibilities are there for the set $S$?
[b]p12.[/b] In the trapezoid $ABCD$, both $\angle B$ and $\angle C$ are right angles, and all four sides of the trapezoid are tangent to the same circle. If $\overline{AB} = 13$ and $\overline{CD} = 33$, find the area of $ABCD$.
[b]p13.[/b] Alice wishes to walk from the point $(0, 0)$ to the point $(6, 4)$ in increments of $(1, 0)$ and $(0, 1)$, and Bob wishes to walk from the point $(0, 1)$ to the point $(6, 5)$ in increments of $(1, 0)$ and $(0,1)$. How many ways are there for Alice and Bob to get to their destinations if their paths never pass through the same point (even at different times)?
[b]p14.[/b] The continuous function $f(x)$ satisfies $9f(x + y) = f(x)f(y)$ for all real numbers $x$ and $y$. If $f(1) = 3$, what is $f(-3)$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2013 Korea National Olympiad, 5
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying
\[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \]
for all positive integer $m,n$.
2014 AMC 8, 7
There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of number of girls to the number of boys in her class?
$\textbf{(A) }3 : 4\qquad\textbf{(B) }4 : 3\qquad\textbf{(C) }3 : 2\qquad\textbf{(D) }7 : 4\qquad \textbf{(E) }2 : 1$
2018 Brazil Team Selection Test, 4
Consider an isosceles triangle $ABC$ with $AB = AC$. Let $\omega(XYZ)$ be the circumcircle of the triangle $XY Z$. The tangents to $\omega(ABC)$ through $B$ and $C$ meet at the point $D$. The point $F$ is marked on the arc $AB$ of $\omega(ABC)$ that does not contain $C$. Let $K$ be the point of intersection of lines $AF$ and $BD$ and $L$ the point of intersection of the lines $AB$ and $CF$. Let $T$ and $S$ be the centers of the circles $\omega(BLC)$ and $\omega(BLK)$, respectively. Suppose that the circles $\omega(BTS)$ and $\omega(CFK)$ are tangent to each other at the point $P$. Prove that $P$ belongs to the line $AB$.
2012 USA TSTST, 8
Let $n$ be a positive integer. Consider a triangular array of nonnegative integers as follows: \[
\begin{array}{rccccccccc}
\text{Row } 1: &&&&& a_{0,1} &&&& \smallskip\\
\text{Row } 2: &&&& a_{0,2} && a_{1,2} &&& \smallskip\\
&&& \vdots && \vdots && \vdots && \smallskip\\
\text{Row } n-1: && a_{0,n-1} && a_{1,n-1} && \cdots && a_{n-2,n-1} & \smallskip\\
\text{Row } n: & a_{0,n} && a_{1,n} && a_{2,n} && \cdots && a_{n-1,n}
\end{array}
\] Call such a triangular array [i]stable[/i] if for every $0 \le i < j < k \le n$ we have \[ a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1. \] For $s_1, \ldots s_n$ any nondecreasing sequence of nonnegative integers, prove that there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.
2021 Thailand TST, 2
Prove that, for all positive integers $m$ and $n$, we have $$\left\lfloor m\sqrt{2} \right\rfloor\cdot\left\lfloor n\sqrt{7} \right\rfloor<\left\lfloor mn\sqrt{14} \right\rfloor.$$
2010 Austria Beginners' Competition, 2
In a national park there is a group of sequoia trees, all of which have a positive integer age. Their average age is $41$ years. After a $2010$ year old building was destroyed by lightning, the average age drops to $40$ years. How many trees were originally in the group? At most, how many of them were exactly $2010$ years old?
(W. Janous, WRG Ursulinen, Innsbruck)
2024 5th Memorial "Aleksandar Blazhevski-Cane", P4
Let $D$ be a point inside $\triangle ABC$ such that $\angle CDA + \angle CBA = 180^{\circ}.$ The line $CD$ meets the circle $\odot ABC$ at the point $E$ for the second time. Let $G$ be the common point of the circle centered at $C$ with radius $CD$ and the arc $\overset{\LARGE \frown}{AC}$ of $\odot ABC$ which does not contain the point $B$. The circle centered at $A$ with radius $AD$ meets $\odot BCD$ for the second time at $F$.
Prove that the lines $GE, FD, CB$ are concurrent or parallel.
Kvant 2019, M2543
Let $a$ and $b$ be 2019-digit numbers. Exactly 12 digits of $a$ are non-zero: the five leftmost and seven rightmost, and exactly 14 digits of $b$ are non-zero: the five leftmost and nine rightmost. Prove that the largest common divisor of $a$ and $b$ has no more than 14 digits.
[i]Proposed by L. Samoilov[/i]
2014 Contests, 1
On a circle there are $99$ natural numbers. If $a,b$ are any two neighbouring numbers on the circle, then $a-b$ is equal to $1$ or $2$ or $ \frac{a}{b}=2 $. Prove that there exists a natural number on the circle that is divisible by $3$.
[i]S. Berlov[/i]
2017 Gulf Math Olympiad, 2
One country consists of islands $A_1,A_2,\cdots,A_N$,The ministry of transport decided to build some bridges such that anyone will can travel by car from any of the islands $A_1,A_2,\cdots,A_N$ to any another island by one or more of these bridges. For technical reasons the only bridges that can be built is between $A_i$ and $A_{i+1}$ where $i = 1,2,\cdots,N-1$ , and between $A_i$ and $A_N$ where $i<N$.
We say that a plan to build some bridges is good if it is satisfies the above conditions , but when we remove any bridge it will not satisfy this conditions. We assume that there is $a_N$ of good plans. Observe that $a_1 = 1$ (The only good plan is to not build any bridge) , and $a_2 = 1$ (We build one bridge).
1-Prove that $a_3 = 3$
2-Draw at least $5$ different good plans in the case that $N=4$ and the islands are the vertices of a square
3-Compute $a_4$
4-Compute $a_6$
5-Prove that there is a positive integer $i$ such that $1438$ divides $a_i$
2006 Vietnam Team Selection Test, 3
In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.
Find the minimum value of a [i]good[/i] number $m$.