Found problems: 85335
2010 Saudi Arabia Pre-TST, 1.1
Using each of the first eight primes exactly once and several algebraic operations, obtain the result $2010$.
2023 European Mathematical Cup, 1
Suppose $a,b,c$ are positive integers such that \[\gcd(a,b)+\gcd(a,c)+\gcd(b,c)=b+c+2023\] Prove that $\gcd(b,c)=2023$.
[i]Remark.[/i] For positive integers $x$ and $y$, $\gcd(x,y)$ denotes their greatest common divisor.
[i]Ivan Novak[/i]
2017 South East Mathematical Olympiad, 5
Let $a, b, c$ be real numbers, $a \neq 0$. If the equation $2ax^2 + bx + c = 0$ has real root on the interval $[-1, 1]$.
Prove that
$$\min \{c, a + c + 1\} \leq \max \{|b - a + 1|, |b + a - 1|\},$$
and determine the necessary and sufficient conditions of $a, b, c$ for the equality case to be achieved.
2010 China Team Selection Test, 1
Let $ABCD$ be a convex quadrilateral with $A,B,C,D$ concyclic. Assume $\angle ADC$ is acute and $\frac{AB}{BC}=\frac{DA}{CD}$. Let $\Gamma$ be a circle through $A$ and $D$, tangent to $AB$, and let $E$ be a point on $\Gamma$ and inside $ABCD$.
Prove that $AE\perp EC$ if and only if $\frac{AE}{AB}-\frac{ED}{AD}=1$.
2025 All-Russian Olympiad, 10.6
What is the smallest value of \( k \) such that for any polynomial \( f(x) \) of degree $100$ with real coefficients, there exists a polynomial \( g(x) \) of degree at most \( k \) with real coefficients such that the graphs of \( y = f(x) \) and \( y = g(x) \) intersect at exactly $100$ points? \\
2016 IMO Shortlist, N1
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
2022 Assara - South Russian Girl's MO, 4
Nadya has $2022$ cards, each with a number one or seven written on it. It is known that there are both cards.Nadya looked at all possible $2022$-digit numbers that can be composed from all these cards. What is the largest value that can take the greatest common divisor of all these numbers?
2021 AMC 12/AHSME Spring, 23
Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability it is tossed into bin $i$ is $2^{-i}$ for $i = 1, 2, 3, \ldots$. More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins $3$, $17$, and $10$.) What is $p+q$?
$\textbf{(A)}\ 55 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 57 \qquad\textbf{(D)}\ 58 \qquad\textbf{(E)}\ 59$
2018 Saint Petersburg Mathematical Olympiad, 5
Can we draw $\triangle ABC$ and points $X,Y$, such that $AX=BY=AB$, $ BX = CY = BC$,
$CX = AY = CA$?
1990 IMO Shortlist, 1
The integer $ 9$ can be written as a sum of two consecutive integers: $ 9 \equal{} 4\plus{}5.$ Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $ 9 \equal{} 4\plus{}5 \equal{} 2\plus{}3\plus{}4.$ Is there an integer that can be written as a sum of $ 1990$ consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly $ 1990$ ways?
1969 IMO Shortlist, 49
$(NET 4)$ A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by $4.$
1998 National High School Mathematics League, 13
Complex number $z=1-\sin\theta+\text{i}\cos\theta\left(\frac{\pi}{2}<\theta<\pi\right)$, find the range value of $\arg{\overline{z}}$.
1998 Croatia National Olympiad, Problem 2
Find all positive integer solutions of the equation $10(m+n)=mn$.
Indonesia MO Shortlist - geometry, g7
A semicircle $(O)$ is drawn with the center $O$, where $O$ lies on a line $\ell$. $C$ and $D$ lie on the circle $(O)$, and the tangent lines of $(O)$ at points $C$ and $D$ intersects the line $\ell$ at points $B$ and $A$, respectively, such that $O$ lies between points $B$ and $A$. Let $E$ be the intersection point between $AC$ and $BD$, and $F$ the point on $\ell$ so that $EF $ is perpendicular to line $\ell$. Prove that $EF$ bisects the angle $\angle CFD$.
2015 Hanoi Open Mathematics Competitions, 15
Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2+d^2 \le 12$.
Determine the maximum value of $M = 4(a^3 + b^3 + c^3+d^3) - (a^4 + b^4 + c^4+d^4)$
2012 AMC 8, 4
Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat?
