Found problems: 85335
2004 VTRMC, Problem 4
A $9\times9$ chess board has two squares from opposite corners and its central square removed. Is it possible to cover the remaining squares using dominoes, where each domino covers two adjacent squares? Justify your answer.
1935 Moscow Mathematical Olympiad, 016
How many real solutions does the following system have ?$\begin{cases} x+y=2 \\
xy - z^2 = 1 \end{cases}$
1950 AMC 12/AHSME, 24
The equation $ x\plus{}\sqrt{x\minus{}2}\equal{}4$ has:
$\textbf{(A)}\ \text{2 real roots} \qquad
\textbf{(B)}\ \text{1 real and 1 imaginary root} \qquad
\textbf{(C)}\ \text{2 imaginary roots} \qquad
\textbf{(D)}\ \text{No roots} \qquad
\textbf{(E)}\ \text{1 real root}$
2010 AMC 8, 10
$6$ pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered with pepperoni?
$ \textbf{(A)}\ \frac 12 \qquad\textbf{(B)}\ \frac 23 \qquad\textbf{(C)}\ \frac 34 \qquad\textbf{(D)}\ \frac 56 \qquad\textbf{(E)}\ \frac 78 $
2024 Princeton University Math Competition, A8
Let $a,b,c$ be pairwise coprime integers such a that $\tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}=\tfrac{N}{a+b+c}$ for some positive integer $N.$ What is the sum of all possible values of $N.$
2017 AMC 12/AHSME, 8
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
$\textbf{(A)} \text{ } \frac{\sqrt{3}-1}{2} \qquad \textbf{(B)} \text{ } \frac{1}{2} \qquad \textbf{(C)} \text{ } \frac{\sqrt{5}-1}{2} \qquad \textbf{(D)} \text{ } \frac{\sqrt{2}}{2} \qquad \textbf{(E)} \text{ } \frac{\sqrt{6}-1}{2}$
1980 IMO, 13
Prove that the integer $145^{n} + 3114\cdot 138^{n}$ is divisible by $1981$ if $n=1981$, and that it is not divisible by $1981$ if $n=1980$.
2014 China Team Selection Test, 6
Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote
$S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$,
$S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$,
Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$
2006 AMC 10, 10
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
$ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$
2025 Taiwan Mathematics Olympiad, 4
Find all positive integers $n$ satisfying the following: there exists a way to fill in $1, \cdots, n^2$ into a $n \times n$ grid so that each block has exactly one number, each number appears exactly once, and:
1. For all positive integers $1 \leq i < n^2$, $i$ and $i + 1$ are neighbors (two numbers neighbor each other if and only if their blocks share a common edge.)
2. Any two numbers among $1^2, \cdots, n^2$ are not in the same row or the same column.
Croatia MO (HMO) - geometry, 2016.3
Given a cyclic quadrilateral $ABCD$ such that the tangents at points $B$ and $D$ to its circumcircle $k$ intersect at the line $AC$. The points $E$ and $F$ lie on the circle $k$ so that the lines $AC, DE$ and $BF$ parallel. Let $M$ be the intersection of the lines $BE$ and $DF$. If $P, Q$ and $R$ are the feet of the altitides of the triangle $ABC$, prove that the points $P, Q, R$ and $M$ lie on the same circle
2015 India PRMO, 2
$2.$ The equations $x^2-4x+k=0$ and $x^2+kx-4=0,$ where $k$ is a real number, have exactly one common root. What is the value of $k ?$
Estonia Open Senior - geometry, 2020.1.5
A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$ . The line $BC$ intersects the circle $c$ for second time at point $F$. The point $G$ on the circle $c$ is chosen such that $| F B | = | FG |$ and $B \ne G$. Prove that the lines $AB, EF$ and $DG$ intersect at one point.
