Found problems: 85335
Suppose a $m \times n$ table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either $+1$ or $-1$ to all of its cells. Prove that if for all arbitrary $3 \times 3$ table we can change all numbers to zero, then we can change all numbers of $m \times n$ table to zero.
([i]Hint[/i]: First of all think about it how we can change number of $ 3 \times 3$ table to zero.)
What is the $101$st smallest integer which can represented in the form $3^a+3^b+3^c$, where $a,b,$ and $c$ are integers?
[i]Proposed by Dilhan Salgado[/i]
A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD+BC=AB$.
Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$
How many numbers greater than $1.000$ but less than $10.000$ have as a product of their digits $256$?
There are $ n\ge 1 $ ordered bulbs controlled by $ n $ ordered switches such that the $ k\text{-th} $ switch controls the $ k\text{-th} $ bulb and also the $ j\text{-th} $ bulb if and only if the $ j\text{-th} $ switch controls the $ k\text{-th} $ bulb, for any $ 1\le k,j\le n. $ If all bulbs are off, show that it can be chosen some switches such that, if pushed simmultaneously, the bulbs turn all on.
For any integer $n\ge2$ evaluate the sum \[\sum_{k=1}^{n^2-1}\bigl\lfloor\sqrt k\bigr\rfloor.\]
An ant starts at the point $(1,1)$. It can travel along the integer lattice, only moving in the positive $x$ and $y$ directions. What is the number of ways it can reach $(5,5)$ without passing through $(3,3)$?
There are $100$ cities in Matland. Every road in Matland connects two cities, does not pass through any other city and does not form crossroads with other roads (although roads can go through tunnels one after the other). Driving in Matlandia by road, it is possible to get from any city to any other. Prove that that it is possible to repair some of the roads of Matlandia so that from an odd number of repaired roads would go in each city.
Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$, where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$.
Let $ABC$ be an acute triangle with $AB < AC$ and inscribed in the circle $(O)$. Denote $I$ as the incenter of $ABC$ and $D$, $E$ as the intersections of $AI$ with $BC$, $(O)$ respectively. Take a point $K$ on $BC$ such that $\angle AIK = 90^o$ and $KA$, $KE$ meet $(O)$ again at M,N respectively. The rays $ND$, $NI$ meet the circle $(O)$ at $Q$,$P$.
1. Prove that the quadrilateral $MPQE$ is a kite.
2. Take $J$ on $IO$ such that $AK \perp AJ$. The line through $I$ and perpendicular to $OI$ cuts $BC$ at $R$ ,cuts $EK$ at $S$ .Prove that $OR \parallel JS$.
A merganser mates every 7th day, a scaup mates every 11th day, and a gadwall mates every 13th day. A merganser, scaup, and gadwall all mate on Day 0. On Days N, N+1, and N+2 the merganser, scaup, and gadwall mate in some order with no two birds mating on the same day. Determine the smallest possible value of N.
[i]2022 CCA Math Bonanza Lightning Round 3.4[/i]
Let $n$ be the number of polynomial functions from the integers modulo $2010$ to the integers modulo $2010$. $n$ can be written as $n = p_1 p_2 \cdots p_k$, where the $p_i$s are (not necessarily distinct) primes. Find $p_1 + p_2 + \cdots + p_n$.
Let $ABC$ be a triangle and $D$ a point in $AC$. If $\angle{CBD} - \angle{ABD} = 60^{\circ}, \hat{BDC} = 30^{\circ}$ and also $AB \cdot BC = BD^{2}$, determine the measure of all the angles of triangle $ABC$.
Kevin is running $1000$ meters. He wants to have an average speed of $10$ meters a second. He runs the first $100$ meters at a speed of $4$ meters a second. Compute how quickly, in meters per second, he must run the last $900$ meters to attain his desired average speed of $10$ meters a second.
Let $ P_1$, $ P_2$, $ P_3$, $ P_4$ be five distinct points on a circle. The distance of $ P$ from the line $ P_iP_k$ is denoted by $ d_{ik}$. Prove that $ d_{12}d_{34} \equal{} d_{13}d_{24}$.
Let $ABC$ be an acute-angled triangle with $\angle ACB = 60^o$ , and its heights $AD$ and $BE$ intersect at point $H$. Prove that the circumcenter of triangle $ABC$ lies on a line bisecting the angles $AHE$ and $BHD$.
In triangle $ABC$, $AD$ is angle bisector ($D$ is on $BC$) if $AB+AD=CD$ and $AC+AD=BC$, what are the angles of $ABC$?
[b]Q5.[/b] How many different 4-digit even integers can be form from the elements of the set $\{ 1,2,3,4,5 \}.$
\[(A) \; 4; \qquad (B) \; 5; \qquad (C ) \; 8; \qquad (D) \; 9; \qquad (E) \; \text{None of the above.}\]
Let $ABCD$ be a quadrilateral circumscribed around a circle $\omega$ with center $I$. Assume $P$ and $Q$ are distinct points and are isogonal conjugates such that $P, Q$, and $I$ are collinear. Show that $ABCD$ is a kite, that is, it has two disjoint pairs of consecutive equal sides.
For all integers $x,y,z$, let \[S(x,y,z) = (xy - xz, yz-yx, zx - zy).\] Prove that for all integers $a$, $b$ and $c$ with $abc>1$, and for every integer $n\geq n_0$, there exists integers $n_0$ and $k$ with $0<k\leq abc$ such that \[S^{n+k}(a,b,c) \equiv S^n(a,b,c) \pmod {abc}.\] ($S^1 = S$ and for every integer $m\geq 1$, $S^{m+1} = S \circ S^m.$
$(u_1, u_2, u_3) \equiv (v_1, v_2, v_3) \pmod M \Longleftrightarrow u_i \equiv v_i \pmod M (i=1,2,3).$)
Without calculating the value of the determinant
$$ \begin{vmatrix}1 &1 &3& 1\\1& 2& 3 &5\\ 3& 0& 5& 5\\ 0& a& -11a& a^{13}+9a\end{vmatrix} , $$
show that it is divisible by $ 26, $ for any integer $ a. $
How many positive integers $a$ with $a\le 154$ are there such that the coefficient of $x^a$ in the expansion of \[(1+x^{7}+x^{14}+ \cdots +x^{77})(1+x^{11}+x^{22}+\cdots +x^{77})\] is zero?
[i]Author: Ray Li[/i]
Let $p \ge 2$ be a prime number and $\frac{a_p}{b_p}= 1 +\frac12+ .. +\frac{1}{p^2 -1}$, where $a_p$ and $b_p$ are two relatively prime positive integers. Compute gcd $(p, b_p)$.
For all $n \geq 1$, define $a_{n}$ to be the fraction $\frac{k}{2^n}$ such that $a_{n}$ is the closest to $\frac{1}{3}$ over all integer values of $k$.
Prove that the sequence $a_{1}, a_{2}, \cdots $satisfies the equation $2a_{i+2} = a_{i+1} + a_{i}$ for all $i \geq 1$.