$\textbf{(A)}\hspace{.05in} \dfrac1{24}\qquad \textbf{(B)}\hspace{.05in}\dfrac1{12} \qquad \textbf{(C)}\hspace{.05in}\dfrac18 \qquad \textbf{(D)}\hspace{.05in}\dfrac16 \qquad \textbf{(E)}\hspace{.05in}\dfrac14 $
2020 Turkey EGMO TST, 3
There are $33!$ empty boxes labeled from $1$ to $33!$. In every move, we find the empty box with the smallest label, then we transfer $1$ ball from every box with a smaller label and we add an additional $1$ ball to that box. Find the smallest labeled non-empty box and the number of the balls in it after $33!$ moves.
2014 Purple Comet Problems, 21
Let $a$, $b$, $c$ be positive integers such that $29a + 30b + 31c = 366$. Find $19a + 20b + 21c$.
2016 All-Russian Olympiad, 6
There are $n>1$ cities in the country, some pairs of cities linked two-way through straight flight. For every pair of cities there is exactly one aviaroute (can have interchanges).
Major of every city X counted amount of such numberings of all cities from $1$ to $n$ , such that on every aviaroute with the beginning in X, numbers of cities are in ascending order. Every major, except one, noticed that results of counting are multiple of $2016$.
Prove, that result of last major is multiple of $2016$ too.
1970 IMO Longlists, 5
Prove that $\sqrt[n]{\sum_{i=1}^{n}{\frac{i}{n+1}}}\ge 1$ for $2 \le n \in \mathbb{N}$.
Kyiv City MO Seniors Round2 2010+ geometry, 2017.10.3
Circles $w_1$ and $w_2$ with centers at points $O_1$ and $O_2$ respectively, intersect at points $A$ and $B$. A line passing through point $B$, intersects the circles $w_1$ and $w_2$ at points $C$ and $D$ other than $B$. Tangents to the circles $w_1$ and $w_2$ at points $C$ and $D$ intersect at point $E$. Line $EA$ intersects the circumscribed circle $w$ of triangle $AO_1O_2$ at point $F$. Prove that the length of the segment is $EF$ is equal to the diameter of the circle $w$.
(Vovchenko V., Plotnikov M.)
2015 Singapore Junior Math Olympiad, 4
Let $A$ be a set of numbers chosen from $1,2,..., 2015$ with the property that any two distinct numbers, say $x$ and $y$, in $A$ determine a unique isosceles triangle (which is non equilateral) whose sides are of length $x$ or $y$. What is the largest possible size of $A$?
2013 Princeton University Math Competition, 3
Let $A$ be the greatest possible value of a product of positive integers that sums to $2014$. Compute the sum of all bases and exponents in the prime factorization of $A$. For example, if $A=7\cdot 11^5$, the answer would be $7+11+5=23$.
2023 JBMO TST - Turkey, 3
Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$f(x+f(x))=f(-x)$ and for all $x \leq y$ it satisfies $f(x) \leq f(y)$
1994 AMC 8, 12
Each of the three large squares shown below is the same size. Segments that intersect the sides of the squares intersect at the midpoints of the sides. How do the shaded areas of these squares compare?
[asy]
unitsize(36);
fill((0,0)--(1,0)--(1,1)--cycle,gray);
fill((1,1)--(1,2)--(2,2)--cycle,gray);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((1,0)--(1,2));
draw((0,0)--(2,2));
fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,gray);
draw((3,0)--(5,0)--(5,2)--(3,2)--cycle);
draw((4,0)--(4,2));
draw((3,1)--(5,1));
fill((6,1)--(6.5,0.5)--(7,1)--(7.5,0.5)--(8,1)--(7.5,1.5)--(7,1)--(6.5,1.5)--cycle,gray);
draw((6,0)--(8,0)--(8,2)--(6,2)--cycle);
draw((6,0)--(8,2));
draw((6,2)--(8,0));
draw((7,0)--(6,1)--(7,2)--(8,1)--cycle);
label("$I$",(1,2),N);
label("$II$",(4,2),N);
label("$III$",(7,2),N); [/asy]
$\text{(A)}\ \text{The shaded areas in all three are equal.}$
$\text{(B)}\ \text{Only the shaded areas of }I\text{ and }II\text{ are equal.}$
$\text{(C)}\ \text{Only the shaded areas of }I\text{ and }III\text{ are equal.}$
$\text{(D)}\ \text{Only the shaded areas of }II\text{ and }III\text{ are equal.}$
$\text{(E)}\ \text{The shaded areas of }I, II\text{ and }III\text{ are all different.}$