Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for $24$ hours. If she is using it constantly, the battery will last for only $3$ hours. Since the last recharge, her phone has been on $9$ hours, and during that time she has used it for $60$ minutes. If she doesn't talk any more but leaves the phone on, how many more hours will the battery last? $\textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 11\qquad \textbf{(D)}\ 14\qquad \textbf{(E)}\ 15$

The polynomial $x^3-kx^2+20x-15$ has $3$ roots, one of which is known to be $3$. Compute the greatest possible sum of the other two roots. [i]2015 CCA Math Bonanza Lightning Round #2.4[/i]

For each ordered pair of real numbers $(x,y)$ satisfying \[ \log_2(2x+y) = \log_4(x^2+xy+7y^2) \] there is a real number $K$ such that \[ \log_3(3x+y) = \log_9(3x^2+4xy+Ky^2). \] Find the product of all possible values of $K$.

The sequence $ (a_{n})_{n\geq 1}$ is defined by $ a_{1} \equal{} 1, a_{2} \equal{} 3$, and $ a_{n \plus{} 2} \equal{} (n \plus{} 3)a_{n \plus{} 1} \minus{} (n \plus{} 2)a_{n}, \forall n \in \mathbb{N}$. Find all values of $ n$ for which $ a_{n}$ is divisible by $ 11$.

An isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ is given. Circles with centers $O_1$ and $O_2$ are inscribed in triangles $ABC$ and $ABD$. Prove that line $O_1O_2$ is perpendicular on $BC$.

Each positive integer should be colored red or green in such a way that the following two conditions are met: - Let $n$ be any red number. The sum of any $n$ (not necessarily different) red numbers is red. - Let $m$ be any green number. The sum of any $m$ (not necessarily different) green numbers is green. Determine all such colorings.

$\vartriangle ABC$ is a scalene triangle with circumcircle $\Omega$. For a arbitrary $X$ in the plane, define $D_x,E_x, F_x$ to be the intersection of tangent line of $X$ (with respect to $BXC$) and $BC,CA,AB$, respectively. Let the intersection of $AX$ with $\Omega$ be $S_x$ and $T_x = D_xS_x \cap \Omega$. Show that $\Omega$ and circumcircle of $\vartriangle T_xE_xF_x$ are tangent to each other. [img]https://2.bp.blogspot.com/-rTMODHbs5Ac/XnYNQYjYzBI/AAAAAAAALeg/576nGDQ6NDA0-W5XqiNczNtI07cEZxPeQCK4BGAYYCw/s1600/imoc2019g4.png[/img]

Let $ABC$ be a triangle and $\omega$ its circumcircle. Let $D$ and $E$ be the feet of the angle bisectors relative to $B$ and $C$, respectively. The line $DE$ meets $\omega$ at $F$ and $G$. Prove that the tangents to $\omega$ through $F$ and $G$ are tangents to the excircle of $\triangle ABC$ opposite to $A$.

A finite set $\mathcal{S}$ of positive integers is called cardinal if $\mathcal{S}$ contains the integer $|\mathcal{S}|$ where $|\mathcal{S}|$ denotes the number of distinct elements in $\mathcal{S}$. Let $f$ be a function from the set of positive integers to itself such that for any cardinal set $\mathcal{S}$, the set $f(\mathcal{S})$ is also cardinal. Here $f(\mathcal{S})$ denotes the set of all integers that can be expressed as $f(a)$ where $a \in \mathcal{S}$. Find all possible values of $f(2024)$ $\quad$ Proposed by Sutanay Bhattacharya

Solve the following equation in $\mathbb{R}$: $$\left(x-\frac{1}{x}\right)^\frac{1}{2}+\left(1-\frac{1}{x}\right)^\frac{1}{2}=x.$$

For a constant $a$, denote $C(a)$ the part $x\geq 1$ of the curve $y=\sqrt{x^2-1}+\frac{a}{x}$. (1) Find the maximum value $a_0$ of $a$ such that $C(a)$ is contained to lower part of $y=x$, or $y<x$. (2) For $0<\theta <\frac{\pi}{2}$, find the volume $V(\theta)$ of the solid $V$ obtained by revoloving the figure bounded by $C(a_0)$ and three lines $y=x,\ x=1,\ x=\frac{1}{\cos \theta}$ about the $x$-axis. (3) Find $\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)$. 1992 Tokyo University entrance exam/Science, 2nd exam

A circle is divided into $2n$ equal sectors, $n \in \mathbb{N}$. Vitya and Masha are playing the following game. At first, Vitya writes one number in every sector from the set $\{1,2,\ldots,n\}$ and every number is used exatly twice. After that Masha chooses $n$ consecutive sectors and writes $1$ in the first sector, $2$ in the second, $n$ in the last. Vitya wins if at least in one sector two equal number will be written, otherwise Masha wins. Find all $n$ for which Vitya can guarantee his win.

Let $x_1, x_2, ..., x_n$ be positive numbers, with $n \ge 2$. Prove that $$\left(x_1+\frac{1}{x_1}\right)\left(x_2+\frac{1}{x_2}\right)...\left(x_n+\frac{1}{x_n}\right)\ge \left(x_1+\frac{1}{x_2}\right)\left(x_2+\frac{1}{x_3}\right)...\left(x_{n-1}+\frac{1}{x_n}\right)\left(x_n+\frac{1}{x_1}\right)$$

Let $ABC$ be a isosceles triangle with $\overline{AB}=\overline{AC}$. Let $D(\neq A, C)$ be a point on the side $AC$, and circle $\Omega$ is tangent to $BD$ at point $E$, and $AC$ at point $C$. Denote by $F(\neq E)$ the intersection of the line $AE$ and the circle $\Omega$, and $G(\neq a)$ the intersection of the line $AC$ and the circumcircle of the triangle $ABF$. Prove that points $D, E, F,$ and $G$ are concyclic.

