Found problems: 85335
Let $a, b, c$ be real numbers satisfying the condition $18ab + 9ca + 29bc = 1$.
Find the minimum value of the expression $T = 42a^2 + 34b^2 + 43c^2$.
Let $N$ be the product of the first nine multiples of $19$ (i.e. $N = 19\times38 \times57\times\cdots\times 152\times 171$). What is the last digit of $N$?
Which of the following numbers is a perfect square?
$\textbf{(A) }4^45^56^6\qquad\textbf{(B) }4^45^66^5\qquad\textbf{(C) }4^55^46^6\qquad\textbf{(D) }4^65^46^5\qquad\textbf{(E) }4^65^56^4$
Determine all $n$ for which the system with of equations can be solved in $\mathbb{R}$:
\[\sum^{n}_{k=1} x_k = 27\]
and
\[\prod^{n}_{k=1} x_k = \left( \frac{3}{2} \right)^{24}.\]
Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively.
(a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$
(b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]
Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.
A grasshopper is jumping along the number line. Initially it is situated at zero. In $k$-th step, the
length of his jump is $k$.
a) If the jump length is even, then it jumps to the left, otherwise it jumps to the right (for example, firstly
it jumps one step to the right, then two steps to the left, then three steps to the right, then four steps to
the left...). Will it visit on every integer at least once?
b) If the jump length is divisible by three, then it jumps to the left, otherwise it jumps to the right (for
example, firstly it jumps one step to the right, then two steps to the right, then three steps to the left,
then four steps to the right...). Will it visit every integer at least once?
Proposed by Matko Ljulj
$m < n$ are positive integers. Let $p=\frac{n^2+m^2}{\sqrt{n^2-m^2}}$.
[b](a)[/b] Find three pairs of positive integers $(m,n)$ that make $p$ prime.
[b](b)[/b] If $p$ is prime, then show that $p \equiv 1 \pmod 8$.
Determine all Functions $f:\mathbb{Z} \to \mathbb{Z}$ such that $f(f(a)-b)+bf(2a)$ is a perfect square for all integers $a$ and $b$.
Convex pentagon $PQRST$ has $PQ = T P = 5$, $QR = RS = ST = 6$, and $\angle QRS = \angle RST = 90^o$. Given that points $U$ and $V$ exist such that $RU =UV = VS = 2$, find the area of pentagon $PQUVT$ .
[i]Proposed by Kira Tang[/i]
Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$.
(1) Find $u(t),\ v(t)$.
(2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$.
(3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$.
[i]2011 Tokyo University entrance exam/Science, Problem 3[/i]
What's more $\sqrt[4]{7}+\sqrt[4]{11}$ or $2\sqrt{\frac{\sqrt{7}+\sqrt{11}}{2}}$ ?
Show that there exists a convex hexagon in the plane such that
(i) all its interior angles are equal;
(ii) its sides are $1,2,3,4,5,6$ in some order.
$I$ is the incenter of a given triangle $\triangle ABC$. The angle bisectors of $ABC$ meet the sides at $D,E,F$, and $EF$ meets $(ABC)$ at $L$ and $T$ ($F$ is on segment $LE$.). Suppose $M$ is the midpoint of $BC$. Prove that if $DT$ is tangent to the incircle of $ABC$, then $IL$ bisects $\angle MLT$.
Do there exist
(a) four
(b) five
distinct positive integers such that the sum of any three of them is a prime number?
(V Senderov)
Let $a_0, a_1, a_2, ... , a_n$ be real numbers, which fulfill the following two conditions:
a) $0 = a_0 \leq a_1 \leq a_2 \leq ... \leq a_n$.
b) For all $0 \leq i < j \leq n$ holds: $a_j - a_i \leq j-i$.
Prove that
$$\left( \displaystyle \sum_{i=0}^n a_i \right)^2 \geq \sum_{i=0}^n a_i^3.$$
There three piles of pebbles, containing 5, 49, and 51 pebbles respectively. It is allowed to combine any two piles into a new one or to split any pile consisting of even number of pebbles into two equal piles. Is it possible to have 105 piles with one pebble in each in the end?
[i](3 points)[/i]
Find the smallest number $k$, such that $ \frac{l_a+l_b}{a+b}<k$ for all triangles with sides $a$ and $b$ and bisectors $l_a$ and $l_b$ to them, respectively.
[i]Proposed by Sava Grodzev, Svetlozar Doichev, Oleg Mushkarov and Nikolai Nikolov[/i]
[b]p4[/b] Define the sequence ${a_n}$ as follows:
1) $a_1 = -1$, and
2) for all $n \ge 2$, $a_n = 1 + 2 + . . . + n - (n + 1)$.
For example, $a_3 = 1+2+3-4 = 2$. Find the largest possible value of $k$ such that $a_k+a_{k+1} = a_{k+2}$.
[b]p5[/b] The taxicab distance between two points $(a, b)$ and $(c, d)$ on the coordinate plane is $|c-a|+|d-b|$. Given that the taxicab distance between points $A$ and $B$ is $8$ and that the length of $AB$ is $k$, find the minimum possible value of $k^2$.
[b]p6[/b] For any two-digit positive integer $\overline{AB}$, let $f(\overline{AB}) = \overline{AB}-A\cdot B$, or in other words, the result of subtracting the product of its digits from the integer itself. For example, $f(\overline{72}) = 72-7\cdot 2 = 58$. Find the maximum possible $n$ such that there exist distinct two-digit integers$ \overline{XY}$ and $\overline{WZ}$ such that $f(\overline{XY} ) = f(\overline{WZ}) = n$.
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
Alicia bought some number of disposable masks, of which she uses one per day. After she uses each of her masks, she throws out half of them (rounding up if necessary) and reuses each of the remaining masks, repeating this process until she runs out of masks. If her masks lasted her $222$ days, how many masks did she start out with?
In a restaurant, a meal consists of one sandwich and one optional drink. In other words, a sandwich is necessary for a meal but a drink is not necessary. There are two types of sandwiches and two types of drinks. How many possible meals can be purchased?
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }12\qquad\textbf{(E) }16$
Let $x_1, x_2, ... , x_n$ be a real numbers such that\\
1) $1 \le x_1, x_2, ... , x_n \le 160$
2) $x^{2}_{i} + x^{2}_{j} + x^{2}_{k} \ge 2(x_ix_j + x_jx_k + x_kx_i)$ for all $1\le i < j < k \le n$
Find the largest possible $n$.
Light of a blue laser (wavelength $\lambda=475 \, \text{nm}$) goes through a narrow slit which has width $d$. After the light emerges from the slit, it is visible on a screen that is $ \text {2.013 m} $ away from the slit. The distance between the center of the screen and the first minimum band is $ \text {765 mm} $. Find the width of the slit $d$, in nanometers.
[i](Proposed by Ahaan Rungta)[/i]
a) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
\[g^n - n\quad\text{ and }\quad g^{n+1} - (n + 1).\]
b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
\[g^n - n^2\quad\text{ and }g^{n+1} - (n + 1)^2.\]