Found problems: 85335
1996 Portugal MO, 1
Consider a square on the hypotenuse of a right triangle $[ABC]$ (right at $B$). Prove that the line segment that joins vertex $B$ with the center of the square makes $45^o$ angles with legs of the triangle.
2007 Vietnam National Olympiad, 3
Let B,C be fixed points and A be roving point. Let H, G be orthecentre and centroid of triagle ABC. Known midpoint of HG lies on BC, find locus of A
2020 Taiwan TST Round 2, 4
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$, $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$.
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$, and Bob will tell Alice the value of $f(k)$. Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.
[i]Proposed by YaWNeeT[/i]
2013 Puerto Rico Team Selection Test, 2
How many 3-digit numbers have the property that the sum of their digits is even?
1999 AMC 12/AHSME, 12
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $ y \equal{} p(x)$ and $ y \equal{} q(x)$, each with leading coefficient $ 1$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$
2006 Bulgaria Team Selection Test, 1
[b]Problem 4.[/b] Let $k$ be the circumcircle of $\triangle ABC$, and $D$ the point on the arc $\overarc{AB},$ which do not pass through $C$. $I_A$ and $I_B$ are the centers of incircles of $\triangle ADC$ and $\triangle BDC$, respectively. Proove that the circumcircle of $\triangle I_AI_BC$ touches $k$ iff \[ \frac{AD}{BD}=\frac{AC+CD}{BC+CD}. \]
[i] Stoyan Atanasov[/i]
1948 Moscow Mathematical Olympiad, 146
Consider two triangular pyramids $ABCD$ and $A'BCD$, with a common base $BCD$, and such that $A'$ is inside $ABCD$. Prove that the sum of planar angles at vertex $A'$ of pyramid $A'BCD$ is greater than the sum of planar angles at vertex $A$ of pyramid $ABCD$.
2008 AMC 10, 9
A quadratic equation $ ax^2\minus{}2ax\plus{}b\equal{}0$ has two real solutions. What is the average of the solutions?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \frac{b}{a} \qquad
\textbf{(D)}\ \frac{2b}{a} \qquad
\textbf{(E)}\ \sqrt{2b\minus{}a}$
2024 HMNT, 3
Points $K,A,L,C,I,T,E$ are such that triangles $CAT$ and $ELK$ are equilateral, share a center $I,$ and points $E,L,K$ lie on sides $CA, AT, TC$ respectively. If the area of triangle $CAT$ is double the area of triangle $ELK$ and $CI = 2,$ compute the minimum possible value of $CK.$
2005 iTest, 27
Find the sum of all non-zero digits that can repeat at the end of a perfect square. (For example, if $811$ were a perfect square, $1$ would be one of these non-zero digits.)
2017 Macedonia National Olympiad, Problem 1
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for each natural integer $n>1$ and for all $x,y \in \mathbb{N}$ the following holds:
$$f(x+y) = f(x) + f(y) + \sum_{k=1}^{n-1} \binom{n}{k}x^{n-k}y^k$$
PEN H Problems, 33
Does there exist an integer such that its cube is equal to $3n^2 +3n+7$, where $n$ is integer?
2024 AMC 10, 8
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at $1{:}00 \text{ PM}$ and were able to pack $4$, $3$, and $3$ packages, respectively, every $3$ minutes. At some later time, Daria joined the group, and Daria was able to pack $5$ packages every $4$ minutes. Together, they finished packing $450$ packages at exactly $2{:}45 \text{ PM}$. At what time did Daria join the group?
$\textbf{(A) }1{:}25\text{ PM}\qquad\textbf{(B) }1{:}35\text{ PM}\qquad\textbf{(C) }1{:}45\text{ PM}\qquad\textbf{(D) }1{:}55\text{ PM}\qquad\textbf{(E) }2{:}05\text{ PM}\qquad$
2013 Poland - Second Round, 5
Let $W(x)$ be a polynomial of integer coefficients such that for any pair of different rational number $r_1$, $r_2$ dependence $W(r_1) \neq W(r_2)$ is true. Decide, whether the assuptions imply that for any pair of different real numbers $t_1$, $t_2$ dependence $W(t_1) \neq W(t_2)$ is true.
