Found problems: 85335
When filled with an cold water using a particular cold water tap, a tank will be full in 14 minutes. In 21 minutes, the full tank could be emptied by opening a hole on the base of the tank. If the cold water tap and the hot water tap are opened simultaneously (allowing hot and cold water fill the tank), and the hole on the base of the tank is opened, the tank will be full in 12.6 minutes. Determine the number of minutes needed to fill the tank with hot water until the tank is full, assuming at first the tank is empty and the hole is closed.
Calculate the sum $$S=\sqrt{1+\frac{8\cdot 1^2-1}{1^2\cdot 3^2}}+\sqrt{1+\frac{8\cdot 2^2-1}{3^2\cdot 5^2}}+...+ \sqrt{1+\frac{8\cdot 1003^2-1}{2005^2\cdot 2007^2}}$$
Given $1991$ distinct positive real numbers, the product of any ten of these numbers is always greater than $1$. Prove that the product of all $1991$ numbers is also greater than $1$.
Find two positive integers $a,b$ such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5$, but $a^5$ does not divide $b^6$
Let $ABC$ be a triangle with symmedian point $K$. Select a point $A_1$ on line $BC$ such that the lines $AB$, $AC$, $A_1K$ and $BC$ are the sides of a cyclic quadrilateral. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear.
[i]Proposed by Sammy Luo[/i]
A trapezoid $ABCD$ lies on the $xy$-plane. The slopes of lines $BC$ and $AD$ are both $\frac 13$, and the slope of line $AB$ is $-\frac 23$. Given that $AB=CD$ and $BC< AD$, the absolute value of the slope of line $CD$ can be expressed as $\frac mn$, where $m,n$ are two relatively prime positive integers. Find $100m+n$.
[i] Proposed by Yannick Yao [/i]
Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $$ f\left(xf\left(y\right)-f\left(x\right)-y\right) = yf\left(x\right)-f\left(y\right)-x $$ holds for all $ x,y \in \mathbb{R} $
There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds.
Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.
The Fibonacci numbers are defined as the sequence $F_n$ with $F_0=1$, $F_1=1$ and $F_{n+2} = F_{n+1} +F_n$. How many ways can $10$ be written as an ordered sum of numbers found in the Fibonacci sequence? For example, $3$ can be written as $1 + 1 + 1$, $2 + 1$, $1 + 2$, and $3$, for a total of $4$ ways.
[i]2015 CCA Math Bonanza Team Round #9[/i]
Prove that the set $\{1, 2, \cdots, 2007\}$ can be expressed as the union of disjoint subsets $A_i$ for $i=1,2,\cdots, 223$ such that each $A_i$ contains nine elements and the sum of all the elements in each $A_i$ is the same.
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $ 71$, $ 76$, $ 80$, $ 82$, and $ 91$. What was the last score Mrs. Walter entered?
$ \textbf{(A)}\ 71 \qquad \textbf{(B)}\ 76 \qquad \textbf{(C)}\ 80 \qquad \textbf{(D)}\ 82 \qquad \textbf{(E)}\ 91$
[u]Round 1[/u]
[b]p1.[/b] Three two-digit numbers are written on a board. One starts with $5$, another with $6$, and the last one with $7$. Annie added the first and the second numbers; Benny added the second and the third numbers; Denny added the third and the first numbers. Could it be that one of these sums is equal to $148$, and the two other sums are three-digit numbers that both start with $12$?
[b]p2.[/b] Three rocks, three seashells, and one pearl are placed in identical boxes on a circular plate in the order shown. The lids of the boxes are then closed, and the plate is secretly rotated. You can open one box at a time. What is the smallest number of boxes you need to open to know where the pearl is, no matter how the plate was rotated?
[img]https://cdn.artofproblemsolving.com/attachments/0/2/6bb3a2a27f417a84ab9a64100b90b8768f7978.png[/img]
[b]p3.[/b] Two detectives, Holmes and Watson, are hunting the thief Raffles in a library, which has the floorplan exactly as shown in the diagram. Holmes and Watson start from the center room marked $D$. Show that no matter where Raffles is or how he moves, Holmes and Watson can find him. Holmes and Watson do not need to stay together. A detective sees Raffles only if they are in the same room. A detective cannot stand in a doorway to see two rooms at the same time.
[img]https://cdn.artofproblemsolving.com/attachments/c/1/6812f615e60a36aea922f145a1ffc470d0f1bc.png[/img]
[b]p4.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken.
[img]https://cdn.artofproblemsolving.com/attachments/4/6/bf0185e142cd3f653d4a9c0882d818c55c64e4.png[/img]
[b]p5.[/b] The numbers $1–14$ are placed around a circle in some order. You can swap two neighbors if they differ by more than $1$. Is it always possible to rearrange the numbers using swaps so they are ordered clockwise from $1$ to $14$?
[u]Round 2[/u]
[b]p6.[/b] A triangulation of a regular polygon is a way of drawing line segments between its vertices so that no two segments cross, and the interior of the polygon is divided into triangles. A flip move erases a line segment between two triangles, creating a quadrilateral, and replaces it with the opposite diagonal through that quadrilateral. This results in a new triangulation.
[img]https://cdn.artofproblemsolving.com/attachments/a/a/657a7cf2382bab4d03046075c6e128374c72d4.png[/img]
Given any two triangulations of a polygon, is it always possible to find a sequence of flip moves that transforms the first one into the second one?
[img]https://cdn.artofproblemsolving.com/attachments/0/9/d09a3be9a01610ffc85010d2ac2f5b93fab46a.png[/img]
[b]p7.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9,..., 121)$ are in one column?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits?