In the following, a [i]word[/i] will mean a finite sequence of letters "$a$" and "$b$". The [i]length[/i] of a word will mean the number of the letters of the word. For instance, $abaab$ is a word of length $5$. There exists exactly one word of length $0$, namely the empty word. A word $w$ of length $\ell$ consisting of the letters $x_1$, $x_2$, ..., $x_{\ell}$ in this order is called a [i]palindrome[/i] if and only if $x_j=x_{\ell+1-j}$ holds for every $j$ such that $1\leq j\leq\ell$. For instance, $baaab$ is a palindrome; so is the empty word. For two words $w_1$ and $w_2$, let $w_1w_2$ denote the word formed by writing the word $w_2$ directly after the word $w_1$. For instance, if $w_1=baa$ and $w_2=bb$, then $w_1w_2=baabb$. Let $r$, $s$, $t$ be nonnegative integers satisfying $r + s = t + 2$. Prove that there exist palindromes $A$, $B$, $C$ with lengths $r$, $s$, $t$, respectively, such that $AB=Cab$, if and only if the integers $r + 2$ and $s - 2$ are coprime.

Six ducklings swim on the surface of a pond, which is in the shape of a circle with radius $5$ m. Show that at every moment, two of the ducklings swim on the distance of at most $5$ m from each other.

a) Let $m$ and $n$ be natural numbers. For some nonnegative integers $k_1, k_2, ... , k_n$ the number $$2^{k_1}+2^{k_2}+...+2^{k_n}$$ is divisible by $(2^m-1)$. Prove that $n \ge m$. b) Can you find a number, divisible by $111...1$ ($m$ times "$1$"), that has the sum of its digits less than $m$?

An ordinary domino tile can be identified as a pair $(k,m)$ where numbers $k$ and $m$ can get values $0, 1, 2, 3, 4, 5$ and $6.$ Pairs $(k,m)$ and $(m, k)$ determine the same tile. In particular, the pair $(k, k)$ determines one tile. We say that two domino tiles [i]match[/i], if they have a common component. [i]Generalized n-domino tiles[/i] $m$ and $k$ can get values $0, 1,... , n.$ What is the probability that two randomly chosen $n$-domino tiles match?

Let $ABCD$ be a quadrilateral circumscribed on the circle $\omega$ with center $I$. Assume $\angle BAD+ \angle ADC <\pi$. Let $M, \ N$ be points of tangency of $\omega $ with $AB, \ CD$ respectively. Consider a point $K \in MN$ such that $AK=AM$. Prove that $ID$ bisects the segment $KN$.

An odd integer $ n \ge 3$ is said to be nice if and only if there is at least one permutation $ a_{1}, \cdots, a_{n}$ of $ 1, \cdots, n$ such that the $ n$ sums $ a_{1} \minus{} a_{2} \plus{} a_{3} \minus{} \cdots \minus{} a_{n \minus{} 1} \plus{} a_{n}$, $ a_{2} \minus{} a_{3} \plus{} a_{3} \minus{} \cdots \minus{} a_{n} \plus{} a_{1}$, $ a_{3} \minus{} a_{4} \plus{} a_{5} \minus{} \cdots \minus{} a_{1} \plus{} a_{2}$, $ \cdots$, $ a_{n} \minus{} a_{1} \plus{} a_{2} \minus{} \cdots \minus{} a_{n \minus{} 2} \plus{} a_{n \minus{} 1}$ are all positive. Determine the set of all `nice' integers.

Let $f_1,f_2$ be nonnegative and concave functions. Then prove that $$(f_1f_2)^{\frac{2^n-1}{n\cdot2^n}}\left(\frac{\displaystyle\prod_{k=1}^n\left(\sqrt[2^k]{f_1}+\sqrt[2^k]{f_2}\right)}{f_1+f_2}\right)^{\frac1n}$$ is concave. [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

A regular hexagon is inscribed in a circle. The ratio of the length of a side of the hexagon to the length of the shorter of the arcs intercepted by the side, is: $ \textbf{(A)}\ 1: 1 \qquad \textbf{(B)}\ 1: 6 \qquad \textbf{(C)}\ 1: \pi \qquad \textbf{(D)}\ 3: \pi \qquad \textbf{(E)}\ 6: \pi$

Point $P$ lies in the interior of a triangle $ABC$. Lines $AP$, $BP$, and $CP$ meet the opposite sides of triangle $ABC$ at $A$', $B'$, and $C'$ respectively. Let $P_A$ the midpoint of the segment joining the incenters of triangles $BPC'$ and $CPB'$, and define points $P_B$ and $P_C$ analogously. Show that if \[ AB'+BC'+CA'=AC'+BA'+CB' \] then points $P,P_A,P_B,$ and $P_C$ are concyclic. [i]Nikolai Beluhov[/i]

How many non-congruent triangles with perimeter $ 7$ have integer side lengths? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Let $k$ and $l$ be two given positive integers and $a_{ij}(1 \leq i \leq k, 1 \leq j \leq l)$ be $kl$ positive integers. Show that if $q \geq p > 0$, then \[(\sum_{j=1}^{l}(\sum_{i=1}^{k}a_{ij}^{p})^{q/p})^{1/q}\leq (\sum_{i=1}^{k}(\sum_{j=1}^{l}a_{ij}^{q})^{p/q})^{1/p}.\]