2019 Romania Team Selection Test, 2
Determine the largest natural number $ N $ having the following property: every $ 5\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\times 2 $ subarray of numbers whose sum is, at least, $ N. $
[i]Demetres Christofides[/i] and [i]Silouan Brazitikos[/i]
2020 Online Math Open Problems, 11
A mahogany bookshelf has four identical-looking books which are $200$, $400$, $600$, and $800$ pages long. Velma picks a random book off the shelf, flips to a random page to read, and puts the book back on the shelf. Later, Daphne also picks a random book off the shelf and flips to a random page to read. Given that Velma read page $122$ of her book and Daphne read page $304$ of her book, the probability that they chose the same book is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
[i]Proposed by Sean Li[/i]
2018 Tajikistan Team Selection Test, 1
Problem 1. Let ω be the incircle of triangle ABC which is tangent to BC,CA,AB at points D,E,F, respectively. The altitudes of triangle DEF with respect to E,F meet AB,AC at points X,Y, respectively. Prove that the second intersection of the circumcircles of triangles AEX,AFY lies on the circle ω.
2014 Postal Coaching, 5
Let $p>3$ be a prime and let $1+\frac 12 +\frac 13 +\cdots+\frac 1p=\frac ab$.Prove that $p^4$ divides $ap-b$.
2025 Ukraine National Mathematical Olympiad, 9.2
On the side \(AC\) of an acute-angled triangle \(ABC\), arbitrary points \(D\) and \(E\) are chosen. The circumcircles of triangles \(BDC\) and \(BEA\) intersect the sides \(BA\) and \(BC\) respectively for the second time at points \(F\) and \(G\). Point \(O\) is the circumcenter of \(\triangle BFG\). Prove that \(OD = OE\).
[i]Proposed by Anton Trygub[/i]
1970 AMC 12/AHSME, 20
Lines $HK$ and $BC$ lie in a plane. $M$ is the midpoint of line segment $BC$, and $BH$ and $CK$ are perpendicular to $HK$. Then we
$\textbf{(A) }\text{always have }MH=MK\qquad\textbf{(B) }\text{always have }MH>BK\qquad$
$\textbf{(C) }\text{sometimes have }MH=MK\text{ but not always}\qquad$
$\textbf{(D) }\text{always have }MH>MB\qquad \textbf{(E) }\text{always have }BH<BC$
2020 Austrian Junior Regional Competition, 4
Find all positive integers $a$ for which the equation $7an -3n! = 2020$ has a positive integer solution $n$.
(Richard Henner)
2020 LMT Fall, 14
Ada and Emily are playing a game that ends when either player wins, after some number of rounds. Each round, either nobody wins, Ada wins, or Emily wins. The probability that neither player wins each round is $\frac{1}{5}$ and the probability that Emily wins the game as a whole is $\frac{3}{4}.$ If the probability that in a given round Emily wins is $\frac{m}{n}$ such that $m$ and $n$ are relatively prime integers, then find $m+n.$
[i]Proposed by Ada Tsui[/i]
2018 Bangladesh Mathematical Olympiad, 1
Solve:
$x^2(2-x)^2=1+2(1-x)^2$
Where $x$ is real number.
2014 JHMMC 7 Contest, 8
A hedgehog has $4$ friends on Day $1$. If the number of friends he has increases by $5$ every day, how many friends will he have on Day $6$?
2000 Tournament Of Towns, 3
Peter plays a solitaire game with a deck of cards, some of which are face-up while the others are face-down. Peter loses if all the cards are face-down. As long as at least one card is face up, Peter must choose a stack of consecutive cards from the deck, so that the top and the bottom cards of the stack are face-up. They may be the same card. Then Peter turns the whole stack over and puts it back into the deck in exactly the same place as before. Prove that Peter always loses.
(A Shapovalov)