2006 Petru Moroșan-Trident, 2
Consider $ n\ge 1 $ complex numbers $ z_1,z_2,\ldots ,z_n $ that have the same nonzero modulus, and which verify
$$ 0=\Re\left( \sum_{a=1}^n\sum_{b=1}^n\sum_{c=1}^n\sum_{d=1}^n \frac{z_bz_c}{z_az_d} \right) . $$
Prove that $ n\left( -1+\left| z_1 \right|^2 \right) =\sum_{k=1}^n\left| 1-z_k \right| . $
[i]Botea Viorel[/i]
1990 Romania Team Selection Test, 1
Let $f : N \to N$ be a function such that the set $\{k | f(k) < k\}$ is finite.
Prove that the set $\{k | g(f(k)) \le k\}$ is infinite for all functions $g : N \to N$.
1987 IMO Longlists, 62
Let $l, l'$ be two lines in $3$-space and let $A,B,C$ be three points taken on $l$ with $B$ as midpoint of the segment $AC$. If $a, b, c$ are the distances of $A,B,C$ from $l'$, respectively, show that $b \leq \sqrt{ \frac{a^2+c^2}{2}}$, equality holding if $l, l'$ are parallel.
2009 Harvard-MIT Mathematics Tournament, 9
Let $ABC$ be a triangle with $AB=16$ and $AC=5$. Suppose that the bisectors of angle $\angle ABC$ and $\angle BCA$ meet at a point $P$ in the triangle's interior. Given that $AP=4$, compute $BC$.
1989 Vietnam National Olympiad, 1
Let $ n$ and $ N$ be natural number. Prove that for any $ \alpha$ , $ 0\le\alpha\le N$, and any real $ x$, it holds that \[{ |\sum_ {k=0}^n}\frac{\sin((\alpha+k)x)}{N+k}|\le\min\{(n+1)|x|, \frac{1}{N|\sin\frac{x}{2}|}\}\]
2018 Junior Regional Olympiad - FBH, 5
It is given square $ABCD$ which is circumscribed by circle $k$. Let us construct a new square so vertices $E$ and $F$ lie on side $ABCD$ and vertices $G$ and $H$ on arc $AB$ of circumcircle. Find out the ratio of area of squares
2016 Iran MO (2nd Round), 4
Let $l_1,l_2,l_3,...,L_n$ be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let $A$ be the intersection point of lines $l_i,l_j$. We call $A$ an "Interior Point" if there are points $C,D$ on $l_i$ and $E,F$ on $l_j$ such that $A$ is between $C,D$ and $E,F$. Prove that there are at least $\frac{(n-2)(n-3)}{2}$ Interior points.($n>2$)
note: by point here we mean the points which are intersection point of two of $l_1,l_2,...,l_n$.
Russian TST 2014, P1
Let $p{}$ be a prime number and $x_1,x_2,\ldots,x_p$ be integers for which $x_1^n+x_2^n+\cdots+x_p^n$ is divisible by $p{}$ for any positive integer $n{}$. Prove that $x_1-x_2$ is divisible by $p{}.$
2003 Czech And Slovak Olympiad III A, 6
a，b，c>0，abc=1，prove that(a/b)+(b/c)+(c/a)≥a+b+c.
2021 HMNT, 10
Let $n$ be the answer to this problem. Suppose square $ABCD$ has side-length $3$. Then, congruent non-overlapping squares $EHGF$ and $IHJK$ of side-length $\frac{n}{6}$ are drawn such that $A$,$C$, and $H$ are collinear, $E$ lies on $BC$ and $I$ lies on $CD$. Given that $AJG$ is an equilateral triangle, then the area of $AJG$ is $a + b\sqrt{c}$, where $a$, $b$, $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$.
Kvant 2019, M2581
In a social network with a fixed finite setback of users, each user had a fixed set of [i]followers[/i] among the other users. Each user has an initial positive integer rating (not necessarily the same for all users). Every midnight, the rating of every user increases by the sum of the ratings that his followers had just before midnight.
Let $m$ be a positive integer. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by $m$. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days.
[i]Vladislav Novikov[/i]
2014 Tournament of Towns., 2
Peter marks several cells on a $5\times 5$ board. Basil wins if he can cover all marked cells with three-cell corners. The corners must be inside the board and not overlap. What is the least number of cells Peter should mark to prevent Basil from winning? (Cells of the corners must coincide with the cells of the board).