Found problems: 85335
OIFMAT III 2013, 1
Find all four-digit perfect squares such that:
$\bullet$ All your figures are less than $9$.
$\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.
2013 Princeton University Math Competition, 7
Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$, where $n$ is either $2012$ or $2013$.
2011 HMNT, 6
Let $ABC$ be an equilateral triangle with $AB = 3$. Circle $\omega$ with diameter $1$ is drawn inside the triangle such that it is tangent to sides $AB$ and $AC$. Let $P$ be a point on $\omega$ and $ $ be a point on segment $BC$. Find the minimum possible length of the segment $PQ$.
2022 ITAMO, 2
Let $ABC$ be an acute triangle with $AB<AC$. Let then
• $D$ be the foot of the bisector of the angle in $A$,
• $E$ be the point on segment $BC$ (different from $B$) such that $AB=AE$,
• $F$ be the point on segment $BC$ (different from $B$) such that $BD=DF$,
• $G$ be the point on segment $AC$ such that $AB=AG$.
Prove that the circumcircle of triangle $EFG$ is tangent to line $AC$.
2014 ASDAN Math Tournament, 5
A positive integer $k$ is $2014$-ambiguous if the quadratics $x^2+kx+2014$ and $x^2+kx-2014$ both have two integer roots. Compute the number of integers which are $2014$-ambiguous.
2011 AMC 12/AHSME, 19
At a competition with $N$ players, the number of players given elite status is equal to \[2^{1+\lfloor\log_2{(N-1)}\rfloor} - N. \] Suppose that $19$ players are given elite status. What is the sum of the two smallest possible values of $N$?
$ \textbf{(A)}\ 38\qquad
\textbf{(B)}\ 90 \qquad
\textbf{(C)}\ 154 \qquad
\textbf{(D)}\ 406 \qquad
\textbf{(E)}\ 1024$
2007 F = Ma, 6
At time $t = 0$ a drag racer starts from rest at the origin and moves along a straight line with velocity given by $v = 5t^2$, where $v$ is in $\text{m/s}$ and $t$ in $\text{s}$. The expression for the displacement of the car from $t = 0$ to time $t$ is
$ \textbf{(A)}\ 5t^3 \qquad\textbf{(B)}\ 5t^3/3\qquad\textbf{(C)}\ 10t \qquad\textbf{(D)}\ 15t^2 \qquad\textbf{(E)}\ 5t/2 $
1998 Chile National Olympiad, 7
When rolling two normal dice, the set of possible outcomes of the sum of the points is $2, 3, 3, 4,4, 4,..., 11, 11,12$. Notice that this sequence can be obtained from the identity $$(x + x^2 + x^3 + x^4 + x^5 + x^6) (x + x^2 + x^3 + x^4 + x^5 + x^6) = x^2 + 2x^3 + 3x^4 +... + 2x^{11} + x^{12}.$$ Design a crazy pair of dice, that is, two other cubes, not necessarily the same, with a natural number indicated on each face, such that the set of possible results of the sum of its points is equal to of two normal dice.
2009 China Second Round Olympiad, 1
Let $\omega$ be the circumcircle of acute triangle $ABC$ where $\angle A<\angle B$ and $M,N$ be the midpoints of minor arcs $BC,AC$ of $\omega$ respectively. The line $PC$ is parallel to $MN$, intersecting $\omega$ at $P$ (different from $C$). Let $I$ be the incentre of $ABC$ and let $PI$ intersect $\omega$ again at the point $T$.
1) Prove that $MP\cdot MT=NP\cdot NT$;
2) Let $Q$ be an arbitrary point on minor arc $AB$ and $I,J$ be the incentres of triangles $AQC,BCQ$. Prove that $Q,I,J,T$ are concyclic.
