Solve the equation: \[4x^2+|9-6x|=|10x-15|+6(2x+1)\]
2022 Bulgarian Autumn Math Competition
It's given a right-angled triangle $ABC (\angle{C}=90^{\circ})$ and area $S$. Let $S_1$ be the area of the circle with diameter $AB$ and $k=\frac{S_1}{S}$ a) Compute the angles of $ABC$, if $k=2\pi$ b) Prove it is not possible for k to be $3$
On a circle are given the points $A_1, B_1, A_2, B_2, \cdots, A_9, B_9$ in this order. All the segments $A_iB_j (i, j=1, 2, \cdots, 9$ must be colored in one of $k$ colors, so that no two segments from the same color intersect (inside the circle) and for every $i$ there is a color, such that no segments with an end $A_i$, nor $B_i$ is colored such. What is the least possible $k$?
Find the number of sequences with $2022$ natural numbers $n_1, n_2, n_3, \ldots, n_{2022}$, such that in every sequence: $\bullet$ $n_{i+1}\geq n_i$ $\bullet$ There is at least one number $i$, such that $n_i=2022$ $\bullet$ For every $(i, j)$ $n_1+n_2+\ldots+n_{2022}-n_i-n_j$ is divisible to both $n_i$ and $n_j$
Given is the equation: \[x^2+mx+2022=0\]a) Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{natural}$ numbers b)Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{integer}$ numbers
Given is the triangle $ABC$ such that $BC=13, CA=14, AB=15$ Prove that $B$, the incenter $J$ and the midpoints of $AB$ and $BC$ all lie on a circle
Find all the pairs of natural numbers $(a, b),$ such that \[a!+1=(a+1)^{(2^b)}\]
Given is $2022\times 2022$ cells table. We can select $4$ cells, such that they make the figure $L$ (rotations, symmetric still count) (left one) and put a ball in each of them, or select $4$ cell which makes up the right figure (rotations, symmetric still count) and get one ball from each of them. For which $k$ is it possible in a given moment to be exactly $k$ points in each of the cells
Solve the equation: \[3\sqrt{3x-1}=x^2+1\]
Fix a triangle $ABC$. The variable point $M$ in its interior is such that $\angle MAC = \angle MBC$ and $N$ is the reflection of $M$ with respect to the midpoint of $AB$. Prove that $|AM| \cdot |BM| + |CM| \cdot |CN|$ is independent of the choice of $M$.
Are there natural number(s) $n$, such that $3^n+1$ has a divisor in the form $24k+20$
The European zoos with exactly $100$ types of species each are separated into two groups $\hat{A}$ and $\hat{B}$ in such a way that every pair of zoos $(A, B)$ $(A\in\hat{A}, B\in\hat{B})$ have some animal in common. Prove that we can colour the cages in $3$ colours (all animals of the same type live in the same cage) such that no zoo has cages of only one colour
Find all real numbers $q$, such that for all real $p \geq 0$, the equation $x^2-2px+q^2+q-2=0$ has at least one real root in $(-1;0)$.
Given is a triangle $ABC$ and a circle through $A, B$. The perpendicular bisector of $AB$ meets the circle at $P, Q$, such that $AP>AQ$. Let $M$ be a point on the segment $AB$. The lines through $M$, parallel to $QA, QB$ meet $PB, PA$ at $R, S$. Prove that $MQ$ bisects $RS$.
Find the largest positive integer $n$ of the form $n=p^{2\alpha}q^{2\beta}r^{2\gamma}$ for primes $p<q, r$ and positive integers $\alpha, \beta, \gamma$, such that $|r-pq|=1$ and $p^{2\alpha}-1, q^{2\beta}-1, r^{2\gamma}-1$ all divide $n$.
The number $2022$ is written on the white board. Ivan and Peter play a game, Ivan starts and they alternate. On a move, Ivan erases the number $b$, written on the board, throws a dice which shows some number $a$, and writes the residue of $(a+b) ^2$ modulo $5$. Similarly, Peter throws a dice which shows some number $a$, and changes the previously written number $b$ to the residue of $a+b$ modulo $3$. The first player to write a $0$ wins. What is the probability of Ivan winning the game?
Find $A=x^5+y^5+z^5$ if $x+y+z=1$, $x^2+y^2+z^2=2$ and $x^3+y^3+z^3=3$.
Point $M$ lies inside an isosceles right $\triangle ABC$ with hypotenuse $AB$ such that $MA=5$, $MB=7$, $MC=4\sqrt{2}$. Find $\angle AMC$.
The sequence $a_{n}$ is defined by $a_{1}\geq 2$ and the recurrence formula \[a_{n+1}=a_{n}\sqrt{\frac{a_{n}^3+2}{2(a_{n}^3+1)}}\]for $n\geq 1$. Prove that for every integer $n$, the inequality $a_{n}>\sqrt{\frac{3}{n}}$ holds.
The European zoos with at least two types of species are separated in two groups $\hat{A}$ and $\hat{B}$ in such a way that every pair of zoos $(A,B)$ $(A\in\hat{A}, B\in\hat{B})$ have some animal in common. What is the least $k$ for which we can color the cages in the zoos (each cage only has all animals of one species) such that no zoo has cages of only one color (with every animal across all zoos having to be colored in the same color)? For the maximal value of $k$, find all possibilities (zoos and species), for which this maximum is achieved.