Found problems: 85335
There are $n$ students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of $n$.
An enormous party has an infinite number of people. Each two people either know or don't know each other. Given a positive integer $n$, prove there are $n$ people in the party such that either they all know each other, or nobody knows each other (so the first possibility means that if $A$ and $B$ are any two of the $n$ people, then $A$ knows $B$, whereas the second possibility means that if $A$ and $B$ are any two of the $n$ people, then $A$ does not know $B$).
Let $p$ - a prime, where $p> 11$. Prove that there exists a number $k$ such that the product $p \cdot k$ can be written in the decimal system with only ones.
The real function $f$ is defined for $\forall$ $x\in \mathbb{R}$ and $f(0)=0$. Also $f(9+x)=f(9-x)$ and $f(x-10)=f(-x-10)$ for $\forall$ $x\in \mathbb{R}$. What’s the least number of zeros $f$ can have in the interval $[0;2014]$? Does this change, if $f$ is also continuous?
Let $ a,\ b$ be real constants. Find the minimum value of the definite integral:
$ I(a,\ b)\equal{}\int_0^{\pi} (1\minus{}a\sin x \minus{}b\sin 2x)^2 dx.$
Prove that among any $ 16 $ numbers smaller than $ 101 $ there are four of them that have the property that the sum of two of them is equal to the sum of the other two.
Let $\mathcal{S}$ be the set of permutations of $\{1,2,\ldots,6\}$, and let $\mathcal{T}$ be the set of permutations of $\mathcal{S}$ that preserve compositions: i.e., if $F\in\mathcal{T}$ then \[F(f_2\circ f_1)=F(f_2)\circ F(f_1)\] for all $f_1,f_2\in\mathcal{S}$. Find the number of elements $F\in\mathcal{T}$ such that if $f\in\mathcal{S}$ satisfies $f(1)=2$ and $f(2)=1$, then $(F(f))(1)=2$ and $(F(f))(2)=1$.
Let $ABCD$ be a convex quadrilateral inscribed in a circle with $AC = 7$, $AB = 3$, $CD = 5$, and $AD - BC = 3$. Then $BD = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Let $n \in N$. The land of Draconis has more than $2^n$ dungeons. Between two different Dungeons have exactly one path, but each path is a one-way street. Total $n$ Dragon cults share the territory; each path is controlled by exactly one cult. It is said that a dragon cult $K$ has established itself in a dungeon $D$ if there is both one a path beginning in $D$ and one a path ending in $D$, both of which are controlled by $K$ . Prove that there is a cult $K$, which has at least one dungeon controlled.
Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.
There are $7$ balls in a jar, numbered from $1$ to $7$, inclusive. First, Richard takes $a$ balls from the jar at once, where $a$ is an integer between $1$ and $6$, inclusive. Next, Janelle takes $b$ of the remaining balls from the jar at once, where $b$ is an integer between $1$ and the number of balls left, inclusive. Finally, Tai takes all of the remaining balls from the jar at once, if any are left. Find the remainder when the number of possible ways for this to occur is divided by $1000$, if it matters who gets which ball.
[i]Proposed by firebolt360 & DeToasty3[/i]
We have $ n $ kinds of puddings. There are $ a_{i} $ puddings which are $ i $-th type and those $ S = a_{1}+\cdots+a_{n} $ puddings are distinct. Now, for a given arrangement of puddings: $ p_{1}, \dots, p_{n} $. Define $ c_{i} $ to be $$ \#\{1 \le j \le S-1 \ \mid \ p_{j}, p_{j+1} \text{ are the same type.}\} $$ Show that if $ S $ is composite, then the sum of $ \prod_{i=1}^{n}{c_{i}} $ over all possible arrangements is a multiple of $ S $.
Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$
i) Show that a regular hexagon, six squares, and six equilateral triangles can be assembled without overlapping to form a regular dodecagon.
ii) Let $P_1 , P_2 ,\ldots, P_{12}$ be the vertices of a regular dodecagon. Prove that the three diagonals $P_{1}P_{9}, P_{2}P_{11}$ and $P_{4}P_{12}$ intersect.
Five numbers 1,2,3,4,5 are written on a blackboard. A student may
erase any two of the numbers a and b on the board and write the
numbers a+b and ab replacing them. If this operation is performed repeatedly, can the numbers 21,27,64,180,540 ever appear on the board?
Use the standard reduction potentials to determine what is observed at the cathode during the electrolysis of a $1.0 \text{M}$ solution of $\ce{KBr}$ that contains phenolphthalein. What observation(s) is(are) made?
