Found problems: 85335
$\frac{2}{25}=$
$\text{(A)}\ .008 \qquad \text{(B)}\ .08 \qquad \text{(C)}\ .8 \qquad \text{(D)} 1.25 \qquad \text{(E)}\ 12.5$
We place the numbers from $1$ to $81$ in a $9\times $ board. Prove that exist $k \in \{1,2,...,9\}$ so that the product of the numbers in the $k$-th column is diferent to the product of the numbers in the $k$-th row.
Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this?
[i]Proposed by Oleksiy Masalitin[/i]
A convex hexagon $ ABCDEF$ satisfies $ AB\equal{}BC, CD\equal{}DE, EF\equal{}FA$ and: $ \angle ABC\plus{}\angle CDE\plus{}\angle EFA \equal{} 360^{\circ}$. Prove that the perpendiculars from $ A,C$ and $ E$ to $ FB,BD$ and $ DF$ respectively are concurrent.
Let be given a semicircle with the diameter $AB$, and points $C,D$ on it such that $AC = CD$. The tangent at $C$ intersects the line $BD$ at $E$. The line $AE$ intersects the arc of the semicircle at $F$. Prove that $CF < FD$.
In three cisterns of milk lies $780$ litres of milk. When we pour off from first cistern quarter of milk, from second cistern fifth of milk and from third cistern $\frac{3}{7}$ of milk, in all cisterns remain same amount of milk. How many milk is in cisterns?
Let $S = \{1, \cdots, 6\}$ and $\mathcal{P}$ be the set of all nonempty subsets of $S$. Let $N$ equal the number of functions $f:\mathcal P \to S$ such that if $A,B\in \mathcal P$ are disjoint, then $f(A)\neq f(B)$.
Determine the number of positive integer divisors of $N$.
Source: 2018 Canadian Open Math Challenge Part C Problem 3
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Consider a convex quadrilateral $ABCD$. Let rays $BA$ and $CD$ intersect at $E$, rays $DA$ and $CB$ intersect at $F$, and the diagonals $AC$ and $BD$ intersect at $G$. It is given that the triangles $DBF$ and $DBE$ have the same area.
$\text{(a)}$ Prove that $EF$ and $BD$ are parallel.
$\text{(b)}$ Prove that $G$ is the midpoint of $BD$.
$\text{(c)}$ Given that the area of triangle $ABD$ is 4 and the area of triangle $CBD$ is 6,
[color=transparent](C.)[/color]compute the area of triangle $EFG$.
In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a $\tfrac{2}{3}$ probability of voting for Alice and a $\tfrac{1}{3}$ probability of voting for Celia.
What is the probability that Alice wins the election (by having more votes than Celia)?
Show that there are no functions $f : R \to R$ satisfying $f(x + f(y)) = f(x) + y^2$ for all real numbers $x$ and $y$
Is it possible to fill the cells of a table of size $2019\times2019$ with pairwise distinct positive integers in such a way that in each rectangle of size $1\times2$ or $2\times1$ the larger number is divisible by the smaller one, and the ratio of the largest number in the table to the smallest one is at most $2019?$
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$. The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$. Let $E$ be the second point of intersection between $AQ$ and $\omega$. Prove that $B$, $E$, $R$ are collinear.
The sequence $(a_n)$ is defined as:
$$a_1=1007$$
$$a_{i+1}\geq a_i+1$$
Prove the inequality:
$$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$
Let $n$ be a positive integer. Each cell on an $n \times n$ board will be filled with a positive integer less than or equal to $2n-1$ such that for each index $i$ with $1 \leq i \leq n$, the $2n-1$ cells in the $i^{\text{th}}$ row or $i^{\text{th}}$ collumn contain distinct integers.
(a) Is this filling possible for $n=4$?
(b) Is this filling possible for $n=5$?
Solve the equation $x^4 + 4x^3 - 8x + 4 = 0$.
Evaluate $\frac{2i}{\frac{1}{30 - 5i} - \frac{1}{30 + 5i}}$ where $i=\sqrt{-1}$.
Let $ABC$ be a triangle such that $AB > BC$ and let $D$ be a variable point on the line segment $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$, lying on the opposite side of $BC$ from $A$ such that $\angle BAE = \angle DAC$. Let $I$ be the incenter of triangle $ABD$ and let $J$ be the incenter of triangle $ACE$. Prove that the line $IJ$ passes through a fixed point, that is independent of $D$.
[i]Proposed by Merlijn Staps[/i]
Three wise men live in an old region. As they do not always agree on their advice to the king, he decided to stay with the wisest of the three, killing the others. To decide which of them was saved, performed the following test: He put each sage a hat, without him seeing its color, then locked them in a common room and told them:
$\bullet$ Only the first to guess the color of his own hat will save his life.
$\bullet$ In total there are five hats, three are white and two are black.
$\bullet$ You cannot communicate with each other, but you can look at each other.
After a long time, one of the wise men says: "I know the color of my hat."
What color did they have? How did you figure it out? What color were the other hats used?
The tetrahedron $ S.ABC$ has the faces $ SBC$ and $ ABC$ perpendicular. The three angles at $ S$ are all $ 60^{\circ}$ and $ SB \equal{} SC \equal{} 1$. Find the volume of the tetrahedron.
Let $ABC$ be an acute triangle with orthocentre $H$. Let $D$ be a point outside the circumcircle of triangle $ABC$ such that $\angle ABD=\angle DCA$. The reflection of $AB$ in $BD$ intersects $CD$ at $X$. The reflection of $AC$ in $CD$ intersects $BD$ at $Y$. The lines through $X$ and $Y$ perpendicular to $AC$ and $AB$, respectively, intersect at $P$. Prove that points $D$, $P$ and $H$ are collinear.
Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
$\textbf{(A)}\ \frac{1}{10}\qquad
\textbf{(B)}\ \frac{1}{6}\qquad
\textbf{(C)}\ \frac{11}{60}\qquad
\textbf{(D)}\ \frac{1}{5}\qquad
\textbf{(E)}\ \frac{7}{30}$
Two concentric spheres have radii $r$ and $R,r < R$. We try to select points $A, B$ and $C$ on the surface of the larger sphere such that all sides of the triangle $ABC$ would be tangent to the surface of the smaller sphere. Show that the points can be selected if and only if $R \le 2r$.
Let $H$ be the orthocenter of an acute-angled triangle $ABC$; $E$, $F$ be points on $AB, AC$ respectively, such that $AEHF$ is a parallelogram; $X, Y$ be the common points of the line $EF$ and the circumcircle $\omega$ of triangle $ABC$; $Z$ be the point of $\omega$ opposite to $A$. Prove that $H$ is the orthocenter of triangle $XYZ$.
Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$, find the modulo $2007$ residue of $m$.
Sahand and Gholam play on a $1403\times 1403$ table. Initially all the unit square cells are white. For each row and column there is a key for it (totally 2806 keys). Starting with Sahand players take turn alternatively pushing a button that has not been pushed yet, until all the buttons are pushed. When Sahand pushes a button all the cells of that row or column become black, regardless of the previous colors. When Gholam pushes a button all the cells of that row or column become red, regardless of the previous colors. Finally, Gholam's score equals the number of red squares minus the number of black squares and Sahand's score equals the number of black squares minus the number of red squares. Determine the minimum number of scores Gholam can guarantee without if both players play their best moves.