Found problems: 85335
$a,b,c$ are reals, such that every pair of equations of $x^3-ax^2+b=0,x^3-bx^2+c=0,x^3-cx^2+a=0$ has common root.
Prove $a=b=c$
A pile contains $2021^{2021}$ stones. In a move any pile can be divided into two piles so that the numbers of stones in them differ by a power
of $2$ with non-negative integer exponent. After some move it turned out that the number of stones in each pile is a power of $2$ with non-negative integer exponent. Prove that the number of moves performed was even.
[i](a)[/i] Determine if the set $ \{1,2,\ldots,96\}$ can be partitioned into 32 sets of equal size and equal sum.
[i](b)[/i] Determine if the set $ \{1,2,\ldots,99\}$ can be partitioned into 33 sets of equal size and equal sum.
Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.
A biologist found a pond with frogs. When classifying them by their mass, he noticed the following:
[i]The $50$ lightest frogs represented $30\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\%$ of the total mass.
[/i]As fate would have it, the frogs escaped and the biologist only has the above information. How many frogs were in the pond?
Let $N$ be the number of ordered 5-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ of positive integers satisfying
$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$
Is $N$ even or odd?
Oh and [b]HINTS ONLY[/b], please do not give full solutions. Thanks.
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ holds $f(mn)=f(m)f(n)$ and $m+n \mid f(m)+f(n)$ .
As shown in the figure, given $\vartriangle ABC$ with $AB \perp AC$, $AB=BC$, $D$ is the midpoint of the side $AB$, $DF\perp DE$, $DE=DF$ and $BE \perp EC$. Prove that $\angle AFD= \angle CEF$.
[img]https://cdn.artofproblemsolving.com/attachments/9/2/f16a8c8c463874f3ccb333d91cdef913c34189.png[/img]
Prove that for any integer $n>1$ there are distinct integers $a,b,c$ between $n^2$ and $(n+1)^2$ such that $c$ divides $a^2+b^2$.
What is the greatest number of regions into which four planes can divide three-dimensional space?
Prove for positive $a,b,c$ that
$$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$
A regular tetrahedron of height $h$ has a tetrahedron of height $xh$ cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.)
Show that $x$ is a root of the equation $x^3 + x^2 + x = 2$.
A two-digit number is [i]nice [/i] if it is both a multiple of the product of its digits and a multiple of the sum of its digits. How many numbers satisfy this property? What is the ratio of the number to the sum of digits for each of the nice numbers?
For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given
a) $A=\sqrt{2}$;
b) $A=1$;
c) $A=2$,
where only non-negative real numbers are admitted for square roots?
In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.
Let $ABC$ be an acute triangle with $AB<AC$. Let then
• $D$ be the foot of the bisector of the angle in $A$,
• $E$ be the point on segment $BC$ (different from $B$) such that $AB=AE$,
• $F$ be the point on segment $BC$ (different from $B$) such that $BD=DF$,
• $G$ be the point on segment $AC$ such that $AB=AG$.
Prove that the circumcircle of triangle $EFG$ is tangent to line $AC$.
The degree measures of the angles of a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.
In a tennis ball factory there are $4$ machines $m_1 , m_2 , m_3 , m_4$, which produce, respectively, $10\%$, $20\%$, $30\%$ and $40\%$ of the balls that come out of the factory. The machine $m_1$ introduces defects in $1\%$ of the balls it manufactures, the machine $m_2$ in $2\%$, $m_3$ in $4\%$ and $m_4$ in $15\%$. Of the balls manufactured In one day, one is chosen at random and it turns out to be defective. What is the probability that Has this ball been made by the machine $ m_3$ ?
Lines \( FD \) and \( BE \) intersect at point \( O \). Rays \( OA \) and \( OC \) are drawn from point \( O \). You are given the following information about the angles:
\[
\angle DOC = 36^\circ, \quad \angle AOC = 90^\circ, \quad \angle AOB = 4x, \quad \angle FOE = 5x,
\]
as shown in the figure below. What is the degree measure of \( x \)?
[img]https://i.ibb.co/m5rwmXm/Kyiv-MO-2025-R1-7.png[/img]
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$,
\[f(f(x)-y^{2})=f(x)^{2}-2f(x)y^{2}+f(f(y)).\]
[b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for
which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.
Let $f(n)$ be the sum of the first $n$ terms of the sequence \[ 0, 1,1, 2,2, 3,3, 4,4, 5,5, 6,6, \ldots\, . \] a) Give a formula for $f(n)$.
b) Prove that $f(s+t)-f(s-t)=st$ where $s$ and $t$ are positive integers and $s>t$.
Solve the system of equations:
$ (1\plus{}4^{2x\minus{}y}).5^{1\minus{}2x\plus{}y}\equal{}1\plus{}2^{2x\minus{}y\plus{}1}$
$ y^3\plus{}4x\plus{}ln(y^2\plus{}2x)\plus{}1\equal{}0$
Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$.
Titu Andreescu, Mathematics Department, College of Texas, USA
$a,b$ are positive integers, $a>b$. $\sin\theta=\frac{2ab}{a^2+b^2}(0<\theta<\frac{\pi}{2})$. If $A_n=(a^2+b^2)\sin n\theta$, prove that $A_n$ is an integer for all $n\in\mathbb{Z}_+$