$\ce{O2 (g)} + \ce{4H^+ (aq)} + 4e^- \rightarrow \ce{2H2O (l) }; \text{ E}^\circ = \text{1.23 V}$
$\ce{Br2 (l)} + 2e^- \rightarrow \ce{2Br^- (aq)} ; \text{ E}^\circ = \text{1.07 V}$
$\ce{2H2O (l)} + 2e^- \rightarrow \ce{H2 (g)} + \ce{2OH^-} ; \text{ E}^\circ = \text{-0.80 V}$
$\ce{K^+ (aq)} + e^- \rightarrow \ce{K (s)} ; \text{ E}^\circ = \text{-2.92 V}$
$ \textbf{(A) }\text{Solid metal forms}\qquad$
$\textbf{(B) }\text{Bubbles form and a pink color appears}\qquad$
$\textbf{(C) }\text{Dark red } \ce{ Br2} \text{ forms}\qquad$
$\textbf{(D) }\text{Bubbles form and the solution remains colorless}\qquad $
Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.
A triangle $ABC$ with $\angle A = 30^\circ$ and $\angle C = 54^\circ$ is given. On $BC$ a point $D$ is chosen such that $ \angle CAD = 12^\circ.$ On $AB$ a point $E$ is chosen such that $\angle ACE = 6^\circ.$ Let $S$ be the point of intersection of $AD$ and $CE.$ Prove that $BS = BC.$
Prove that there are infinitely many positive integers $m$ for which there exists consecutive odd positive integers $p_m<q_m$ such that $p_m^2+p_mq_m+q_m^2$ and $p_m^2+m\cdot p_mq_m+q_m^2$ are both perfect squares. If $m_1, m_2$ are two positive integers satisfying this condition, then we have $p_{m_1}\neq p_{m_2}$
Find all positive integers $k<202$ for which there exist a positive integers $n$ such that
$$\bigg {\{}\frac{n}{202}\bigg {\}}+\bigg {\{}\frac{2n}{202}\bigg {\}}+\cdots +\bigg {\{}\frac{kn}{202}\bigg {\}}=\frac{k}{2}$$
Statement 1: In a cuboid, we can always find a point, its distances to eight vertexes are equal.
Statement 2: In a cuboid, we can always find a point, its distances to twelve edges are equal.
Statement 3: In a cuboid, we can always find a point, its distances to six surfaces are equal.
In the statements above, how many of them are true?
$\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}2\qquad\text{(D)}3$
The incircle of a triangle $ABC$ touches $AC, BC$ , and $AB$ at $M , N$, and $R$, respectively. Let $S$ be a point on the smaller arc $MN$ and $t$ be the tangent to this arc at $S$ . The line $t$ meets $NC$ at $P$ and $MC$ at $Q$. Prove that the lines $AP, BQ, SR, MN$ have a common point.
Let $ f:[-1,1]\longrightarrow\mathbb{R} $ a derivable function and a non-negative integer $ n. $ Show that there is a $ c\in [-1,1] $ so that:
$$ \int_{-1}^1 x^{2n+1} f(x)dx =\frac{2}{2n+3}f'(c). $$
Let $A_1A_2\dotsm A_{2025}$ be a convex 2025-gon, and let $A_i = A_{i+2025}$ for all integers $i$. Distinct points $P$ and $Q$ lie in its interior such that $\angle A_{i-1}A_iP = \angle QA_iA_{i+1}$ for all $i$. Define points $P^{j}_{i}$ and $Q^{j}_{i}$ for integers $i$ and positive integers $j$ as follows:
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[*] For all $i$, $P^1_i = Q^1_i = A_i$.
[*] For all $i$ and $j$, $P^{j+1}_{i}$ and $Q^{j+1}_i$ are the circumcenters of $PP^j_iP^j_{i+1}$ and $QQ^j_iQ^{j}_{i+1}$, respectively.
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Let $\mathcal{P}$ and $\mathcal{Q}$ be the polygons $P^{2025}_{1}P^{2025}_{2}\dotsm P^{2025}_{2025}$ and $Q^{2025}_{1}Q^{2025}_{2}\dotsm Q^{2025}_{2025}$, respectively.
[list=a]
[*] Prove that $\mathcal{P}$ and $\mathcal{Q}$ are cyclic.
[*] Let $O_P$ and $O_Q$ be the circumcenters of $\mathcal{P}$ and $\mathcal{Q}$, respectively. Assuming that $O_P\neq O_Q$, show that $O_PO_Q$ is parallel to $PQ$.
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[i]Ruben Carpenter[/i]
Let $f$ be an irreducible (in $Z[x]$) monic polynomial with integer coefficients and of odd degree greater than $1$. Suppose that the modules of the roots of $f$ are greater than $1$ and that $f(0)$ is a square-free number. Prove that
the polynomial $g(x) = f(x^3)$ is also irreducible