1985 AMC 8, 14
The difference between a $ 6.5 \%$ sales tax and a $ 6 \%$ sales tax on an item priced at $ \$20$ before tax is
\[ \textbf{(A)}\ \$.01 \qquad
\textbf{(B)}\ \$.10 \qquad
\textbf{(C)}\ \$.50 \qquad
\textbf{(D)}\ \$1 \qquad
\textbf{(E)}\ \$10
\]
2004 Italy TST, 3
Given real numbers $x_i,y_i (i=1,2,\ldots ,n)$, let $A$ be the $n\times n$ matrix given by $a_{ij}=1$ if $x_i\ge y_j$ and $a_{ij}=0$ otherwise. Suppose $B$ is a $n\times n$ matrix whose entries are $0$ and $1$ such that the sum of entries in any row or column of $B$ equals the sum of entries in the corresponding row or column of $A$. Prove that $B=A$.
2006 Bulgaria Team Selection Test, 2
[b] Problem 5. [/b]Denote with $d(a,b)$ the numbers of the divisors of natural $a$, which are greater or equal to $b$. Find all natural $n$, for which
$d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006.$
[i]Ivan Landgev[/i]
2018 Ukraine Team Selection Test, 2
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.
2024 Malaysian Squad Selection Test, 3
Given $n$ students in the plane such that the $\frac{n(n-1)}{2}$ distances are pairwise distinct. Each student gives a candy each to the $k$ students closest to him. Given that each student receives the same amount of candies, determine all possible values of $n$ in terms of $k$.
[i]Proposed by Wong Jer Ren[/i]
2013 Costa Rica - Final Round, 5
Determine the number of polynomials of degree $5$ with different coefficients in the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ such that they are divisible by $x^2-x + 1$. Justify your answer.
1977 Bundeswettbewerb Mathematik, 3
The number $50$ is written as a sum of several positive integers (not necessarily distinct) whose product is divisible by $100.$ What is the largest possible value of this product?
2015 Mexico National Olympiad, 3
Let $\mathbb{N} =\{1, 2, 3, ...\}$ be the set of positive integers. Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a function that gives a positive integer value, to every positive integer. Suppose that $f$ satisfies the following conditions:
$f(1)=1$
$f(a+b+ab)=a+b+f(ab)$
Find the value of $f(2015)$
Proposed by Jose Antonio Gomez Ortega
Mathematical Minds 2023, P6
Let $ABC$ be a triangle, $O{}$ be its circumcenter, $I{}$ its incenter and $I_A,I_B,I_C$ the excenters. Let $M$ be the midpoint of $BC$ and $H_1$ and $H_2$ be the orthocenters of the triangles $MII_A$ and $MI_BI_C$. Prove that the parallel to $BC$ through $O$ passes through the midpoint of the segment $H_1H_2$.
[i]Proposed by David Anghel[/i]
2008 Brazil Team Selection Test, 4
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.
2018 USA Team Selection Test, 2
Find all functions $f\colon \mathbb{Z}^2 \to [0, 1]$ such that for any integers $x$ and $y$,
\[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\]
[i]Proposed by Yang Liu and Michael Kural[/i]
2001 Cuba MO, 2
Let $ABCD$ be a square. On the sides $BC$ and $CD$ the points $M$ and $K$ respectively, so that $MC = KD$. Let $P$ the intersection point of of segments $MD$ and $BK$. Prove that $AP \perp MK$.
2016 SGMO, Q5
Let $d_{m} (n)$ denote the last non-zero digit of $n$ in base $m$ where $m,n$ are naturals. Given distinct odd primes $p_1,p_2,\ldots,p_k$, show that there exists infinitely many natural $n$ such that
$$d_{2p_i} (n!) \equiv 1 \pmod {p_i}$$
for all $i = 1,2,\ldots,k$.
1984 IMO Longlists, 7
Prove that for any natural number $n$, the number $\dbinom{2n}{n}$ divides the least common multiple of the numbers $1, 2,\cdots, 2n -1, 2n$.
2015 IMO Shortlist, N5
Find all positive integers $(a,b,c)$ such that
$$ab-c,\quad bc-a,\quad ca-b$$ are all powers of $2$.
[i]Proposed by Serbia[/i]
2016 AMC 10, 5
The mean age of Amanda's $4$ cousins is $8$, and their median age is $5$. What is the sum of the ages of Amanda's youngest and oldest cousins?
$\textbf{(A)}\ 13\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 25$