Found problems: 85335
Find all functions $f:\mathbb{R}\to \mathbb{R}$ so that for any reals $x,y$ the following holds:
\[f(x\cdot f(x+y))+f(f(y)\cdot f(x+y))=(x+y)^2\]
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
(1) $D$ is an arbitary point in $\triangle{ABC}$. Prove that:
\[ \frac{BC}{\min{AD,BD,CD}} \geq \{ \begin{array}{c} \displaystyle 2\sin{A}, \ \angle{A}< 90^o \\ \\ 2, \ \angle{A} \geq 90^o \end{array} \]
(2)$E$ is an arbitary point in convex quadrilateral $ABCD$. Denote $k$ the ratio of the largest and least distances of any two points among $A$, $B$, $C$, $D$, $E$. Prove that $k \geq 2\sin{70^o}$. Can equality be achieved?
Let $p$ be a permutation of the set $S_n = \{1, 2, \dots, n\}$. An element $j \in S_n$ is called a fixed point of $p$ if $p(j) = j$. Let $f_n$ be the number of permutations having no fixed points, and $g_n$ be the number with exactly one fixed point. Show that $|f_n - g_n| = 1$.
A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. What is $\cos(\angle AVB)$?
$ \textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad
\textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad
\textbf{(C)}\ -\frac{4}{5} \qquad
\textbf{(D)}\ -\frac{3}{5} \qquad
\textbf{(E)}\ -\frac{1}{2} $
Suppose that the vertices of a regular polygon of $20$ sides are coloured with three colours - red, blue and green - such that there are exactly three red vertices. Prove that there are three vertices $A,B,C$ of the polygon having the same colour such that triangle $ABC$ is isosceles.
Find all functions $f: \mathbb{R}\to{R}$ such that for any $a<b$, $f([a,b])$ is an interval of length $b-a$
Let $x=\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ}.$ What is the greatest integer that does not exceed $100x$?
On a large chessboard $2n$ of its $1 \times 1$ squares have been marked such thar the rook (which moves only horizontally or vertically) can visit all the marked squares without jumpin over any unmarked ones. Prove that the figure consisting of all the marked squares can be cut into rectangles.
(A Shapovalov)
Different points $A,B,C,D,E,F$ lie on circle $k$ in this order. The tangents to $k$ in the points $A$ and $D$ and the lines $BF$ and $CE$ have a common point $P$. Prove that the lines $AD,BC$ and $EF$ also have a common point or are parallel.
Consider an unbiased coin which is tossed infinitely many times. Let $A_n$ be the event that no two successive heads occur in the first $n$ tosses of this experiment. Then which of the following is incorrect :
(A) $\lim_{n \to \infty} P(A_n)=0$
(B) $\lim_{n \to \infty}3^n P(A_n)=0$
(C) $2^nP(A_n) +2^{n+1}P(A_{n+1})=2^{n+2}P(A_{n+2}$
(D) $\lim_{n \to \infty} \frac{P(A_n)}{P(A_{n+1})}$ is lesser than $1.2$
Let $ABC$ be a scalene triangle. Given the center $I$ of the inscribe circle and the points $K_1$, $K_2$ and $K_3$ where the inscribed circle is tangent to the sides $BC$, $AC$ and $AB$. Using only a ruler, construct the center of the circumscribed circle of triangle $ABC$.
(Hryhorii Filippovskyi)
For positive integer $n,$ let $s(n)$ be the sum of the digits of n when n is expressed in base ten. For example, $s(2022) = 2 + 0 + 2 + 2 = 6.$ Find the sum of the two solutions to the equation $n - 3s(n) = 2022.$
Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle.
[b]a.)[/b] Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle.
[b]b.)[/b] Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.
$x$ and $y$ are chosen at random (with uniform density) from the interval $(0, 1)$. What is the probability that the closest integer to $x/y$ is even?
There is a pile of eggs. Joan counted the eggs, but her count was way off by $1$ in the $1$'s place. Tom counted in the eggs, but his count was off by $1$ in the $10$'s place. Raoul counted the eggs, but his count was off by $1$ in the $100$'s place. Sasha, Jose, Peter, and Morris all counted the eggs and got the correct count. When these seven people added their counts together, the sum was $3162$. How many eggs were in the pile?
Let $n$ be a positive integer. A [i]Nordic[/i] square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a [i]valley[/i]. An [i]uphill path[/i] is a sequence of one or more cells such that:
(i) the first cell in the sequence is a valley,
(ii) each subsequent cell in the sequence is adjacent to the previous cell, and
(iii) the numbers written in the cells in the sequence are in increasing order.
Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square.
Author: Nikola Petrović
Prove that if $ M $, $ N $, $ P $ are the feet of the altitudes of acute-angled triangle $ ABC $, then the ratio of the perimeter of triangle $ MNP $ to the perimeter of triangle $ ABC $ is equal to the ratio of the radius of the circle inscribed in triangle $ ABC $ to the radius of the circle circumscribed about triangle $ ABC $.
Let there be $n$ students, numbered $1$ through $n$. Let there be $n$ cards with numbers $1$ through $n$ written on them. Each student picks a card from the stack, and two students are called a pair if they pick each other's number. Let the probability that there are no pairs be $p_n$.
Prove that $p_n - p_{n-1}=0$ if $n$ is odd, and
prove that $p_n - p_{n-1}= \frac{1}{(-2)^kk^{1-k}}$ if $n = 2k$.
Let one of the intersection points of two circles with centres $O_1,O_2$ be $P$. A common tangent touches the circles at $A,B$ respectively. Let the perpendicular from $A$ to the line $BP$ meet $O_1O_2$ at $C$. Prove that $AP\perp PC$.
The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that
\[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\]
for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that
\[\varphi(x,y,z) = h(x+y+z)\]
for all real numbers $x,y$ and $z.$
A four-digit number $n$ is said to be [i]literally 1434[/i] if, when every digit is replaced by its remainder when divided by $5$, the result is $1434$. For example, $1984$ is [i]literally 1434[/i] because $1$ mod $5$ is $1$, $9$ mod $5$ is $4$, $8$ mod $5$ is $3$, and $4$ mod $5$ is $4$. Find the sum of all four-digit positive integers that are [i]literally 1434[/i].
[i]Proposed by Evin Liang[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{67384}$
The possible numbers are $\overline{abcd}$ where $a$ is $1$ or $6$, $b$ is $4$ or $9$, $c$ is $3$ or $8$, and $d$ is $4$ or $9$. There are $16$ such numbers and the average is $\dfrac{8423}{2}$, so the total in this case is $\boxed{67384}$.
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Prove that there exist a natural number $a$, for which 999 divides $2^{5n}+a.5^n$ for $\forall$ odd $n\in \mathbb{N}$ and find the smallest such $a$.
Find the real numbers $a$, $b$, $c$ and $d$ that satisfy the following equations:
$$\left \{\begin{matrix}
a\cdot b+c+d & = & 6, \\
b\cdot c+d+a & = & 2, \\
c\cdot d+a+b & = & 5, \\
d\cdot a+b+c & = & 3.
\end{matrix}\right .$$
Inside a quadrilateral $ABCD$ there exists a point $P$ such that $AP$ is perpendicular to $AD$ and the line $BP$ is perpendicular to $DC$. Besides, $AB = 7$, $AP = 3$, $BP = 6$, $AD = 5 $ and $CD = 10$. Calculate the area of the triangle $ABC$.