[i]Proposed by A. Shapovalov[/i]
Given a triangle $ABC$. Let $S$ be the circle passing through $C$, centered at $A$. Let $X$ be a variable point on $S$ and let $K$ be the midpoint of the segment $CX$ . Find the locus of the midpoints of $BK$, when $X$ moves along $S$.
(I. Gorodnin)
Joao Salta-Pocinhas jumps $1$ meter in the first jump, $2$ meters in the second, $4$ meters in the third, . . ., $2^{n-1}$ meters in jump number $n$. Is there any possibility for Joao to choose the directions of his jumps in order to get back to the starting point?
Let $n>1$ be an integer. Given a simple graph $G$ on $n$ vertices $v_1, v_2, \dots, v_n$ we let $k(G)$ be the minimal value of $k$ for which there exist $n$ $k$-dimensional rectangular boxes $R_1, R_2, \dots, R_n$ in a $k$-dimensional coordinate system with edges parallel to the axes, so that for each $1\leq i<j\leq n$, $R_i$ and $R_j$ intersect if and only if there is an edge between $v_i$ and $v_j$ in $G$.
Define $M$ to be the maximal value of $k(G)$ over all graphs on $n$ vertices. Calculate $M$ as a function of $n$.
Let $M$ be a connected, compact $C^{\infty}$-differentiable manifold, and denote the vector space of smooth real functions on $M$ by $C^{\infty}(M)$. Let the subspace $V\le C^{\infty}(M)$ be invariant under $C^{\infty}$-diffeomorphisms of $M$, that is, let $f\circ h\in V$ for every $f\in V$ and for every $C^{\infty}$-diffeomorphism $h\colon M\rightarrow M$. Prove that if $V$ is different from the subspaces $\{ 0\}$ and $C^{\infty}(M)$ then $V$ only contains the constant functions.
Calculate the least integer greater than $ 5^{(\minus{}6)(\minus{}5)(\minus{}4)...(2)(3)(4)}$.
Let $n$ be a positive integer.Consider all $2^n$ binary strings of length $n$.We say two of these strings are neighbors if they differ in exactly 1 digit.We have colored $m$ strings.In each moment,we can color any uncolored string which is neighbor with at least 2 colored strings.After some time,all the strings are colored.Find the minimum possible value of $m$.
Let $k$ be a positive even integer. Show that
$$\sum_{n=0}^{k/2}(-1)^n\binom{k+2}n\binom{2(k-n)+1}{k+1}=\frac{(k+1)(k+2)}2.$$
For prime number $q$ the polynomial $P(x)$ with integer coefficients is said to be factorable if there exist non-constant polynomials $f_q,g_q$ with integer coefficients such that all of the coefficients of the polynomial $Q(x)=P(x)-f_q(x)g_q(x)$ are dividable by $q$ ; and we write:
$$P(x)\equiv f_q(x)g_q(x)\pmod{q}$$
For example the polynomials $2x^3+2,x^2+1,x^3+1$ can be factored modulo $2,3,p$ in the following way:
$$\left\{\begin{array}{lll}
X^2+1\equiv (x+1)(-x+1)\pmod{2}\\
2x^3+2\equiv (2x-1)^3\pmod{3}\\
X^3+1\equiv (x+1)(x^2-x+1)
\end{array}\right.$$
Also the polynomial $x^2-2$ is not factorable modulo $p=8k\pm 3$.
a) Find all prime numbers $p$ such that the polynomial $P(x)$ is factorable modulo $p$:
$$P(x)=x^4-2x^3+3x^2-2x-5$$
b) Does there exist irreducible polynomial $P(x)$ in $\mathbb{Z}[x]$ with integer coefficients such that for each prime number $p$ , it is factorable modulo $p$?
[b]p1.[/b] A [i]multimagic [/i] square is a $3 \times 3$ array of distinct positive integers with the property that the product of the $3$ numbers in each row, each column, and each of the two diagonals of the array is always the same.
(a) Prove that the numbers $1, 2, 3, . . . , 9$ cannot be used to form a multimagic square.
(b) Give an example of a multimagic square.
[b]p2.[/b] A sequence $a_1, a_2, a_3, ... , a_n$ of real numbers is called an arithmetic progression if $$a_1 - a_2 = a_2 - a_3 = ... = a_{n-1} - a_n.$$
Prove that there exist distinct positive integers $n_1, n_2, n_3, ... , n_{2014}$ such that $$\frac{1}{n_1},\frac{1}{n_2}, ... ,\frac{1}{n_{2014}}$$ is an arithmetic progression.
[b]p3.[/b] Let $\lfloor x \rfloor$ be the largest integer that is less than or equal to $x$. For example, $\lfloor 3.9 \rfloor = 3$ and $\lfloor 4\rfloor = 4$. Determine (with proof) all real solutions of the equation $$x^2 - 25 \lfloor x\rfloor + 100 = 0.$$
[b]p4.[/b] An army has $10$ cannons and $8$ carts. Each cart can carry at most one cannon. It takes one day for a cart to cross the desert. What is the least number of days that it takes to get the cannons across the desert? (Cannons can be left part way and picked up later during the procedure.) Prove that the amount of time that your solution requires to move the cannons across the desert is the smallest possible.
[b]p5.[/b] Let $C$ be a convex polygon with $4031$ sides. Let $p$ be the length of its perimeter and let $d$ be the sum of the lengths of its diagonals. Show that $$\frac{d}{p}> 2014.$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Suppose $Z, Y$ , and $W$ are points on a circle such that lengths $ZY = Y W$. Extend $ZY$ and let $X$ be a point on $ZY$ where $ZY = Y X$. If $XW$ is a tangent of the circle, what is $\angle W XY$ ?
Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.